Art of Reasoning

A Priori Knowledge

In a nutshell, the term “a priori” refers to knowledge that is gained logically-prior to, or independent of, experience. Two questions immediately emerge: 1) what exactly do we mean by “experience;” and do we actually have any knowledge independent of experience?

To answer the first question, let’s think about the common denominator in all of our experiences. From your experience of a pain to your experience of the sensations of eating ice cream to your experience of the passage of time itself, the common denominator is that your experiences are sensory experiences.

Now, by “sensory,” we mean both the five “typical” senses and what is most commonly called “inner sense.” We are, of course, familiar with the “five senses” that are associated with the body’s connections with the world. But a bit of self reflection will reveal that you are also intimately familiar with “inner sense,” that sense of self we all have, which includes our sense of time, our awareness of our own thoughts, our sense of our own emotional states, and so on. Inner sense is just as much a connection with “the world” as are the “outer senses.” We experience “the world of experience” through both inner and outer sense. So, by “experience” we mean literally “experience of the world” via “the senses.”

To answer the second question, we must first clarify that by “prior” we do not mean “prior in time.” We mean strictly logical priority rather than temporal priority. Here is an important example:

When we come to understand, say, the Pythagorean Theorem, we usually learn it in school. We learn it by being taught, which is, strictly speaking, “by experience.” Indeed, thinking of the process in time, we learn the Pythagorean Theorem temporally after we have experience. We are taught (experience), and then we come to know the Theorem. So it is easy to think of that learning process as being entirely dependent on experience.

Yet, knowledge of the Pythagorean Theorem is actually a priori knowledge. You, personally, might learn the Pythagorean Theorem via the experience of being taught it in time, but that is not what we mean by “logically prior.” To see that this is the case, let’s think more deeply about the process.

Consider what the Pythagorean Theorem is really about. It expresses a necessary relation between the lengths of the sides of a right triangle, actually any right triangle. Regardless of what right triangle you imagine, the Pythagorean Theorem accurately expresses the relation of the lengths of its sides. Now, there are two important things to notice about this fact.

1) Necessity: the Pythagorean Theorem is never an incorrect expression of the relation between the lengths of the sides. Never. It is always correct, and this is because it is logically necessary that it is correct. The Theorem can be deductively proved because that sides-length-relation necessarily holds.

2) Universality: the Pythagorean Theorem is universally correct; it correctly expresses that sides-length-relation for all right triangles without exception. This notion closely correlates with that of generality, which is to say that a priori knowledge is generally applicable rather than only in particular cases. Here the term “general” does not mean “most of the time;” it means all of the time and in all scenarios.

Now, notice an important point about these attributes. We cannot in principle derive these attributes from experience.

Yes, we come to learn the Pythagorean Theorem through being taught, which is a function of experience. But we do not know the truth of the Pythagorean Theorem by being taught it. The truth of it can only be known completely apart from experience. The geometry teacher is only making us aware of a geometric relation, but that awareness is not the same thing as knowing the Theorem to be true. That knowledge must be gained completely apart from experience. And here is how we know that this knowledge is independent of experience.

Remember that the Pythagorean Theorem applies to all right triangles, universally. And, remember that the geometrical objects to which the Theorem applies are not actually empirical objects that we can ever measure.

Of what width is a geometrical line? It is of zero width.

Of what precision is a geometrical angle? It is of perfect precision that can never be truly replicated by the crude representations we make with our drawings and CAD programs and houses and so forth.

We cannot in principle measure every possible right triangle to see if the Pythagorean Theorem turns out to be accurate of all of them! We cannot do that because we cannot have access to all of them, even given an infinite time in which to perform measurements!

And even if we could somehow do an infinite measuring project, we cannot measure the actual geometrical objects because they are abstractions. We cannot access them in our experience, because we cannot have any possible sensory experience of these abstract objects.

