Logic is the systematic study of inference: how conclusions follow from premises and which patterns of reasoning preserve truth or otherwise justify conclusions. It examines arguments, logical consequence, and the formal languages used to express them. Historically a branch of philosophy, logic also forms a major field within mathematics and provides foundational methods for computer science. Its subject is not simply what people believe, but the relationships between statements and the conditions under which reasoning succeeds.
Arguments, validity, and truth
An argument consists of premises offered in support of a conclusion. In deductive reasoning, the intended support is conclusive: if the premises are true, the conclusion must also be true. Validity concerns this relationship rather than the actual truth of the premises. A valid argument may contain false premises; a sound argument is valid and has true premises.
For example, “All mammals are animals; all whales are mammals; therefore all whales are animals” is valid. Its form ensures that any interpretation making both premises true also makes the conclusion true. By contrast, a true conclusion can follow an invalid argument.
A standard valid inference is modus ponens: from “If P, then Q” and “P,” infer “Q.” Inferring P from those same conditional premises together with Q instead is affirming the consequent, an invalid pattern. Such distinctions help identify a fallacy, although some reasoning errors depend on context, ambiguity, or relevance rather than formal structure.
Inductive reasoning provides support without guaranteeing truth. Observations may make a generalization probable while leaving exceptions possible. Its analysis overlaps with probability and epistemology. Deductive validity, inductive strength, and the credibility of premises are therefore different dimensions of argument assessment.
Formal languages and interpretation
Formal logic represents reasoning using explicitly defined symbols and rules. A formal language specifies which expressions are well formed; a semantics explains how those expressions are interpreted. Formalization makes patterns of inference easier to compare, though translating ordinary language can involve substantive choices.
Propositional logic treats whole statements as units and combines them with connectives such as negation, conjunction, disjunction, and implication. In classical propositional logic, each statement receives one of two truth values. Truth tables determine the values of compound expressions and can test whether an inference is valid.
First-order logic adds variables, predicates, relations, and quantifiers. Universal quantification expresses claims about every object in a domain; existential quantification expresses claims about at least one object. Thus “Every whale is a mammal” can be represented as ∀x(Whale(x) → Mammal(x)). Quantifier order matters: “Everyone admires someone” need not mean that one person is admired by everyone.
Logical consequence is commonly defined through interpretations: a conclusion follows from premises if it is true in every interpretation in which those premises are true. The domain and interpretation of predicates are therefore essential parts of the analysis.
Proof and metatheory
A proof system supplies rules for deriving expressions from assumptions. A formal proof records a derivation whose steps can be checked against those rules. An axiom is a statement accepted as a starting point within a specified theory.
Proof theory studies derivations and their structure, while model theory studies mathematical structures satisfying sentences or theories. Metatheory examines properties of logical systems themselves. A sound proof system derives only semantic consequences; a complete one can derive every semantic consequence of the relevant kind.
Classical first-order logic has sound and complete proof systems, but no algorithm decides validity for every first-order sentence. Propositional validity, by contrast, is decidable.
Gödel’s incompleteness theorems establish limits for consistent, effectively axiomatized theories capable of expressing sufficient arithmetic. Such theories cannot decide every sentence in their language. This does not contradict first-order completeness: completeness of a logical calculus differs from a particular theory’s ability to prove or refute every sentence.
Historical development
In ancient Greece, Aristotle developed a systematic theory of the syllogism, analyzing deductions involving categorical statements. Stoic thinkers investigated patterns closer to propositional inference. Distinct traditions of argument analysis also developed in India, notably Nyāya and Buddhist logic, and in ancient China among Mohist thinkers.
Medieval scholars in Islamic and Latin intellectual traditions extended logical analysis, including studies of terms, consequences, and modal statements. Nineteenth-century mathematical developments transformed the field. George Boole’s algebraic treatment of reasoning contributed to Boolean algebra, while Gottlob Frege developed a powerful quantified logical notation in 1879.
Twentieth-century work connected logic with the foundations of mathematics, set theory, and computation. Alan Turing helped establish precise limits on mechanical procedures, linking questions about logical decision methods to computability.
Alternative systems and applications
Classical logic is not the only framework. Modal logic introduces operators for necessity and possibility, with related systems analyzing knowledge, obligation, or time. Intuitionistic logic reflects constructive approaches to proof and does not accept excluded middle as an unrestricted principle. Paraconsistent logic permits contradictions without automatically allowing every statement to follow.
These systems differ in their rules and interpretations, not merely in notation. Their suitability depends on the phenomena being modeled and the consequences desired.
Logic supports software verification, database querying, circuit design, and automated theorem proving. Symbolic artificial intelligence uses explicit representations and inference rules, while a proof assistant checks mathematical derivations within a specified formal framework. In ordinary discourse, logical analysis separates an argument’s structure from questions about evidence, meaning, and the factual reliability of its premises.