define hypothesis in logic

What Is a Hypothesis? (Science)

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A hypothesis (plural hypotheses) is a proposed explanation for an observation. The definition depends on the subject.

In science, a hypothesis is part of the scientific method. It is a prediction or explanation that is tested by an experiment. Observations and experiments may disprove a scientific hypothesis, but can never entirely prove one.

In the study of logic, a hypothesis is an if-then proposition, typically written in the form, "If X , then Y ."

In common usage, a hypothesis is simply a proposed explanation or prediction, which may or may not be tested.

Writing a Hypothesis

Most scientific hypotheses are proposed in the if-then format because it's easy to design an experiment to see whether or not a cause and effect relationship exists between the independent variable and the dependent variable . The hypothesis is written as a prediction of the outcome of the experiment.

Null Hypothesis and Alternative Hypothesis

Statistically, it's easier to show there is no relationship between two variables than to support their connection. So, scientists often propose the null hypothesis . The null hypothesis assumes changing the independent variable will have no effect on the dependent variable.

In contrast, the alternative hypothesis suggests changing the independent variable will have an effect on the dependent variable. Designing an experiment to test this hypothesis can be trickier because there are many ways to state an alternative hypothesis.

For example, consider a possible relationship between getting a good night's sleep and getting good grades. The null hypothesis might be stated: "The number of hours of sleep students get is unrelated to their grades" or "There is no correlation between hours of sleep and grades."

An experiment to test this hypothesis might involve collecting data, recording average hours of sleep for each student and grades. If a student who gets eight hours of sleep generally does better than students who get four hours of sleep or 10 hours of sleep, the hypothesis might be rejected.

But the alternative hypothesis is harder to propose and test. The most general statement would be: "The amount of sleep students get affects their grades." The hypothesis might also be stated as "If you get more sleep, your grades will improve" or "Students who get nine hours of sleep have better grades than those who get more or less sleep."

In an experiment, you can collect the same data, but the statistical analysis is less likely to give you a high confidence limit.

Usually, a scientist starts out with the null hypothesis. From there, it may be possible to propose and test an alternative hypothesis, to narrow down the relationship between the variables.

Example of a Hypothesis

Examples of a hypothesis include:

• If you drop a rock and a feather, (then) they will fall at the same rate.
• Plants need sunlight in order to live. (if sunlight, then life)
• Eating sugar gives you energy. (if sugar, then energy)
• White, Jay D.  Research in Public Administration . Conn., 1998.
• Schick, Theodore, and Lewis Vaughn.  How to Think about Weird Things: Critical Thinking for a New Age . McGraw-Hill Higher Education, 2002.
• Null Hypothesis Examples
• Examples of Independent and Dependent Variables
• Difference Between Independent and Dependent Variables
• Definition of a Hypothesis
• Null Hypothesis Definition and Examples
• What Are the Elements of a Good Hypothesis?
• Six Steps of the Scientific Method
• What Are Examples of a Hypothesis?
• Independent Variable Definition and Examples
• Understanding Simple vs Controlled Experiments
• Scientific Method Flow Chart
• What Is a Testable Hypothesis?
• Scientific Method Vocabulary Terms
• What 'Fail to Reject' Means in a Hypothesis Test
• How To Design a Science Fair Experiment
• What Is an Experiment? Definition and Design

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Definition of hypothesis

Did you know.

The Difference Between Hypothesis and Theory

A hypothesis is an assumption, an idea that is proposed for the sake of argument so that it can be tested to see if it might be true.

In the scientific method, the hypothesis is constructed before any applicable research has been done, apart from a basic background review. You ask a question, read up on what has been studied before, and then form a hypothesis.

A hypothesis is usually tentative; it's an assumption or suggestion made strictly for the objective of being tested.

A theory , in contrast, is a principle that has been formed as an attempt to explain things that have already been substantiated by data. It is used in the names of a number of principles accepted in the scientific community, such as the Big Bang Theory . Because of the rigors of experimentation and control, it is understood to be more likely to be true than a hypothesis is.

In non-scientific use, however, hypothesis and theory are often used interchangeably to mean simply an idea, speculation, or hunch, with theory being the more common choice.

Since this casual use does away with the distinctions upheld by the scientific community, hypothesis and theory are prone to being wrongly interpreted even when they are encountered in scientific contexts—or at least, contexts that allude to scientific study without making the critical distinction that scientists employ when weighing hypotheses and theories.

The most common occurrence is when theory is interpreted—and sometimes even gleefully seized upon—to mean something having less truth value than other scientific principles. (The word law applies to principles so firmly established that they are almost never questioned, such as the law of gravity.)

This mistake is one of projection: since we use theory in general to mean something lightly speculated, then it's implied that scientists must be talking about the same level of uncertainty when they use theory to refer to their well-tested and reasoned principles.

The distinction has come to the forefront particularly on occasions when the content of science curricula in schools has been challenged—notably, when a school board in Georgia put stickers on textbooks stating that evolution was "a theory, not a fact, regarding the origin of living things." As Kenneth R. Miller, a cell biologist at Brown University, has said , a theory "doesn’t mean a hunch or a guess. A theory is a system of explanations that ties together a whole bunch of facts. It not only explains those facts, but predicts what you ought to find from other observations and experiments.”

While theories are never completely infallible, they form the basis of scientific reasoning because, as Miller said "to the best of our ability, we’ve tested them, and they’ve held up."

• proposition
• supposition

hypothesis , theory , law mean a formula derived by inference from scientific data that explains a principle operating in nature.

hypothesis implies insufficient evidence to provide more than a tentative explanation.

theory implies a greater range of evidence and greater likelihood of truth.

law implies a statement of order and relation in nature that has been found to be invariable under the same conditions.

Examples of hypothesis in a Sentence

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'hypothesis.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

Greek, from hypotithenai to put under, suppose, from hypo- + tithenai to put — more at do

1641, in the meaning defined at sense 1a

Phrases Containing hypothesis

• counter - hypothesis
• nebular hypothesis
• null hypothesis
• planetesimal hypothesis
• Whorfian hypothesis

Articles Related to hypothesis

This is the Difference Between a...

This is the Difference Between a Hypothesis and a Theory

In scientific reasoning, they're two completely different things

Dictionary Entries Near hypothesis

hypothermia

hypothesize

Cite this Entry

“Hypothesis.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/hypothesis. Accessed 27 May. 2024.

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9.1: Hypothetical Reasoning

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Suppose I’m going on a picnic and I’m only selecting items that fit a certain rule. You want to find out what rule I’m using, so you offer up some guesses at items I might want to bring:

An Egg Salad Sandwich

A grape soda

Suppose now that I tell you that I’m okay with the first two, but I won’t bring the third. Your next step is interesting: you look at the first two, figure out what they have in common, and then you take a guess at the rule I’m using. In other words, you posit a hypothesis. You say something like

Do you only want to bring things that are yellow or tan?

Notice how at this point your hypothesis goes way beyond the evidence. Bananas and egg salad sandwiches have so much more in common than being yellow/tan objects. This is how hypothetical reasoning works: you look at the evidence, add a hypothesis that makes sense of that evidence (one among many hypotheses available), and then check to be sure that your hypothesis continues to make sense of new evidence as it is collected.

Suppose I now tell you that you haven’t guessed the right rule. So, you might throw out some more objects:

A key lime pie

A jug of orange juice

I then tell you that the first two are okay, but again the last item is not going with me on this picnic.

It’s solid items! Solid items are okay, but liquid items are not.

Again, not quite. Try another set of items. You are still convinced that it has to do with the soda and the juice being liquid, so you try out an interesting tactic:

An ice cube

Some liquid water

Some water Vapor

The first and last items are okay, but not the middle one. Now you think you’ve got me. You guess that the rule is “anything but liquids,” but I refuse to tell you whether you got it right. You’re pretty confident at this point, but perhaps you’re not certain . In principle, there could always be more evidence that upsets your hypothesis. I might say that the ocean is okay but a fresh water lake isn’t, and that would be very confusing for you. You’ll never be quite certain that you’ve guessed my rule correctly because it’s always in principle possible that I have a super complex rule that is more complex than your hypothesis.

So in hypothetical reasoning what we’re doing is making a leap from the evidence we have available to the rule or principle or theory which explains that evidence. The hypothesis is the link between the two. We have some finite evidence available to us, and we hypothesize an explanation. The explanation we posit either is or is not the true explanation, and so we’re using the hypothesis as a bridge to get onto the true explanation of what is happening in the world.

The hypothetical method has four stages. Let’s illustrate each with an example. You are investigating a murder and have collected a lot of evidence but do not yet have a guess as to who the killer might be.

1. The occurrence of a problem

Example \(\PageIndex{1}\)

Someone has been murdered and we need to find out who the killer is so that we might bring them to justice.

2. Formulating a hypothesis

Example \(\PageIndex{2}\)

After collecting some evidence, you weigh the reasons in favor of thinking that each suspect is indeed the murderer, and you decide that the spouse is responsible.

3. Drawing implications from the hypothesis

Example \(\PageIndex{3}\)

If the spouse was the murderer, then a number of things follow. The spouse must have a weak alibi or their alibi must rest on some falsehood. There is likely to be some evidence on their property or among their belongings that links the spouse to the murder. The spouse likely had motive. etc., etc., etc.

We can go on for ages, but the basic point is that once we’ve got an idea of what the explanation for the murder is (in this case, the hypothesis is that the spouse murdered the victim), we can ask ourselves what the world would have to be like for that to have been true. Then we move onto the final step:

4. Test those implications.

Example \(\PageIndex{4}\)

We can search the murder scene, try to find a murder weapon, run DNA analysis on the organic matter left at the scene, question the spouse about their alibi and possible motives, check their bank accounts, talk to friends and neighbors, etc. Once we have a hypothesis, in other words, that hypothesis drives the search for new evidence—it tells us what might be relevant and what irrelevant and therefore what is worth our time and what is not.

The Logic of Hypothetical Reasoning

If the spouse did it, then they must have a weak alibi. Their alibi is only verifiable by one person: the victim. So they do have a weak alibi. Therefore...they did it? Not quite.

Just because they have a weak alibi doesn’t mean they did it. If that were true, anyone with a weak alibi would be guilty for everything bad that happened when they weren’t busy with a verifiable activity.

Similarly, if your car’s battery is dead, then it won’t start. This doesn’t mean that whenever your car doesn’t start, the battery is dead. That would be a wild and bananas claim to make (and obviously false), but the original conditional (the first sentence in this paragraph) isn’t wild and bananas. In fact, it’s a pretty normal claim to make and it seems obviously true.

Let’s talk briefly about the logic of hypothetical reasoning so we can discover an important truth.

If the spouse did it, then their alibi will be weak

Their alibi is weak

So, the spouse did it

This is bad reasoning. How do we know? Well, here’s the logical form:

If A, then B

Therefore, A

This argument structure—called “affirming the consequent”—is invalid because there are countless instances of this general structure that have true premises and a false conclusion. Consider the following examples:

Example \(\PageIndex{5}\)

If I cook, I eat well

I ate well tonight, so I cooked.

Example \(\PageIndex{6}\)

If Eric runs for student president, he’ll become more popular.

Eric did become more popular, so he must’ve run for student president.

Maybe I ate well because I’m at the finest restaurant in town. Maybe I ate well because my brother cooked for me. Any of these things is possible, which is the root problem with this argument structure. It infers that one of the many possible antecedents to the conditional is the true antecedent without giving any reason for choosing or preferring this antecedent.

More concretely, affirming the consequent is the structure of an argument that states that a) one thing will explain an event, and b) that the event in question in fact occurred, and then concludes that c) the one thing that would’ve explained the event is the correct explanation of the event.

More concretely still, here’s yet another example of affirming the consequent:

Example \(\PageIndex{7}\)

My being rich would explain my being popular

I am in fact popular,

Therefore I am in fact rich

I might be popular without having a penny to my name. People sometimes root for underdogs, or respond to the right kind of personality regardless of their socioeconomic standing, or respect a good sense of humor or athletic prowess.

If I were rich, though, that would be one potential explanation for my being popular. Rich people have nice clothes, cool cars, nice houses, and get to have the kinds of experiences that make someone a potentially popular person because everyone wants to hear the cool stories or be associated with the exciting life they lead. Perhaps, people often seem to think, they’ll get to participate in the next adventure if they cozy up to the rich people. Rich kids in high school can also throw the best parties (if we’re honest, and that’s a great source of popularity).

But If I’m not rich, that doesn’t mean I’m not popular. It only means that I’m not popular because I’m rich .

Okay, so we’ve established that hypothetical reasoning has the logical structure of affirming the consequent. We’ve further established that affirming the consequent is an invalid deductive argumentative structure. Where does this leave us? Is the hypothetical method bad reasoning ?!?!?!? Nope! Luckily not all reasoning is deductive reasoning.

Remember that we’re discussing inductive reasoning in this chapter. Inductive reasoning doesn’t obey the rules of deductive logic. So it’s no crime for a method of inductive reasoning to be deductively invalid. The crime against logic would be to claim that we have certain knowledge when we only use inductive reasoning to justify that knowledge. The upshot? Science doesn’t produce certain knowledge—it produces justified knowledge, knowledge to a more or less high degree of certitude, knowledge that we can rely on and build bridges on, knowledge that almost certainly won’t let us down (but it doesn’t produce certain knowledge).

We can, though, with deductive certainty, falsify a hypothesis. Consider the murder case: if the spouse did it, then they’d have a weak alibi. That is, if the spouse did it, then they wouldn’t have an airtight alibi because they’d have to be lying about where they were when the murder took place. If it turns out that the spouse does have an airtight alibi, then your hypothesis was wrong.

Let’s take a look at the logic of falsification:

If the spouse did it, then they won’t have an airtight alibi

They have an airtight alibi

So the spouse didn’t do it

Now it’s possible that the conditional premise (the first premise) isn’t true, but we’ll assume it’s true for the sake of the illustration. The hypothesis was that the spouse did it and so the spouse’s alibi must have some weakness.

It’s also possible that our detective work hasn’t been thorough enough and so the second premise is false. These are important possibilities to keep in mind. Either way, here’s the logical form (a bit cleaned up and simplified):

Therefore not A

This is what argument pattern? That’s right! You’re so smart! It’s modus tollens or “the method of denying”. It’s a type of argument where you deny the implications of something and thereby deny that very thing. It’s a deductively valid argument form (remember from our unit on natural deduction?), so we can falsify hypotheses with deductive certainty: if your hypothesis implies something with necessity, and that something doesn’t come to pass, then your hypothesis is wrong.

Your hypothesis is wrong. That is, your hypothesis as it stands was wrong. You might be like one of those rogue and dogged detectives in the television shows that never gives up on a hunch and ultimately discovers the truth through sheer stubbornness and determination. You might think that the spouse did it, even though they’ve got an airtight alibi. In that case, you’ll have to alter your hypothesis a bit.

The process of altering a hypothesis to react to potentially falsifying evidence typically involves adding extra hypotheses onto your original hypothesis such that the original hypothesis no longer has the troubling implications which turned out not to be true. These extra hypotheses are called ad hoc hypotheses.

As an example, Newton’s theory of gravity had one problem: it made a sort of wacky prediction. So the idea was that gravity was an instantaneous attractive force exerted by all massive bodies on all other bodies. That is, all bodies attract all other bodies regardless of distance or time. The result of this should be that all massive bodies should smack into each other over time (after all, they still have to travel towards one another). But we don’t witness this. We should see things crashing towards the center of gravity of the universe at incredible speeds, but that’s not what’s happening. So, by the logic of falsification, Newton’s theory is simply false.

But Newton had a trick up his sleeve: he claimed that God arranged things such that the heavenly bodies are so far apart from one another that they are prevented from crashing into one another. Problem solved! God put things in the right spatial orientation such that the theory of gravity is saved: they won’t crash into each other because they’re so far apart! Newton employed an ad hoc hypothesis to save his theory from falsification.

Abductive Reasoning

There’s one more thing to discuss while we’re still on the topic of hypothetical reasoning or reasoning using hypotheses. ‘Abduction’ is a fancy word for a process or method sometimes called “inference to the best explanation. The basic idea is that we have a bunch of evidence, we try to explain it, and we find that we could explain it in multiple ways. Then we find the “best” explanation or hypothesis and infer that this is the true explanation.

For example, say we’re playing a game that’s sort of like the picnic game from before. I give you a series of numbers, and then you give me more series of numbers so that I can confirm or deny that each meets the rule I have in mind. So I say:

And then you offer the following series (serieses?):

60, 90, 120

Each of these series tests a particular hypothesis. The first tests whether the important thing is that the numbers start with 2, 3, and 4. The second tests whether the rule is to add 10 each successive number in the series. The third tests a more complicated hypothesis: add half of the first number to itself to get the second number, then add one third of the second number to itself to get the third number.

