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Propositional calculus. Not to be confused with Propositional analysis. The propositional calculus[a] is a branch of logic. [1] It is also called propositional logic, [2] statement logic, [1] sentential calculus, [3] sentential logic, [1] or sometimes zeroth-order logic. [4][5] It deals with propositions [1] (which can be true or false) [6] and ...
The intuitionistic propositional calculus and some of its extensions exhibit the disjunction property instead, implying one of the disjuncts of any disjunction individually would have to be derivable as well. The converse variants of those two, and the equivalent variants with double-negated antecedents, had already been mentioned above.
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. [1] This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.
Knowledge representation and reasoning. Knowledge representation and reasoning (KRR, KR&R, KR²) is the field of artificial intelligence (AI) dedicated to representing information about the world in a form that a computer system can use to solve complex tasks such as diagnosing a medical condition or having a dialog in a natural language.
Automated theorem proving. Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major motivating factor for the development of computer science.
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A → B, it is sufficient to assume A as a hypothesis and then proceed to derive B. Deduction theorems exist for both propositional logic ...
The implicational propositional calculus is semantically complete with respect to the usual two-valued semantics of classical propositional logic. That is, if Γ is a set of implicational formulas, and A is an implicational formula entailed by Γ, then Γ ⊢ A {\displaystyle \Gamma \vdash A} .
Rules of inference. Implication introduction / elimination (modus ponens) Biconditional introduction / elimination. Conjunction introduction / elimination. Disjunction introduction / elimination. Disjunctive / hypothetical syllogism. Constructive / destructive dilemma. Absorption / modus tollens / modus ponendo tollens.