Propositional logic

Abstract

In propositional logic, atomic formulas are propositions. Any assertion will do. For example, . . . A = “Aristotle is dead,” B = “Barcelona is on the Seine,” and C = “Courtney Love is tall” . . . are atomic formulas. Atomic formulas are the building blocks used to construct sentences. In any logic, a sentence is regarded as a particular type of formula. In propositional logic, there is no distinction between these two terms. We use “formula” and “sentence” interchangeably. In propositional logic, as with all logics we study, each sentence is either true or false. A truth value of 1 or 0 is assigned to the sentence accordingly. In the above example, we may assign truth value 1 to formula A and truth value 0 to formula B. If we take proposition C literally, then its truth is debatable. Perhaps it would make more sense to allow truth values between 0 and 1. We could assign 0.75 to statement C if Miss Love is taller than 75% of American women. Fuzzy logic allows such truth values, but the classical logics we study do not. In fact, the content of the propositions is not relevant to propositional logic. Henceforth, atomic formulas are denoted only by the capital letters A, B, C,. . . (possibly with subscripts) without referring to what these propositions actually say. The veracity of these formulas does not concern us. Propositional logic is not the study of truth, but of the relationship between the truth of one statement and that of another. The language of propositional logic contains words for “not,” “and,” “or,” “implies,” and “if and only if.” These words are represented by symbols: . . . ￢ for “not,” ∧ for “and,” ∨ for “or,” → for “implies,” and ↔ for “if and only if.” . . . As is always the case when translating one language into another, this correspondence is not exact. Unlike their English counterparts, these symbols represent concepts that are precise and invariable. The meaning of an English word, on the other hand, always depends on the context.