Conditional logic
Conditional logic refers to a family of formal systems for reasoning with statements of the form "if A, 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 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.
Overview
Conditional logics form a family of formal systems for reasoning with sentences of the form "if A, B". They aim to explain when such conditionals are acceptable, how they interact with other logical operators, and which arguments involving them should count as valid or rationally compelling. In contrast to the classical material conditional, most conditional logics build some notion of dependence between antecedent and consequent into their semantics, and allow the validity of inferences with conditionals to be sensitive to context, background information, or normality assumptions.Rather than a single canonical calculus, conditional logic is now used as an umbrella term for several families of interrelated systems. Prominent traditions include: three-valued and other many-valued logics that treat conditionals with false antecedents as void; possible worlds and selection function logics inspired by Stalnaker and Lewis; premise or ordering source semantics that work with sets of background assumptions; probabilistic and suppositional accounts that tie acceptability to conditional probability; and belief revision and nonmonotonic approaches grounded in the Ramsey test. A common slogan is that to evaluate "if A, then B" one should add ''A hypothetically to one's information state and ask whether B'' would then be accepted.
Most authors distinguish between indicative conditionals, which concern open possibilities relative to an information state, and counterfactual conditionals, which describe ways things would or might have been contrary to fact. Many formal frameworks nevertheless treat these as variants of a common semantic core, parametrized by the kind of modality or background premises involved. This article follows that practice and surveys a range of systems that have been proposed for both indicative and counterfactual readings.
Notation
The conditional studied by a given conditional logic is here notated with, for instance represents "if A, then B". This conditional "corner", not to be confused with the indentically written greater than sign, was the notation used by Robert Stalnaker's paper that started the field, and which has been followed in various other reference publications. Some conditional logics have been studied in the specific context of counterfactual conditionals, and in that context, David Lewis's notation for counterfactuals is also common.Interaction with modality and speech acts
In many languages, if-clauses interact closely with overt or covert modal expressions such as must, might, or would. On the influential "restrictor" view, an if-clause does not itself introduce a binary connective, but instead restricts the domain of a following modal or quantificational operator; this helps explain the apparent commutation of conditionals and modals in sentences like "if A, it must be that B" and "it must be that if A, then B".Dynamic and expressivist accounts build on this idea by treating an indicative or counterfactual conditional as an update on an information state or conversational scoreboard. On such views, incompatibilities between "if A, B" and "if A, might not B" arise because the two updates place conflicting constraints on the same body of shared information. The same machinery extends naturally to conditional questions and to conditional imperatives, which are modelled as transformations of inquiry states or decision problems rather than as simple truth evaluations.
In this way, conditional logics are used not only to model the truth conditions of if-sentences, but also to capture their role in conversation, planning and decision, and to connect the semantics of conditionals with the pragmatics of assertion, questioning and commanding.
Logical principles and their failure
Classical material implication validates a number of principles that look plausible at first sight but seem problematic for ordinary if-sentences. Early work by Adams, Stalnaker and Lewis highlighted counterexamples to strengthening the antecedent, to transitivity, and to contraposition.Modern conditional logics typically invalidate at least some of these principles. They also differ with respect to more specific schemas such as:
- Or-to-If ;
- Import–export ;
- Simplification of disjunctive antecedents.
Relation to nonmonotonic reasoning and AI
Conditional logics are closely connected to nonmonotonic logics and to AI formalisms for default and defeasible reasoning. Preferential and ranked models for nonmonotonic consequence relations, introduced by Kraus, Lehmann and Magidor, can be understood as abstracting away from the explicit conditional connective and instead working directly with a consequence relation that encodes defaults of the form "normally, if A then B". Their system P of preferential entailment, and the stronger rational system R, correspond to the "flat" fragments of several variably strict conditional logics, including Burgess's system B and Lewis's systems for counterfactuals.From the perspective of artificial intelligence, these logics support representation and automated reasoning with rules that have exceptions: for example, that birds normally fly, or that a component normally functions unless it is known to be faulty. The correspondence between conditional logics and nonmonotonic consequence relations has been used to transfer results and techniques between the two areas, to design theorem provers for conditional logics, and to apply them in diagnosis, planning and knowledge representation.
