. An inference or an argument is valid if its conclusion follows deductively from its premises. The premises may be false, but if they are true the conclusion must be true. An inference is invalid if it is not valid. It is contravalid if an inference from the same premises to the opposite conclusion would be valid. Sometimes, however, a valid argument is simply defined as one where it is logically impossible for all the premises to be true and the conclusion false. This raises the same ‘paradoxes’ as strict IMPLICATION. With inductive, etc., inferences, ‘valid’ may be used as above, in which case they are all invalid, but it may mean simply ‘meeting the standards proper to them’. A formula (propositional FUNCTION, open SENTENCE) is valid if it is true for every value of its VARIABLES. Otherwise it is invalid. It is contravalid if it is false for every value. Logically true propositions, i.e.
propositions instantiating valid propositional functions, are sometimes called valid, and logically false ones contravalid. Sound, applied to an inference, means either ‘valid, and having all its premises true’ or just ‘valid’. An interpretation of an AXIOM SYSTEM is sound if, under it, all the axioms and theorems are truths. Alternatively, it is sound if whatever is derivable in it from certain premises really follows from those premises. A proof system of any kind can similarly be called sound. Soundness is similar to but not identical with CONSISTENCY.
A.Church, Introduction to Mathematical Logic, vol. 1, Princeton UP, 1956, p. 55. (Soundness.)
B.Mates, Elementary Logic, Oxford UP, 1965. (See index. Soundness and consistency.)
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