Logic

I think that set theory and formal logic need to be re-done rationally. This is a big project. I will list out some of my problems with formal logic below:

Material implication

$A\implies B$ is held to equivalent to $\neg A\vee B$, so anything follows from a falsehood. This doesn't fit natural language reasoning. For example, suppose that it is not raining outside right now, and I say "if it were raining outside right now, then communism would be true." The former statement is false, so the implication is true. Clearly that's absurd.

Essence

In plain English, we have a lot of sentences like "bread nourishes," which don't fit into "for all" or "there exists." Indeed, "bread nourishes" doesn't just mean that there exists some bread that nourishes (it means way more than that), nor does it mean that ALL bread, even bread laced with arsenic, nourishes. It means that bread, in essence, nourishes.

Examples of universal statements that fit this pattern:

• Lamps provide light (but not ALL of them provide light, because some lamps are broken)
• Mirrors reflect light (but not ALL mirrors reflect, because some mirrors are covered in soot)
• Leaves absorb light through photosynthesis (but not ALL leaves do, because some are dead)
• Birds have feathers (but not ALL birds have feathers because some have been plucked)

Maybe examples:

Non examples:

• Birds can fly (isn't this just a false sentence? Not true in essence, and not true for all birds.)

Conjecture(?): It appears that the above is to "for all" as statements about particulars, like "Bob is mortal," are to "there exists."

Subjunctive quantifiers

In English, there is a difference between "all" and "any," and between "there exists" and "there could exist." Linguists call the former "indicative" and the latter "subjunctive."

This seems very important to me, especially the difference between "there exists" and "there could exist." The current theory of logic has a "block universe" picture built into it. Either something exists in the block universe, or it doesn't; there is no "could."

The "all" vs "any" distinction comes up when dealing with the infinite. With the doubling function $\mathbb {N} \rightarrow \mathbb {N}$ sending $n\mapsto 2n$, it's not right to think about it as "all" natural numbers. You should think about it as doubling any natural number that may be presented.

There's some connection between this and the material implication. Usually, though not always, when we say "if X then Y," we have in mind that X could be true and that Y could be true.

Terms vs. predicates

[TODO]