History
Early modern logic identified "if A then B" with the material implication true in all cases except when A is true and B false. While attractive for mathematical proof, this leads to the paradoxes of material implication and counterintuitive behavior of negation and denial. Classical analyses also vacuously validate counterfactuals with false antecedents. These issues motivated richer accounts distinguishing indicative conditionals from counterfactual conditionals and modeling the dependency between antecedent and consequent.Although systems for reasoning with conditionals go back at least to C. I. Lewis's strict implication and related modal logics, the contemporary field of conditional logic is usually traced to Robert Stalnaker's 1968 paper A Theory of Conditionals. Drawing on Ramsey's idea of hypothetically adding the antecedent to one's belief state, Stalnaker proposed a possible worlds semantics in which an indicative or counterfactual conditional is true just in case its consequent holds at the "closest" antecedent worlds. This provided a unified account of ordinary and counterfactual conditionals, explained the failure of principles such as Transitivity and Strengthening the Antecedent, and suggested a corresponding modal logic of a non-material conditional connective.
Stalnaker's framework was soon generalized and made more systematic. In a joint paper, Stalnaker and Richmond Thomason gave a fully explicit semantics and axiomatization for conditional logics in terms of selection functions that pick, for each world and antecedent proposition, a set of closest antecedent worlds, thereby turning Stalnaker's informal proposal into a family of well-behaved formal systems. Around the same time, David Lewis developed an alternative but closely related "variably strict" account using comparative similarity orderings over possible worlds, culminating in his monograph Counterfactuals and a hierarchy of systems for counterfactual conditionals. These works established the now standard picture of conditionals evaluated relative to similarity or selection among possible worlds, and they framed many of the central questions about which structural principles such logics should validate.
During the mid-1970s and early 1980s, researchers working in the Stalnaker–Lewis tradition developed a shared axiomatic and semantic toolkit. Brian Chellas introduced the system CK as a "basic conditional logic", corresponding to the core selection function semantics and serving as an analogue of the normal modal logic K for conditionals. John Burgess later supplied simplified completeness proofs for a range of conditional logics and introduced system B, tied to ordering semantics. Donald Nute's monograph Topics in Conditional Logic synthesized these developments, systematizing selection function, ordering, and relative modality frameworks and mapping out the space of normal conditional logics that extend CK in different ways.
By the late 1980s and 1990s, conditional logic had become tightly connected with work on nonmonotonic reasoning and probabilistic conditionals. Ernest W. Adams had already proposed a high probability consequence relation for simple indicative conditionals, based on identifying the acceptability of "if A then B" with a high conditional probability P. Building on both the Stalnaker–Lewis semantics and Adams's ideas, Kraus, Lehmann, and Magidor introduced preferential and cumulative models for defeasible conditionals, and defined the nonmonotonic consequence system P, which can be seen as corresponding to the "flat" fragment of several variably strict conditional logics and of Burgess's system B. This work cemented the role of conditional logics as a link between philosophical analyses of "if... then..." and AI formalisms for default and defeasible inference.
In parallel, possible worlds approaches spawned alternative but equivalent perspectives. Premise based and "ordering source" semantics, originating with Frank Veltman and further developed by Lewis and Angelika Kratzer, reinterpreted conditionals in terms of sets of background premises or ordering sources that are updated or restricted by the antecedent, showing how similarity based models could be recast in more explicitly epistemic and dynamic terms. At the same time, belief revision theorists such as Peter Gärdenfors investigated conditionals via AGM revision and the Ramsey test, while proof theorists developed sequent and tableau calculi for many of the main systems of conditional logic.
More recently, survey articles and handbooks have emphasized that conditional logic is not a single system but a cluster of related frameworks: trivalent, possible worlds, premise based, probabilistic, and belief revision based that can often be intertranslated or shown to share a common "flat" fragment. The Stanford Encyclopedia of Philosophy entry on the logic of conditionals, for example, situates Stalnaker's 1968 proposal as a central ancestor of modern conditional logics and highlights how later work by Lewis, Nute, Burgess, Kratzer, Kraus–Lehmann–Magidor, and others has diversified and refined the field over the past half century.