Please explain their relationship, and explain how they reduce the question of completeness to a proof of the inconsistency lemma reproduced below.

Question

Statements

a. Γ ⊢ P

b. Γ ⊨ P

c. Γ ∪ ~{P} is inconsistent in SD

d. Γ ∪ ~{P} is truth-functionally inconsistent

Consistency lemma

If a set Γ of sentences of SL is consistent in SD, then Γ is also

truth-functionally consistent.

Answer 1

Based on the question, I assume you are taking an intro logic course, probably using The Logic Book by Bergmann, Moor, and Nelson. Note that terminology differs somewhat across different textbooks, but what I say below should hold pretty broadly.

The difference between (a) and (b) is the single turnstile versus the double turnstile. (That's the little symbol that looks like a T turned on its side.) The single turnstile in (a) is the relation of syntactic consequence, while the double turnstile in (b) is semantic consequence. In terms of basic propositional logic, syntactic consequence is usually unpacked in terms of provability in an appropriate axiom system -- like The Logic Book's system SD. So Γ ⊢ P means that given a set of sentences Γ, you can prove P, in the relevant system (SD).

Turning to (b), Γ ⊨ P -- using the double turnstile -- means that whenever all of the sentences Γ are true, so is P. In propositional logic, you usually demonstrate that by constructing a truth table.

(c) Γ ∪ ~{P} is inconsistent in SD. This means that from the combined set of sentences including everything in Γ and ~P, we can derive a contradiction (in the relevant system, SD). This ends up being equivalent to (a), in the sense that (a) implies (c), and (c) implies (a).

Likewise, (d) Γ ∪ ~{P} is truth-functionally inconsistent means that we can show that the big conjunction of everything in Γ and ~P is always false. That's how you show something is a contradiction, using a truth table. It also means that (d) is equivalent to (b), as above.

In terms of consistence and completeness, here's what you need to know:

1. Consistency (of a system) means whatever is provable is also true. That is, (Γ ⊢ P) implies (Γ ⊨ P).
2. Completeness (of a system) is the converse: whatever is true is provable, so (Γ ⊨ P) implies (Γ ⊢ P).

So, coming back to the original question about the consistency lemma, it looks like there may be a couple of typos in your question. For example, you don't mean "inconsistency lemma", I think. But at any rate, the statement as written just says that (Γ ⊢ P) implies (Γ ⊨ P), which as I noted above is pretty much the definition of consistency. So if you prove that lemma for the system SD, you have proven that SD is consistent. For completeness, you need to prove the converse. Proving consistency does not prove completeness, by itself.