Now let’s say I tell you that only the third series is acceptable. What now?

Well, our hypothesis was pretty complex, but it seems pretty good. I can infer that this is the correct rule. Alternatively, I might look at other hypotheses which fit the evidence equally well: 1x, 1.5x, 2x? or maybe it’s 2x, 3x, 4x? What about x, 1.5x, x\(^2\)? These all make sense of the data, but are they equal apart from that?

Let’s suppose we can’t easily get more data with which to test our various hypotheses. We’ve got 4 to choose from and nothing in the evidence suggests that one of the hypotheses is better than the others—they all fit the evidence perfectly. What do we do?

One thing we could do is choose which hypothesis is best for reasons other than fit with the evidence. Maybe we want a simpler hypothesis, or maybe we want a more elegant hypothesis, or one which suggests more routes for investigation. These are what we might call “theoretical virtues”—they’re the things we want to see in a theory. The process of abduction is the process of selecting the hypothesis that has the most to offer in terms of theoretical virtues: the simplest, most elegant, most fruitful, most general, and so on.

In science in particular, we value a few theoretical virtues over others: support by the empirical evidence available, replicability of the results in a controlled setting by other scientists, ideally mathematical precision or at least a lack of vagueness, and parsimony or simplicity in terms of the sorts of things the hypothesis requires us to believe in.

Confirmation Bias

This is a great opportunity to discuss confirmation bias, or the natural tendency we have to seek out evidence which supports our beliefs and to ignore evidence which gets in the way of our beliefs. We’ll discuss cognitive biases more in Chapter 10, but since we’re dealing with the relationship between evidence and belief, this seems like a good spot to pause and reflect on how our minds work.

The way our minds work naturally, it seems, is to settle on a belief and then work hard to maintain that belief whatever happens. We come to believe that global warming is anthropogenic—is caused by human activities—and then we’re happy to accept a wide variety of evidence for the claim. If the evidence supports our belief, in other words, we don’t take the time or energy to really investigate exactly how convincing that evidence is. If we already believe the conclusion of an inference, in other words, we are much less likely to test or analyze the inference.

Alternatively, when we see pieces of evidence or arguments that appear to point to the contrary, we are either more skeptical of that evidence or more critical of that argument. For instance, if someone notes that the Earth goes through normal cycles of warming and ice ages and warming again, we immediately will look for ways to explain how this warming period is different than others in the past. Or we might look at the period of the cycles to find out if this is happening at the “right” time in our geological history for it not to be caused by humankind. In other words, we’re more skeptical of arguments or evidence that would defeat or undermine our beliefs, but we’re less skeptical and critical of arguments and evidence that supports our beliefs.

Here are some questions to reflect on as you try to decide how guilty you are of confirmation bias in your own reasoning:

Questions for Reflection:

1. Which news sources do you trust? Why?

2. What’s your process for exploring a topic—say a political or scientific or news topic?

3. How do you decide what to believe about a new subject?

4. When other people express an opinion about someone you don’t know, do you withhold judgment? How well do you do so?

5. Are you harder on arguments and evidence that would shake up your beliefs?

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Understanding Science

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• Understanding Science 101
• Scientific arguments are logical descriptions of a scientific idea and the evidence for or against it.
• Sometimes a scientific idea precedes any evidence relevant to it, and other times the evidence helps inspire the idea.

The logic of scientific arguments

Taken together, the expectations generated by a scientific idea and the actual observations relevant to those expectations form what we’ll call a scientific argument . This is a bit like an argument in a court case — a logical description of what we think and why we think it. A scientific argument uses evidence to make a case for whether a scientific idea is accurate or inaccurate. For example, the idea that illness in new mothers can be caused by doctors’ dirty hands generates the expectation that illness rates should go down when doctors are required to wash their hands before attending births. When this test was actually performed in the 1800s, the results matched the expectations, forming a strong scientific argument in support of the idea — and hand-washing!

Testing ideas with evidence may seem like plain old common sense — and at its core, it is! — but there are some subtleties to the process:

• Ideas can be tested in many ways. Some tests are relatively straightforward (e.g., raising 1000 fruit flies and counting how many have red eyes), but some require a lot of time (e.g., waiting for the next appearance of Halley’s Comet), effort (e.g., painstakingly sorting through thousands of microfossils), and/or the development of specialized tools (like a particle accelerator). To explore further, jump to Tactics for testing .
• Evidence can reflect on ideas in many different ways. To explore further, jump to Reviewing your test results .
• There are multiple lines-of-evidence and many criteria to consider in evaluating an idea. To explore further, jump to Competing ideas: A perfect fit for the evidence? or Competing ideas: Other considerations .
• All testing involves making some assumptions . To explore further, jump to Making assumptions .

Despite these details, it’s important to remember that, in the end, hypotheses and theories live and die by whether or not they work — in other words, whether they are useful in explaining data , generating expectations, providing explanations, inspiring research questions, answering questions, and solving problems. Science filters through many ideas and builds on those that work !

• Take a sidetrip
• Teaching resources

To learn more about the logic of scientific arguments, visit our unit on  The core of science: Relating evidence and ideas .

• Forming hypotheses — scientific explanations — can be difficult for students. It is often easier for students to generate an expectation (what they think will happen or what they expect to observe) based on prior experience than to formulate a potential explanation for that phenomena. You can help students go beyond expectations to generate real, explanatory hypotheses by providing sentence stems for them to fill in: “I expect to observe A because B.” Once students have filled in this sentence you can explain that B is a hypothesis and A is the expectation generated by that hypothesis.
• You can help students learn to distinguish between hypotheses and the expectations generated by them by regularly asking students to analyze lecture material, text, or video. Students should try to figure out which aspects of the content were hypotheses and which were expectations.

Testing scientific ideas

Tactics for testing ideas

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Science and Hypothesis

• First Online: 13 June 2021

Cite this chapter

• Satya Sundar Sethy 2  

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In this chapter, we will discuss the significance of a ‘hypothesis’ in a logical inquiry, a scientific investigation, and research work. We will enumerate some of the definitions of ‘hypothesis’. We will elaborate on the nature and scope of the ‘hypothesis’ and the sources to obtain a hypothesis. Further, we will explain the kinds of hypothesis with suitable examples. In the end, we will illustrate methods to verify a hypothesis in a logical inquiry and a scientific investigation.

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Lundberg, G.A. (1968). Social research: A study in methods of gathering data . New York: Greenwood Press.

Black, J. A., and Champion, D.J. (1976). Method and issues in social research . New York: John Wiley & Sons.

Goode, W.J., and Hatt, P.K. (1971). Methods in social research . New York: McGraw-Hill Publication.

https://www.merriam-webster.com/dictionary/hypothesis .

Sarantakos, S. (2005) (3rd Edition). Social research . New York: Palgrave Macmillan.

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Sethy, S.S. (2021). Science and Hypothesis. In: Introduction to Logic and Logical Discourse. Springer, Singapore. https://doi.org/10.1007/978-981-16-2689-0_17

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DOI : https://doi.org/10.1007/978-981-16-2689-0_17

Published : 13 June 2021

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A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false.

In statistics, a hypothesis (sometimes called a statistical hypothesis) refers to a statement on which hypothesis testing will be based. Particularly important statistical hypotheses include the null hypothesis and alternative hypothesis .

In symbolic logic , a hypothesis is the first part of an implication (with the second part being known as the predicate ).

In general mathematical usage, "hypothesis" is roughly synonymous with " conjecture ."

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Hypothesis is a testable statement that explains what is happening or observed. It proposes the relation between the various participating variables. Hypothesis is also called Theory, Thesis, Guess, Assumption, or Suggestion. Hypothesis creates a structure that guides the search for knowledge.

In this article, we will learn what is hypothesis, its characteristics, types, and examples. We will also learn how hypothesis helps in scientific research.

What is Hypothesis?

A hypothesis is a suggested idea or plan that has little proof, meant to lead to more study. It’s mainly a smart guess or suggested answer to a problem that can be checked through study and trial. In science work, we make guesses called hypotheses to try and figure out what will happen in tests or watching. These are not sure things but rather ideas that can be proved or disproved based on real-life proofs. A good theory is clear and can be tested and found wrong if the proof doesn’t support it.

Hypothesis Meaning

A hypothesis is a proposed statement that is testable and is given for something that happens or observed.

• It is made using what we already know and have seen, and it’s the basis for scientific research.
• A clear guess tells us what we think will happen in an experiment or study.
• It’s a testable clue that can be proven true or wrong with real-life facts and checking it out carefully.
• It usually looks like a “if-then” rule, showing the expected cause and effect relationship between what’s being studied.

Characteristics of Hypothesis

Here are some key characteristics of a hypothesis:

• Testable: An idea (hypothesis) should be made so it can be tested and proven true through doing experiments or watching. It should show a clear connection between things.
• Specific: It needs to be easy and on target, talking about a certain part or connection between things in a study.
• Falsifiable: A good guess should be able to show it’s wrong. This means there must be a chance for proof or seeing something that goes against the guess.
• Logical and Rational: It should be based on things we know now or have seen, giving a reasonable reason that fits with what we already know.
• Predictive: A guess often tells what to expect from an experiment or observation. It gives a guide for what someone might see if the guess is right.
• Concise: It should be short and clear, showing the suggested link or explanation simply without extra confusion.
• Grounded in Research: A guess is usually made from before studies, ideas or watching things. It comes from a deep understanding of what is already known in that area.
• Flexible: A guess helps in the research but it needs to change or fix when new information comes up.
• Relevant: It should be related to the question or problem being studied, helping to direct what the research is about.
• Empirical: Hypotheses come from observations and can be tested using methods based on real-world experiences.

Sources of Hypothesis

Hypotheses can come from different places based on what you’re studying and the kind of research. Here are some common sources from which hypotheses may originate:

• Existing Theories: Often, guesses come from well-known science ideas. These ideas may show connections between things or occurrences that scientists can look into more.
• Observation and Experience: Watching something happen or having personal experiences can lead to guesses. We notice odd things or repeat events in everyday life and experiments. This can make us think of guesses called hypotheses.
• Previous Research: Using old studies or discoveries can help come up with new ideas. Scientists might try to expand or question current findings, making guesses that further study old results.
• Literature Review: Looking at books and research in a subject can help make guesses. Noticing missing parts or mismatches in previous studies might make researchers think up guesses to deal with these spots.
• Problem Statement or Research Question: Often, ideas come from questions or problems in the study. Making clear what needs to be looked into can help create ideas that tackle certain parts of the issue.
• Analogies or Comparisons: Making comparisons between similar things or finding connections from related areas can lead to theories. Understanding from other fields could create new guesses in a different situation.
• Hunches and Speculation: Sometimes, scientists might get a gut feeling or make guesses that help create ideas to test. Though these may not have proof at first, they can be a beginning for looking deeper.
• Technology and Innovations: New technology or tools might make guesses by letting us look at things that were hard to study before.
• Personal Interest and Curiosity: People’s curiosity and personal interests in a topic can help create guesses. Scientists could make guesses based on their own likes or love for a subject.

Types of Hypothesis

Here are some common types of hypotheses:

Simple Hypothesis

Complex hypothesis, directional hypothesis.

• Non-directional Hypothesis

Null Hypothesis (H0)

Alternative hypothesis (h1 or ha), statistical hypothesis, research hypothesis, associative hypothesis, causal hypothesis.

Simple Hypothesis guesses a connection between two things. It says that there is a connection or difference between variables, but it doesn’t tell us which way the relationship goes.

Complex Hypothesis tells us what will happen when more than two things are connected. It looks at how different things interact and may be linked together.

Directional Hypothesis says how one thing is related to another. For example, it guesses that one thing will help or hurt another thing.

Non-Directional Hypothesis

Non-Directional Hypothesis are the one that don’t say how the relationship between things will be. They just say that there is a connection, without telling which way it goes.

Null hypothesis is a statement that says there’s no connection or difference between different things. It implies that any seen impacts are because of luck or random changes in the information.

Alternative Hypothesis is different from the null hypothesis and shows that there’s a big connection or gap between variables. Scientists want to say no to the null hypothesis and choose the alternative one.

Statistical Hypotheis are used in math testing and include making ideas about what groups or bits of them look like. You aim to get information or test certain things using these top-level, common words only.

Research Hypothesis comes from the research question and tells what link is expected between things or factors. It leads the study and chooses where to look more closely.

Associative Hypotheis guesses that there is a link or connection between things without really saying it caused them. It means that when one thing changes, it is connected to another thing changing.

Causal Hypothesis are different from other ideas because they say that one thing causes another. This means there’s a cause and effect relationship between variables involved in the situation. They say that when one thing changes, it directly makes another thing change.

Hypothesis Examples

Following are the examples of hypotheses based on their types:

Simple Hypothesis Example

• Studying more can help you do better on tests.
• Getting more sun makes people have higher amounts of vitamin D.

Complex Hypothesis Example

• How rich you are, how easy it is to get education and healthcare greatly affects the number of years people live.
• A new medicine’s success relies on the amount used, how old a person is who takes it and their genes.

Directional Hypothesis Example

• Drinking more sweet drinks is linked to a higher body weight score.
• Too much stress makes people less productive at work.

Non-directional Hypothesis Example

• Drinking caffeine can affect how well you sleep.
• People often like different kinds of music based on their gender.
• The average test scores of Group A and Group B are not much different.
• There is no connection between using a certain fertilizer and how much it helps crops grow.

Alternative Hypothesis (Ha)

• Patients on Diet A have much different cholesterol levels than those following Diet B.
• Exposure to a certain type of light can change how plants grow compared to normal sunlight.
• The average smarts score of kids in a certain school area is 100.
• The usual time it takes to finish a job using Method A is the same as with Method B.
• Having more kids go to early learning classes helps them do better in school when they get older.
• Using specific ways of talking affects how much customers get involved in marketing activities.
• Regular exercise helps to lower the chances of heart disease.
• Going to school more can help people make more money.
• Playing violent video games makes teens more likely to act aggressively.
• Less clean air directly impacts breathing health in city populations.

Functions of Hypothesis

Hypotheses have many important jobs in the process of scientific research. Here are the key functions of hypotheses:

• Guiding Research: Hypotheses give a clear and exact way for research. They act like guides, showing the predicted connections or results that scientists want to study.
• Formulating Research Questions: Research questions often create guesses. They assist in changing big questions into particular, checkable things. They guide what the study should be focused on.
• Setting Clear Objectives: Hypotheses set the goals of a study by saying what connections between variables should be found. They set the targets that scientists try to reach with their studies.
• Testing Predictions: Theories guess what will happen in experiments or observations. By doing tests in a planned way, scientists can check if what they see matches the guesses made by their ideas.
• Providing Structure: Theories give structure to the study process by arranging thoughts and ideas. They aid scientists in thinking about connections between things and plan experiments to match.
• Focusing Investigations: Hypotheses help scientists focus on certain parts of their study question by clearly saying what they expect links or results to be. This focus makes the study work better.
• Facilitating Communication: Theories help scientists talk to each other effectively. Clearly made guesses help scientists to tell others what they plan, how they will do it and the results expected. This explains things well with colleagues in a wide range of audiences.
• Generating Testable Statements: A good guess can be checked, which means it can be looked at carefully or tested by doing experiments. This feature makes sure that guesses add to the real information used in science knowledge.
• Promoting Objectivity: Guesses give a clear reason for study that helps guide the process while reducing personal bias. They motivate scientists to use facts and data as proofs or disprovals for their proposed answers.
• Driving Scientific Progress: Making, trying out and adjusting ideas is a cycle. Even if a guess is proven right or wrong, the information learned helps to grow knowledge in one specific area.

How Hypothesis help in Scientific Research?

Researchers use hypotheses to put down their thoughts directing how the experiment would take place. Following are the steps that are involved in the scientific method:

• Initiating Investigations: Hypotheses are the beginning of science research. They come from watching, knowing what’s already known or asking questions. This makes scientists make certain explanations that need to be checked with tests.
• Formulating Research Questions: Ideas usually come from bigger questions in study. They help scientists make these questions more exact and testable, guiding the study’s main point.
• Setting Clear Objectives: Hypotheses set the goals of a study by stating what we think will happen between different things. They set the goals that scientists want to reach by doing their studies.
• Designing Experiments and Studies: Assumptions help plan experiments and watchful studies. They assist scientists in knowing what factors to measure, the techniques they will use and gather data for a proposed reason.
• Testing Predictions: Ideas guess what will happen in experiments or observations. By checking these guesses carefully, scientists can see if the seen results match up with what was predicted in each hypothesis.
• Analysis and Interpretation of Data: Hypotheses give us a way to study and make sense of information. Researchers look at what they found and see if it matches the guesses made in their theories. They decide if the proof backs up or disagrees with these suggested reasons why things are happening as expected.
• Encouraging Objectivity: Hypotheses help make things fair by making sure scientists use facts and information to either agree or disagree with their suggested reasons. They lessen personal preferences by needing proof from experience.
• Iterative Process: People either agree or disagree with guesses, but they still help the ongoing process of science. Findings from testing ideas make us ask new questions, improve those ideas and do more tests. It keeps going on in the work of science to keep learning things.

