Epistemic Possibility Semantics #
@cite{holliday-mandelkern-2024}
Epistemic extension of possibility semantics. Adds an accessibility relation R to a compatibility frame, yielding □ and ◇ operators whose ortholattice validates Wittgenstein's Law (¬A ∧ ◇A = ∅) while invalidating distributivity across epistemic levels.
The Epistemic Scale #
The central example is the 5-point Epistemic Scale, constructed from a 2-world possible worlds model by "possibilizing" it. The scale runs:
□p — p — ◇p ∧ ◇¬p — ¬p — □¬p
x₁ x₂ x₃ x₄ x₅
Each point represents a degree of epistemic commitment: from certainty that p, through partial information, to certainty that ¬p. The partial possibility x₃ verifies ◇p ∧ ◇¬p without verifying either p or ¬p.
Linguistic applications #
- Wittgenstein sentences: "It's raining and it might not be" is contradictory: ¬A ∩ ◇A = ∅ (Proposition 4.27).
- Epistemic possibility ≠ negation: ◇¬p does not entail ¬p. "It might not be raining" does not mean "It's not raining" (§1, p. 2).
- Knowledge ≠ truth: p does not entail □p. "It's raining" does not mean "It must be raining" (§2, p. 3).
- Distributivity failure: reasoning by cases fails across epistemic levels (Example 3.20, Example 4.33).
A modal compatibility frame: a compatibility frame equipped with an
accessibility relation R satisfying reflexivity (Definition 4.24).
The paper's full Definition 4.20 also requires R-regularity; the
epistemic compatibility frame (Definition 4.26) adds Knowability.
Our epistemicScale satisfies all three conditions by construction
(Example 4.30).
- access : S → S → Bool
Instances For
Box operator: □A = {x | R(x) ⊆ A}. eq. (III).
Equations
Instances For
Diamond operator: ◇A = ¬□¬A (via orthocomplement, NOT Boolean dual). eq. (IV).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The Epistemic Scale is constructed from a 2-world model W = {0, 1} with V(p) = {0}. The possibilities are pairs (A, I) where ∅ ≠ A ⊆ I ⊆ W: - x₁ = ({0}, {0}) — settled that p, knows p - x₂ = ({0}, {0,1}) — settled that p, doesn't know - x₃ = ({0,1}, {0,1}) — nothing settled (full uncertainty) - x₄ = ({1}, {0,1}) — settled that ¬p, doesn't know - x₅ = ({1}, {1}) — settled that ¬p, knows ¬p
Compatibility: (a,i) ◇ (a',i') iff a ∩ a' ≠ ∅ ∧ a ⊆ i' ∧ a' ⊆ i.
Accessibility: (a,i) R (a',i') iff a ⊆ a' ∧ i' ⊆ i.
Definition 5.1, Example 5.3.
The epistemic scale frame. Compatibility is the path graph; accessibility captures epistemic access (refining information). Example 4.30, Example 4.33.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Verification of the truth values listed in Example 4.33.
□p = {x₁}: only x₁ knows p (R(x₁) = {x₁} ⊆ V(p)).
¬p = {x₄, x₅}: the orthocomplement of V(p).
□¬p = {x₅}: only x₅ knows ¬p.
◇p = {x₁, x₂, x₃}: p might be true at x₁, x₂, and x₃.
◇¬p = {x₃, x₄, x₅}: ¬p might be true at x₃, x₄, and x₅.
◇p ∧ ◇¬p = {x₃}: only the point of full uncertainty.
The motivating examples from §1–2 of the paper: epistemic possibility does not collapse to classical negation, and truth does not collapse to knowledge.
◇¬p does NOT entail ¬p: x₃ makes "it might not be raining" true but does not settle "it's not raining." This is the core motivation for the entire paper — in classical logic, ◇¬p → ¬p, but this fails in possibility semantics. §1, p. 2.
Wittgenstein's Law: ¬A ∧ ◇A = ∅. "It's raining and it might not be raining" is contradictory. In possibility semantics, if x settles ¬A (all compatible possibilities fail A), then x cannot also make ◇A true (which requires a compatible possibility in □¬A's complement). Proposition 4.27.
