The distinction between analytic and synthetic propositions is often associated with logical positivism, a philosophy propounded by A.J. Ayer, among others.

Synthetic propositions are those that are associated with some level of uncertainty. Their referents are data that can be gathered using the senses.

Analytic propositions, on the other hand, are true because of the way that we have defined them. If a synthetic proposition has some probability $p$, where $p < 1$, then an analytic proposition has a probability of 1.

One might object to the notion that anything can ever be fully certain. Those objections are, of course, well-founded; however, to build upward, certain necessary assumptions must provide us with a foundation.

That we defend analytic propositions is not to mire ourselves in dogma but to show that they enable a vast array of cherished propositions.

Whether mathematical operations are uncertain

Suppose you did try to incorporate uncertainty into your math, and you did so using the language of probability.

For every probability calculation, you would multiply each posterior by $1- \epsilon$, where $\epsilon$ is a penalty constant. Here, I want to emphasize that it would be the operation itself that we are skeptical of, not hardware limitations, i.e., errors due to limited precision floating point numbers.

Then, since that multiplication by $1 - \epsilon$ is also a probability calculation, we would need to multiply the resulting posterior by $1 - \epsilon$ again, and again, resulting in an infinite regress.

Note that this is not the same as saying that there could have been a misstep in a body of calculations or a proof. In that case, we simply say that the author made a mistake, not that math itself is somehow suspect. The written result either does or does not follow from the written premises, and we either do or do not assume that it follows until we have verified for ourselves that it does not. Such cases are legion, but they are always the fallible human.

Whether anything in nature cannot be explained by math

The objection to all of this might just be that the uncertainty does not have to be distributed over the mathematical operations. Instead, what we are uncertain about is whether there is anything in nature that could be explained by math.

I literally cannot conceive of what that would be. What thought could anyone have that could not be formulated as mathematics? What property would it have to have that it would not simply be one more relation added to humanity’s knowledge of mathematics, and yet still be capable of being formulated as a synthetic proposition?

Whether all of this is nonsense because we are exalting math as some kind of higher being

From a purely conventionalist standpoint, we have said nothing inconsistent. “Math” is not a term that describes the scribbles in the notebook of a calculus student. It is not the syntax. It is the semantics.

To have a coherent theory of logic and language, we must employ the concept of meaning. How else would math fit into this picture except how we have painted it when we have already assumed that math can imply nothing about the real world?

“Ah, but you should not have assumed that.”

Then let’s claim that math can imply observations and tell a little parable to illustrate the possible consequences.

In ancient Greece, two philosophers decided to set out to find the result of $1 + 1$. One traveled north, and the other went south.

In the north, one philosopher came to stay as a guest in the home of a couple for most of a year. In that time, the couple conceived ($1 + 1$) and the woman birthed a baby ($= 3$).

The other philosopher came to a pond with water so crystal clear that you could see right to the bottom. He spent the better part of a year watching the fish. The fish had large mouths that they could open wide and use to gobble up smaller fish. He saw a fish ($1$) eat a smaller fish ($+ 1$), and the result was only one fish ($= 1$). This was a pattern that he saw repeat itself many times.

The two philosophers journeyed back to their starting point and reconvened.

The northern philosopher said, “I have discovered the answer. One plus one equals three.”

“Why, no,” the southern philosopher said. “One plus one equals one.”

To reconcile this difference, they decided that in the north $1 + 1 = 3$ and in the south $1 + 1 = 1$. In other words, the result depended on the geographic location.

Claims regarding analytic propositions are seen as suspect because of instincts that carry over from examining synthetic propositions.

A synthetic proposition, p, is one for which there is an observation we could encounter for which we would have to adjust our credence regarding p.

We will illustrate two examples in which a proposition might at first be taken to be synthetic but is in fact nonsense.

The dragon in the garage

This hypothetical famously comes to us from Carl Sagan in A Demon-Haunted World. Suppose someone claims that there is a dragon in his garage.

I follow the claimant to his home to examine this alleged dragon. Within, however, I see no dragon. Subsequently, the claimant explains my observation away by saying that this dragon is invisible.

