Unedited Notes with Practice Activities I use in Class


Video Version


Studying epistemology can deepen your understanding of knowledge and the types of beliefs you hold. In this lesson, we will explore some common ways of categorizing your beliefs: a priori vs. a posteriori, analytic vs. synthetic, and necessary vs. contingent. Studying these can deepen your epistemology, clarify your ideas, help you better understand the philosophers and discover truth.


A Priori vs. A posteriori

A priori claims are those you can know independent of experience. For example, the interior angles of a triangle will always add up to 180 degrees. You do not have to measure all triangles to know this; it is an a priori claim. You can know it independently of (or prior to) experience. Here are some other examples of a priori claims:

Bob is taller than Jane and Jane is taller than Fred. So, Bob is taller than Fred.

All bachelors are single.

Whereas a priori claims seem to be justified based on pure thought or reason, a posteriori claims are justified based on experience. We can only know a posteriori claims after experience. Here are some a posteriori claims:

The triangle is blue.

Bob is over six feet tall.

The boat is sinking.

60% of Americans are clinically overweight.

To review,

“All crows are birds” is a priori.

“All crows are black” is a posteriori.

“Green is a color” is a priori.

“Grass is green” is a posteriori.

"A house is an abode for living” is a priori.

“A house undermined will fall” is a posteriori.

“2+2=4” is a priori.

“2 quarts of any liquid added to 2 more quarts of any liquid= 4 quarts of liquid.” Is a posteriori.

“If you know something, you believe it is true” is a priori.

“I know the earth is the third planet from the sun” is a posteriori.


Ok, let’s do a practice activity to make sure you understand this distinction.

Practice 1: Identify the following statements as a priori or posteriori

  1. All bachelors are unmarried.
  2. It is raining in Austin, TX right now.
  3. If today is Tuesday, then today is not Thursday.
  4. Most people act self-interestedly most of the time.
  5. Water is composed of hydrogen and oxygen.
  6. 7+5=12
  7. Thomas Jefferson once lived but is now dead.
  8. It is false that, “A is B and A is not B.”
  9. I was born in 1861.
  10. If I was born in 1861, and Bob was born in 1841, then I was born after Bob.
  11. God, by definition, is a being that must exist. Therefore, God exists.
  12. God is about 4 feet tall and is sitting behind that tree.


Answers: 1. A priori: true by definition. I do not have to research all bachelors to know this. 2. A posteriori 3. A priori 4. A posteriori (see Batson Research) 5. A posteriori 6. A priori (for now) 7. A posteriori 8. A priori 9. A posteriori 10. A priori 11. A priori (see Ontological Argument) 12. A posteriori

Some of these answers are controversial, but I will explore that a bit later.

Now, people sometimes get confused because we learn about triangles from math teachers and math classes. That is, we learn about triangles from experience. These people therefore think that math should be a posteriori. But this is a confusion between origin and method of proof. “What makes something a priori is not the means by which it came to be first known, but the means by which it can be shown to be true or false” (Baggini). We may need experience to furnish ourselves with the concept of triangle, but once we have that concept, we do not need to refer to experience to determine what the properties of triangles are. A priori knowledge is thus distinguished by its method of proof, not by how we came to acquire it” (Baggini, 142).

Discussion 1: A posteriori knowledge is based on experience, but what exactly do they mean by experience?

Discussion 2: Why are geometric claims (triangles =180 degrees) a priori?

One answer is that triangles are not real objects. They are idealized in the mind. We live in a three-dimensional world, but triangles are two-dimensional. If you look microscopically at any three-dimensional object, you will see it is vibrating, moving, wiggling. But two-dimensional triangles in Euclidian Geometry are perfect. If this is right, then triangles can be known without looking out at the empirical world. We can think of them and know/deduce their truths without observing objects out there.

To deepen our epistemology and explore these points more rigorously, let’s turn to the next distinction: the analytic vs. synthetic distinction.


Part II: Analytic vs. Synthetic

Kant clearly explained that analytic propositions are those in which the predicate is contained in the subject. For example, “all bachelors are single” because the predicate (single) is contained in the subject (bachelor). So, you can think of analytic statements as those that are true by definition. Here are some other examples:

  • All Texans are North Americans.
  • All dogs are animals.
  • Triangles have three sides.

North American is in the definition of Texan, animal is in the definition of dog, and three sides is in the definition of triangle.

Notice analytic statements are not truths about the world, they are truths about words. The bachelor is unmarried is true because of the meaning of bachelor. You don’t have to go out and look at the world to know bachelors are unmarried. Analytic propositions are what Hume calls “a mere relation of ideas.”

Synthetic statements are true by experience; the predicate is not contained in the subject.

  • People from Texas are usually more obese than people from Colorado.
  • My dog is sick.
  • The triangle is red.

