# Argue that if a sentence has a proof, then it is a tautology

This is a corollary of the soundness theorem, which states that for a set of formulas $\Phi$ (of propositional logic) and a formula $\alpha$ : $$\Phi\vdash\alpha\Longrightarrow\Phi\vDash\alpha$$ What I'm trying to show is the case where $\Phi=\emptyset$, i.e. that if $\alpha$ is a proof then it must be a tautology, but without applying the soundness theorem directly.

A proof for the soundness theorem could go like this: let $\Phi\vdash\phi$, then there is a deduction $\phi_1,\ldots,\phi_n$ from $\Phi$ of $\alpha=\phi_n$. Each $\phi_k$ of the deduction is either a tautology, a member of $\Phi$ or the result of Modus Ponens of two previous $\phi_i$ and $\phi_j$. We can then show by strong induction on $k$ that $\Phi\vDash\phi_k$.

However, I don't quite see how we could apply this scheme to the case $\Phi=\emptyset$. Any idea? Or another way of showing it?

• $\vdash \phi$ means that : there is a sequence of formulae such that the last formula is $\phi$ and that each formula in the sequence is a logical axiom or is derived from previous formulae in the sequence by modus ponens. So you have to apply induction on the derivation proving that (i) every log ax is a tautology and (ii) an application of mp preserves "tautologuesness". – Mauro ALLEGRANZA May 10 '15 at 9:34
• Thanks @MauroALLEGRANZA, that answers my question - if you type it up as answer, you'll get my vote :) As I said in my comment to Bruno below, I'm aware of the abbreviation you mention, and of the so-called "weak soundness" theorem. I was just wondering if we could apply the same sketch of proof as for the "strong" soundness theorem, and you cleared that up for me. Thanks! – Demosthene May 10 '15 at 23:27

If we agree that "to have a proof of $\alpha$", i.e. $\vdash \alpha$, means :

there is a sequence of formulae $\phi_1, \ldots, \phi_n$ such that $\alpha=\phi_n$ and that formula in the sequence is a logical axiom or is derived from previous formulae in the sequence by modus ponens,

this amounts to the "special case" of derivation from assumptions : $\Phi \vdash \alpha$ with $\Phi=\emptyset$.

So, to prove the desired result, we have to apply induction on the derivation showing that :

(i) every logical axioms is a tautology, and

(ii) an application of modus ponens preserves "tautologuesness".

In conclusion, the proof is basically the same we have for the general case : we have only to "skip" from the induction the part : "... or is a member of $\Phi$".