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Trying to make my way through the book Gödel's Proof (Nagel & Newman, edited by Hofstadter).

In chapter V, the authors are showing that the axioms of sentential calculus are not contradictory. In short, they say (or seem to say):

Let's assume that the calculus is contradictory, such that $S$ and its contradiction $\sim S$ are both true. Then the theorem $p \supset ( \sim p \supset q )$ can be substituted with $S$ and $ \sim S$ to force $q$ to become true! Hence, if the calculus is contradictory, then any formula whatsoever is deducible from the axioms.

The way to show that the calculus is not contradictory is to find a formula that's not always true. In the text they use the example $p \vee q$. It's not always true, hence, it is shown that the calculus is consistent.

However... how can you assume that $p \vee q$ cannot be true? If the calculus is inconsistent, why couldn't there be a lengthy transformation that shows that $p \vee q$ is true?

It seems like a somewhat circular argument... what am I missing?

Thanks!

Mauro ALLEGRANZA
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Derek Ross
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1 Answers1

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The truth of a formula in propositional calculus is defined through the semantics for the language, based on valuations.

A formula of propositional logic is called a tautology when it is true for every valuation, like : $p \lor \lnot p$.

A simple way to check if a formula of propositional logic is a tautology or not is to use the truth-table method.

For propositional logic, we can prove two meta-theorems :

(i) Soundness : if a formula $\varphi$ is provable in the propositional calculus, it is a tautology (i.e. if $\vdash \varphi$, then $\vDash \varphi$).

(ii) A logical calculus is consistent iff there is at least one formula $\varphi$ not provable in it (i.e. $\nvdash \varphi$).

Now, we can cook them together in the following way : $p \lor q$ is not a tautology, because with the valuation $v$ such that $v(p)=v(q)=$f we have that $v(p \lor q)=$f. Thus, having found a valuation that does not satisfy it, we conclude that the formula is not true for all valuations, and thus it is not a tautology.

Thus, by (i) (soundness), it is not provable in the calculus.

Having found a formula not provable in the propositional calculus, we conclude by (ii) that the calculus is consistent.

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Mauro ALLEGRANZA
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    As always, a very clear exposition! – André Nicolas Sep 26 '15 at 18:05
  • Thanks, I agree it's clear that $p \vee $q is not a tautology. However, if we ignore the truth table analysis of $p \vee $q, how do we know there's not a way to transform $p \vee $q into a tautology, step by step? For example, we might start with $p \vee q$, then go to $(p \land p) \lor (q \lor q)$. Then DeMorganize that to $\lnot(\lnot(p \land p) \land \lnot(q \lor q))$. Then continue like that for, say, 5000 steps... how do we know we will never arrive at some actual tautology, such as $p \lor \lnot p$? – Derek Ross Sep 27 '15 at 15:15
  • @DerekRoss - $p \lor q$ is not a tautology, fullstop. But assuming that we can derive a tautology from it, so what ? In inconsistent calculus is a claculus taht prove absolutely everything : this does not mean that all formulae are tautology or that all formulae are true. – Mauro ALLEGRANZA Sep 27 '15 at 16:59
  • @MauroALLEGRANZA $v$ is the truth functional, correct? $v(\phi) = \text{true}$ if proposition/wff $\phi$ evaluates to true under some model, and $\text{false}$ otherwise? – user525966 Sep 24 '18 at 17:03
  • @user525966 - $v$ is a truth valuation (an interpretation). If $v(\phi) = \text {true]$, then $v$ satisfies formula $\phi$. – Mauro ALLEGRANZA Sep 24 '18 at 19:11