Booth (2022b): Necessity modals, disjunctions, and collectivity #
[Boo22b] resolves Ross's Puzzle by requiring the alternatives of a necessity modal's
prejacent to form a minimal cover of the relevant worlds, not merely a cover: each alternative
must be needed, so □(p ∨ q) licenses the Independence inferences ◇(p ∧ ¬q) and ◇(q ∧ ¬p),
and the Ross inference from □p to □(p ∨ q) is strongly invalid. The semantics is the bilateral
minimal covering semantics of [Boo22a] (the theory-layer MinimalCovering), whose results
on atoms this paper restates; its new content is the generalization of the Ross and Independence
facts from atoms to arbitrary non-Hurford disjunctions, resting on the compactness of alternatives.
The closing reading of □ as a collective predicate of the plurality of propositions a disjunction
denotes (§4) is not formalized.
Main results #
Entailment over a model (Def 18) is inclusion of truth sets, and strong invalidity (Def 21) is disjointness of the premises' truth set from the conclusion's.
kratzer_monotone: Fact 1;not_monotone_necessity: Booth's necessity is not upward monotonic.truth_necessity_subset: every Booth necessity is a Kratzerian one (Def 1).Formula.eval_neg_neg: Fact 3, double negation.Formula.fg_eval: Fact 5, both interpretations of every sentence finitely generated (Question.FG);Formula.truth_necessity_eq: hence Def 14's necessity clause agrees with [Boo22a]'s on every sentence.ross_strongly_invalid_of_alt,inter_diff_truth_nonempty_of_altand their sentence formsFormula.ross_strongly_invalid,Formula.independence: the meta-language Facts 7 and 6.not_forall_disjoint_of_nonHurford,not_forall_independence_of_nonHurford: Facts 7 and 6 as printed, for every non-Hurford disjunction of Def 22, are false.ross_strongly_invalid,extended_ross_strongly_invalid: Facts 7 and 8 for atoms.independence_left,independence_right(Fact 9),free_choice_left,free_choice_right(Fact 10),independence_conditional_left,independence_conditional_right(Fact 11),unnecessity_distribution_left,unnecessity_distribution_right(Fact 12),impossibility_distribution_left,impossibility_distribution_right(Fact 13): [Boo22a]'s Facts 1, 4, 2, 7 and 6.
Implementation notes #
- The semantics is
MinimalCovering:Question Wsupplies Def 10's subset-closed families (Question.ofSetis↓{·}of Def 11,Question.infoisinfoof Def 12,Question.altisaltof Def 13), Def 10's bilateral propositions areMinimalCovering.BilatInqProp, whose third bulletP⁺ ∩ P⁻ = {∅}is disjointness inQuestion W, and Def 14's ¬-, ∧- and ∨-clauses are itsᶜ,⊓and⊔. Def 14 defines◇φas¬□¬φ, which isMinimalCovering.possibility_eq_compl_necessity_compl. - Def 14's □⁺ clause adds the conjunct
R(w) ≠ ∅, which [Boo22a]'s clause lacks. The two differ only on propositions without alternatives, and every sentence has one (Fact 5), so they agree on sentences (Formula.truth_necessity_eq) and this file uses the substrate's clause. - Def 8's language is the
!-free fragment ofMinimalCovering.Formula; Fact 5 is proved for the whole language. - Def 17 writes
w ∈ ⟦φ⟧⁺for a worldwand a set of states; it is read asw ∈ info ⟦φ⟧⁺, equivalently{w} ∈ ⟦φ⟧⁺(BilatInqProp.mem_truth_iff). - Def 22 is mathlib's
IncompRelfor the positive interpretations; for atoms it is [Boo22a]'s admissibility (nonHurford_atom_iff). The meta-language Facts 6 and 7 are stated for every such disjunction, but their proofs assume that no alternative of either disjunct is a state of the other,Disjoint (alt ⟦φ⟧⁺) ⟦ψ⟧⁺, which is strictly stronger. Under Def 22 both facts fail; under the alternative-wise condition the proofs go through, and for atoms the two conditions coincide, so the object-language Facts 7–13 stand. The correction is this file's, not the paper's. - Facts 12 and 13 need no non-Hurford hypothesis.
- Figures 1 and 2 reprint those of [Boo22a] and are formalized there
(
Booth2022a.Figures).
References #
Kratzer's semantics (Def 1) #
Booth Fact 1: Kratzer's necessity, true at w when R w ⊆ ⟦φ⟧ (Def 1), is upward
monotonic, so it validates the Ross inference. Booth's is not (not_monotone_necessity).
A Booth necessity is a Kratzerian one (Def 1): the relevant worlds lie in the truth set of the prejacent.
