Conditional logic

Background and motivations

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 (any true B or false A makes A ⊃ B true) and counterintuitive behavior of negation and denial. Classical analyses also vacuously validate counterfactuals with false antecedents.^[3]^[4] These issues motivated richer accounts distinguishing indicative conditionals from counterfactual conditionals and modeling the dependency between antecedent and consequent.

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Main frameworks

Summarize

Perspective

Conditional logics are often grouped by their semantic framework.

Truth-functional and trivalent approaches

Some systems remain truth-functional but use more than two truth values. Łukasiewicz introduced a three-valued logic; de Finetti later argued that "if A then B" should be void when A is false (a bet is called off). Modern trivalent systems (e.g., Cooper/Cantwell's and de Finetti–style logics) typically validate modus ponens but invalidate modus tollens and contraposition. They aim to block material-implication paradoxes and to connect to suppositional/probabilistic intuitions.^[5]^[6]^[7]

Possible-worlds semantics

A dominant intensional tradition treats conditionals as quantifying over accessible or similar worlds.

Strict conditional (C. I. Lewis): $A\mathbin {>} B\;\equiv \;\Box (A\supset B)$ . Powerful, but too strong for natural language (validates strengthening, transitivity, and contraposition).^[8]
Variably strict (Stalnaker–Lewis): truth depends on the closest A-worlds (given a contextually supplied similarity or selection function). These invalidate strengthening, transitivity, and contraposition, better fitting linguistic data.^[9]^[10]

Within this family, axiomatic systems range from basic normal logics (CK) to Burgess's B, Lewis's V/VW/VC, and Stalnaker's C2. Many retain rules like Right Weakening and Conditional K but reject classical schemas such as Strengthening the Antecedent, Transitivity, and Contraposition.^[11]^[12]^[13]

Nonmonotonic and preferential models

In AI, "if A then normally B" is modeled by preference orderings over states. Kraus, Lehmann, and Magidor's systems C/P/R characterize rational, defeasible consequence; the flat (non-nested) fragment of several conditional logics coincides with their P system.^[14]^[15]

Premise semantics

Premise (or ordering-source) semantics evaluates "if A then C" by keeping a maximal subset of background premises consistent with A and checking whether C follows. It captures context sensitivity and is equivalent (via theorems) to similarity-based ordering semantics.^[16]^[17]^[18]

Probabilistic approaches

Adams proposed that for simple (non-nested) indicatives the probability of "if A then B" equals the conditional probability $P(B\mid A)$ . His consequence relation (preserving high probability) validates the KLM system P and rejects strengthening, transitivity, and contraposition.^[19] Lewis's triviality results show that identifying all conditional probabilities with probabilities of (possibly nested) conditionals collapses to trivial constraints; under Stalnaker semantics the probability of a conditional instead matches imaging on the antecedent, not Bayesian conditioning.^[20] Subsequent work explores product-space models and coherence-based previsions for compounds of conditionals.^[21]^[22]

Belief revision and the Ramsey test

On the AGM model of belief revision, "if A then B" is accepted if and only if, after minimally revising a belief set by A, B is believed. This can be understood as a formal version of the Ramsey test. Combined with standard AGM postulates this yields a Gärdenfors-style triviality theorem, prompting proposals to weaken the postulates, restrict compounds, or replace revision with update/imaging.^[23]

Relevance and difference-making

Beyond topic-relevance in relevance logic, many theorists require that A make a (probabilistic or doxastic) difference to B for a conditional to be assertable or valid. This motivates confirmational scores (e.g., P(B|A) > P(B)) and connexive-style principles, often at the cost of rules like Right Weakening.^[24]^[25]

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Interaction with modality and speech acts

In many languages, if-clauses restrict the domain of (overt or covert) modals (e.g., "if A then must/possibly B"), explaining commutation effects between conditionals and modals. Dynamic/expressivist accounts derive incompatibilities such as "if A, might not B" versus "if A, B", and extend to conditional questions and imperatives.^[26]^[27]

Controversial principles

Several plausible-sounding schemas typically fail in modern conditional logics:

Or-to-If (from A ∨ B infer ¬A > B)
Import–Export (A > (B > C) ≡ (A ∧ B) > C)
Simplification of disjunctive antecedents ((A ∨ B) > C ⇒ (A > C) ∧ (B > C))

Each interacts in complex ways with modus ponens, context, and discourse structure.^[9]^[10]