Conditional Logic

Origin. C.I. Lewis introduced strict conditional to fix paradoxes of material implication (1912). Robert Stalnaker (1968) and David Lewis (1973) developed possible worlds semantics for counterfactuals. Conditional logic generalizes: "If A were the case, then B" — not just "A materially implies B." Foundation for counterfactual reasoning and causal analysis.

Models. Conditionals beyond material implication. Material conditional A → B is true whenever A is false or B is true — too weak. Strict conditional □(A → B) requires necessity — too strong. Conditional logic captures: "In the most similar A-worlds, B holds." Different similarity orderings give different conditional logics.

Formalism.

Conditional operator: A > B (or A □→ B): "If A were the case, B would be the case"

Stalnaker's semantics: Selection function f(w, A) picks the closest A-world to w. A > B true at w iff B true at f(w, A) (or no A-worlds exist).

Assumes uniqueness: exactly one closest A-world (or none).

Lewis's semantics: System of spheres: nested neighborhoods around w ordered by similarity. A > B true at w iff:

• No A-worlds exist, OR
• Some sphere S contains A-worlds, and all A-worlds in S are B-worlds

Allows ties: multiple equally-close worlds.

Key validities:

• A > A (identity)
• (A > B) ∧ (A > C) → (A > (B ∧ C))
• A > B equivalent to ¬A ∨ B? NO — conditionals are non-truth-functional

Key non-validities (in Lewis's system):

• (A → B) → (A > B) — material doesn't imply conditional (the converse is valid in VC, by centering)
• Strengthening: (A > C) → ((A ∧ B) > C) — FAILS
• Contraposition: (A > B) → (¬B > ¬A) — FAILS
• Transitivity: (A > B) ∧ (B > C) → (A > C) — FAILS

Stalnaker vs Lewis: Stalnaker: Conditional excluded middle: (A > B) ∨ (A > ¬B) Lewis: Rejected (there may be ties).

Might-counterfactual: A ◇→ B: "If A were the case, B might be the case" A ◇→ B ≡ ¬(A > ¬B)

Symbols.

Metatheory. Stalnaker's C2 is sound and complete for selection-function models satisfying the uniqueness assumption (Stalnaker and Thomason, 1970), and Lewis's VC is sound and complete for systems of spheres ordered by comparative similarity (Lewis, 1973) — a proof that deliberately avoids the limit assumption, so infinitely descending chains of ever-closer antecedent worlds cause no difficulty. The two systems stand in an exact relation: C2 is VC plus conditional excluded middle, which is valid precisely on models where each antecedent selects at most one closest world. Strengthening of the antecedent, contraposition and transitivity all fail in VC, but restricted forms survive and are valid: cautious monotonicity (A > B) ∧ (A > C) → ((A ∧ B) > C), and restricted transitivity (A > B) ∧ ((A ∧ B) > C) → (A > C). Friedman and Halpern (1994) settled the complexity: satisfiability is PSPACE-complete for arbitrarily nested conditionals and NP-complete once nesting depth is bounded, rising to EXPTIME-complete under a uniformity assumption and falling back to NP under absoluteness.

Applies to. Counterfactual reasoning. Causal analysis. Legal reasoning (but-for causation). Decision theory. AI planning. Natural language semantics. Philosophy of science (laws as conditionals).

Limitations. The similarity ordering is a free parameter rather than a constraint: the completeness theorems hold for arbitrary comparative similarity relations, so the semantics fixes the logic of conditionals without fixing any particular conditional's truth value, and Fine (1975) showed that filling the parameter in by overall resemblance delivers the wrong verdict on the Nixon-button case. Two triviality results bound the available readings. Gärdenfors (1986) proved the Ramsey test inconsistent with the AGM preservation postulate, so conditionals cannot in general be read as belief-revision instructions; Lewis (1976) proved that no binary connective can have probability equal to conditional probability across all probability functions, so they cannot be read probabilistically either. Because > is a binary operator rather than a box over a fixed accessibility relation, Sahlqvist correspondence and the canonicity apparatus do not apply, and causal structure requires structural equations rather than similarity.

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