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Summary – Hypothesis

A hypothesis is a testable statement serving as an initial explanation for phenomena, based on observations, theories, or existing knowledge. It acts as a guiding light for scientific research, proposing potential relationships between variables that can be empirically tested through experiments and observations. The hypothesis must be specific, testable, falsifiable, and grounded in prior research or observation, laying out a predictive, if-then scenario that details a cause-and-effect relationship. It originates from various sources including existing theories, observations, previous research, and even personal curiosity, leading to different types, such as simple, complex, directional, non-directional, null, and alternative hypotheses, each serving distinct roles in research methodology. The hypothesis not only guides the research process by shaping objectives and designing experiments but also facilitates objective analysis and interpretation of data, ultimately driving scientific progress through a cycle of testing, validation, and refinement.

FAQs on Hypothesis

What is a hypothesis.

A guess is a possible explanation or forecast that can be checked by doing research and experiments.

What are Components of a Hypothesis?

The components of a Hypothesis are Independent Variable, Dependent Variable, Relationship between Variables, Directionality etc.

What makes a Good Hypothesis?

Testability, Falsifiability, Clarity and Precision, Relevance are some parameters that makes a Good Hypothesis

Can a Hypothesis be Proven True?

You cannot prove conclusively that most hypotheses are true because it’s generally impossible to examine all possible cases for exceptions that would disprove them.

How are Hypotheses Tested?

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data

Can Hypotheses change during Research?

Yes, you can change or improve your ideas based on new information discovered during the research process.

What is the Role of a Hypothesis in Scientific Research?

Hypotheses are used to support scientific research and bring about advancements in knowledge.

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A scientific hypothesis is a tentative, testable explanation for a phenomenon in the natural world. It's the initial building block in the scientific method . Many describe it as an "educated guess" based on prior knowledge and observation. While this is true, a hypothesis is more informed than a guess. While an "educated guess" suggests a random prediction based on a person's expertise, developing a hypothesis requires active observation and background research. 

The basic idea of a hypothesis is that there is no predetermined outcome. For a solution to be termed a scientific hypothesis, it has to be an idea that can be supported or refuted through carefully crafted experimentation or observation. This concept, called falsifiability and testability, was advanced in the mid-20th century by Austrian-British philosopher Karl Popper in his famous book "The Logic of Scientific Discovery" (Routledge, 1959).

A key function of a hypothesis is to derive predictions about the results of future experiments and then perform those experiments to see whether they support the predictions.

A hypothesis is usually written in the form of an if-then statement, which gives a possibility (if) and explains what may happen because of the possibility (then). The statement could also include "may," according to California State University, Bakersfield .

Here are some examples of hypothesis statements:

• If garlic repels fleas, then a dog that is given garlic every day will not get fleas.
• If sugar causes cavities, then people who eat a lot of candy may be more prone to cavities.
• If ultraviolet light can damage the eyes, then maybe this light can cause blindness.

A useful hypothesis should be testable and falsifiable. That means that it should be possible to prove it wrong. A theory that can't be proved wrong is nonscientific, according to Karl Popper's 1963 book " Conjectures and Refutations ."

An example of an untestable statement is, "Dogs are better than cats." That's because the definition of "better" is vague and subjective. However, an untestable statement can be reworded to make it testable. For example, the previous statement could be changed to this: "Owning a dog is associated with higher levels of physical fitness than owning a cat." With this statement, the researcher can take measures of physical fitness from dog and cat owners and compare the two.

Types of scientific hypotheses

In an experiment, researchers generally state their hypotheses in two ways. The null hypothesis predicts that there will be no relationship between the variables tested, or no difference between the experimental groups. The alternative hypothesis predicts the opposite: that there will be a difference between the experimental groups. This is usually the hypothesis scientists are most interested in, according to the University of Miami .

For example, a null hypothesis might state, "There will be no difference in the rate of muscle growth between people who take a protein supplement and people who don't." The alternative hypothesis would state, "There will be a difference in the rate of muscle growth between people who take a protein supplement and people who don't."

If the results of the experiment show a relationship between the variables, then the null hypothesis has been rejected in favor of the alternative hypothesis, according to the book " Research Methods in Psychology " (​​BCcampus, 2015). 

There are other ways to describe an alternative hypothesis. The alternative hypothesis above does not specify a direction of the effect, only that there will be a difference between the two groups. That type of prediction is called a two-tailed hypothesis. If a hypothesis specifies a certain direction — for example, that people who take a protein supplement will gain more muscle than people who don't — it is called a one-tailed hypothesis, according to William M. K. Trochim , a professor of Policy Analysis and Management at Cornell University.

Sometimes, errors take place during an experiment. These errors can happen in one of two ways. A type I error is when the null hypothesis is rejected when it is true. This is also known as a false positive. A type II error occurs when the null hypothesis is not rejected when it is false. This is also known as a false negative, according to the University of California, Berkeley . 

A hypothesis can be rejected or modified, but it can never be proved correct 100% of the time. For example, a scientist can form a hypothesis stating that if a certain type of tomato has a gene for red pigment, that type of tomato will be red. During research, the scientist then finds that each tomato of this type is red. Though the findings confirm the hypothesis, there may be a tomato of that type somewhere in the world that isn't red. Thus, the hypothesis is true, but it may not be true 100% of the time.

Scientific theory vs. scientific hypothesis

The best hypotheses are simple. They deal with a relatively narrow set of phenomena. But theories are broader; they generally combine multiple hypotheses into a general explanation for a wide range of phenomena, according to the University of California, Berkeley . For example, a hypothesis might state, "If animals adapt to suit their environments, then birds that live on islands with lots of seeds to eat will have differently shaped beaks than birds that live on islands with lots of insects to eat." After testing many hypotheses like these, Charles Darwin formulated an overarching theory: the theory of evolution by natural selection.

"Theories are the ways that we make sense of what we observe in the natural world," Tanner said. "Theories are structures of ideas that explain and interpret facts." 

• Read more about writing a hypothesis, from the American Medical Writers Association.
• Find out why a hypothesis isn't always necessary in science, from The American Biology Teacher.
• Learn about null and alternative hypotheses, from Prof. Essa on YouTube .

Encyclopedia Britannica. Scientific Hypothesis. Jan. 13, 2022. https://www.britannica.com/science/scientific-hypothesis

Karl Popper, "The Logic of Scientific Discovery," Routledge, 1959.

California State University, Bakersfield, "Formatting a testable hypothesis." https://www.csub.edu/~ddodenhoff/Bio100/Bio100sp04/formattingahypothesis.htm  

Karl Popper, "Conjectures and Refutations," Routledge, 1963.

Price, P., Jhangiani, R., & Chiang, I., "Research Methods of Psychology — 2nd Canadian Edition," BCcampus, 2015.‌

University of Miami, "The Scientific Method" http://www.bio.miami.edu/dana/161/evolution/161app1_scimethod.pdf  

William M.K. Trochim, "Research Methods Knowledge Base," https://conjointly.com/kb/hypotheses-explained/  

University of California, Berkeley, "Multiple Hypothesis Testing and False Discovery Rate" https://www.stat.berkeley.edu/~hhuang/STAT141/Lecture-FDR.pdf  

University of California, Berkeley, "Science at multiple levels" https://undsci.berkeley.edu/article/0_0_0/howscienceworks_19

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Aristotle’s Logic

Aristotle’s logic, especially his theory of the syllogism, has had an unparalleled influence on the history of Western thought. It did not always hold this position: in the Hellenistic period, Stoic logic, and in particular the work of Chrysippus, took pride of place. However, in later antiquity, following the work of Aristotelian Commentators, Aristotle’s logic became dominant, and Aristotelian logic was what was transmitted to the Arabic and the Latin medieval traditions, while the works of Chrysippus have not survived.

This unique historical position has not always contributed to the understanding of Aristotle’s logical works. Kant thought that Aristotle had discovered everything there was to know about logic, and the historian of logic Prantl drew the corollary that any logician after Aristotle who said anything new was confused, stupid, or perverse. During the rise of modern formal logic following Frege and Peirce, adherents of Traditional Logic (seen as the descendant of Aristotelian Logic) and the new mathematical logic tended to see one another as rivals, with incompatible notions of logic. More recent scholarship has often applied the very techniques of mathematical logic to Aristotle’s theories, revealing (in the opinion of many) a number of similarities of approach and interest between Aristotle and modern logicians.

This article is written from the latter perspective. As such, it is about Aristotle’s logic, which is not always the same thing as what has been called “Aristotelian” logic.

1. Introduction

2. aristotle’s logical works: the organon, 3.1 induction and deduction, 3.2 aristotelian deductions and modern valid arguments, 4.2 affirmations, denials, and contradictions, 4.3 all, some, and none, 5.1 the figures, 5.2 methods of proof: “perfect” deductions, conversion, reduction, 5.3 disproof: counterexamples and terms, 5.4 the deductions in the figures (“moods”), 5.5 metatheoretical results, 5.6 syllogisms with modalities, 6.1 aristotelian sciences, 6.2 the regress problem, 6.3 aristotle’s solution: “it eventually comes to a stop”, 6.4 knowledge of first principles: nous, 7.1 definitions and essences, 7.2 species, genus, and differentia, 7.3 the categories, 7.4 the method of division, 7.5 definition and demonstration, 8.1 dialectical premises: the meaning of endoxos, 8.2 the two elements of the art of dialectic, 8.3 the uses of dialectic and dialectical argument, 9. dialectic and rhetoric, 10. sophistical arguments, 11. non-contradiction and metaphysics, 12. time and necessity: the sea-battle, 13. glossary of aristotelian terminology, other internet resources, related entries.

Aristotle’s logical works contain the earliest formal study of logic that we have. It is therefore all the more remarkable that together they comprise a highly developed logical theory, one that was able to command immense respect for many centuries: Kant, who was ten times more distant from Aristotle than we are from him, even held that nothing significant had been added to Aristotle’s views in the intervening two millennia.

In the last century, Aristotle’s reputation as a logician has undergone two remarkable reversals. The rise of modern formal logic following the work of Frege and Russell brought with it a recognition of the many serious limitations of Aristotle’s logic; today, very few would try to maintain that it is adequate as a basis for understanding science, mathematics, or even everyday reasoning. At the same time, scholars trained in modern formal techniques have come to view Aristotle with new respect, not so much for the correctness of his results as for the remarkable similarity in spirit between much of his work and modern logic. As Jonathan Lear has put it, “Aristotle shares with modern logicians a fundamental interest in metatheory”: his primary goal is not to offer a practical guide to argumentation but to study the properties of inferential systems themselves.

The ancient commentators grouped together several of Aristotle’s treatises under the title Organon (“Instrument”) and regarded them as comprising his logical works:

• On Interpretation
• Prior Analytics
• Posterior Analytics
• On Sophistical Refutations

In fact, the title Organon reflects a much later controversy about whether logic is a part of philosophy (as the Stoics maintained) or merely a tool used by philosophy (as the later Peripatetics thought); calling the logical works “The Instrument” is a way of taking sides on this point. Aristotle himself never uses this term, nor does he give much indication that these particular treatises form some kind of group, though there are frequent cross-references between the Topics and the Analytics . On the other hand, Aristotle treats the Prior and Posterior Analytics as one work, and On Sophistical Refutations is a final section, or an appendix, to the Topics ). To these works should be added the Rhetoric , which explicitly declares its reliance on the Topics .

3. The Subject of Logic: “Syllogisms”

All Aristotle’s logic revolves around one notion: the deduction ( sullogismos ). A thorough explanation of what a deduction is, and what they are composed of, will necessarily lead us through the whole of his theory. What, then, is a deduction? Aristotle says:

A deduction is speech ( logos ) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so. ( Prior Analytics I.2, 24b18–20)

Each of the “things supposed” is a premise ( protasis ) of the argument, and what “results of necessity” is the conclusion ( sumperasma ).

The core of this definition is the notion of “resulting of necessity” ( ex anankês sumbainein ). This corresponds to a modern notion of logical consequence: \(X\) results of necessity from \(Y\) and \(Z\) if it would be impossible for \(X\) to be false when \(Y\) and \(Z\) are true. We could therefore take this to be a general definition of “valid argument”.

Deductions are one of two species of argument recognized by Aristotle. The other species is induction ( epagôgê ). He has far less to say about this than deduction, doing little more than characterize it as “argument from the particular to the universal”. However, induction (or something very much like it) plays a crucial role in the theory of scientific knowledge in the Posterior Analytics : it is induction, or at any rate a cognitive process that moves from particulars to their generalizations, that is the basis of knowledge of the indemonstrable first principles of sciences.

Despite its wide generality, Aristotle’s definition of deduction is not a precise match for a modern definition of validity. Some of the differences may have important consequences:

• Aristotle explicitly says that what results of necessity must be different from what is supposed. This would rule out arguments in which the conclusion is identical to one of the premises. Modern notions of validity regard such arguments as valid, though trivially so.
• The plural “certain things having been supposed” was taken by some ancient commentators to rule out by definition arguments with only one premise, and Aristotle himself says in some places that nothing new follows from just one premise.
• The force of the qualification “because of their being so” has sometimes been seen as ruling out arguments in which the conclusion is not ‘relevant’ to the premises, e.g., arguments in which the premises are inconsistent, arguments with conclusions that would follow from any premises whatsoever, or arguments with superfluous premises.

Of these three possible restrictions, the most interesting would be the third. This could be (and has been) interpreted as committing Aristotle to something like a relevance logic . In fact, there are passages that appear to confirm this. However, this is too complex a matter to discuss here.

However the definition is interpreted, it is clear that Aristotle does not mean to restrict it only to a subset of the valid arguments. This is why I have translated sullogismos with ‘deduction’ rather than its English cognate. In modern usage, ‘syllogism’ means an argument of a very specific form. Moreover, modern usage distinguishes between valid syllogisms (the conclusions of which follow from their premises) and invalid syllogisms (the conclusions of which do not follow from their premises). The second of these is inconsistent with Aristotle’s use: since he defines a sullogismos as an argument in which the conclusion results of necessity from the premises, “invalid sullogismos ” is a contradiction in terms. The first is also at least highly misleading, since Aristotle does not appear to think that the sullogismoi are simply an interesting subset of the valid arguments. Moreover (see below), Aristotle expends great efforts to argue that every valid argument, in a broad sense, can be “reduced” to an argument, or series of arguments, in something like one of the forms traditionally called a syllogism. If we translate sullogismos as “syllogism”, this becomes the trivial claim “Every syllogism is a syllogism”,

4. Premises: The Structures of Assertions

Syllogisms are structures of sentences each of which can meaningfully be called true or false: assertions ( apophanseis ), in Aristotle’s terminology. According to Aristotle, every such sentence must have the same structure: it must contain a subject ( hupokeimenon ) and a predicate and must either affirm or deny the predicate of the subject. Thus, every assertion is either the affirmation kataphasis or the denial ( apophasis ) of a single predicate of a single subject.

In On Interpretation , Aristotle argues that a single assertion must always either affirm or deny a single predicate of a single subject. Thus, he does not recognize sentential compounds, such as conjunctions and disjunctions, as single assertions. This appears to be a deliberate choice on his part: he argues, for instance, that a conjunction is simply a collection of assertions, with no more intrinsic unity than the sequence of sentences in a lengthy account (e.g. the entire Iliad , to take Aristotle’s own example). Since he also treats denials as one of the two basic species of assertion, he does not view negations as sentential compounds. His treatment of conditional sentences and disjunctions is more difficult to appraise, but it is at any rate clear that Aristotle made no efforts to develop a sentential logic. Some of the consequences of this for his theory of demonstration are important.

Subjects and predicates of assertions are terms . A term ( horos ) can be either individual, e.g. Socrates , Plato or universal, e.g. human , horse , animal , white . Subjects may be either individual or universal, but predicates can only be universals: Socrates is human , Plato is not a horse , horses are animals , humans are not horses .

The word universal ( katholou ) appears to be an Aristotelian coinage. Literally, it means “of a whole”; its opposite is therefore “of a particular” ( kath’ hekaston ). Universal terms are those which can properly serve as predicates, while particular terms are those which cannot.

This distinction is not simply a matter of grammatical function. We can readily enough construct a sentence with “Socrates” as its grammatical predicate: “The person sitting down is Socrates”. Aristotle, however, does not consider this a genuine predication. He calls it instead a merely accidental or incidental ( kata sumbebêkos ) predication. Such sentences are, for him, dependent for their truth values on other genuine predications (in this case, “Socrates is sitting down”).

