Conditional Modality Bridge — @cite{kratzer-2012} §2.9 Derivations #
Proves that specific Kratzer parameter settings yield specific conditional types
(material, strict, ordered), and that a past-tense antecedent composes
transparently with the modal analysis via atTime.
Section A: Conditional parameter derivations #
- Totally realistic base + empty ordering = material implication
- Empty base + empty ordering = strict implication (fails: w1 counterexample)
- Empty base + normalcy ordering = ordered conditional (succeeds: w1 eliminated)
Section B: The ordering source resolves the counterexample #
Section C: Tensed conditional forces tense-modal composition #
Section D: Tensed conditional matches atemporal conditional #
Reference: Kratzer, A. (2012). Modals and Conditionals. Oxford University Press. Ch. 2 §2.9.
Section A: Conditional parameter derivations (§2.9) #
Material implication (§2.9): With a totally realistic modal base and empty ordering source, necessity over the restricted base equals material implication. At w0/w1 the single accessible world decides; at w2/w3 the restricted base is vacuously satisfied (no rain-worlds accessible).
Strict implication fails (§2.9): With empty base and empty ordering,
all worlds are accessible. Restricting by rained gives {w0, w1}, and
since streetWet is false at w1, necessity fails at every evaluation world.
Ordering conditional succeeds (§2.9): With empty base and normalcy ordering, the rain-worlds {w0, w1} are ordered by normalcySource. Since w0 satisfies the normalcy proposition (rain → wet) and w1 does not, w0 is strictly better. The best worlds are {w0}, and streetWet w0 = true.
Section B: The ordering source makes the difference #
The ordering source resolves the counterexample. Strict implication fails because w1 (rained ∧ ¬streetWet) is accessible, but the normalcy ordering eliminates w1 as non-best. This is Kratzer's central insight: the ordering source handles graded possibility and anomalous worlds.
Section C: Tensed conditional (forces tense-modal composition) #
Tensed counterfactual succeeds: "If it had rained (yesterday), the
street would be wet (now)." The past-tense antecedent atTime rainedAt (-1)
enters the Kratzer modal base via the type bridge atTime, and the
normalcy-ordered analysis gives the correct prediction.
Section D: Composition bridge #
Tensed matches atemporal: The tensed conditional gives the same result
as the atemporal one, witnessing that temporal projection via atTime is
transparent to the modal analysis.