Booth's necessity is not upward monotonic, unlike Kratzer's (kratzer_monotone): on the model
of Figure 1, ⟦p⟧ ⊆ ⟦p ∨ q⟧, yet □p is true and □(p ∨ q) is not.
Non-Hurford disjunctions (Def 22) #
Booth Def 22: φ ∨ ψ is non-Hurford when the positive interpretations of the disjuncts
are incomparable.
Equations
- Booth2022b.NonHurford φ ψ = IncompRel (fun (x1 x2 : Question W) => x1 ≤ x2) (Kalman.pro φ) (Kalman.pro ψ)
Instances For
For atoms, Def 22 is [Boo22a]'s admissibility: the truth sets are incomparable.
When no alternative of either disjunct is a state of the other, the alternatives of the disjunction are those of the disjuncts.
The language (Def 8) and compactness (Fact 5) #
Booth Fact 3, double negation.
Booth Fact 5, compactness of alternatives: both interpretations of a sentence are
finitely generated, so each has finitely many alternatives and is generated by them,
⟦φ⟧° = ↓alt°(⟦φ⟧) (Question.fg_iff_finite_alt_and_isNormal).
Def 14's □⁺ clause, with its conjunct R(w) ≠ ∅, agrees with [Boo22a]'s on every
sentence: the prejacent has an alternative (Fact 5), and a nonempty family minimally covers only
nonempty sets.
The meta-language Facts 6 and 7 #
Booth states both facts for every non-Hurford disjunction (Def 22). Their proofs use that no alternative of either disjunct is a state of the other, which is strictly stronger; the facts hold under that condition and fail under Def 22.
Booth Fact 7 (the Ross inference is strongly invalid) under the alternative-wise
non-Hurford condition, once ψ has an alternative: alt⁺(⟦φ⟧) is then a proper subfamily of
alt⁺(⟦φ ∨ ψ⟧) that still covers the relevant worlds.
Booth Fact 6 (Independence, meta-language) under the alternative-wise non-Hurford
condition: if □(φ ∨ ψ) is true, some relevant world is in the truth set of φ but not of ψ.
The alternatives other than a chosen alternative of φ fail to cover the relevant worlds, and the
world they miss lies in no state of ψ.
Booth Fact 7 for sentences under the alternative-wise non-Hurford condition; Fact 5
supplies the alternative of ψ.
Booth Fact 6 for sentences under the alternative-wise non-Hurford condition; Fact 5 supplies the alternatives and the normality of the disjuncts.
Booth Fact 7 as printed, for every non-Hurford disjunction of Def 22, is false: with
worlds 0, 1, 2, φ = r₁ ∨ r₂ for V(r₁) = {0}, V(r₂) = {1}, ψ = q for V(q) = {0, 2} and
R(0) = {0, 1}, both □φ and □(φ ∨ ψ) are true at 0. The alternative {0} of φ is a state
of ψ, which the paper's proof overlooks.
Booth Fact 6 as printed, for every non-Hurford disjunction of Def 22, is false: with
worlds 0, 1, 2, φ = r for V(r) = {0, 1}, ψ = p ∨ q for V(p) = {0}, V(q) = {1, 2} and
R(0) = W, □(φ ∨ ψ) is true at 0 but every world in the truth set of φ is in that of ψ.
The object-language Facts 7–13 #
Facts 9–13 are [Boo22a]'s Facts 1, 4, 2, 7 and 6, stated here under Def 22's hypothesis.
Booth Fact 7 for atoms: the Ross inference □p ∴ □(p ∨ q) is strongly invalid; indeed
□p makes □(p ∨ q) neither true nor false (Booth2022a.ross_gap).
Booth Fact 8: the Extended Ross inference □p, ◇q ∴ □(p ∨ q) is strongly invalid.
Booth Fact 9, first Independence inference: □(p ∨ q) entails ◇(p ∧ ¬q).
Booth Fact 9, second Independence inference: □(p ∨ q) entails ◇(q ∧ ¬p).
Booth Fact 10, Free Choice: ◇(p ∨ q) entails ◇p.
Booth Fact 10, Free Choice: ◇(p ∨ q) entails ◇q.
Booth Fact 11, Independence Conditionals: □(p ∨ q) entails ¬p → □q.
Booth Fact 11, Independence Conditionals: □(p ∨ q) entails ¬q → □p.
Booth Fact 12, Unnecessity Distribution: ¬□(p ∨ q) entails ¬□p.
Booth Fact 12, Unnecessity Distribution: ¬□(p ∨ q) entails ¬□q.
Booth Fact 13, Impossibility Distribution: ¬◇(p ∨ q) entails ¬◇p.
Booth Fact 13, Impossibility Distribution: ¬◇(p ∨ q) entails ¬◇q.