Consequently, predication for Aristotle is as much a matter of metaphysics as a matter of grammar. The reason that the term Socrates is an individual term and not a universal is that the entity which it designates is an individual, not a universal. What makes white and human universal terms is that they designate universals.

Further discussion of these issues can be found in the entry on Aristotle’s metaphysics .

Aristotle takes some pains in On Interpretation to argue that to every affirmation there corresponds exactly one denial such that that denial denies exactly what that affirmation affirms. The pair consisting of an affirmation and its corresponding denial is a contradiction ( antiphasis ). In general, Aristotle holds, exactly one member of any contradiction is true and one false: they cannot both be true, and they cannot both be false. However, he appears to make an exception for propositions about future events, though interpreters have debated extensively what this exception might be (see further discussion below). The principle that contradictories cannot both be true has fundamental importance in Aristotle’s metaphysics (see further discussion below).

One major difference between Aristotle’s understanding of predication and modern (i.e., post-Fregean) logic is that Aristotle treats individual predications and general predications as similar in logical form: he gives the same analysis to “Socrates is an animal” and “Humans are animals”. However, he notes that when the subject is a universal, predication takes on two forms: it can be either universal or particular . These expressions are parallel to those with which Aristotle distinguishes universal and particular terms, and Aristotle is aware of that, explicitly distinguishing between a term being a universal and a term being universally predicated of another.

Whatever is affirmed or denied of a universal subject may be affirmed or denied of it it universally ( katholou or “of all”, kata pantos ), in part ( kata meros , en merei ), or indefinitely ( adihoristos ).

4.3.1 The “Square of Opposition”

In On Interpretation , Aristotle spells out the relationships of contradiction for sentences with universal subjects as follows:

Simple as it appears, this table raises important difficulties of interpretation (for a thorough discussion, see the entry on the square of opposition ).

In the Prior Analytics , Aristotle adopts a somewhat artificial way of expressing predications: instead of saying “\(X\) is predicated of \(Y\)” he says “\(X\) belongs ( huparchei ) to \(Y\)”. This should really be regarded as a technical expression. The verb huparchein usually means either “begin” or “exist, be present”, and Aristotle’s usage appears to be a development of this latter use.

4.3.2 Some Convenient Abbreviations

For clarity and brevity, I will use the following semi-traditional abbreviations for Aristotelian categorical sentences (note that the predicate term comes first and the subject term second ):

5. The Syllogistic

Aristotle’s most famous achievement as logician is his theory of inference, traditionally called the syllogistic (though not by Aristotle). That theory is in fact the theory of inferences of a very specific sort: inferences with two premises, each of which is a categorical sentence, having exactly one term in common, and having as conclusion a categorical sentence the terms of which are just those two terms not shared by the premises. Aristotle calls the term shared by the premises the middle term ( meson ) and each of the other two terms in the premises an extreme ( akron ). The middle term must be either subject or predicate of each premise, and this can occur in three ways: the middle term can be the subject of one premise and the predicate of the other, the predicate of both premises, or the subject of both premises. Aristotle refers to these term arrangements as figures ( schêmata ):

Aristotle calls the term which is the predicate of the conclusion the major term and the term which is the subject of the conclusion the minor term. The premise containing the major term is the major premise , and the premise containing the minor term is the minor premise .

Aristotle then systematically investigates all possible combinations of two premises in each of the three figures. For each combination, he either demonstrates that some conclusion necessarily follows or demonstrates that no conclusion follows. The results he states are correct.

Aristotle’s proofs can be divided into two categories, based on a distinction he makes between “perfect” or “complete” ( teleios ) deductions and “imperfect” or “incomplete” ( atelês ) deductions. A deduction is perfect if it “needs no external term in order to show the necessary result” (24b23–24), and it is imperfect if it “needs one or several in addition that are necessary because of the terms supposed but were not assumed through premises” (24b24–25). The precise interpretation of this distinction is debatable, but it is at any rate clear that Aristotle regards the perfect deductions as not in need of proof in some sense. For imperfect deductions, Aristotle does give proofs, which invariably depend on the perfect deductions. Thus, with some reservations, we might compare the perfect deductions to the axioms or primitive rules of a deductive system.

In the proofs for imperfect deductions, Aristotle says that he “reduces” ( anagein ) each case to one of the perfect forms and that they are thereby “completed” or “perfected”. These completions are either probative ( deiktikos : a modern translation might be “direct”) or through the impossible ( dia to adunaton ).

A direct deduction is a series of steps leading from the premises to the conclusion, each of which is either a conversion of a previous step or an inference from two previous steps relying on a first-figure deduction. Conversion, in turn, is inferring from a proposition another which has the subject and predicate interchanged. Specifically, Aristotle argues that three such conversions are sound:

He undertakes to justify these in An. Pr. I.2. From a modern standpoint, the third is sometimes regarded with suspicion. Using it we can get Some monsters are chimeras from the apparently true All chimeras are monsters ; but the former is often construed as implying in turn There is something which is a monster and a chimera , and thus that there are monsters and there are chimeras. In fact, this simply points up something about Aristotle’s system: Aristotle in effect supposes that all terms in syllogisms are non-empty. (For further discussion of this point, see the entry on the square of opposition ).

As an example of the procedure, we may take Aristotle’s proof of Camestres . He says:

If \(M\) belongs to every \(N\) but to no \(X\), then neither will \(N\) belong to any \(X\). For if \(M\) belongs to no \(X\), then neither does \(X\) belong to any \(M\); but \(M\) belonged to every \(N\); therefore, \(X\) will belong to no \(N\) (for the first figure has come about). And since the privative converts, neither will \(N\) belong to any \(X\). ( An. Pr. I.5, 27a9–12)

From this text, we can extract an exact formal proof, as follows:

A completion or proof “through the impossible” shows that a certain conclusion follows from a pair of premises by assuming as a third premise the denial of that conclusion and giving a deduction, from it and one of the original premises, the denial (or the contrary) of the other premises. This is the deduction of an “impossible”, and Aristotle’s proof ends at that point. An example is his proof of Baroco in 27a36–b1:

Aristotle proves invalidity by constructing counterexamples. This is very much in the spirit of modern logical theory: all that it takes to show that a certain form is invalid is a single instance of that form with true premises and a false conclusion. However, Aristotle states his results not by saying that certain premise-conclusion combinations are invalid but by saying that certain premise pairs do not “syllogize”: that is, that, given the pair in question, examples can be constructed in which premises of that form are true and a conclusion of any of the four possible forms is false.

When possible, he does this by a clever and economical method: he gives two triplets of terms, one of which makes the premises true and a universal affirmative “conclusion” true, and the other of which makes the premises true and a universal negative “conclusion” true. The first is a counterexample for an argument with either an \(E\) or an \(O\) conclusion, and the second is a counterexample for an argument with either an \(A\) or an \(I\) conclusion.

In Prior Analytics I.4–6, Aristotle shows that the premise combinations given in the following table yield deductions and that all other premise combinations fail to yield a deduction. In the terminology traditional since the middle ages, each of these combinations is known as a mood Latin modus , “way”, which in turn is a translation of Greek tropos ). Aristotle, however, does not use this expression and instead refers to “the arguments in the figures”.

In this table, “\(\vdash\)” separates premises from conclusion; it may be read “therefore”. The second column lists the medieval mnemonic name associated with the inference (these are still widely used, and each is actually a mnemonic for Aristotle’s proof of the mood in question). The third column briefly summarizes Aristotle’s procedure for demonstrating the deduction.

Table of the Deductions in the Figures

Having established which deductions in the figures are possible, Aristotle draws a number of metatheoretical conclusions, including:

• No deduction has two negative premises
• No deduction has two particular premises
• A deduction with an affirmative conclusion must have two affirmative premises
• A deduction with a negative conclusion must have one negative premise.
• A deduction with a universal conclusion must have two universal premises

He also proves the following metatheorem:

All deductions can be reduced to the two universal deductions in the first figure.

His proof of this is elegant. First, he shows that the two particular deductions of the first figure can be reduced, by proof through impossibility, to the universal deductions in the second figure:

He then observes that since he has already shown how to reduce all the particular deductions in the other figures except Baroco and Bocardo to Darii and Ferio , these deductions can thus be reduced to Barbara and Celarent . This proof is strikingly similar both in structure and in subject to modern proofs of the redundancy of axioms in a system.

Many more metatheoretical results, some of them quite sophisticated, are proved in Prior Analytics I.45 and in Prior Analytics II. As noted below, some of Aristotle’s metatheoretical results are appealed to in the epistemological arguments of the Posterior Analytics .

Aristotle follows his treatment of “arguments in the figures” with a much longer, and much more problematic, discussion of what happens to these figured arguments when we add the qualifications “necessarily” and “possibly” to their premises in various ways. In contrast to the syllogistic itself (or, as commentators like to call it, the assertoric syllogistic), this modal syllogistic appears to be much less satisfactory and is certainly far more difficult to interpret. Here, I only outline Aristotle’s treatment of this subject and note some of the principal points of interpretive controversy.

5.6.1 The Definitions of the Modalities

Modern modal logic treats necessity and possibility as interdefinable: “necessarily P” is equivalent to “not possibly not P”, and “possibly P” to “not necessarily not P”. Aristotle gives these same equivalences in On Interpretation . However, in Prior Analytics , he makes a distinction between two notions of possibility. On the first, which he takes as his preferred notion, “possibly P” is equivalent to “not necessarily P and not necessarily not P”. He then acknowledges an alternative definition of possibility according to the modern equivalence, but this plays only a secondary role in his system.

5.6.2 Aristotle’s General Approach

Aristotle builds his treatment of modal syllogisms on his account of non-modal ( assertoric ) syllogisms: he works his way through the syllogisms he has already proved and considers the consequences of adding a modal qualification to one or both premises. Most often, then, the questions he explores have the form: “Here is an assertoric syllogism; if I add these modal qualifications to the premises, then what modally qualified form of the conclusion (if any) follows?”. A premise can have one of three modalities: it can be necessary, possible, or assertoric. Aristotle works through the combinations of these in order:

• Two necessary premises
• One necessary and one assertoric premise
• Two possible premises
• One assertoric and one possible premise
• One necessary and one possible premise

Though he generally considers only premise combinations which syllogize in their assertoric forms, he does sometimes extend this; similarly, he sometimes considers conclusions in addition to those which would follow from purely assertoric premises.

Since this is his procedure, it is convenient to describe modal syllogisms in terms of the corresponding non-modal syllogism plus a triplet of letters indicating the modalities of premises and conclusion: \(N\) = “necessary”, \(P\) = “possible”, \(A\) = “assertoric”. Thus, “Barbara \(NAN\)” would mean “The form Barbara with necessary major premise, assertoric minor premise, and necessary conclusion”. I use the letters “\(N\)” and “\(P\)” as prefixes for premises as well; a premise with no prefix is assertoric. Thus, Barbara \(NAN\) would be \(NAab, Abc \vdash NAac\).

5.6.3 Modal Conversions

As in the case of assertoric syllogisms, Aristotle makes use of conversion rules to prove validity. The conversion rules for necessary premises are exactly analogous to those for assertoric premises:

Possible premises behave differently, however. Since he defines “possible” as “neither necessary nor impossible”, it turns out that \(x\) is possibly \(F\) entails, and is entailed by, \(x\) is possibly not \(F\). Aristotle generalizes this to the case of categorical sentences as follows:

In addition, Aristotle uses the intermodal principle \(N\rightarrow A\): that is, a necessary premise entails the corresponding assertoric one. However, because of his definition of possibility, the principle \(A\rightarrow P\) does not generally hold: if it did, then \(N\rightarrow P\) would hold, but on his definition “necessarily \(P\)” and “possibly \(P\)” are actually inconsistent (“possibly \(P\)” entails “possibly not \(P\)”).

This leads to a further complication. The denial of “possibly \(P\)” for Aristotle is “either necessarily \(P\) or necessarily not \(P\)”. The denial of “necessarily \(P\)” is still more difficult to express in terms of a combination of modalities: “either possibly \(P\) (and thus possibly not \(P\)) or necessarily not \(P\)” This is important because of Aristotle’s proof procedures, which include proof through impossibility. If we give a proof through impossibility in which we assume a necessary premise, then the conclusion we ultimately establish is simply the denial of that necessary premise, not a “possible” conclusion in Aristotle’s sense. Such propositions do occur in his system, but only in exactly this way, i.e., as conclusions established by proof through impossiblity from necessary assumptions. Somewhat confusingly, Aristotle calls such propositions “possible” but immediately adds “ not in the sense defined”: in this sense, “possibly \(Oab\)” is simply the denial of “necessarily \(Aab\)”. Such propositions appear only as premises, never as conclusions.

5.6.4 Syllogisms with Necessary Premises

Aristotle holds that an assertoric syllogism remains valid if “necessarily” is added to its premises and its conclusion: the modal pattern \(NNN\) is always valid. He does not treat this as a trivial consequence but instead offers proofs; in all but two cases, these are parallel to those offered for the assertoric case. The exceptions are Baroco and Bocardo , which he proved in the assertoric case through impossibility: attempting to use that method here would require him to take the denial of a necessary \(O\) proposition as hypothesis, raising the complication noted above, and he uses the procedure he calls ecthesis instead (see Smith 1982).

5.6.5 The Problem of the “Two Barbaras” and Other Problems of Interpretation

Since a necessary premise entails an assertoric premise, every \(AN\) or \(NA\) combination of premises will entail the corresponding \(AA\) pair, and thus the corresponding \(A\) conclusion. Thus, \(ANA\) and \(NAA\) syllogisms are always valid. However, Aristotle holds that some, but not all, \(ANN\) and \(NAN\) combinations are valid. Specifically, he accepts Barbara \(NAN\) but rejects Barbara \(ANN\). Almost from Aristotle’s own time, interpreters have found his reasons for this distinction obscure, or unpersuasive, or both, and often have not followed his view. His close associated Theophrastus, for instance, adopted the simpler rule that the modality of the conclusion of a syllogism was always the “weakest” modality found in either premise, where \(N\) is stronger than \(A\) and \(A\) is stronger than \(P\) (and where \(P\) probably has to be defined as “not necessarily not”).

Beginning with Albrecht Becker, interpreters using the methods of modern formal logic to interpret Aristotle’s modal logic have seen the Two-Barbaras problem as only one of a series of difficulties in giving a coherent interpretation of the modal syllogistic. A very wide range of reconstructions has been proposed: see Becker 1933, McCall 1963, Nortmann 1996, Van Rijen 1989, Patterson 1995, Thomason 1993, Thom 1996, Rini 2012, Malink 2013. The majority of reconstructions do not attempt to reproduce every detail of Aristotle’s exposition but instead produce modified reconstructions that abandon some of those results. Malink 2013, however, offers a reconstruction that reproduces everything Aristotle says, although the resulting model introduces a high degree of complexity. (This subject quickly becomes too complex for summarizing in this brief article.

6. Demonstrations and Demonstrative Sciences

A demonstration ( apodeixis ) is “a deduction that produces knowledge”. Aristotle’s Posterior Analytics contains his account of demonstrations and their role in knowledge. From a modern perspective, we might think that this subject moves outside of logic to epistemology. From Aristotle’s perspective, however, the connection of the theory of sullogismoi with the theory of knowledge is especially close.

The subject of the Posterior Analytics is epistêmê . This is one of several Greek words that can reasonably be translated “knowledge”, but Aristotle is concerned only with knowledge of a certain type (as will be explained below). There is a long tradition of translating epistêmê in this technical sense as science , and I shall follow that tradition here. However, readers should not be misled by the use of that word. In particular, Aristotle’s theory of science cannot be considered a counterpart to modern philosophy of science, at least not without substantial qualifications.

We have scientific knowledge, according to Aristotle, when we know:

the cause why the thing is, that it is the cause of this, and that this cannot be otherwise. ( Posterior Analytics I.2)

This implies two strong conditions on what can be the object of scientific knowledge:

• Only what is necessarily the case can be known scientifically
• Scientific knowledge is knowledge of causes

He then proceeds to consider what science so defined will consist in, beginning with the observation that at any rate one form of science consists in the possession of a demonstration ( apodeixis ), which he defines as a “scientific deduction”:

by “scientific” ( epistêmonikon ), I mean that in virtue of possessing it, we have knowledge.

The remainder of Posterior Analytics I is largely concerned with two tasks: spelling out the nature of demonstration and demonstrative science and answering an important challenge to its very possibility. Aristotle first tells us that a demonstration is a deduction in which the premises are:

• primary ( prota )
• immediate ( amesa , “without a middle”)
• better known or more familiar ( gnôrimôtera ) than the conclusion
• prior to the conclusion
• causes ( aitia ) of the conclusion

The interpretation of all these conditions except the first has been the subject of much controversy. Aristotle clearly thinks that science is knowledge of causes and that in a demonstration, knowledge of the premises is what brings about knowledge of the conclusion. The fourth condition shows that the knower of a demonstration must be in some better epistemic condition towards them, and so modern interpreters often suppose that Aristotle has defined a kind of epistemic justification here. However, as noted above, Aristotle is defining a special variety of knowledge. Comparisons with discussions of justification in modern epistemology may therefore be misleading.

