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4. Formal Logic / B. Propositional Logic PL / 4. Soundness of PL

[any formula which has a proof is a valid formula]

2 ideas

9536	If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology [Lemmon]
	Full Idea: If any application of the nine derivation rules of propositional logic is made on tautologous sequents, we have demonstrated that the result is always a tautologous sequent. Thus the system is consistent.
	From: E.J. Lemmon (Beginning Logic [1965], 2.4)
	A reaction: The term 'sound' tends to be used now, rather than 'consistent'. See Lemmon for the proofs of each of the nine rules.

9543	The rules preserve validity from the axioms, so no thesis negates any other thesis [Hughes/Cresswell]
	Full Idea: An axiomatic system is most naturally consistent iff no thesis is the negation of another thesis. It can be shown that every axiom is valid, that the transformation rules are validity-preserving, and if a wff α is valid, then ¬α is not valid.
	From: GE Hughes/M Cresswell (An Introduction to Modal Logic [1968], Ch.1)
	A reaction: [The labels 'soundness' and 'consistency' seem interchangeable here, with the former nowadays preferred]