Classical Logic

Classical logic is a family of systems of logic for representing deductive reasoning, especially classical propositional logic and first-order logic. Its standard semantics assigns each sentence exactly one of two truth values, true or false, and interprets logical operations accordingly. An argument is valid when no admissible interpretation makes all its premises true and its conclusion false. “Classical” identifies a logical framework, not merely logic originating in antiquity. (plato.stanford.edu)

Historical development

Classical logic has historical antecedents in Aristotle’s theory of the syllogism, which systematically studied inferences involving categorical statements. Aristotelian logic nevertheless differs substantially from modern predicate logic: its characteristic subject–predicate forms do not provide the latter’s general treatment of relations and nested quantification. (plato.stanford.edu)

A decisive development was Gottlob Frege’s Begriffsschrift, published in 1879. Frege introduced a formal notation capable of analyzing quantified statements and representing complex deductive relationships. His treatment of predication, quantification, and proof helped establish modern mathematical logic. The contemporary classical framework developed through subsequent work on formal languages, deductive calculi, and mathematical interpretations, rather than through the unchanged transmission of ancient rules. (plato.stanford.edu)

Language and interpretation

A logical language separates syntax, which specifies well-formed expressions, from semantics, which determines their interpretation. Propositional logic treats elementary statements as units represented by letters such as $P$ and $Q$. Compound statements are formed using negation $\neg$, conjunction $\land$, disjunction $\lor$, implication $\rightarrow$, and biconditional $\leftrightarrow$. (forallx.openlogicproject.org)

These connectives are truth-functional: the truth value of a compound depends only on the truth values of its components. Conjunction is true when both components are true; inclusive disjunction is true when at least one is true. Material implication $P\rightarrow Q$ is false only when $P$ is true and $Q$ false. It therefore does not, by itself, express a causal relationship or an explanatory connection. A truth table displays these conditions for every possible assignment. (forallx.openlogicproject.org)

First-order logic adds variables, predicates, individual constants, and quantifiers. The universal quantifier $\forall x$ ranges over every object in a domain, while the existential quantifier $\exists x$ asserts that at least one object satisfies a condition. A standard interpretation supplies a nonempty domain and meanings for nonlogical symbols. For example, $\forall x(H(x)\rightarrow M(x))$ states that every object satisfying $H$ also satisfies $M$. Quantification ranges over individuals, not directly over properties or sets. (builds.openlogicproject.org)

Characteristic principles

Classical logic validates the law of excluded middle, $P\lor\neg P$, and the principle of noncontradiction, $\neg(P\land\neg P)$. It also permits double-negation elimination: from $\neg\neg P$, infer $P$. Bivalence is a semantic condition on truth values, whereas excluded middle is a formula schema; although closely connected in standard classical semantics, they are not interchangeable definitions in every logical framework. (plato.stanford.edu)

Another characteristic is explosion: contradictory premises $P$ and $\neg P$ entail any conclusion $Q$. Semantically, no classical interpretation makes both premises true, so there is no counterexample to the inference. Explosion does not mean that an isolated false premise entails everything; the crucial condition is the joint inconsistency of the premises. (plato.stanford.edu)

Classical consequence is also monotonic. If a conclusion follows from a collection of premises, adding further premises cannot invalidate that consequence. This concerns deductive entailment, not whether new evidence should change a person’s beliefs. (builds.openlogicproject.org)

Validity and proof

Logical validity concerns truth preservation across interpretations. By contrast, a formal proof is a finite derivation governed by specified rules of inference. The notation $\Gamma\models A$ expresses semantic consequence, while $\Gamma\vdash A$ expresses derivability in a chosen calculus. A tautology is a propositional formula true under every truth assignment. (forallx.openlogicproject.org)

Classical logic can be presented through axiomatic calculi, natural deduction, or sequent calculus. These organize proofs differently while capturing the same consequence relation when appropriately formulated. Modus ponens, for example, permits the inference from $P$ and $P\rightarrow Q$ to $Q$. Classical proof by contradiction permits deriving $P$ by showing that assuming $\neg P$ leads to contradiction. (builds.openlogicproject.org)

Metatheoretical properties

A proof system is sound if every derivable consequence is semantically valid, and complete if every semantic consequence is derivable. Gödel’s completeness theorem establishes this correspondence for classical first-order logic with suitable deductive systems. Completeness does not say that every sentence is provable or refutable from any given theory. (builds.openlogicproject.org)

The compactness theorem states that a set of first-order sentences has a model if every finite subset has a model. These results connect formal derivation with the study of mathematical structures in model theory. (builds.openlogicproject.org)

Propositional validity is decidable by checking finitely many truth assignments. General first-order validity is undecidable: no algorithm always terminates with the correct answer for every sentence. It is nevertheless semidecidable, because proofs can be systematically enumerated; a valid sentence will eventually be recognized, although the search need not terminate for an invalid sentence. (builds.openlogicproject.org)

Relations to other logics

Intuitionistic logic does not accept unrestricted excluded middle or double-negation elimination. Adding either principle to its standard calculus yields classical logic. It still validates explosion, so that principle alone does not distinguish classical from intuitionistic reasoning. Paraconsistent logic, by contrast, rejects explosion and can distinguish inconsistency from the derivability of every statement. These differences concern precise semantic conditions and inference rules, rather than a distinction between formal and informal reasoning. (plato.stanford.edu)