The same can be said of the terms “primary”, “immediate” and “better known”. Modern interpreters sometimes take “immediate” to mean “self-evident”; Aristotle does say that an immediate proposition is one “to which no other is prior”, but (as I suggest in the next section) the notion of priority involved is likely a notion of logical priority that it is hard to detach from Aristotle’s own logical theories. “Better known” has sometimes been interpreted simply as “previously known to the knower of the demonstration” (i.e., already known in advance of the demonstration). However, Aristotle explicitly distinguishes between what is “better known for us” with what is “better known in itself” or “in nature” and says that he means the latter in his definition. In fact, he says that the process of acquiring scientific knowledge is a process of changing what is better known “for us”, until we arrive at that condition in which what is better known in itself is also better known for us.

In Posterior Analytics I.2, Aristotle considers two challenges to the possibility of science. One party (dubbed the “agnostics” by Jonathan Barnes) began with the following two premises:

• Whatever is scientifically known must be demonstrated.
• The premises of a demonstration must be scientifically known.

They then argued that demonstration is impossible with the following dilemma:

• If the premises of a demonstration are scientifically known, then they must be demonstrated.
• The premises from which each premise are demonstrated must be scientifically known.
• Either this process continues forever, creating an infinite regress of premises, or it comes to a stop at some point.
• If it continues forever, then there are no first premises from which the subsequent ones are demonstrated, and so nothing is demonstrated.
• On the other hand, if it comes to a stop at some point, then the premises at which it comes to a stop are undemonstrated and therefore not scientifically known; consequently, neither are any of the others deduced from them.
• Therefore, nothing can be demonstrated.

A second group accepted the agnostics’ view that scientific knowledge comes only from demonstration but rejected their conclusion by rejecting the dilemma. Instead, they maintained:

• Demonstration “in a circle” is possible, so that it is possible for all premises also to be conclusions and therefore demonstrated.

Aristotle does not give us much information about how circular demonstration was supposed to work, but the most plausible interpretation would be supposing that at least for some set of fundamental principles, each principle could be deduced from the others. (Some modern interpreters have compared this position to a coherence theory of knowledge.) However their position worked, the circular demonstrators claimed to have a third alternative avoiding the agnostics’ dilemma, since circular demonstration gives us a regress that is both unending (in the sense that we never reach premises at which it comes to a stop) and finite (because it works its way round the finite circle of premises).

Aristotle rejects circular demonstration as an incoherent notion on the grounds that the premises of any demonstration must be prior (in an appropriate sense) to the conclusion, whereas a circular demonstration would make the same premises both prior and posterior to one another (and indeed every premise prior and posterior to itself). He agrees with the agnostics’ analysis of the regress problem: the only plausible options are that it continues indefinitely or that it “comes to a stop” at some point. However, he thinks both the agnostics and the circular demonstrators are wrong in maintaining that scientific knowledge is only possible by demonstration from premises scientifically known: instead, he claims, there is another form of knowledge possible for the first premises, and this provides the starting points for demonstrations.

To solve this problem, Aristotle needs to do something quite specific. It will not be enough for him to establish that we can have knowledge of some propositions without demonstrating them: unless it is in turn possible to deduce all the other propositions of a science from them, we shall not have solved the regress problem. Moreover (and obviously), it is no solution to this problem for Aristotle simply to assert that we have knowledge without demonstration of some appropriate starting points. He does indeed say that it is his position that we have such knowledge ( An. Post. I.2,), but he owes us an account of why that should be so.

Aristotle’s account of knowledge of the indemonstrable first premises of sciences is found in Posterior Analytics II.19, long regarded as a difficult text to interpret. Briefly, what he says there is that it is another cognitive state, nous (translated variously as “insight”, “intuition”, “intelligence”), which knows them. There is wide disagreement among commentators about the interpretation of his account of how this state is reached; I will offer one possible interpretation. First, Aristotle identifies his problem as explaining how the principles can “become familiar to us”, using the same term “familiar” ( gnôrimos ) that he used in presenting the regress problem. What he is presenting, then, is not a method of discovery but a process of becoming wise. Second, he says that in order for knowledge of immediate premises to be possible, we must have a kind of knowledge of them without having learned it, but this knowledge must not be as “precise” as the knowledge that a possessor of science must have. The kind of knowledge in question turns out to be a capacity or power ( dunamis ) which Aristotle compares to the capacity for sense-perception: since our senses are innate, i.e., develop naturally, it is in a way correct to say that we know what e.g. all the colors look like before we have seen them: we have the capacity to see them by nature, and when we first see a color we exercise this capacity without having to learn how to do so first. Likewise, Aristotle holds, our minds have by nature the capacity to recognize the starting points of the sciences.

In the case of sensation, the capacity for perception in the sense organ is actualized by the operation on it of the perceptible object. Similarly, Aristotle holds that coming to know first premises is a matter of a potentiality in the mind being actualized by experience of its proper objects: “The soul is of such a nature as to be capable of undergoing this”. So, although we cannot come to know the first premises without the necessary experience, just as we cannot see colors without the presence of colored objects, our minds are already so constituted as to be able to recognize the right objects, just as our eyes are already so constituted as to be able to perceive the colors that exist.

It is considerably less clear what these objects are and how it is that experience actualizes the relevant potentialities in the soul. Aristotle describes a series of stages of cognition. First is what is common to all animals: perception of what is present. Next is memory, which he regards as a retention of a sensation: only some animals have this capacity. Even fewer have the next capacity, the capacity to form a single experience ( empeiria ) from many repetitions of the same memory. Finally, many experiences repeated give rise to knowledge of a single universal ( katholou ). This last capacity is present only in humans.

See Section 7 of the entry on Aristotle’s psychology for more on his views about mind.

7. Definitions

The definition ( horos , horismos ) was an important matter for Plato and for the Early Academy. Concern with answering the question “What is so-and-so?” are at the center of the majority of Plato’s dialogues, some of which (most elaborately the Sophist ) propound methods for finding definitions. External sources (sometimes the satirical remarks of comedians) also reflect this Academic concern with definitions. Aristotle himself traces the quest for definitions back to Socrates.

For Aristotle, a definition is “an account which signifies what it is to be for something” ( logos ho to ti ên einai sêmainei ). The phrase “what it is to be” and its variants are crucial: giving a definition is saying, of some existent thing, what it is, not simply specifying the meaning of a word (Aristotle does recognize definitions of the latter sort, but he has little interest in them).

The notion of “what it is to be” for a thing is so pervasive in Aristotle that it becomes formulaic: what a definition expresses is “the what-it-is-to-be” ( to ti ên einai ), or in modern terminology, its essence.

Since a definition defines an essence, only what has an essence can be defined. What has an essence, then? That is one of the central questions of Aristotle’s metaphysics; once again, we must leave the details to another article. In general, however, it is not individuals but rather species ( eidos : the word is one of those Plato uses for “Form”) that have essences. A species is defined by giving its genus ( genos ) and its differentia ( diaphora ): the genus is the kind under which the species falls, and the differentia tells what characterizes the species within that genus. As an example, human might be defined as animal (the genus) having the capacity to reason (the differentia).

Essential Predication and the Predicables

Underlying Aristotle’s concept of a definition is the concept of essential predication ( katêgoreisthai en tôi ti esti , predication in the what it is). In any true affirmative predication, the predicate either does or does not “say what the subject is”, i.e., the predicate either is or is not an acceptable answer to the question “What is it?” asked of the subject. Bucephalus is a horse, and a horse is an animal; so, “Bucephalus is a horse” and “Bucephalus is an animal” are essential predications. However, “Bucephalus is brown”, though true, does not state what Bucephalus is but only says something about him.

Since a thing’s definition says what it is, definitions are essentially predicated. However, not everything essentially predicated is a definition. Since Bucephalus is a horse, and horses are a kind of mammal, and mammals are a kind of animal, “horse” “mammal” and “animal” are all essential predicates of Bucephalus. Moreover, since what a horse is is a kind of mammal, “mammal” is an essential predicate of horse. When predicate \(X\) is an essential predicate of \(Y\) but also of other things, then \(X\) is a genus ( genos ) of \(Y\).

A definition of \(X\) must not only be essentially predicated of it but must also be predicated only of it: to use a term from Aristotle’s Topics , a definition and what it defines must “counterpredicate” ( antikatêgoreisthai ) with one another. \(X\) counterpredicates with \(Y\) if \(X\) applies to what \(Y\) applies to and conversely. Though X’s definition must counterpredicate with \(X\), not everything that counterpredicates with \(X\) is its definition. “Capable of laughing”, for example, counterpredicates with “human” but fails to be its definition. Such a predicate (non-essential but counterpredicating) is a peculiar property or proprium ( idion ).

Finally, if \(X\) is predicated of \(Y\) but is neither essential nor counterpredicates, then \(X\) is an accident ( sumbebêkos ) of \(Y\).

Aristotle sometimes treats genus, peculiar property, definition, and accident as including all possible predications (e.g. Topics I). Later commentators listed these four and the differentia as the five predicables , and as such they were of great importance to late ancient and to medieval philosophy (e.g., Porphyry).

The notion of essential predication is connected to what are traditionally called the categories ( katêgoriai ). In a word, Aristotle is famous for having held a “doctrine of categories”. Just what that doctrine was, and indeed just what a category is, are considerably more vexing questions. They also quickly take us outside his logic and into his metaphysics. Here, I will try to give a very general overview, beginning with the somewhat simpler question “What categories are there?”

We can answer this question by listing the categories. Here are two passages containing such lists:

We should distinguish the kinds of predication ( ta genê tôn katêgoriôn ) in which the four predications mentioned are found. These are ten in number: what-it-is, quantity, quality, relative, where, when, being-in-a-position, having, doing, undergoing. An accident, a genus, a peculiar property and a definition will always be in one of these categories. ( Topics I.9, 103b20–25) Of things said without any combination, each signifies either substance or quantity or quality or a relative or where or when or being-in-a-position or having or doing or undergoing. To give a rough idea, examples of substance are man, horse; of quantity: four-foot, five-foot; of quality: white, literate; of a relative: double, half, larger; of where: in the Lyceum, in the market-place; of when: yesterday, last year; of being-in-a-position: is-lying, is-sitting; of having: has-shoes-on, has-armor-on; of doing: cutting, burning; of undergoing: being-cut, being-burned. ( Categories 4, 1b25–2a4, tr. Ackrill, slightly modified)

These two passages give ten-item lists, identical except for their first members. What are they lists \(of\)? Here are three ways they might be interpreted:

The word “category” ( katêgoria ) means “predication”. Aristotle holds that predications and predicates can be grouped into several largest “kinds of predication” ( genê tôn katêgoriôn ). He refers to this classification frequently, often calling the “kinds of predication” simply “the predications”, and this (by way of Latin) leads to our word “category”.

• First, the categories may be kinds of predicate : predicates (or, more precisely, predicate expressions) can be divided into ten separate classes, with each expression belonging to just one class. This comports well with the root meaning of the word katêgoria (“predication”). On this interpretation, the categories arise out of considering the most general types of question that can be asked about something: “ What is it?”; “ How much is it?”; “ What sort is it?”; “ Where is it?”; “ What is it doing ?” Answers appropriate to one of these questions are nonsensical in response to another (“When is it?” “A horse”). Thus, the categories may rule out certain kinds of question as ill-formed or confused. This plays an important role in Aristotle’s metaphysics.
• Second, the categories may be seen as classifications of predications , that is, kinds of relation that may hold between the predicate and the subject of a predication. To say of Socrates that he is human is to say what he \(is\), whereas to say that he is literate is not to say what he is but rather to give a quality that he has . For Aristotle, the relation of predicate to subject in these two sentences is quite different (in this respect he differs both from Plato and from modern logicians). The categories may be interpreted as ten different ways in which a predicate may be related to its subject. This last division has importance for Aristotle’s logic as well as his metaphysics.
• Third, the categories may be seen as kinds of entity , as highest genera or kinds of thing that are. A given thing can be classified under a series of progressively wider genera: Socrates is a human, a mammal, an animal, a living being. The categories are the highest such genera. Each falls under no other genus, and each is completely separate from the others. This distinction is of critical importance to Aristotle’s metaphysics.

Which of these interpretations fits best with the two passages above? The answer appears to be different in the two cases. This is most evident if we take note of point in which they differ: the Categories lists substance ( ousia ) in first place, while the Topics list what-it-is ( ti esti ). A substance, for Aristotle, is a type of entity, suggesting that the Categories list is a list of types of entity.

On the other hand, the expression “what-it-is” suggests most strongly a type of predication. Indeed, the Topics confirms this by telling us that we can “say what it is” of an entity falling under any of the categories:

an expression signifying what-it-is will sometimes signify a substance, sometimes a quantity, sometimes a quality, and sometimes one of the other categories.

As Aristotle explains, if I say that Socrates is a man, then I have said what Socrates is and signified a substance; if I say that white is a color, then I have said what white is and signified a quality; if I say that some length is a foot long, then I have said what it is and signified a quantity; and so on for the other categories. What-it-is, then, here designates a kind of predication, not a kind of entity.

This might lead us to conclude that the categories in the Topics are only to be interpreted as kinds of predicate or predication, those in the Categories as kinds of being. Even so, we would still want to ask what the relationship is between these two nearly-identical lists of terms, given these distinct interpretations. However, the situation is much more complicated. First, there are dozens of other passages in which the categories appear. Nowhere else do we find a list of ten, but we do find shorter lists containing eight, or six, or five, or four of them (with substance/what-it-is, quality, quantity, and relative the most common). Aristotle describes what these lists are lists of in different ways: they tell us “how being is divided”, or “how many ways being is said”, or “the figures of predication” (ta schêmata tês katêgorias). The designation of the first category also varies: we find not only “substance” and “what it is” but also the expressions “this” or “the this” ( tode ti , to tode , to ti ). These latter expressions are closely associated with, but not synonymous with, substance. He even combines the latter with “what-it-is” ( Metaphysics Z 1, 1028a10: “… one sense signifies what it is and the this, one signifies quality …”).

Moreover, substances are for Aristotle fundamental for predication as well as metaphysically fundamental. He tells us that everything that exists exists because substances exist: if there were no substances, there would not be anything else. He also conceives of predication as reflecting a metaphysical relationship (or perhaps more than one, depending on the type of predication). The sentence “Socrates is pale” gets its truth from a state of affairs consisting of a substance (Socrates) and a quality (whiteness) which is in that substance. At this point we have gone far outside the realm of Aristotle’s logic into his metaphysics, the fundamental question of which, according to Aristotle, is “What is a substance?”. (For further discussion of this topic, see the entry on Aristotle’s Categories and the entry on Aristotle’s metaphysics , ( Section 2 ).

See Frede 1981, Ebert 1985 for additional discussion of Aristotle’s lists of categories.

For convenience of reference, I include a table of the categories, along with Aristotle’s examples and the traditional names often used for them. For reasons explained above, I have treated the first item in the list quite differently, since an example of a substance and an example of a what-it-is are necessarily (as one might put it) in different categories.

In the Sophist , Plato introduces a procedure of “Division” as a method for discovering definitions. To find a definition of \(X\), first locate the largest kind of thing under which \(X\) falls; then, divide that kind into two parts, and decide which of the two \(X\) falls into. Repeat this method with the part until \(X\) has been fully located.

This method is part of Aristotle’s Platonic legacy. His attitude towards it, however, is complex. He adopts a view of the proper structure of definitions that is closely allied to it: a correct definition of \(X\) should give the genus ( genos : kind or family) of \(X\), which tells what kind of thing \(X\) is, and the differentia ( diaphora : difference) which uniquely identifies \(X\) within that genus. Something defined in this way is a species ( eidos : the term is one of Plato’s terms for “Form”), and the differentia is thus the “difference that makes a species” ( eidopoios diaphora , “specific difference”). In Posterior Analytics II.13, he gives his own account of the use of Division in finding definitions.

However, Aristotle is strongly critical of the Platonic view of Division as a method for establishing definitions. In Prior Analytics I.31, he contrasts Division with the syllogistic method he has just presented, arguing that Division cannot actually prove anything but rather assumes the very thing it is supposed to be proving. He also charges that the partisans of Division failed to understand what their own method was capable of proving.