Wittgenstein's Law for ALL regular propositions in the epistemic scale: ¬A ∧ ◇A = ∅. There are 2⁵ = 32 Boolean functions on Poss5, of which 10 are ◇-regular (Figure 8); the theorem checks all 160 cases. Proposition 4.27.
Distributivity fails across epistemic levels: (p ∨ ¬p) ∧ (◇p ∧ ◇¬p) is true at x₃, but (p ∧ ◇p ∧ ◇¬p) ∨ (¬p ∧ ◇p ∧ ◇¬p) is not. The partial possibility x₃ verifies the disjunction without committing to either disjunct — both disjuncts are empty by Wittgenstein's Law (p ∧ ◇¬p = ∅ and ¬p ∧ ◇p = ∅). Example 3.20, Example 4.33.
Free choice disjunction: ◇(A ∨ B) entails ◇A ∧ ◇B.
The full free choice entailment holds for propositions in the image
of the embedding e_B in epistemic extensions of Boolean algebras
(Proposition 5.12.3, inheritance
principle). The path frame is non-Boolean (distributivity fails),
so free choice does NOT hold in general in the epistemic scale.
It does hold at x₃ (full uncertainty), where both p and ¬p remain
epistemically possible. It fails at x₁ (knows p), where ◇(p ∨ ¬p)
is trivially true but ◇¬p is false.
Free choice holds at x₃: ◇(p ∨ ¬p) → ◇p ∧ ◇¬p.
Free choice FAILS at x₁: ◇(p ∨ ¬p) is true but ◇¬p is false. x₁ knows p, so while the disjunction is trivially possible, the individual disjunct ¬p is not epistemically accessible. Proposition 5.12.3.
□p entails ◇p (knowledge implies epistemic possibility).
When the compatibility relation is identity (every possibility is a
world), the modal operators reduce to standard Kripke semantics
from Core.ModalLogic. The compat relation only affects negation
(orthocomplement) — box is Kripke necessity regardless.
Conceptual parallel to `Semantics.Supervaluation`: supervaluation's
D operator is □ with universal access among specification points,
and its fidelity theorem shows that singleton specification spaces
yield classical logic. Here, when all possibilities are worlds
(compat = identity), the ortholattice collapses to a Boolean algebra
— the same classical-collapse phenomenon from opposite directions.
Remark 4.9.
Box = Kripke necessity: the compatibility frame's box operator is definitionally Kripke necessity evaluation. The compat relation only affects negation, not the modal operators themselves.
Diamond = Kripke possibility when compat = identity. The orthocomplement reduces to Boolean negation, so ◇A = ¬□¬A becomes the standard ¬∀¬ = ∃ dual. Remark 4.9.
The T axiom for modal compatibility frames follows from the general
Kripke T axiom (Core.ModalLogic.T_of_refl). Reflexive accessibility
- any compat relation → □A entails A.
Disjunctive syllogism ({p ∨ q, ¬q} ⊨ p) fails for epistemic modals: "Either the dog is inside or it must be outside; it's not the case that it must be outside; therefore it is inside." The tautological first premise carries no information. §2.3.
Disjunctive syllogism fails: p ∨ □¬p and ¬□¬p both hold at x₃ (full uncertainty) but p does not.
Orthomodularity (if φ ⊨ ψ then ψ ⊨ φ ∨ (¬φ ∧ ψ)) fails. Since p ⊨ ◇p, orthomodularity would give ◇p ⊨ p ∨ (¬p ∧ ◇p). But ¬p ∧ ◇p = ⊥ by Wittgenstein's Law, so this collapses to ◇p ⊨ p — absurd. §2.4.
Orthomodularity fails: ◇p holds at x₃ but p ∨ (¬p ∧ ◇p) does not (since ¬p ∧ ◇p = ⊥ by Wittgenstein, this reduces to p).
Pseudocomplementation (a ∧ b = 0 → b ≤ ¬a) fails. In a Boolean algebra, ