We now turn our attention to the fundamental notion of a logical deduction or a proof. As mentioned earlier, in mathematical parlance, statements valid in all models of a theory are called the theorems of the theory. But any statement is a well-formed finite sequence of symbols of the language. So, it is natural to expect a finitary definition of a theorem depending only on its subformulas and the syntactical construction. Note that while computing the truth value of a statement in a structure, one uses some rules of inference depending only on the syntactical construction of the statement. For instance, if A or B is true in a structure, we infer that A ∨ B is true in the structure. We have also noted that statements with some specific syntactical structures are valid in all structures. For instance, a statement of the form ¬A∨A is true in all structures. Statements true in all structures of a language are called tautologies. So all tautologies ought to be theorems. Is there a conveniently nice list of tautologies (to be called logical axioms) and a list of rules of inference such that a statement is valid if and only if it can be inferred from logical and nonlogical axioms using the rules of inference from our list? It is indeed the case.
In this chapter we first develop a simpler but important form of logic called propositional logic. The main objective of propositional logic is to formalize reasoning involving logical connectives ∨ and ¬ only.
In this chapter we shall introduce propositional logic in details.
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