[CS18] — 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) #
[CS18] 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 [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" — conditional credence 1/2 (paper ex. (31)/eq. (32), §8) | 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 |
| 11 | Selectional will-conditional validates Compositional CEM (§7) | cap_will_conditional_cem |
| 12 | Universal-base will-conditional refutes CEM (the Lewis side) | universal_will_conditional_cem_fails |
Equations
- CarianiSantorio2018.instDecidableEqW x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Equations
- CarianiSantorio2018.instReprW.repr CarianiSantorio2018.W.cw prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "CarianiSantorio2018.W.cw")).group prec✝
- CarianiSantorio2018.instReprW.repr CarianiSantorio2018.W.cg prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "CarianiSantorio2018.W.cg")).group prec✝
- CarianiSantorio2018.instReprW.repr CarianiSantorio2018.W.cn prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "CarianiSantorio2018.W.cn")).group prec✝
Instances For
Equations
- CarianiSantorio2018.instReprW = { reprPrec := CarianiSantorio2018.instReprW.repr }
Equations
Equations
- CarianiSantorio2018.instFintypeW = { elems := { val := ↑CarianiSantorio2018.W.enumList, nodup := CarianiSantorio2018.W.enumList_nodup }, complete := CarianiSantorio2018.instFintypeW._proof_1 }
Cynthia's modal parameter: every cap-choice is historically open. [CS18] 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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Instances For
Proposition: "Robin wears a Warriors cap."
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Instances For
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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}.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
- CarianiSantorio2018.cynthiaSel = { sel := CarianiSantorio2018.selFn, inclusion := CarianiSantorio2018.selFn_inclusion, centering := CarianiSantorio2018.selFn_centering }
Instances For
Coherence [CS18] §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.
Instances For
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.
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 [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).
Equations
- CarianiSantorio2018.cynthiaPMF = PMF.ofFintype (fun (x : CarianiSantorio2018.W) => 1 / 3) CarianiSantorio2018.cynthiaPMF._proof_2
Instances For
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 [CS18] §8.1.
Prediction 8 (will-conditional, paper ex. (31)/eq. (32) §8):
If Robin wears a cap, Robin'll wear a Warriors cap. The Kratzer
restriction (willConditional) sends the modal parameter from
histAlt = {cw, cg, cn} to histAlt ∩ ‖cap‖ = {cw, cg} — the
cap-wearing alternatives.
The theorem records the conditional (Bayesian) credence
P(Warriors-cap | cap) = 1/2: of the cap-wearing worlds (mass 2/3),
the Warriors-cap world has mass 1/3, so 1/3 ÷ 2/3 = 1/2. This is
the intuitive Adams-thesis value.
Note [CS18] §8 are explicit that the selectional will-conditional proposition itself does not reach 1/2 in this 3-world model — it gets 1/3 or 2/3, since "no proposition can have probability 1/2" when every world has probability 1/3. Standard conditional semantics "fails to vindicate the intuitive assignments of probabilities to conditional sentences"; recovering 1/2 for the proposition needs the §8 refinement to a finer world-algebra. The theorem here captures the Bayesian conditional ratio (the value that refinement targets), not the selectional proposition's probability.
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.
[CS18] §5.3.2's claim that would = past-
tense will lifts to conditionals: wouldConditional and
willConditional agree definitionally.
The Stalnaker/Lewis CEM fault line, lifted to will-conditionals #
[CS18] §7 derive Compositional CEM —
(if A, will B) ∨ (if A, will ¬B) — from selection single-valuedness.
The Lewis-style universal-base reading (universalWillConditional)
refutes it, exactly as Lewis's universal counterfactual refutes
Conditional Excluded Middle for counterfactuals (cf.
Stalnaker1981.bizet_cem_fails_universal). The contrast below exhibits
that fault line at the future-modal layer, on the Sports Fan model: the
restricted parameter histAlt ∩ ‖cap‖ = {cw, cg} contains both a
Warriors-cap world (cw) and a non-Warriors-cap world (cg).
Selectional will-conditionals validate Compositional CEM
(paper §7): for the cap-conditional on the Sports Fan model,
(if cap, will Warriors) ∨ (if cap, will ¬Warriors) holds. Inherited
from the generic WillConditional.compositional_CEM.
The universal-base reading refutes Compositional CEM — the
will-conditional analogue of Stalnaker1981.bizet_cem_fails_universal.
On the restricted parameter histAlt ∩ ‖cap‖ = {cw, cg}, neither
(if cap, will Warriors) nor (if cap, will ¬Warriors) is
universally true: cg is a cap-world that is not a Warriors-world
(killing the first disjunct) and cw is a Warriors-world (killing the
second). So the Lewis-style universal future-conditional falsifies the
CEM that the selectional analysis validates.
The CEM split at the future-modal layer: the selectional
will-conditional validates Compositional CEM while the universal-base
reading refutes it. This is the future-tense image of the Stalnaker /
Lewis dispute that Stalnaker1981 records for counterfactuals — one
structural divergence, surfaced at both the counterfactual and the
will-conditional layer.
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}.
[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 English.Auxiliaries.will
and 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 ([Con02], [Kra81], etc.) record
their own morphological preconditions parallel to these. To enumerate
every analysis that touches a given entry, grep for
English.Auxiliaries.<entry> across the library.
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
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
[Con02] 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.