Closely related to this is the discussion, in Posterior Analytics II.3–10, of the question whether there can be both definition and demonstration of the same thing (that is, whether the same result can be established either by definition or by demonstration). Since the definitions Aristotle is interested in are statements of essences, knowing a definition is knowing, of some existing thing, what it is. Consequently, Aristotle’s question amounts to a question whether defining and demonstrating can be alternative ways of acquiring the same knowledge. His reply is complex:

• Not everything demonstrable can be known by finding definitions, since all definitions are universal and affirmative whereas some demonstrable propositions are negative.
• If a thing is demonstrable, then to know it just is to possess its demonstration; therefore, it cannot be known just by definition.
• Nevertheless, some definitions can be understood as demonstrations differently arranged.

As an example of case 3, Aristotle considers the definition “Thunder is the extinction of fire in the clouds”. He sees this as a compressed and rearranged form of this demonstration:

• Sound accompanies the extinguishing of fire.
• Fire is extinguished in the clouds.
• Therefore, a sound occurs in the clouds.

We can see the connection by considering the answers to two questions: “What is thunder?” “The extinction of fire in the clouds” (definition). “Why does it thunder?” “Because fire is extinguished in the clouds” (demonstration).

As with his criticisms of Division, Aristotle is arguing for the superiority of his own concept of science to the Platonic concept. Knowledge is composed of demonstrations, even if it may also include definitions; the method of science is demonstrative, even if it may also include the process of defining.

8. Dialectical Argument and the Art of Dialectic

Aristotle often contrasts dialectical arguments with demonstrations. The difference, he tells us, is in the character of their premises, not in their logical structure: whether an argument is a sullogismos is only a matter of whether its conclusion results of necessity from its premises. The premises of demonstrations must be true and primary , that is, not only true but also prior to their conclusions in the way explained in the Posterior Analytics . The premises of dialectical deductions, by contrast, must be accepted ( endoxos ).

Recent scholars have proposed different interpretations of the term endoxos . Aristotle often uses this adjective as a substantive: ta endoxa , “accepted things”, “accepted opinions”. On one understanding, descended from the work of G. E. L. Owen and developed more fully by Jonathan Barnes and especially Terence Irwin, the endoxa are a compilation of views held by various people with some form or other of standing: “the views of fairly reflective people after some reflection”, in Irwin’s phrase. Dialectic is then simply “a method of argument from [the] common beliefs [held by these people]”. For Irwin, then, endoxa are “common beliefs”. Jonathan Barnes, noting that endoxa are opinions with a certain standing, translates with “reputable”.

My own view is that Aristotle’s texts support a somewhat different understanding. He also tells us that dialectical premises differ from demonstrative ones in that the former are questions , whereas the latter are assumptions or assertions : “the demonstrator does not ask, but takes”, he says. This fits most naturally with a view of dialectic as argument directed at another person by question and answer and consequently taking as premises that other person’s concessions. Anyone arguing in this manner will, in order to be successful, have to ask for premises which the interlocutor is liable to accept, and the best way to be successful at that is to have an inventory of acceptable premises, i.e., premises that are in fact acceptable to people of different types.

In fact, we can discern in the Topics (and the Rhetoric , which Aristotle says depends on the art explained in the Topics ) an art of dialectic for use in such arguments. My reconstruction of this art (which would not be accepted by all scholars) is as follows.

Given the above picture of dialectical argument, the dialectical art will consist of two elements. One will be a method for discovering premises from which a given conclusion follows, while the other will be a method for determining which premises a given interlocutor will be likely to concede. The first task is accomplished by developing a system for classifying premises according to their logical structure. We might expect Aristotle to avail himself here of the syllogistic, but in fact he develops quite another approach, one that seems less systematic and rests on various “common” terms. The second task is accomplished by developing lists of the premises which are acceptable to various types of interlocutor. Then, once one knows what sort of person one is dealing with, one can choose premises accordingly. Aristotle stresses that, as in all arts, the dialectician must study, not what is acceptable to this or that specific person, but what is acceptable to this or that type of person, just as the doctor studies what is healthful for different types of person: “art is of the universal”.

8.2.1 The “Logical System” of the Topics

The method presented in the Topics for classifying arguments relies on the presence in the conclusion of certain “common” terms ( koina ) — common in the sense that they are not peculiar to any subject matter but may play a role in arguments about anything whatever. We find enumerations of arguments involving these terms in a similar order several times. Typically, they include:

• Contraries ( enantia )
• Contradictories ( apophaseis )
• Possession and Privation ( hexis kai sterêsis )
• Relatives ( pros ti )
• Cases ( ptôseis )
• “More and Less and Likewise”

The four types of opposites are the best represented. Each designates a type of term pair, i.e., a way two terms can be opposed to one another. Contraries are polar opposites or opposed extremes such as hot and cold, dry and wet, good and bad. A pair of contradictories consists of a term and its negation: good, not good. A possession (or condition) and privation are illustrated by sight and blindness. Relatives are relative terms in the modern sense: a pair consists of a term and its correlative, e.g. large and small, parent and child.

The argumentative patterns Aristotle associated with cases generally involve inferring a sentence containing adverbial or declined forms from another sentence containing different forms of the same word stem: “if what is useful is good, then what is done usefully is done well and the useful person is good”. In Hellenistic grammatical usage, ptôsis meant “case” (e.g. nominative, dative, accusative); Aristotle’s use here is obviously an early form of that.

Under the heading more and less and likewise , Aristotle groups a somewhat motley assortment of argument patterns all involving, in some way or other, the terms “more”, “less”, and “likewise”. Examples: “If whatever is \(A\) is \(B\), then whatever is more (less) \(A\) is more (less) \(B\)”; “If \(A\) is more likely \(B\) than \(C\) is, and \(A\) is not \(B\), then neither is \(C\)”; “If \(A\) is more likely than \(B\) and \(B\) is the case, then \(A\) is the case”.

8.2.2 The Topoi

At the heart of the Topics is a collection of what Aristotle calls topoi , “places” or “locations”. Unfortunately, though it is clear that he intends most of the Topics (Books II–VI) as a collection of these, he never explicitly defines this term. Interpreters have consequently disagreed considerably about just what a topos is. Discussions may be found in Brunschwig 1967, Slomkowski 1996, Primavesi 1997, and Smith 1997.

An art of dialectic will be useful wherever dialectical argument is useful. Aristotle mentions three such uses; each merits some comment.

8.3.1 Gymnastic Dialectic

First, there appears to have been a form of stylized argumentative exchange practiced in the Academy in Aristotle’s time. The main evidence for this is simply Aristotle’s Topics , especially Book VIII, which makes frequent reference to rule-governed procedures, apparently taking it for granted that the audience will understand them. In these exchanges, one participant took the role of answerer, the other the role of questioner. The answerer began by asserting some proposition (a thesis : “position” or “acceptance”). The questioner then asked questions of the answerer in an attempt to secure concessions from which a contradiction could be deduced: that is, to refute ( elenchein ) the answerer’s position. The questioner was limited to questions that could be answered by yes or no; generally, the answerer could only respond with yes or no, though in some cases answerers could object to the form of a question. Answerers might undertake to answer in accordance with the views of a particular type of person or a particular person (e.g. a famous philosopher), or they might answer according to their own beliefs. There appear to have been judges or scorekeepers for the process. Gymnastic dialectical contests were sometimes, as the name suggests, for the sake of exercise in developing argumentative skill, but they may also have been pursued as a part of a process of inquiry.

8.3.2 Dialectic That Puts to the Test

Aristotle also mentions an “art of making trial”, or a variety of dialectical argument that “puts to the test” (the Greek word is the adjective peirastikê , in the feminine: such expressions often designate arts or skills, e.g. rhêtorikê , “the art of rhetoric”). Its function is to examine the claims of those who say they have some knowledge, and it can be practiced by someone who does not possess the knowledge in question. The examination is a matter of refutation, based on the principle that whoever knows a subject must have consistent beliefs about it: so, if you can show me that my beliefs about something lead to a contradiction, then you have shown that I do not have knowledge about it.

This is strongly reminiscent of Socrates’ style of interrogation, from which it is almost certainly descended. In fact, Aristotle often indicates that dialectical argument is by nature refutative.

8.3.3 Dialectic and Philosophy

Dialectical refutation cannot of itself establish any proposition (except perhaps the proposition that some set of propositions is inconsistent). More to the point, though deducing a contradiction from my beliefs may show that they do not constitute knowledge, failure to deduce a contradiction from them is no proof that they are true. Not surprisingly, then, Aristotle often insists that “dialectic does not prove anything” and that the dialectical art is not some sort of universal knowledge.

In Topics I.2, however, Aristotle says that the art of dialectic is useful in connection with “the philosophical sciences”. One reason he gives for this follows closely on the refutative function: if we have subjected our opinions (and the opinions of our fellows, and of the wise) to a thorough refutative examination, we will be in a much better position to judge what is most likely true and false. In fact, we find just such a procedure at the start of many of Aristotle’s treatises: an enumeration of the opinions current about the subject together with a compilation of “puzzles” raised by these opinions. Aristotle has a special term for this kind of review: a diaporia , a “puzzling through”.

He adds a second use that is both more difficult to understand and more intriguing. The Posterior Analytics argues that if anything can be proved, then not everything that is known is known as a result of proof. What alternative means is there whereby the first principles of sciences are known? Aristotle’s own answer as found in Posterior Analytics II.19 is difficult to interpret, and recent philosophers have often found it unsatisfying since (as often construed) it appears to commit Aristotle to a form of apriorism or rationalism both indefensible in itself and not consonant with his own insistence on the indispensability of empirical inquiry in natural science.

Against this background, the following passage in Topics I.2 may have special importance:

It is also useful in connection with the first things concerning each of the sciences. For it is impossible to say anything about the science under consideration on the basis of its own principles, since the principles are first of all, and we must work our way through about these by means of what is generally accepted about each. But this is peculiar, or most proper, to dialectic: for since it is examinative with respect to the principles of all the sciences, it has a way to proceed.

A number of interpreters (beginning with Owen 1961) have built on this passage and others to find dialectic at the heart of Aristotle’s philosophical method. Further discussion of this issue would take us far beyond the subject of this article (the fullest development is in Irwin 1988; see also Nussbaum 1986 and Bolton 1990; for criticism, Hamlyn 1990, Smith 1997).

Aristotle says that rhetoric, i.e., the study of persuasive speech, is a “counterpart” ( antistrophos ) of dialectic and that the rhetorical art is a kind of “outgrowth” ( paraphues ti ) of dialectic and the study of character types. The correspondence with dialectical method is straightforward: rhetorical speeches, like dialectical arguments, seek to persuade others to accept certain conclusions on the basis of premises they already accept. Therefore, the same measures useful in dialectical contexts will, mutatis mutandis, be useful here: knowing what premises an audience of a given type is likely to believe, and knowing how to find premises from which the desired conclusion follows.

The Rhetoric does fit this general description: Aristotle includes both discussions of types of person or audience (with generalizations about what each type tends to believe) and a summary version (in II.23) of the argument patterns discussed in the Topics . For further discussion of his rhetoric see Aristotle’s rhetoric .

Demonstrations and dialectical arguments are both forms of valid argument, for Aristotle. However, he also studies what he calls contentious ( eristikos ) or sophistical arguments: these he defines as arguments which only apparently establish their conclusions. In fact, Aristotle defines these as apparent (but not genuine) dialectical sullogismoi . They may have this appearance in either of two ways:

• Arguments in which the conclusion only appears to follow of necessity from the premises (apparent, but not genuine, sullogismoi ).
• Genuine sullogismois the premises of which are merely apparently, but not genuinely, acceptable.

Arguments of the first type in modern terms, appear to be valid but are really invalid. Arguments of the second type are at first more perplexing: given that acceptability is a matter of what people believe, it might seem that whatever appears to be endoxos must actually be endoxos . However, Aristotle probably has in mind arguments with premises that may at first glance seem to be acceptable but which, upon a moment’s reflection, we immediately realize we do not actually accept. Consider this example from Aristotle’s time:

• Whatever you have not lost, you still have.
• You have not lost horns.
• Therefore, you still have horns

This is transparently bad, but the problem is not that it is invalid: the problem is rather that the first premise, though superficially plausible, is false. In fact, anyone with a little ability to follow an argument will realize that at once upon seeing this very argument.

Aristotle’s study of sophistical arguments is contained in On Sophistical Refutations , which is actually a sort of appendix to the Topics .

To a remarkable extent, contemporary discussions of fallacies reproduce Aristotle’s own classifications. See Dorion 1995 for further discussion.

Two frequent themes of Aristotle’s account of science are (1) that the first principles of sciences are not demonstrable and (2) that there is no single universal science including all other sciences as its parts. “All things are not in a single genus”, he says, “and even if they were, all beings could not fall under the same principles” ( On Sophistical Refutations 11). Thus, it is exactly the universal applicability of dialectic that leads him to deny it the status of a science.

In Metaphysics IV (Γ), however, Aristotle takes what appears to be a different view. First, he argues that there is, in a way, a science that takes being as its genus (his name for it is “first philosophy”). Second, he argues that the principles of this science will be, in a way, the first principles of all (though he does not claim that the principles of other sciences can be demonstrated from them). Third, he identifies one of its first principles as the “most secure” of all principles: the principle of non-contradiction. As he states it,

It is impossible for the same thing to belong and not belong simultaneously to the same thing in the same respect ( Met. )

This is the most secure of all principles, Aristotle tells us, because “it is impossible to be in error about it”. Since it is a first principle, it cannot be demonstrated; those who think otherwise are “uneducated in analytics”. However, Aristotle then proceeds to give what he calls a “refutative demonstration” ( apodeixai elenktikôs ) of this principle.

Further discussion of this principle and Aristotle’s arguments concerning it belong to a treatment of his metaphysics (see Aristotle: Metaphysics ). However, it should be noted that: (1) these arguments draw on Aristotle’s views about logic to a greater extent than any treatise outside the logical works themselves; (2) in the logical works, the principle of non-contradiction is one of Aristotle’s favorite illustrations of the “common principles” ( koinai archai ) that underlie the art of dialectic.

See Aristotle’s Metaphysics , Aristotle on non-contradiction , Dancy 1975, and Code 1986 for further discussion.

The passage in Aristotle’s logical works which has received perhaps the most intense discussion in recent decades is On Interpretation 9, where Aristotle discusses the question whether every proposition about the future must be either true or false. Though something of a side issue in its context, the passage raises a problem of great importance to Aristotle’s near contemporaries (and perhaps contemporaries).

A contradiction ( antiphasis ) is a pair of propositions one of which asserts what the other denies. A major goal of On Interpretation is to discuss the thesis that, of every such contradiction, one member must be true and the other false. In the course of his discussion, Aristotle allows for some exceptions. One case is what he calls indefinite propositions such as “A man is walking”: nothing prevents both this proposition and “A man is not walking” being simultaneously true. This exception can be explained on relatively simple grounds.

A different exception arises for more complex reasons. Consider these two propositions:

• There will be a sea-battle tomorrow
• There will not be a sea-battle tomorrow

It seems that exactly one of these must be true and the other false. But if (1) is now true, then there must be a sea-battle tomorrow, and there cannot fail to be a sea-battle tomorrow. The result, according to this puzzle, is that nothing is possible except what actually happens: there are no unactualized possibilities.

Such a conclusion is, as Aristotle is quick to note, a problem both for his own metaphysical views about potentialities and for the commonsense notion that some things are up to us. He therefore proposes another exception to the general thesis concerning contradictory pairs.

This much would probably be accepted by most interpreters. What the restriction is, however, and just what motivates it are matters of wide disagreement. It has been proposed, for instance, that Aristotle adopted, or at least flirted with, a three-valued logic for future propositions, or that he countenanced truth-value gaps, or that his solution includes still more abstruse reasoning. The literature is much too complex to summarize: see Anscombe, Hintikka, D. Frede, Whitaker, Waterlow.

Historically, at least, it is likely that Aristotle is responding to an argument originating with the Megarian philosophers. He ascribes the view that only that which happens is possible to the Megarians in Metaphysics IX (Θ). The puzzle with which he is concerned strongly recalls the “Master Argument” of Diodorus Cronus especially in certain further details. For instance, Aristotle imagines the statement about tomorrow’s sea battle having been uttered ten thousand years ago. If it was true, then its truth was a fact about the past; if the past is now unchangeable, then so is the truth value of that past utterance. This recalls the Master Argument’s premise that “what is past is necessary”. Diodorus Cronus was active a little after Aristotle, and he was certainly influenced by Megarian views, whether or not it is correct to call him a Megarian (David Sedley 1977 argues that he was instead a member of the Dialectical School which was in any event an offshoot of the Megarians; see Dorion 1995 and Döring 1989, Ebert 2008 and the article Dialectical School). It is therefore likely that Aristotle’s target here is some Megarian argument, perhaps a forerunner of Diodorus’ Master Argument.

