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I am currently taking an introductory course to mathematical logic. We have started with propositional logic and today introduced the Gentzen style proof calculus. In order to prove that the soundness property of this we first needed the Principle of Rule Induction. My professor stated it as follows:

Theorem: Let P be a property closed under all basic rules of derivability. Also, let X be a set of sentences and $\alpha$ be a sentence. Then X $\vdash$ $\alpha$ implies P holds for the sequent (X,$\alpha$).

I understood the proof of this theorem, it was fairly simple proof by induction on the length of formulas. Where I am struggling is the intuition behind the application of the principle. Is it saying that if I can show that some property holds for all basic rules of the propositional calculus, then it will hold for all rules that are derivable in the propositional calculus? For example, in proving soundness, would I be showing that for all basic rules their syntactic provability implies they have semantic truth?

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You want to prove that the calculus only produces true statements. To do this, you first prove that all the axioms are true statements, and then prove that, for each rule, if the rule takes as input true statements, then it produces a true statement. Then, by induction, this implies that the calculus produces only true statements, since every statement is obtained in a finite number of steps.

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    $\begingroup$ So essentially if I can prove that all basic rules produce semantic truth all possible derivations from these rules will be semantically true since all derivations must either be an initial sequent or derived from the rules in a finite number of steps. Is this correct? $\endgroup$ – Matt Dyer Feb 2 '16 at 17:00
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    $\begingroup$ You have to prove that the rules preserve semantic truth. Meaning, that if the input sequents to the rule are semantically true then the output sequent is semantically true. $\endgroup$ – Gabriel Nivasch Feb 2 '16 at 17:27
  • $\begingroup$ I think I have it now, thank you very much! $\endgroup$ – Matt Dyer Feb 2 '16 at 17:59
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The intuition is:

the rules of the calculus are "designed" in order to preserve (semantic) validity (this means soundness).

Thus, if you start from a true sequent, you can never derive an "untrue" one applying the rules of the calculus.

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