On the question of what logic is about, what it’s supposed to help out with, I’ve often thought that the most fundamental answer is probably integration. Logic is the study of how to integrate your beliefs, how to be consistent, how to not contradict yourself. Although it’s not always the most useful social move to enforce integration in yourself and others—the social world often rewards inconsistency, contradiction—the tools of logic are there for that purpose. When you do want to integrate your beliefs, logic is there to help.
That way of explaining what logic is about, though—that logic is supposed to help out with something sometimes useful, sometimes not—is a way of zooming out and explaining what logic’s place is in life. If instead we just take for granted, as most scientists and philosophers do, that we shouldn’t contradict ourselves, then there’s perhaps a better way of explaining what logic is about:
Consider the distinction between the linguistic form of a word and its logical substance. In Japanese, along with Japanese and all of the other natural languages of civilization, there are at least two forms: (a) the spoken form, and (b) the written form. For logic, though, the linguistic form doesn’t matter. What matters is the logical substance. For example: Whether the word “moon” is spoken or written, the literal meaning is the same: 🌙. And in the same way, for logic it doesn’t matter whether we write “moon” or 月. Spoken or written English, spoken or written Japanese, all of those are just different linguistic forms associated with the same logical substance.
But it doesn’t stop with vocabulary. Consider grammar too:
- “English men and women”
- “English men and English women”
Famously, logic is about such equivalencies. Why? For the same reason: because here again the linguistic form is different but the logical substance is the same.
Consider also:
- “English men and women, together, make up more than half of everybody here.”
- “English men and English women, together, make up more than half of everybody here.”
It’s impossible for #1 to be true without #2 also being true, and it’s impossible for #1 to be false without #2 also being false—logically impossible. That’s because logically speaking, #1 and #2 are the same. They’re different linguistic forms associated with the same proposition.
Logic maps the complexity of the surface forms of natural language to something simpler and deeper: the underlying logic, the underlying literal meaning. The surface-grammatical form “[adjective X] [noun X] and [noun Y],” exemplified in the phrase “English men and women,” is logically equivalent, equivalent in its literal meaning, to the surface-grammatical form “[adjective X] [noun X] and [adjective X] [noun Y],” exemplified in the phrase “English men and English women.”
To follow George Boole in adapting the traditional notation of algebra to logic:
- “English men and women.” x(y + z)
- “English men and English women.” xy + xz
Logic, then, (a) takes as an axiom that we shouldn’t contradict ourselves, presupposes that we shouldn’t contradict ourselves, and then (b) ignores everything except literal meaning. That’s what logic is about.