• Accept: tithenai (in a dialectical argument)
• Accepted: endoxos (also ‘reputable’ ‘common belief’)
• Accident: sumbebêkos (see incidental )
• Accidental: kata sumbebêkos
• Affirmation: kataphasis
• Affirmative: kataphatikos
• Assertion: apophansis (sentence with a truth value, declarative sentence)
• Assumption: hupothesis
• Belong: huparchein
• Category: katêgoria (see the discussion in Section 7.3).
• Contradict: antiphanai
• Contradiction: antiphasis (in the sense “contradictory pair of propositions” and also in the sense “denial of a proposition”)
• Contrary: enantion
• Deduction: sullogismos
• Definition: horos , horismos
• Demonstration: apodeixis
• Denial (of a proposition): apophasis
• Dialectic: dialektikê (the art of dialectic)
• Differentia: diaphora ; specific difference, eidopoios diaphora
• Direct: deiktikos (of proofs; opposed to “through the impossible”)
• Essence: to ti esti , to ti ên einai
• Essential: en tôi ti esti (of predications)
• Extreme: akron (of the major and minor terms of a deduction)
• Figure: schêma
• Form: eidos (see also Species)
• Genus: genos
• Immediate: amesos (“without a middle”)
• Impossible: adunaton ; “through the impossible” ( dia tou adunatou ), of some proofs.
• Incidental: see Accidental
• Induction: epagôgê
• Middle, middle term (of a deduction): meson
• Negation (of a term): apophasis
• Objection: enstasis
• Particular: en merei , epi meros (of a proposition); kath’hekaston (of individuals)
• Peculiar, Peculiar Property: idios , idion
• Possible: dunaton , endechomenon ; endechesthai (verb: “be possible”)
• Predicate: katêgorein (verb); katêegoroumenon (“what is predicated”)
• Predication: katêgoria (act or instance of predicating, type of predication)
• Primary: prôton
• Principle: archê (starting point of a demonstration)
• Quality: poion
• Reduce, Reduction: anagein , anagôgê
• Refute: elenchein ; refutation, elenchos
• Science: epistêmê
• Species: eidos
• Specific: eidopoios (of a differentia that “makes a species”, eidopoios diaphora )
• Subject: hupokeimenon
• Substance: ousia
• Term: horos
• Universal: katholou (both of propositions and of individuals)
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Aristotle, General Topics: aesthetics | Aristotle, General Topics: metaphysics | Aristotle, General Topics: rhetoric | Aristotle, Special Topics: mathematics | Aristotle, Special Topics: on non-contradiction | -->Chrysippus --> | Diodorus Cronus | future contingents | logic: ancient | logic: relevance | -->Megaric School --> | square of opposition | Stoicism

Acknowledgments

I am indebted to Alan Code, Marc Cohen, and Theodor Ebert for helpful criticisms of earlier versions of this article. I thank Franz Fritsche, Nikolai Biryukov, Ralph E. Kenyon, Johann Dirry, Ben Greenberg, Hasan Masoud, Marc Michael Hämmerling, James Whitely, and [email protected] for calling my attention to errors.

Copyright © 2022 by Robin Smith < rasmith @ tamu . edu >

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1.1: Statements and Conditional Statements

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• Ted Sundstrom
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Much of our work in mathematics deals with statements. In mathematics, a statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition . The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both. So a sentence such as "The sky is beautiful" is not a statement since whether the sentence is true or not is a matter of opinion. A question such as "Is it raining?" is not a statement because it is a question and is not declaring or asserting that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation 2\(x\)+5 = 10 is not a statement since we do not know what \(x\) represents. If we substitute a specific value for \(x\) (such as \(x\) = 3), then the resulting equation, 2\(\cdot\)3 +5 = 10 is a statement (which is a false statement). Following are some more examples:

• There exists a real number \(x\) such that 2\(x\)+5 = 10. This is a statement because either such a real number exists or such a real number does not exist. In this case, this is a true statement since such a real number does exist, namely \(x\) = 2.5.
• For each real number \(x\), \(2x +5 = 2 \left( x + \dfrac{5}{2}\right)\). This is a statement since either the sentence \(2x +5 = 2 \left( x + \dfrac{5}{2}\right)\) is true when any real number is substituted for \(x\) (in which case, the statement is true) or there is at least one real number that can be substituted for \(x\) and produce a false statement (in which case, the statement is false). In this case, the given statement is true.
• Solve the equation \(x^2 - 7x +10 =0\). This is not a statement since it is a directive. It does not assert that something is true.
• \((a+b)^2 = a^2+b^2\) is not a statement since it is not known what \(a\) and \(b\) represent. However, the sentence, “There exist real numbers \(a\) and \(b\) such that \((a+b)^2 = a^2+b^2\)" is a statement. In fact, this is a true statement since there are such integers. For example, if \(a=1\) and \(b=0\), then \((a+b)^2 = a^2+b^2\).
• Compare the statement in the previous item to the statement, “For all real numbers \(a\) and \(b\), \((a+b)^2 = a^2+b^2\)." This is a false statement since there are values for \(a\) and \(b\) for which \((a+b)^2 \ne a^2+b^2\). For example, if \(a=2\) and \(b=3\), then \((a+b)^2 = 5^2 = 25\) and \(a^2 + b^2 = 2^2 +3^2 = 13\).

Progress Check 1.1: Statements

Which of the following sentences are statements? Do not worry about determining whether a statement is true or false; just determine whether each sentence is a statement or not.

• 2\(\cdot\)7 + 8 = 22.
• \((x-1) = \sqrt(x + 11)\).
• \(2x + 5y = 7\).
• There are integers \(x\) and \(y\) such that \(2x + 5y = 7\).
• There are integers \(x\) and \(y\) such that \(23x + 27y = 52\).
• Given a line \(L\) and a point \(P\) not on that line, there is a unique line through \(P\) that does not intersect \(L\).
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\).
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) for all real numbers \(a\) and \(b\).
• The derivative of \(f(x) = \sin x\) is \(f' (x) = \cos x\).
• Does the equation \(3x^2 - 5x - 7 = 0\) have two real number solutions?
• If \(ABC\) is a right triangle with right angle at vertex \(B\), and if \(D\) is the midpoint of the hypotenuse, then the line segment connecting vertex \(B\) to \(D\) is half the length of the hypotenuse.
• There do not exist three integers \(x\), \(y\), and \(z\) such that \(x^3 + y^2 = z^3\).

Add texts here. Do not delete this text first.

How Do We Decide If a Statement Is True or False?

In mathematics, we often establish that a statement is true by writing a mathematical proof. To establish that a statement is false, we often find a so-called counterexample. (These ideas will be explored later in this chapter.) So mathematicians must be able to discover and construct proofs. In addition, once the discovery has been made, the mathematician must be able to communicate this discovery to others who speak the language of mathematics. We will be dealing with these ideas throughout the text.

For now, we want to focus on what happens before we start a proof. One thing that mathematicians often do is to make a conjecture beforehand as to whether the statement is true or false. This is often done through exploration. The role of exploration in mathematics is often difficult because the goal is not to find a specific answer but simply to investigate. Following are some techniques of exploration that might be helpful.

Techniques of Exploration

• Guesswork and conjectures . Formulate and write down questions and conjectures. When we make a guess in mathematics, we usually call it a conjecture.

For example, if someone makes the conjecture that \(\sin(2x) = 2 \sin(x)\), for all real numbers \(x\), we can test this conjecture by substituting specific values for \(x\). One way to do this is to choose values of \(x\) for which \(\sin(x)\)is known. Using \(x = \frac{\pi}{4}\), we see that

\(\sin(2(\frac{\pi}{4})) = \sin(\frac{\pi}{2}) = 1,\) and

\(2\sin(\frac{\pi}{4}) = 2(\frac{\sqrt2}{2}) = \sqrt2\).

Since \(1 \ne \sqrt2\), these calculations show that this conjecture is false. However, if we do not find a counterexample for a conjecture, we usually cannot claim the conjecture is true. The best we can say is that our examples indicate the conjecture is true. As an example, consider the conjecture that

If \(x\) and \(y\) are odd integers, then \(x + y\) is an even integer.

We can do lots of calculation, such as \(3 + 7 = 10\) and \(5 + 11 = 16\), and find that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.)

• Use of prior knowledge. This also is very important. We cannot start from square one every time we explore a statement. We must make use of our acquired mathematical knowledge. For the conjecture that \(\sin (2x) = 2 \sin(x)\), for all real numbers \(x\), we might recall that there are trigonometric identities called “double angle identities.” We may even remember the correct identity for \(\sin (2x)\), but if we do not, we can always look it up. We should recall (or find) that for all real numbers \(x\), \[\sin(2x) = 2 \sin(x)\cos(x).\]
• We could use this identity to argue that the conjecture “for all real numbers \(x\), \(\sin (2x) = 2 \sin(x)\)” is false, but if we do, it is still a good idea to give a specific counterexample as we did before.
• Cooperation and brainstorming . Working together is often more fruitful than working alone. When we work with someone else, we can compare notes and articulate our ideas. Thinking out loud is often a useful brainstorming method that helps generate new ideas.

Progress Check 1.2: Explorations

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

• \((a + b)^2 = a^2 + b^2\), for all real numbers a and b.
• There are integers \(x\) and \(y\) such that \(2x + 5y = 41\).
• If \(x\) is an even integer, then \(x^2\) is an even integer.
• If \(x\) and \(y\) are odd integers, then \(x \cdot y\) is an odd integer.

Conditional Statements

One of the most frequently used types of statements in mathematics is the so-called conditional statement. Given statements \(P\) and \(Q\), a statement of the form “If \(P\) then \(Q\)” is called a conditional statement . It seems reasonable that the truth value (true or false) of the conditional statement “If \(P\) then \(Q\)” depends on the truth values of \(P\) and \(Q\). The statement “If \(P\) then \(Q\)” means that \(Q\) must be true whenever \(P\) is true. The statement \(P\) is called the hypothesis of the conditional statement, and the statement \(Q\) is called the conclusion of the conditional statement. Since conditional statements are probably the most important type of statement in mathematics, we give a more formal definition.

A conditional statement is a statement that can be written in the form “If \(P\) then \(Q\),” where \(P\) and \(Q\) are sentences. For this conditional statement, \(P\) is called the hypothesis and \(Q\) is called the conclusion .

Intuitively, “If \(P\) then \(Q\)” means that \(Q\) must be true whenever \(P\) is true. Because conditional statements are used so often, a symbolic shorthand notation is used to represent the conditional statement “If \(P\) then \(Q\).” We will use the notation \(P \to Q\) to represent “If \(P\) then \(Q\).” When \(P\) and \(Q\) are statements, it seems reasonable that the truth value (true or false) of the conditional statement \(P \to Q\) depends on the truth values of \(P\) and \(Q\). There are four cases to consider:

• \(P\) is true and \(Q\) is true.
• \(P\) is false and \(Q\) is true.
• \(P\) is true and \(Q\) is false.
• \(P\) is false and \(Q\) is false.

The conditional statement \(P \to Q\) means that \(Q\) is true whenever \(P\) is true. It says nothing about the truth value of \(Q\) when \(P\) is false. Using this as a guide, we define the conditional statement \(P \to Q\) to be false only when \(P\) is true and \(Q\) is false, that is, only when the hypothesis is true and the conclusion is false. In all other cases, \(P \to Q\) is true. This is summarized in Table 1.1 , which is called a truth table for the conditional statement \(P \to Q\). (In Table 1.1 , T stands for “true” and F stands for “false.”)

Table 1.1: Truth Table for \(P \to Q\)

The important thing to remember is that the conditional statement \(P \to Q\) has its own truth value. It is either true or false (and not both). Its truth value depends on the truth values for \(P\) and \(Q\), but some find it a bit puzzling that the conditional statement is considered to be true when the hypothesis P is false. We will provide a justification for this through the use of an example.

Example 1.3:

Suppose that I say

“If it is not raining, then Daisy is riding her bike.”

We can represent this conditional statement as \(P \to Q\) where \(P\) is the statement, “It is not raining” and \(Q\) is the statement, “Daisy is riding her bike.”

Although it is not a perfect analogy, think of the statement \(P \to Q\) as being false to mean that I lied and think of the statement \(P \to Q\) as being true to mean that I did not lie. We will now check the truth value of \(P \to Q\) based on the truth values of \(P\) and \(Q\).

• Suppose that both \(P\) and \(Q\) are true. That is, it is not raining and Daisy is riding her bike. In this case, it seems reasonable to say that I told the truth and that\(P \to Q\) is true.
• Suppose that \(P\) is true and \(Q\) is false or that it is not raining and Daisy is not riding her bike. It would appear that by making the statement, “If it is not raining, then Daisy is riding her bike,” that I have not told the truth. So in this case, the statement \(P \to Q\) is false.
• Now suppose that \(P\) is false and \(Q\) is true or that it is raining and Daisy is riding her bike. Did I make a false statement by stating that if it is not raining, then Daisy is riding her bike? The key is that I did not make any statement about what would happen if it was raining, and so I did not tell a lie. So we consider the conditional statement, “If it is not raining, then Daisy is riding her bike,” to be true in the case where it is raining and Daisy is riding her bike.
• Finally, suppose that both \(P\) and \(Q\) are false. That is, it is raining and Daisy is not riding her bike. As in the previous situation, since my statement was \(P \to Q\), I made no claim about what would happen if it was raining, and so I did not tell a lie. So the statement \(P \to Q\) cannot be false in this case and so we consider it to be true.

Progress Check 1.4: xplorations with Conditional Statements

1 . Consider the following sentence:

If \(x\) is a positive real number, then \(x^2 + 8x\) is a positive real number.

Although the hypothesis and conclusion of this conditional sentence are not statements, the conditional sentence itself can be considered to be a statement as long as we know what possible numbers may be used for the variable \(x\). From the context of this sentence, it seems that we can substitute any positive real number for \(x\). We can also substitute 0 for \(x\) or a negative real number for x provided that we are willing to work with a false hypothesis in the conditional statement. (In Chapter 2 , we will learn how to be more careful and precise with these types of conditional statements.)

(a) Notice that if \(x = -3\), then \(x^2 + 8x = -15\), which is negative. Does this mean that the given conditional statement is false?

(b) Notice that if \(x = 4\), then \(x^2 + 8x = 48\), which is positive. Does this mean that the given conditional statement is true?

(c) Do you think this conditional statement is true or false? Record the results for at least five different examples where the hypothesis of this conditional statement is true.

2 . “If \(n\) is a positive integer, then \(n^2 - n +41\) is a prime number.” (Remember that a prime number is a positive integer greater than 1 whose only positive factors are 1 and itself.) To explore whether or not this statement is true, try using (and recording your results) for \(n = 1\), \(n = 2\), \(n = 3\), \(n = 4\), \(n = 5\), and \(n = 10\). Then record the results for at least four other values of \(n\). Does this conditional statement appear to be true?

Further Remarks about Conditional Statements

Suppose that Ed has exactly $52 in his wallet. The following four statements will use the four possible truth combinations for the hypothesis and conclusion of a conditional statement.

• If Ed has exactly $52 in his wallet, then he has $20 in his wallet. This is a true statement. Notice that both the hypothesis and the conclusion are true.
• If Ed has exactly $52 in his wallet, then he has $100 in his wallet. This statement is false. Notice that the hypothesis is true and the conclusion is false.
• If Ed has $100 in his wallet, then he has at least $50 in his wallet. This statement is true regardless of how much money he has in his wallet. In this case, the hypothesis is false and the conclusion is true.

This is admittedly a contrived example but it does illustrate that the conventions for the truth value of a conditional statement make sense. The message is that in order to be complete in mathematics, we need to have conventions about when a conditional statement is true and when it is false.

If \(n\) is a positive integer, then \((n^2 - n + 41)\) is a prime number.

Perhaps for all of the values you tried for \(n\), \((n^2 - n + 41)\) turned out to be a prime number. However, if we try \(n = 41\), we ge \(n^2 - n + 41 = 41^2 - 41 + 41\) \(n^2 - n + 41 = 41^2\) So in the case where \(n = 41\), the hypothesis is true (41 is a positive integer) and the conclusion is false \(41^2\) is not prime. Therefore, 41 is a counterexample for this conjecture and the conditional statement “If \(n\) is a positive integer, then \((n^2 - n + 41)\) is a prime number” is false. There are other counterexamples (such as \(n = 42\), \(n = 45\), and \(n = 50\)), but only one counterexample is needed to prove that the statement is false.

• Although one example can be used to prove that a conditional statement is false, in most cases, we cannot use examples to prove that a conditional statement is true. For example, in Progress Check 1.4 , we substituted values for \(x\) for the conditional statement “If \(x\) is a positive real number, then \(x^2 + 8x\) is a positive real number.” For every positive real number used for \(x\), we saw that \(x^2 + 8x\) was positive. However, this does not prove the conditional statement to be true because it is impossible to substitute every positive real number for \(x\). So, although we may believe this statement is true, to be able to conclude it is true, we need to write a mathematical proof. Methods of proof will be discussed in Section 1.2 and Chapter 3 .

