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Semantic methods for checking logical entailment have the merit of being conceptually simple; they directly manipulate interpretations of sentences
Unfortunately, the number of interpretations of a language grows
with the number of logical constants.
When the number of logical constants in a propositional language is large, the number of interpretations may be impossible to manipulate.
Deductive (proof) systems provide an alternative way of checking and communicating logical entailment that addresses this problem
In many cases, it is possible to create a “proof” of a conclusion from a set of premises that is much smaller than the truth table for the language;
Moreover, it is often possible to find such proofs with less work than is necessary to check the entire truth table
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