Conditional logic
Conditional logic (also: the logic of conditionals) refers to a family of formal systems for reasoning with statements of the form "if A, (then) B". Conditional logics are intended to capture the meaning and patterns of inference associated with natural language conditionals more faithfully than the classical material conditional, which gives rise to well-known paradoxes. Conditional logics are used in philosophical logic, formal semantics of natural language, artificial intelligence, and the psychology of reasoning. They are used to model everyday and scientific reasoning about hypothetical, causal, modal, and counterfactual scenarios.
The material conditional is a truth function, which is always true except when the antecedent is true and the consequent false. Most conditional logics introduce an additional conditional connective (often written or, in the context of counterfactuals, ) whose truth or acceptability can depend on similarity between possible worlds, on context and background information, or on probabilistic support rather than on a simple two-valued truth table. These systems are designed to validate basic principles such as modus ponens, while restricting or invalidating classical schemas like strengthening the antecedent, transitivity, and contraposition, which are not always correct for ordinary "if ... then ..." sentences. It is common to distinguish between indicative conditionals, which concern open possibilities relative to what is currently known, and counterfactual conditionals, which describe ways things would or might have been contrary to fact. However, many conditional logics treat both of these as variants of a common underlying framework.
A wide range of semantic approaches has been developed for such logics, including three-valued and other many-valued systems that treat conditionals with false antecedents as void, possible worlds and selection function models in the tradition of Stalnaker and Lewis, premise or ordering source semantics, probabilistic and suppositional accounts tying acceptability to conditional probability, and belief revision frameworks based on the Ramsey test. Corresponding proof-theoretic systems range from Chellas's basic logics Ck and CK to stronger systems such as Burgess's B and Lewis's V, VW and VC, or Stalnaker's C2, which validate different structural principles for the conditional connective and are often related by soundness and completeness theorems to the underlying semantic frameworks. Conditional logics are also closely linked to nonmonotonic consequence relations and default reasoning systems, notably the cumulative and preferential systems C and P of Kraus, Lehmann and Magidor, which are widely used in AI to formalize rules with exceptions.
Historically, conditional logic grew out of attempts to refine earlier notions such as C. I. Lewis's strict implication, and the contemporary field is usually traced to Robert Stalnaker's 1968 possible worlds theory of conditionals and David Lewis's subsequent development of variably strict logics for counterfactuals. Later work by Nute, Burgess, Kratzer, Gärdenfors and many others has broadened the landscape, yielding a cluster of interrelated frameworks rather than a single canonical system and connecting the logic of "if ... then ..." with topics such as nonmonotonic logic, belief revision, conditional probability and the pragmatics of assertion, questioning and decision-making.