Progress Check 1.5: Working with a Conditional Statement

The following statement is a true statement, which is proven in many calculus texts.

If the function \(f\) is differentiable at \(a\), then the function \(f\) is continuous at \(a\).

Using only this true statement, is it possible to make a conclusion about the function in each of the following cases?

• It is known that the function \(f\), where \(f(x) = \sin x\), is differentiable at 0.
• It is known that the function \(f\), where \(f(x) = \sqrt[3]x\), is not differentiable at 0.
• It is known that the function \(f\), where \(f(x) = |x|\), is continuous at 0.
• It is known that the function \(f\), where \(f(x) = \dfrac{|x|}{x}\) is not continuous at 0.

Closure Properties of Number Systems

The primary number system used in algebra and calculus is the real number system . We usually use the symbol R to stand for the set of all real numbers. The real numbers consist of the rational numbers and the irrational numbers. The rational numbers are those real numbers that can be written as a quotient of two integers (with a nonzero denominator), and the irrational numbers are those real numbers that cannot be written as a quotient of two integers. That is, a rational number can be written in the form of a fraction, and an irrational number cannot be written in the form of a fraction. Some common irrational numbers are \(\sqrt2\), \(\pi\) and \(e\). We usually use the symbol \(\mathbb{Q}\) to represent the set of all rational numbers. (The letter \(\mathbb{Q}\) is used because rational numbers are quotients of integers.) There is no standard symbol for the set of all irrational numbers.

Perhaps the most basic number system used in mathematics is the set of natural numbers . The natural numbers consist of the positive whole numbers such as 1, 2, 3, 107, and 203. We will use the symbol \(\mathbb{N}\) to stand for the set of natural numbers. Another basic number system that we will be working with is the set of integers . The integers consist of zero, the positive whole numbers, and the negatives of the positive whole numbers. If \(n\) is an integer, we can write \(n = \dfrac{n}{1}\). So each integer is a rational number and hence also a real number.

We will use the letter \(\mathbb{Z}\) to stand for the set of integers. (The letter \(\mathbb{Z}\) is from the German word, \(Zahlen\), for numbers.) Three of the basic properties of the integers are that the set \(\mathbb{Z}\) is closed under addition , the set \(\mathbb{Z}\) is closed under multiplication , and the set of integers is closed under subtraction. This means that

• If \(x\) and \(y\) are integers, then \(x + y\) is an integer;
• If \(x\) and \(y\) are integers, then \(x \cdot y\) is an integer; and
• If \(x\) and \(y\) are integers, then \(x - y\) is an integer.

Notice that these so-called closure properties are defined in terms of conditional statements. This means that if we can find one instance where the hypothesis is true and the conclusion is false, then the conditional statement is false.

Example 1.6: Closure

• In order for the set of natural numbers to be closed under subtraction, the following conditional statement would have to be true: If \(x\) and \(y\) are natural numbers, then \(x - y\) is a natural number. However, since 5 and 8 are natural numbers, \(5 - 8 = -3\), which is not a natural number, this conditional statement is false. Therefore, the set of natural numbers is not closed under subtraction.
• We can use the rules for multiplying fractions and the closure rules for the integers to show that the rational numbers are closed under multiplication. If \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\) are rational numbers (so \(a\), \(b\), \(c\), and \(d\) are integers and \(b\) and \(d\) are not zero), then \(\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}.\) Since the integers are closed under multiplication, we know that \(ac\) and \(bd\) are integers and since \(b \ne 0\) and \(d \ne 0\), \(bd \ne 0\). Hence, \(\dfrac{ac}{bd}\) is a rational number and this shows that the rational numbers are closed under multiplication.

Progress Check 1.7: Closure Properties

Answer each of the following questions.

• Is the set of rational numbers closed under addition? Explain.
• Is the set of integers closed under division? Explain.
• Is the set of rational numbers closed under subtraction? Explain.
• Which of the following sentences are statements? (a) \(3^2 + 4^2 = 5^2.\) (b) \(a^2 + b^2 = c^2.\) (c) There exists integers \(a\), \(b\), and \(c\) such that \(a^2 + b^2 = c^2.\) (d) If \(x^2 = 4\), then \(x = 2.\) (e) For each real number \(x\), if \(x^2 = 4\), then \(x = 2.\) (f) For each real number \(t\), \(\sin^2t + \cos^2t = 1.\) (g) \(\sin x < \sin (\frac{\pi}{4}).\) (h) If \(n\) is a prime number, then \(n^2\) has three positive factors. (i) 1 + \(\tan^2 \theta = \text{sec}^2 \theta.\) (j) Every rectangle is a parallelogram. (k) Every even natural number greater than or equal to 4 is the sum of two prime numbers.
• Identify the hypothesis and the conclusion for each of the following conditional statements. (a) If \(n\) is a prime number, then \(n^2\) has three positive factors. (b) If \(a\) is an irrational number and \(b\) is an irrational number, then \(a \cdot b\) is an irrational number. (c) If \(p\) is a prime number, then \(p = 2\) or \(p\) is an odd number. (d) If \(p\) is a prime number and \(p \ne 2\) or \(p\) is an odd number. (e) \(p \ne 2\) or \(p\) is a even number, then \(p\) is not prime.
• Determine whether each of the following conditional statements is true or false. (a) If 10 < 7, then 3 = 4. (b) If 7 < 10, then 3 = 4. (c) If 10 < 7, then 3 + 5 = 8. (d) If 7 < 10, then 3 + 5 = 8.
• Determine the conditions under which each of the following conditional sentences will be a true statement. (a) If a + 2 = 5, then 8 < 5. (b) If 5 < 8, then a + 2 = 5.
• Let \(P\) be the statement “Student X passed every assignment in Calculus I,” and let \(Q\) be the statement “Student X received a grade of C or better in Calculus I.” (a) What does it mean for \(P\) to be true? What does it mean for \(Q\) to be true? (b) Suppose that Student X passed every assignment in Calculus I and received a grade of B-, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (c) Suppose that Student X passed every assignment in Calculus I and received a grade of C-, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (d) Now suppose that Student X did not pass two assignments in Calculus I and received a grade of D, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (e) How are Parts ( 5b ), ( 5c ), and ( 5d ) related to the truth table for \(P \to Q\)?

Theorem If f is a quadratic function of the form \(f(x) = ax^2 + bx + c\) and a < 0, then the function f has a maximum value when \(x = \dfrac{-b}{2a}\). Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g (x) = -8x^2 + 5x - 2\) (b) \(h (x) = -\dfrac{1}{3}x^2 + 3x\) (c) \(k (x) = 8x^2 - 5x - 7\) (d) \(j (x) = -\dfrac{71}{99}x^2 +210\) (e) \(f (x) = -4x^2 - 3x + 7\) (f) \(F (x) = -x^4 + x^3 + 9\)

Theorem If \(f\) is a quadratic function of the form \(f(x) = ax^2 + bx + c\) and ac < 0, then the function \(f\) has two x-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g (x) = -8x^2 + 5x - 2\) (b) \(h (x) = -\dfrac{1}{3}x^2 + 3x\) (c) \(k (x) = 8x^2 - 5x - 7\) (d) \(j (x) = -\dfrac{71}{99}x^2 +210\) (e) \(f (x) = -4x^2 - 3x + 7\) (f) \(F (x) = -x^4 + x^3 + 9\)

Theorem A. If \(f\) is a cubic function of the form \(f (x) = x^3 - x + b\) and b > 1, then the function \(f\) has exactly one \(x\)-intercept. Following is another theorem about \(x\)-intercepts of functions: Theorem B . If \(f\) and \(g\) are functions with \(g (x) = k \cdot f (x)\), where \(k\) is a nonzero real number, then \(f\) and \(g\) have exactly the same \(x\)-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas? (a) \(f (x) = x^3 -x + 7\) (b) \(g (x) = x^3 + x +7\) (c) \(h (x) = -x^3 + x - 5\) (d) \(k (x) = 2x^3 + 2x + 3\) (e) \(r (x) = x^4 - x + 11\) (f) \(F (x) = 2x^3 - 2x + 7\)

• (a) Is the set of natural numbers closed under division? (b) Is the set of rational numbers closed under division? (c) Is the set of nonzero rational numbers closed under division? (d) Is the set of positive rational numbers closed under division? (e) Is the set of positive real numbers closed under subtraction? (f) Is the set of negative rational numbers closed under division? (g) Is the set of negative integers closed under addition? Explorations and Activities
• Exploring Propositions . In Progress Check 1.2 , we used exploration to show that certain statements were false and to make conjectures that certain statements were true. We can also use exploration to formulate a conjecture that we believe to be true. For example, if we calculate successive powers of \(2, (2^1, 2^2, 2^3, 2^4, 2^5, ...)\) and examine the units digits of these numbers, we could make the following conjectures (among others): \(\bullet\) If \(n\) is a natural number, then the units digit of \(2^n\) must be 2, 4, 6, or 8. \(\bullet\) The units digits of the successive powers of 2 repeat according to the pattern “2, 4, 8, 6.” (a) Is it possible to formulate a conjecture about the units digits of successive powers of \(4 (4^1, 4^2, 4^3, 4^4, 4^5,...)\)? If so, formulate at least one conjecture. (b) Is it possible to formulate a conjecture about the units digit of numbers of the form \(7^n - 2^n\), where \(n\) is a natural number? If so, formulate a conjecture in the form of a conditional statement in the form “If \(n\) is a natural number, then ... .” (c) Let \(f (x) = e^(2x)\). Determine the first eight derivatives of this function. What do you observe? Formulate a conjecture that appears to be true. The conjecture should be written as a conditional statement in the form, “If n is a natural number, then ... .”

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11: Logic of Hypothesis Testing

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• Page ID 2151

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When interpreting an experimental finding, a natural question arises as to whether the finding could have occurred by chance. Hypothesis testing is a statistical procedure for testing whether chance is a plausible explanation of an experimental finding. Misconceptions about hypothesis testing are common among practitioners as well as students. To help prevent these misconceptions, this chapter goes into more detail about the logic of hypothesis testing than is typical for an introductory-level text.

• 11.1: Introduction to Hypothesis Testing
• 11.2: Significance Testing It is conventional to conclude the null hypothesis is false if the probability value is less than 0.05. More conservative researchers conclude the null hypothesis is false only if the probability value is less than 0.01. When a researcher concludes that the null hypothesis is false, the researcher is said to have rejected the null hypothesis. The probability value below which the null hypothesis is rejected is called the α (alpha) level or simply α. It is also called the significance level.
• 11.3: Type I and II Errors A Type I error occurs when a significance test results in the rejection of a true null hypothesis.  A Type II error that can be made in significance testing is failing to reject a false null hypothesis. Unlike a Type I error, a Type II error is not really an error. When a statistical test is not significant, it means that the data do not provide strong evidence that the null hypothesis is false. Lack of significance does not support the conclusion that the null hypothesis is true.
• 11.4: One- and Two-Tailed Tests A probability calculated in only one tail of the distribution is called a "one-tailed probability" and probability calculated in both tails of a distribution is called a "two-tailed probability." You should always decide whether you are going to use a one-tailed or a two-tailed probability before looking at the data.
• 11.5: Significant Results When a probability value is below the α level, the effect is statistically significant and the null hypothes is is rejected. However, not all statistically significant effects should be treated the same way. For example, you should have less confidence that the null hypothesis is false if p = 0.049 than p = 0.003. Thus, rejecting the null hypothesis is not an all-or-none proposition.
• 11.6: Non-Significant Results
• 11.7: Steps in Hypothesis Testing
• 11.8: Significance Testing and Confidence Intervals
• 11.9: Misconceptions of Hypothesis Testing
• 11.10: Statistical Literacy
• 11.E: Logic of Hypothesis Testing (Exercises)

• Scientific Methods

What is Hypothesis?

We have heard of many hypotheses which have led to great inventions in science. Assumptions that are made on the basis of some evidence are known as hypotheses. In this article, let us learn in detail about the hypothesis and the type of hypothesis with examples.

A hypothesis is an assumption that is made based on some evidence. This is the initial point of any investigation that translates the research questions into predictions. It includes components like variables, population and the relation between the variables. A research hypothesis is a hypothesis that is used to test the relationship between two or more variables.

Characteristics of Hypothesis

Following are the characteristics of the hypothesis:

• The hypothesis should be clear and precise to consider it to be reliable.
• If the hypothesis is a relational hypothesis, then it should be stating the relationship between variables.
• The hypothesis must be specific and should have scope for conducting more tests.
• The way of explanation of the hypothesis must be very simple and it should also be understood that the simplicity of the hypothesis is not related to its significance.

Sources of Hypothesis

Following are the sources of hypothesis:

• The resemblance between the phenomenon.
• Observations from past studies, present-day experiences and from the competitors.
• Scientific theories.
• General patterns that influence the thinking process of people.

Types of Hypothesis

There are six forms of hypothesis and they are:

• Simple hypothesis
• Complex hypothesis
• Directional hypothesis
• Non-directional hypothesis
• Null hypothesis
• Associative and casual hypothesis

Simple Hypothesis

It shows a relationship between one dependent variable and a single independent variable. For example – If you eat more vegetables, you will lose weight faster. Here, eating more vegetables is an independent variable, while losing weight is the dependent variable.

Complex Hypothesis

It shows the relationship between two or more dependent variables and two or more independent variables. Eating more vegetables and fruits leads to weight loss, glowing skin, and reduces the risk of many diseases such as heart disease.

Directional Hypothesis

It shows how a researcher is intellectual and committed to a particular outcome. The relationship between the variables can also predict its nature. For example- children aged four years eating proper food over a five-year period are having higher IQ levels than children not having a proper meal. This shows the effect and direction of the effect.

Non-directional Hypothesis

It is used when there is no theory involved. It is a statement that a relationship exists between two variables, without predicting the exact nature (direction) of the relationship.

Null Hypothesis

It provides a statement which is contrary to the hypothesis. It’s a negative statement, and there is no relationship between independent and dependent variables. The symbol is denoted by “H O ”.

Associative and Causal Hypothesis

Associative hypothesis occurs when there is a change in one variable resulting in a change in the other variable. Whereas, the causal hypothesis proposes a cause and effect interaction between two or more variables.

Examples of Hypothesis

Following are the examples of hypotheses based on their types:

• Consumption of sugary drinks every day leads to obesity is an example of a simple hypothesis.
• All lilies have the same number of petals is an example of a null hypothesis.
• If a person gets 7 hours of sleep, then he will feel less fatigue than if he sleeps less. It is an example of a directional hypothesis.

Functions of Hypothesis

Following are the functions performed by the hypothesis:

• Hypothesis helps in making an observation and experiments possible.
• It becomes the start point for the investigation.
• Hypothesis helps in verifying the observations.
• It helps in directing the inquiries in the right direction.

How will Hypothesis help in the Scientific Method?

Researchers use hypotheses to put down their thoughts directing how the experiment would take place. Following are the steps that are involved in the scientific method:

• Formation of question
• Doing background research
• Creation of hypothesis
• Designing an experiment
• Collection of data
• Result analysis
• Summarizing the experiment
• Communicating the results

Frequently Asked Questions – FAQs

What is hypothesis.

A hypothesis is an assumption made based on some evidence.

Give an example of simple hypothesis?

What are the types of hypothesis.

Types of hypothesis are:

• Associative and Casual hypothesis

State true or false: Hypothesis is the initial point of any investigation that translates the research questions into a prediction.

Define complex hypothesis..

A complex hypothesis shows the relationship between two or more dependent variables and two or more independent variables.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

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Understanding the Role of Hypotheses and Conclusions in Mathematical Reasoning

Hypothesis and conclusion.

In the context of mathematics and logic, a hypothesis is a statement or proposition that is assumed to be true for the purpose of a logical argument or investigation. It is usually denoted by “H” or “P” and is the starting point for many mathematical proofs.

For example, let’s consider the hypothesis: “If it is raining outside, then the ground is wet.” This statement assumes that whenever it rains, the ground will be wet.

The conclusion, on the other hand, is the statement or proposition that is inferred or reached by logical reasoning, based on the hypothesis or given information. It is typically denoted by “C” or “Q”.

Using the same example, the conclusion derived from the hypothesis could be: “It is currently raining outside, so the ground is wet.” This conclusion is based on the assumption that the given condition of rain implies a wet ground.

In mathematics, hypotheses and conclusions are commonly used in proofs and logical arguments. By stating a hypothesis and then deducing a conclusion from it, mathematicians can demonstrate the validity of certain mathematical concepts, theorems, or formulas.

It’s important to note that in mathematics, a hypothesis is not the same as a guess or a prediction. It is a statement that is assumed to be true and serves as the basis for logical reasoning, while the conclusion is the logical consequence or outcome that is drawn from the hypothesis.

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