“No problem,” I say. “We will simply spread flour on the floor to capture the footprints.”

Except our claimant says that the dragon constantly floats in the air, never touching the ground.

I go on to propose other tests, such as using an infrared detector and spray paint on the dragon. After each such proposal, I am met by some reason why that experiment will not result in observations other than what would in the case where the dragon didn’t exist at all.

There is no observation we could make that would change our minds about whether there is a dragon in his garage.

Exists and only exists

In Language, Truth, and Logic, A.J. Ayer supports the idea that “existence” is not a predicate. This is because any predicate must have the property that it can stand alone. This sets us up for a contradiction in the following way:

Imagine that we have a database of propositions consisting of predicates and atoms. In the proposition p(a), p is a predicate and a is an atom. If the atom a appears in p(a) and q(a), then p and q are properties of the same thing.

Suppose we add to the database existence(b) without the atom b ever appearing in any other proposition. Then this is to be interpreted in English as, “b exists and yet does not have any other properties.”

In other words, it is an invisible, floating, incorporeal dragon. Our senses could not gather any data about this b, but we suppose that it exists anyway.

Making “existence” a predicate makes possible a proposition that flies in the face of common sense.

These two examples are both relevant to our argument in the following way.

“An analytic proposition is not a synthetic proposition” accords with our usage of language

An analytic proposition is not a synthetic proposition in that no observation we could make could ever change our mind about an analytic proposition, and this is consistent with the way we use language.

“Dragon” refers to a member of a class of fearsome, scaly creatures. On the other hand, x as a variable is not a symbol that conventionally refers to any member of any class. Neither does any mathematical relation.

“But wait!” you say. “What if I am solving a problem where we suppose that x is a quantity of money in a certain situation and there is enclosing that x a formula relating several other variables and constants?”

Then you merely suppose x refers to that quantity and quality situationally. That does mean that x has the same characteristics of a symbol like “dragon”, which can call to mind a definite set of sensory experiences even when it is spoken in isolation.

In that situation, your usage of the formula is an empirical proposition. You are asserting that the real-world quantities and qualities can be related in such a way that they can be described by the formula. When an observation is encountered that does not accord with this proposition, you would be well-advised to discard it in favor of another. You do not, however, discard any analytic proposition because no such proposition could ever have implied anything about your observation.

You come to me with a proof beautifully typed in LaTeX. Do I conclude that the conclusion of this proof is 100% certain to follow from the premises? Not necessarily. I could have a severe sinus infection that day, and that skews my judgment.

Is it the analytic proposition referred to by the syntax of the proof that is uncertain, then?

No, what is uncertain is whether the syntax, the physical marks on the page, agrees with my brain about which analytic propositions are referred to therein.

To use an example that hits a bit closer to home for us programmers, suppose that I wrote an app and pushed it to GitHub. My users download it. One of them opens issue #1 in the Issues pane of GitHub. This issue documents that the behavior of the software is other than what the user would expect. In other words, it failed the test t, where t is whatever test the user made of the software.

To break this down precisely, the code represents one analytic proposition p; there’s another empirical proposition, q, in my brain represented by whatever I think the code represents; and there’s another analytic proposition, r, corresponding to a Minimum Viable Product that would pass test t.

Whether the software will subsequently pass test t depends on whether I successfully analyze the code, updating my q to q’; modify the code, updating it to analytic proposition p’; and then update my q’ to q’‘, or the empirical proposition referring to whatever analytic proposition that the code is.

If the user closes the issue, then they believe that the software represents r. In other words, the user holds s(r), or the empirical proposition that the program passes t.

Why care about all of this?

Eventually, I aim to create a probabilistic logic programming language with a general-purpose knowledge base. When I do so, some productive assumptions will need to be made, such as

  • Some propositions are associated with a certain probability.
  • Some propositions are true because they are true by definition.

This corresponds to the synthetic/analytic distinction.

Bibliography

Ayer, Alfred J. Language, Truth and Logic. Reprinted. Penguin Books, 1990.

Sagan, Carl. The Demon-Haunted World: Science As a Candle in the Dark. With Ann Druyan. Random House Publishing Group, 2011.