@cite{cariani-santorio-2018} — Will done Better #
Cariani, F. & Santorio, P. (2018). Will done Better: Selection Semantics, Future Credence, and Indeterminacy. Mind 127(505): 129–165.
Core Claim #
The future modal will is best analyzed as a selectional operator:
will A is true at w iff A holds at the unique world picked out by
a selection function from a set of historical alternatives. This rejects
both the pure-tense view (will A = A holds at a future time) and the
universal view (will A = A at every historical alternative).
Three Constraints (the desiderata) #
@cite{cariani-santorio-2018} argue that an adequate theory must satisfy:
- Modal character — will embeds, takes scope, and interacts with negation/quantifiers. Pure tense fails.
- Scopelessness —
¬ will A ↔ will ¬ Ain matrix uses. Universal quantification over a non-trivial modal base fails (the asymmetry between¬∀and∀¬). - Cognitive role — sincere assertion of
will Arequires non-extreme credence, not credence 1. Universal-base accounts make the assertion conditions too strong.
The selectional analysis satisfies all three by construction.
The Sports Fan model (paper §2.3, §3 figure 2) #
Cynthia is wondering what hat Robin will wear tomorrow to the game.
She considers three open historical alternatives — Robin will wear a
Warriors cap (cw), a Giants cap (cg), or no cap (cn) —
and assigns each credence 1/3. The example is designed to make every
predicate of interest land on a probability in {0, 1/3, 2/3, 1},
which is what blocks @cite{hajek-1989}'s triviality argument from
applying (paper §8.2 footnote 32).
Verified predictions #
| # | Prediction | Theorem |
|---|---|---|
| 1 | Sports Fan: Cynthia thinks Robin will wear a Warriors cap | cynthia_will_warriors_cap |
| 2 | Will Excluded Middle holds at every world | will_em_at_cw |
| 3 | Negation Swap: ¬will A ↔ will ¬A | swap_at_cw |
| 4 | Speaker w/o w in modal base ≠ collapse | nonmember_no_collapse |
| 5 | Speaker with w in modal base ⇒ collapse | member_collapses |
| 6 | Selectional will: μ(‖will Warriors-cap‖) = 1/3 | cynthia_credence_one_third |
| 7 | Universal will: μ(‖∀Warriors-cap‖) = 0 (collapse) | universal_will_credence_zero |
| 8 | "If Robin wears a cap, Robin'll wear a Warriors cap" — credence 1/2 (paper eq. 30, §8.1) | cap_warriors_credence_one_half |
| 9 | Hájek triviality fails: no proposition has probability 1/2 unconditionally (§8.2 fn 32) | no_unconditional_one_half |
| 10 | cynthiaSel is coherent (§5.1: selection induces a well-ordering) | cynthiaSel_coherent |
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Cynthia's modal parameter: every cap-choice is historically open. @cite{cariani-santorio-2018} treat the Sports Fan as a case where all three alternatives are live; nothing is settled at the time Cynthia forms her credences.
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Proposition: "Robin wears a Warriors cap."
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Proposition: "Robin wears some cap" (Warriors or Giants).
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The underlying selection function: prefer w if w ∈ A,
otherwise the first available element in the order cw, cg, cn.
This is total because W is exhausted by {cw, cg, cn}.
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Coherence @cite{cariani-santorio-2018} §5.1: cynthiaSel
induces a well-ordering on {cw, cg, cn} — its pairwise preference
is transitive. The order, by construction of selFn, is
cw < cg < cn from any centre that is not itself in the candidate
pair (Centering forces the centre to win when it is present).
Proved by exhaustive enumeration over 3⁴ = 81 quadruples.
Prediction 1: From the Warriors-cap world cw,
Cynthia's assertion Robin will wear a Warriors cap is true.
willSem cynthiaSel warriorsCap histAlt cw reduces by Centering
(since cw ∈ histAlt) to warriorsCap cw = True.
Prediction 2 (Will Excluded Middle): at every world,
will warriorsCap ∨ will ¬warriorsCap holds.
Prediction 3 (Negation Swap): under selectional semantics,
¬ will A ↔ will ¬ A. This is what makes will "scopeless"
in matrix uses — failing under universal quantification.
A modal parameter that excludes the actual world cw (here taken
as the world from which Cynthia evaluates): the speaker is reasoning
about a counterfactual continuation in which Robin wears no cap.
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Prediction 4: when the evaluation world cw is not in the
modal parameter, no collapse — will A may diverge from A w.
Here will warriorsCap evaluated at cw against counterfactualAlt
selects cn (by Inclusion + the construction of cynthiaSel),
where warriorsCap is False. So the assertion is False even
though warriorsCap cw = True.
Prediction 5 (= @cite{cariani-santorio-2018} eq. (17) collapse):
when w is in the modal parameter, will A collapses to A w.
Cynthia's credence over the historical alternatives. Uniform
on histAlt = {cw, cg, cn} — each cap choice gets 1/3.
The uniform-over-three structure is what blocks the @cite{hajek-1989} triviality argument: no proposition lands on probability 1/2 unconditionally, so the selectional account survives Hájek's objection by construction (paper §8.2 footnote 32).
