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) — conditional doesn't imply material
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.

Symbol	Unicode	Name	Meaning
>	—	Conditional	If...would
□→	—	Counterfactual	If...were...would
◇→	—	Might-counterfactual	If...were...might
→	U+2192	Material	Material conditional
⥽	U+297D	Strict	Strict conditional
f(w,A)	—	Selection function	Closest A-world
≤w	—	Similarity	Closer to w

Metatheory. Stalnaker's C2 has conditional excluded middle; Lewis's VC does not. Both are decidable (PSPACE-complete). Complete axiomatizations exist. Non-monotonic: adding premises can defeat conditionals. Not truth-functional: can't compute A > B from truth values of A, B. Related to belief revision (Ramsey test: accept A > B iff revising by A leads to B).

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. What is "similarity"? Lewis gave criteria but they're debatable. Context-dependence: similarity standards vary. Doesn't directly model causation (needs interventionist or structural additions). Multiple competing systems (Stalnaker, Lewis, others). Conditional excluded middle is controversial. Transitivity failures can be counterintuitive.