Let T be an theorem and S be the sequence of steps required to prove T in whatever system: If disjunctive syllogism is a primitive inference rule:Ī general justification for "verum ex quodlibet": Particular instances of "ex falso quodlibet" and "verum ex quodlibet" have quite short proofs on the myriad of Natural Deduction, Axiomatic, Sequent Calculus and other systems disposable. Consequently, one can valid infer a tautology from any set of propositions. Similarly, since a tautology is never false, therefore it's never false whilst the premises are true.
Consequently, one can validly infer any arbitrary proposition form a set of contradictory premises. Since a contradictory set of premises is never true, therefore it's never true whilst the conclusion is false. Well saying that "whenever the premises are true, so is the conclusion" is equivalent to say that "never the premises are true whilst the conclusion is false". In classical logic, any set of inconsistent propositions entails an arbitrary proposition and any tautology is logical consequence of any set of premises.īy definition, a proposition p is logical consequence of a set P of premises ( P⊨ p) if, and only if, in every valuation/model/universe of discourse/interpretation/situation where all elements of P are true, so is p.įor simplicity, let's say "an inference is valid iff, whenever the premises are true, so is the conclusion". Observe that "A∧¬A⊨B" and "B⊨A∨¬A" are just instances of "ex falso quodlibet" and "verum ex quodlibet", respectively. One can provide quite straightforward justifications for them, both on semantic and proof-theoretic grounds. The reasons underlying "ex falso quodlibet" and "verum ex quodlibet" are quite trivial.
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