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cynthiaPMF is supported on histAlt: the support lies inside the
modal parameter. Vacuously true here, since every world is in
histAlt — but the discipline matches the cognitive_role
interface, which takes μ.support ⊆ f.
Prediction 6 (selectional cognitive role, paper §8.1): Cynthia's credence in Robin will wear a Warriors cap equals her credence in Robin wears a Warriors cap. Both are 1/3 — non-extreme credence licenses the will-assertion.
Direct application of Selectional.cognitive_role.
The universal-quantifier reading of will Warriors-cap is false at
every world: histAlt contains the Giants-cap world cg where
warriorsCap is False, so the universal cannot hold.
Prediction 7 (universal-base credence collapse, paper §8.1):
under the universal-quantifier reading, Cynthia's credence in
will Warriors-cap is 0, because the universal is false at
every world (the Giants-cap world cg is in histAlt).
Contrast with the selectional reading (cynthia_credence_one_third),
which gives 1/3 — the empirically attested value. The
selectional/universal split here is the substantive cognitive-role
argument from @cite{cariani-santorio-2018} §8.1.
Prediction 8 (will-conditional, paper eq. (30) §8.1):
If Robin wears a cap, Robin'll wear a Warriors cap. The Kratzer
restriction sends the modal parameter from histAlt = {cw, cg, cn}
to histAlt ∩ ‖cap‖ = {cw, cg} — the cap-wearing alternatives.
Inside this restricted parameter, both cw and cg are equally
open, but the antecedent's truth-set assigns positive mass to cw
only.
Cynthia's credence in this conditional is 1/2 (paper §8.1):
of the cap-wearing worlds (total mass 2/3), the Warriors-cap world
has mass 1/3, so the conditional credence is 1/3 ÷ 2/3 = 1/2. The
next theorem proves the unconditional credence in the world
where the cap-conditional is true: the world cw, which has
mass 1/3 ÷ (1/3 + 1/3) = 1/2 of the cap-wearing mass.
This exercises @cite{cariani-santorio-2018} §5.3.1 + §5.3.2: the
conditional uses willConditional, which Kratzer-restricts the
modal parameter.
The morphological identity in action: the would-conditional
if Robin had worn a cap, Robin would have worn a Warriors cap
is the same proposition as the corresponding will-conditional.
@cite{cariani-santorio-2018} §5.3.2's claim that would = past-
tense will lifts to conditionals: wouldConditional and
willConditional agree definitionally.
Prediction 9 (paper §8.2 footnote 32): on the 3-cap Sports Fan
model with uniform credence 1/3, no predicate B : W → Bool has
cynthiaPMF-probability 1/2. The probabilities all land in
{0, 1/3, 2/3, 1}.
@cite{hajek-1989}'s triviality argument requires a proposition with probability 1/2 to construct its problematic conditional. Cariani & Santorio observe that the Sports Fan deliberately avoids any such predicate — no proposition gets probability 1/2 here, so Hájek's argument cannot get off the ground in this model. The selectional account is therefore not undermined by the triviality result on this paradigm.
Proved by exhaustive enumeration over 2³ = 8 decidable subsets.
Fragment binding #
C&S analyse the English auxiliaries Fragments.English.Auxiliaries.will
and Fragments.English.Auxiliaries.would. The Fragment is the source
of truth for those entries' morphology; this section records the
morphological facts the C&S analysis depends on, as per-entry rfl
preconditions. If anyone later changes the morphological classification
of will or would in the Fragment (e.g., flips the tense field
on would away from some .Past), the corresponding precondition
theorem here breaks — making the cascading consequence for C&S visible
at compile time.
The Auxiliaries Fragment is a hub: other studies that analyse the
same entries (@cite{condoravdi-2002}, @cite{kratzer-1981}, etc.) record
their own morphological preconditions parallel to these. To enumerate
every analysis that touches a given entry, grep for
Fragments.English.Auxiliaries.<entry> across Phenomena/.
This section records morphological preconditions only. The C&S
semantic clauses (willSem, wouldSem) and their downstream theorems
live in the rest of this file and in
Theories/Semantics/Modality/Selectional.lean. The signature mismatch
between C&S's atemporal-propositional willSem and Condoravdi's
time-indexed-eventive woll means their predictions cannot be
compared by direct equation; a divergence-witness theorem against
@cite{condoravdi-2002} is left for follow-up.
C&S precondition: the Fragment classifies will as a modal auxiliary. C&S's selectional analysis presupposes modal status — constraint #1 (modal character) requires will to embed, scope, and interact with negation/quantifiers.
C&S precondition: the Fragment marks will as morphologically
non-past (tense = none). C&S analyse will as the present-tense
member of the future-modal pair; the wouldSem-as-past-shifted-
willSem argument (§5.3.2) presumes this.
C&S precondition: the Fragment marks would as morphologically
past (tense = some .Past). C&S §5.3.2 derives the would clause
by past-shifting the modal parameter on will; if the Fragment
later reclassified would as non-past, the §5.3.2 argument would
no longer apply at the surface-form level.
C&S precondition: will and would are morphologically
distinguished by their tense fields. The selectional analysis
would collapse vacuously if the Fragment treated them as
morphologically identical.