So, scientific statements are synthetic statements; they tell us about the world.

Ok, let’s practice this distinction before exploring it more deeply.


Practice 2: Identify the following statements as analytic or synthetic.

  1. It is snowing right now in Colorado.
  2. Circles are shapes.
  3. Barns are structures.
  4. Daisies are flowers.
  5. The president is tall.
  6. Water boils at 100 C.
  7. The Earth revolves around the sun.

Answers: Analytic (2, 3, 4), Synthetic (1, 5, 6, 7).

For the last one, notice that the judgment about “the boiling point of water goes beyond what is contained in the concept of water, whereas the judgment that a bachelor is unmarried does not go beyond what is already contained in the concept of bachelor” (Baggini, 148).

Question: Are all a priori claims analytic?

At first, it does seem that way. We could say that we know all a priori claims independently of experience because they are simply analytic claims (i.e. claims in which the predicate is contained in the subject). That is, a priori claims are priori simply because they are analytic. If you review the two practice activities, it seems all a priori statements are analytic and all a posteriori claims are synthetic. Take a moment and test that for yourself.

If that were correct, we could say a priori and analytic claims are pretty much the same. The only difference being that a priori is about why we believe the claim and analytic is about how the predicate of the sentence (e.g. single) is related to the subject (e.g. bachelor). That is, a priori and a posteriori claims are about epistemology (i.e. on what basis we can believe a claim) while analytic and synthetic claims are about language. I know a priori claims just by thinking, but they are analytic if mere definitions make them true.

Based on what we have seen so far, all a priori claims are analytic and all a posteriori claims are synthetic.

However, this point- and the distinctions we just learned- are actually quite controversy. Let’s take a moment to deepen and confuse.

First, in the Critique of Pure Reason, I believe Kant clearly showed that not all a priori claims are analytic. For example, Kant believed the mathematical claim that “2+2=4” is synthetic a priori. “2+2=4” is synthetic because it tells us about the empirical world and our intuitions of space and time are needed to fully grasp such mathematical truths. They are not merely relations of ideas. Yet it is a priori because we can grasp this truth without testing it in the world. See my videos on Kant or mathematical realism for more on this. In the Philosopher’s Toolkit, Baggini and Fossl give this chart for the different ways philosophers have conceived of these terms.

1. “All experienced events have causes.”

   a. Descartes: analytic a priori

   b. Hume: synthetic a posteriori

   c. Kant: synthetic a priori

2. “7+5=12.”

     a. Descartes and Hume: analytic a priori

     b. Kant: synthetic a priori

3. “Paris is the capital of France.”

     a. Leibniz: analytic a priori.

     b. Descartes, Hume, Kant: synthetic a posteriori.

*Page 143, The Philosopher’s Toolkit (Baggini & Fosl).


In short, it is controversial as to where we should draw the line between a priori and posteriori and analytic and synthetic. The debate rages on today and understanding the points up to now will help you better understand both the modern and older philosophers mentioned above. It will also help you better evaluate some modern attempts of trying to reduce philosophy to science and empirical observations/claims.

Second, another objection comes from Quine. He did not believe in a priori knowledge because all a priori claims are in principle revisable in the light of experience. Look back at Practice Activity 1. Do you agree with him that all the a priori claims listed there are revisable in the light of experience? If you are a materialist like Quine, you may agree with him.

You could read Quine’s essay, “Two Dogmas of Empiricism” (1951) if you are enjoying this. In this essay, he questions the idea of containment, of how the subject can contain the predicate in analytic statements. He wanted to undermine these distinctions, I believe, so he could make philosophy a part of science. Yet even Quine acknowledges that there must be a difference between explaining the meaning of a concept and connecting new information to it. But I am going to deep at this point…

It’s also interesting to note that some people believe all knowledge comes from empirical experience. So, how do they explain analytic propositions like 2+2=4. Well, empiricists like Hume simply say they are “mere relations of ideas” and can only tell us how we use words/concepts. According to Hume, only synthetic propositions give us knowledge. Of course, there are deep problems with this reply. You can see my video on Kant’s Critique or Pure Reason or the one on Numbers for more.

Ok, those are some of the controversies. Let’s review for a moment why these distinctions are important. To quote Baggini and Fosl, “the a priori/a posteriori distinction is concerned with whether any reference to experience is required in order to legitimate judgments. The analytic/synthetic distinction is concerned with whether thinkers add anything to concepts when they formulate their judgments, thereby possibly expanding rather than simply elaborating upon their knowledge” (149). Also, crudely put, thinking through these distinctions simply deepens your understanding of knowledge and the types of claims floating around in your head.


