Although propositional logic (also called propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic (Stoic logic) by Chrysippus in the 3rd century BC [20] and expanded by his successor Stoics. The logic was focused on propositions.

2025年1月27日 · Propositional logic is a fundamental branch of mathematical logic that deals with propositions (statements that are either true or false) and their relationships. It uses logical connectives such as AND (∧), OR (∨), NOT (¬), IMPLIES (→), and IF AND ONLY IF (↔) to form compound propositions.

2025年1月20日 · Propositional calculus is the formal basis of logic dealing with the notion and usage of words such as "NOT," "OR," "AND," and "implies." Many systems of propositional calculus have been devised which attempt to achieve consistency, completeness, and independence of axioms.

Propositional Calculus Here is a definition of the formal system for propositional logic. 1. Symbols: A,B,C,D,...,Z(and optionally, allow primes, A0,A00, etc.) ∼,∨,∧,(,) and additionally the symbols ⇒ and ⇔, which are only shorthand for their equivalent forms (defined below). 2. Well Formed Formulas

2025年1月17日 · The simplest and most basic branch of logic is the propositional calculus, hereafter called PC, so named because it deals only with complete, unanalyzed propositions and certain combinations into which they enter. Various notations for PC are used in the literature.

As opposed to the predicate calculus, the propositional calculus employs simple, unanalyzed propositions rather than terms or noun expressions as its atomic units; and, as opposed to the functional calculus, it treats only propositions that do not contain variables.

Propositional Calculus Throughout our treatment of formal logic it is important to distinguish between syntax and semantics. Syntax is concerned with the structure of strings of symbols (e.g. formulas and formal proofs), and rules for manipulating them, without regard to their meaning. Semantics is concerned with their meaning. 1

1. Propositional Calculus The Completeness Theorem. 1.1. Theorem (Completeness Theorem). Let Γ be s set of formulas, and let ψbe a formula. Then Γ ⊢ ψif and only if Γ |= ψ. We will prove the ⇒ direction directly. Proof that if Γ ⊢ ψthen Γ |= ψ. Let β¯ = (β 1,...,βn) be a proof of ψ from Γ.

Study of this calculus reveals the ways in which computers can be used to build formulas, construct and check proofs and even find proofs, and transform them from one style to another.

Notes on Propositional Calculus Learning goals 1. Distinguish between inductive and deductive inference. Provides examples to illustrate each one. 2. Provide de nitions for Propositional Calculus (PC) terminology. See list below. 3. Translate propositions from English into PC. 4. Check consistency of axioms. 5.