Conditional logic

Conditional logic (sometimes: logic of conditionals) is any formal system designed to capture the meaning and inference patterns of natural-language sentences of the form "if A, (then) B". It is a central topic in philosophical logic, formal semantics, artificial intelligence, and the psychology of reasoning. Unlike the classical material conditional A ⊃ B, most conditional logics reject some classically valid principles (e.g. strengthening the antecedent, transitivity, contraposition), and many analyze conditionals in terms of similarity among possible worlds, revision of information, or probabilistic support rather than purely truth-functional tables.^[1]^[2]

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.

Main frameworks

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]

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]

Relation to nonmonotonic reasoning and AI

Flat fragments of many conditional logics align with preferential consequence (system P), enabling algorithms and theorem provers for defeasible rules, diagnosis, and planning.^[28]

References

^ Edgington, Dorothy (2020). "Indicative Conditionals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
^ Starr, William B. (2019). "Counterfactuals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
^ Whitehead, Alfred N.; Russell, Bertrand (1910). Principia Mathematica. CUP.
^ Quine, W. V. O. (1950). Methods of Logic. Holt.
^ Łukasiewicz, Jan (1920). "On Three-Valued Logic". Ruch Filozoficzny.
^ de Finetti, Bruno (1936). "La logique de la probabilité". Actes du Congrès International de Philosophie Scientifique.
^ Égré, Paul; Rossi, Lorenzo; Sprenger, Jan (2021). "De Finettian Logics of Indicative Conditionals: Trivalent Semantics and Validity". Journal of Philosophical Logic. doi:10.1007/s10992-020-09549-6. hdl:2318/1851003.
^ Lewis, C. I. (1912). "Implication and the Algebra of Logic". Mind (84): 522–531. doi:10.1093/mind/XXI.84.522.
^ ^a ^b Stalnaker, Robert C. (1968). "A Theory of Conditionals". Studies in Logical Theory: 98–112.
^ ^a ^b Lewis, David (1973). Counterfactuals. Blackwell.
^ Chellas, Brian F. (1975). "Basic Conditional Logic". Journal of Philosophical Logic. 4 (2): 133–153. doi:10.1007/BF00693270.
^ Burgess, John P. (1981). "Quick Completeness Proofs for Some Logics of Conditionals". Notre Dame Journal of Formal Logic. 22. doi:10.1305/ndjfl/1093883341.
^ Nute, Donald (1980). Topics in Conditional Logic. Reidel.
^ Kraus, Sarit; Lehmann, Daniel; Magidor, Menachem (1990). "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics". Artificial Intelligence. 44 (1–2): 167–207. doi:10.1016/0004-3702(90)90101-5.
^ Lehmann, Daniel; Magidor, Menachem (1992). "What does a conditional knowledge base entail?". Artificial Intelligence. 55: 1–60. doi:10.1016/0004-3702(92)90041-U.
^ Veltman, Frank (1976). "Prejudices, Presuppositions and the Theory of Counterfactuals". Amsterdam Papers in Formal Grammar.
^ Kratzer, Angelika (2012). Modals and Conditionals: New and Revised Perspectives. OUP.
^ Lewis, David (1981). "Ordering Semantics and Premise Semantics for Counterfactuals". Journal of Philosophical Logic. 10 (2). doi:10.1007/BF00248850.
^ Adams, Ernest W. (1975). The Logic of Conditionals. Reidel.
^ Lewis, David (1976). "Probabilities of Conditionals and Conditional Probabilities". Philosophical Review.
^ van Fraassen, Bas C. (1976). "Probabilities of Conditionals". Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science.
^ McGee, Vann (1989). "Conditional Probabilities and Compounds of Conditionals". Philosophical Review. 98 (4): 485–541. doi:10.2307/2185116. JSTOR 2185116.
^ Gärdenfors, Peter (1988). Knowledge in Flux. MIT Press.
^ Mares, Edwin (2020). "Relevance Logic". The Stanford Encyclopedia of Philosophy.
^ Douven, Igor (2016). The Epistemology of Indicative Conditionals. CUP.
^ Kratzer, Angelika (2012). Modals and Conditionals. OUP.
^ von Fintel, Kai (2001). "Counterfactuals in a Dynamic Context". In "Ken Hale: A Life in Language".
^ Olivetti, Nicola; Pozzato, Gian Luca; Schwind, Camilla B. (2007). "A Sequent Calculus and a Theorem Prover for Standard Conditional Logics". ACM Transactions on Computational Logic. 8 (4): 22. doi:10.1145/1276920.1276924.