Part III: Necessary vs. Contingent

A necessary truth is one that cannot be false. The denial leads to a contradiction. Examples: The desk is either black or not black. Cats are mammals. “It is simply not possible for claims that are necessarily true to be false-and for those that are necessarily false to be true” (170, Baggini).

Contingent truths are those that are not necessary and whose opposite or contradiction is possible. Contingent truths could have been different. Examples: I ate a taco for breakfast. Prostate Cancer is killing more people now than it did 10 years ago. The dog is on the cat’s mat.

It could have been the case that I ate cereal instead of a taco this morning. It could have been the case that the prostate cancer went down. It could have been the case that the dog was on the table instead of the mat. Since it seems reasonable to believe these could have been the case, it seems reasonable to believe they are contingent.

If you think about it, you probably see that a priori and analytic seem closely connected to necessary while a posteriori and synthetic seem closely connected to contingent. On the Carneades Channel, he illustrates the distinction like this:


Necessary & Contingent


A priori & A posteriori


Analytic & Synthetic


Group 1: Necessary, A Priori, & Analytic

Group 2: Contingent, A Posteriori, & Synthetic


This is a nice clear way to think of these distinctions. However, as we saw in the last section, there is much controversy. Kant believed some claims are synthetic a priori, so not all a priori statements are analytic. Quine and others have also brought up many objections. Before exploring those, let’s practice to make sure we understand.


Practice 3: Identify the following as necessary or contingent.

  1. It is not the case that it is raining and not raining.
  2. All bachelors are unmarried.
  3. Some men are obese.
  4. Napoleon won that battle.
  5. The cat is on the mat.
  6. George W. Bush was president in the 21st Century.

Answers: 1. Necessary 2. Necessary 3. Contingent 4. Contingent 5. Contingent 6. Contingent.


You may have had problems answering these. The distinction between necessary and contingent is easy to define, but can be difficult to apply.

For example, if you are a hard determinist then you may believe every event that occurs is necessary. In your worldview, there “is no room for luck or free will” (171, Baggini). For example, #6 above is necessary; George W. Bush must have been president; events could not have been otherwise. In a deterministic universe, this result was inevitable. Spinoza is an interesting philosopher who thought all events are necessary. So, as a hard determinist, you might disagree with the answers in Practice 3. You might think all are necessary.

On the other hand, there is W.V. Quine and his semantic holism. He believed all are contingent because even statements like 2+2=4 are not necessarily true; new facts or reasons may emerge that cause us to revise our judgment that 2+2=4. Do you agree with him? I don’t, but perhaps you do? In short, it is easy to define contingent and necessary, but quite difficult to get agreement on which claims (or events) are necessary and which are contingent.

As a sidenote, you can tell a lot about a person’s metaphysics or worldview based on how they think of these distinctions. For example, some philosophers get very angry with me because I agree with Kant that synthetic a priori knowledge is possible.

Problems also arise in Philosophy of Religion. In the ontological argument, defenders present God as a necessary being because he is a being who must exist. That is, it is part of the concept of God that he necessarily exists. I will not explore that here, but simply state that we need not only speak of necessary claims or events, but necessary beings. You can see my video “Cosmological Argument from Contingency” for more on that.

So, these are simple distinctions in theory, but there is much controversy as to how to apply them. Some epistemologists no longer use the analytic/synthetic distinction (since Quine), though it is still useful for studying older philosophers and contemplating your own beliefs. Again, I believe it is useful to deeply understand these distinctions because it will help us more deeply understand each philosopher and the nature of our own beliefs.

Here is a chart to help you understand the distinctions we learned:





Cannot be false

*Bachelors are unmarried men

A Priori

Knowable without experience

*Bachelors are unmarried men



True by Definition

*Bachelors are unmarried men



Can be false

*Bachelors are unhappy

A Posteriori

Knowledge requires experience

*Bachelors are unhappy


True by experience

*Bachelors are unhappy


Derived from Carneades Youtube Channel


Of course, as we have seen, these distinctions do not always line up. Not all synthetic truths are a posteriori, for example. Kant demonstrated that. Quine later questioned these associations in other ways. And so on.

Here’s a Question the leads to a deeper exploration; Classify this statement (Internet Encyclopedia of Philosophy)

  • The standard meter bar in Paris is one meter long.

“This claim appears to be knowable a priori since the bar in question defines the length of a meter. And yet it also seems that there are possible worlds in which this claim would be false (e.g., worlds in which the meter bar is damaged or exposed to extreme heat)”. So is it a priori and contingent? (Internet Encyclopedia of Philosophy).

One last one: consider this statement from Kripke:

Water is H2O.

This statement seems necessary, but also a posteriori?

It’s also interesting to note that Quine is a materialist, but Kripke is not. Does this influence their logical systems or vice versa? Or both?

The End. 51393


See lucidphilosophy.com or logic course on YouTube