Modal Logic

Introductory textbooks in logic devote considerable attention to a part of logic we have not given separate consideration: sentential logic. In this part of logic, the only nonlogical symbols are an enumerable infinity of sentence letters, and the only logical operators are negation, conjunction, and disjunction: ∼,&,∨. Alternatively, the operators may be taken to be the constant false (⊥) and the conditional (→). The syntax of sentential logic is very simple: sentence letters are sentences, the constant ⊥ is a sentence, and if A and B are sentences, so is (A → B).

The semantics is also simple: an interpretation is simply an assignment ω of truth values, true (represented by 1) or false (represented by 0), to the sentence letters. The valuation is extended to formulas by letting ω(⊥)=0, and letting ω(A → B) = 1 if and only if, if ω(A) = 1, then ω(B) = 1. In other words, ω(A → B) = 1 if ω(A) = 0 or ω(B) = 1 or both, and ω(A → B) = 0 if ω(A) = 1 and ω(B) = 0.