Socrates must be mortal

All men are mortal,
Socrates is a man,
Therefore, Socrates must be mortal.

This is the classic Socrates syllogism that has been used to illustrate deductive logic for ages.

Deductive logic takes general rules and comes up with new specifics. “All men are mortal” is a general rule. That Socrates is mortal is a specific fact.

When I was learning logic, it seemed like logic was an arcane set of rules and relationships. The way we teach logic emphasizes the rules, and masks the function of the rules. The simple function of the rules is to avoid contradictions. If Socrates were a man who is immortal, that would contradict the premise that all men are mortal. Deductive logic is based on the principle of non-contradiction: contradictions should be avoided.

Another confusion that can arise is to think that statements in deductive logic refer to beliefs. Generally, they do not. This is a subtle point, but an important one. We can see the validity of the Socrates syllogism, even if we don’t believe that all men are mortal, or that Socrates is a man. The statement “Socrates is a man” is a proposition, i.e., a statement that could be true or false. Propositions are not beliefs, but more like statements we can have a belief in. This is why I like to write the Socrates syllogism with the conditional “if” as a prefix on the premises:

If all men are mortal,
and if Socrates is a man,
Then, Socrates must be mortal.

Deductive logic tells us which sets of statements are logically coherent. That is, it tells us which sets of statements we can believe in at the same time without contradiction. There are countless logically coherent sets of propositions. For example, another valid syllogism goes like this:

If all men are immortal,
and if Socrates is a man,
Then, Socrates must be immortal.

It’s just as valid as the classic syllogism, but we can’t believe in the propositions in both syllogisms at the same time without contradiction.

Given that there are countless sets of logically coherent, but mutually exclusive propositions about the world, how do we decide which set to believe in?

That’s where induction and experience comes in.