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# Propositional Logic: partIIo

Summary: Reducing statements to true/false propositions allows us to craft sentences with precise meaning, and to make precise arguments which can be checked automatically.

## Aside:

N.B. Can be more minimal by only allowing ,,→; use "(afalse)(a)" to mean "¬a¬a".

Question: is it possible for me to ask for a proof, but the sheet doesn't have enough domain axioms (or, enough Boolean identities) for you to successfully prove it? The technical way to phrase this question is Is our system complete?''

In the late 1800s and early 1900s, mathematicians became more concerned with logic and proofs, realizing it was the foundation of all mathematics. A couple of disturbing problems had been posed, with almost childish ease: If a set is defined simply by being able to say, for any item, whether or not the item is in the set, then we can talk about UU, the set of everything. (When asked if something is in UU, the answer will be yes.) Thus, UU contains itself. Okay, but what about the set RR, defined as the set of: all sets which don't contain themselves. Does RR contain itself? There is no consistent answer to this question, called Russell's Paradox'', after Bertrand Russell. This caused people to ask, is mathematics sound? That is, can it be used to prove things which aren't true? (Such as a set which both does and doesn't contain itself.)

This led to the development of a set theory which avoided the possibility of paradoxical sets, and at the same time led people to wonder what is a proof system which both sound and complete, and can we prove it so? Russell, along with Alfred North Whitehead, spent years writing the Principia Mathematica, a multi-volume tome of formal proofs, trying to secure the foundation of all mathematics. Unfortunately, they were unable to prove the soundness of mathematics, using those same proofs.

Some light reading: Hilbert's 23 problems: Here's a Scientific American article which provides some nice context, and a straight listing of Hilbert's 23 problems and their status.

Frege-Lukasiewicz introduced an inference system for Propositional Logic (not first-order logic) which has only three rules:

• (a(ba))(a(ba))
• ((a(bc))((ab)(ac)))((a(bc))((ab)(ac)))
• ((¬a¬b)(ba))((¬a¬b)(ba))
[lemma: (aa)(aa)] Can prove this complete: any tautology has a proof! (Idea: first show it hold for premises aa,¬b¬b,¬c¬c,dd,… [need a lemma for this]; then eliminate the premises one by one (getting 2 n-12 n-1 of them, then 2 n-22 n-2, …).

## Aside:

The proof in Hein's book looks short, but he goofed up a bit: "for any truth assignment, there's a proof from premises q1q1,…", and then later "if q1q1 was true, then …", but q1q1 wasn't a specific. He swaps "for any truth assignment" with "for some specific truth assign". Patching his approach leads to the very long formula described here.]
)

After Hilbert's 1900 address, many talented mathematicians worked hard on proving the completeness of first-order logic. It was not a problem that turned out to be easy, and it became an entire research program, many people investing their entire career in this. The answer came in two surprising parts,

• in 1929, Gödel's completeness theorem (his Ph.D.) showed an inference system that was complete for first-order logic. However, it was a very bare-bones system, with no axioms specific to (say) numbers. Thus, a statement which is true in all domains can be proven, but the question of whether true-statements-about-numbers could be proven by this system remained open.
• In 1935, he followed his feat with Gödel's Incompleteness Theorem, which showed that first-order logic is not complete, when restricted to arithmetic (or more generally, any inference system powerful enough to express the axioms of arithmetic).

This latter was a particular rude shock to many mathematicians; overnight their life's work of incremental progress was shown to have been advancing down a wrong alley. In fact, the idea that something can be true yet not provable is disturbing all by itself.

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