So, there is no possible measurement project by which we could “test” the Pythagorean Theorem via experience to discover its necessary and universal truth regarding right triangles. And when we use the Pythagorean Theorem to measure the crude representations of right triangles that we can produce by drawings or house foundations or other such things, we find that the Theorem is only “relatively” accurate! It is not perfectly accurate when we measure actual objects of experience, simply because we cannot produce perfect representations of geometrical right triangles.

Thus, our learning about and our application of the Pythagorean Theorem is a matter of experience, but our knowledge of the truth of the Pythagorean Theorem cannot in principle result from or in any way be dependent upon experience. The proof of the Pythagorean Theorem does not rely in the slightest upon experience, nor in principle could it!

There are other fields of a priori knowledge, such as mathematics, logic, and (Kant would say) ethics. A priori knowledge is denoted by its necessity and universality. (I am well aware that some philosophers dispute these attributes, but that discussion is far beyond the scope of this course, and I do not find the arguments compelling. So, I present a very “standard” and most widely-accepted account of these knowledge types.)

A Posteriori Knowledge

In stark contrast with a priori knowledge, a posteriori knowledge is knowledge we can only gain through experience. Again, we are referring to sensory experience, where “sensory” includes the five senses plus inner sense.

Another term for a posteriori knowledge, and the one we will use most frequently, is “empirical.” The term “empirical” is quite widely used, even by people who are not scientists or philosophers. We hear such phrases, for example, as “the empirical sciences.”

The term “empirical” refers to “of the senses” and references what we come to know through sensory experience.

Examples are legion!

How much money do you have on you at this moment? To know the answer, you must check, and checking will be based upon sensory experience.

How many chairs are in a given room? To know the answer, you must count, and counting will depend upon sensory experience.

And even these examples immediately reveal that empirical knowledge has exactly the opposite attributes of a priori knowledge:

1) Contingency: empirical knowledge is not necessary. The state of affairs could be other than it happens to be.

2) Relativity: empirical knowledge is not universal. The state of affairs as I presently know it can change depending upon time and place (and even perception itself).

If I tell you that there are 32 chairs in a given classroom, and you then go in to double-check, you might count a different number of chairs. Perhaps at the time I counted there were indeed 32 chairs. Perhaps I miscounted. Perhaps between my count and your count somebody else added or removed chairs from the room.

There is no necessary state of affairs that makes it the case that a given classroom must have a particular number of chairs in it. At any given moment, the only way to know how many there are is to count them.

Also I cannot know just from a definition of “classroom” that the classroom in question must have a particular number of chairs. It is certainly not the case that it is generally/universally the case that classrooms all have a particular number of chairs just in virtue of being classrooms. Thus, the number of chairs in a given classroom is relative to the particular classroom. Empirical knowledge is particular rather than general/universal.

Notice what happens if you and I enter into a dispute about the number of chairs in the classroom. Imagine our conversation, and pay attention to when my depiction of it goes wildly and obviously astray!

Me: There are 32 chairs in ADM201.

You: No. I was just in there and I counted 31. I’m not sure if that will be enough.

Me: What? I’m sure that there are 32 chairs in there. I counted carefully!

You: There are not 32. There are 31. I also counted carefully!

Me: Don’t you trust my count? Do you think I’m lying?

You: Of course I trust your count and your integrity. I believe that you are accurately reporting what you found.

Me: Okay, so, wow! Here we have a clear-cut counterexample to the principle of addition! Cool! Sometimes 1 + 1 does not equal 2. So, sometimes 31 + 1 does not always equal 32. For me it does, at least it did when I did that count; but for you apparently it does not.

You: Yes, that must be it. I’ve always believed in the principle of addition, thinking it to be universally reliable. But now I see that it doesn’t always work as I was taught. Yes, for you it worked as we would typically expect. But for me, just now, I found that 31 + 1 somehow still equals 31.

Obviously this is a ridiculous conversation, but it clearly denotes the vast gulf between a priori and empirical knowledge! When empirical knowledge becomes questionable, notice that what we do not do is question the a priori principles! We know intuitively that empirical knowledge is unreliable, and we know (even if we never articulated it before) that this is because all empirical knowledge is contingent and relative.