Holliday and Icard (2013): Measure semantics and qualitative semantics for epistemic modals #
Holliday and Icard compare semantics for the comparative epistemic modal at least as likely as against the inference patterns of their Figure 1: Yalcin's V1–V7, V11, V12 and I1–I3, and V13, which Lassiter suggested. Kratzer's world-ordering semantics, which lifts a preorder on worlds to propositions by Lewis's l-lifting, validates the invalid patterns and misses V11 and V13; Kratzer's later k-lifting does no better for disjoint propositions. Finitely additive measures validate exactly the intended patterns, but so do the paper's two qualitative alternatives, qualitatively additive measures and the m-lifting. What separates the two kinds of additivity is Kraft, Pratt and Seidenberg's ordering on five worlds, the paper's World Cup comparisons (3)–(6), which the paper doubts speakers find inconsistent.
Main results #
lLift_validities,disjunction_problem,lLift_refutes_V11_V13: Fact 1. The l-lifting's V12, which Yalcin counts among the account's failures, follows from its I2.measure_validities,qualAddMeasure_validitiesand the refutations of I1–I3: Facts 2, 3.mLift_validities,mLift_refutes_I_patterns: Fact 5. Bymeasure_le_of_matchingLiftthe m-lifting is sound for any measure that agrees with the world ordering on singletons.worldCup_not_finitelyAdditive,fa_representable_iff_card_lt_five: Theorem 8.
Implementation notes #
- The paper's finitely additive measures are mathlib's probability measures on a discrete measurable space; on the paper's finite spaces the two coincide.
□and◇are the paper's quantifiers over the epistemic space,A = Set.univandA ≠ ∅, with every world accessible, so V6 comes from non-triviality and V7 from monotonicity. The order-internal◇A := ¬ ∅ ⩾ Aagrees with this for the liftings but, over the measures, only for regular ones.- V8–V10, which Fact 1 names but Figure 1 leaves out, are not formalized.
- The refutations use the uniform measure on three worlds and the indiscriminate world order on two or three worlds. Theorems 2, 6 and 8 are represented by the model-theoretic results they rest on; Theorems 3–5 and 7 and Fact 4 concern the logics themselves.
- Figure 7's chain of four worlds shows the two liftings apart: only the l-lifting makes
{a}more likely than{b, c}, and the m-lifting leaves{a, d}and{b, c}incomparable.
TODO #
- The FA-consistency of (3)–(6) is witnessed by the Kraft–Pratt–Seidenberg order of the substrate under a different labelling of the worlds; state it for the World Cup labelling.
References #
Fact 1: Kratzer's world-ordering semantics #
The l-lifting of any preorder on worlds validates the three measure-invalid patterns I1–I3, since it is right-union closed. This is Fact 1's disjunction problem.
Over a nonempty preorder on worlds the l-lifting validates V1–V7 and V12 (Fact 1), the last because it validates I2.
The l-lifting refutes V11 and V13 (Fact 1). On two indiscriminate worlds, W is probable
and {0} is at least as likely as W, yet {0} is not probable (V11); {0} \ {1} is
strictly more likely than ∅, yet {0} ∪ {1} is not strictly more likely than {1}
(V13).
Facts 2 and 3: finitely and qualitatively additive measures #
Every probability measure validates V1–V7 and V11–V13 (Fact 2).
The uniform measure on three worlds refutes each of I1–I3 (Fact 2).
Qualitative additivity already yields V1–V7 and V11–V13 (Fact 3).
The uniform measure, read as a qualitatively additive measure, refutes each of I1–I3 (Fact 3).
Every FA order on a finite set of worlds is represented by a qualitatively additive measure (Theorem 6, after van der Hoek).
Fact 5: the m-lifting #
The m-lifting validates V13 on a finite domain: B ⊆ A ∪ B gives the weak half, and a
matching of A ∪ B into B would contradict |B| < |A ∪ B|, which holds as A \ B is
nonempty.
On a finite nonempty preorder the m-lifting validates V1–V7 and V11–V13 (Fact 5).
The m-lifting refutes I1–I3, dissolving the disjunction problem (Fact 5). Between
indiscriminate worlds only cardinality matters, so one world matches {0} and {1} but not
{0, 1} (I1), and on three worlds {0, 1} is strictly more likely than its complement but
cannot match all of W (I2, I3).
If the world ordering agrees with a measure on singletons, the m-lifting is sound for the
measure order (footnote 13), since the injection carries the sum of the singleton masses of B
into a sum over a subset of A.
Figure 7: the chain a ≻ b ≻ c ≻ d #
Worlds are Fin 4, with 0 the best; ge_w u v is u ≤ v.
Both liftings make {a, b} more likely than {b, c}.
Only the l-lifting makes {a} more likely than {b, c}. The m-lifting leaves the two
incomparable, since two worlds cannot be matched injectively into one.
The m-lifting is not total, even over a linear order on worlds, as the paper remarks before
Fact 5. On the chain, {a, d} and {b, c} are incomparable: only a can match c, which
leaves nothing for b, and nothing in {b, c} matches a.
Theorem 8: what separates FA from finite additivity #
Every FA order on Fin n is representable by a probability measure iff n < 5
(Theorem 8, after Kraft, Pratt and Seidenberg).
No finite measure satisfies the World Cup comparisons (3)–(6), with Argentina, Brazil,
China, Denmark and England as the worlds 0–4: Argentina-or-England more likely than
China-or-Denmark, Brazil-or-China more likely than Argentina-or-Denmark, Denmark more likely
than Argentina-or-China, and Argentina-or-China-or-Denmark more likely than Brazil-or-England.
The four left-hand sides and the four right-hand sides have the same total mass.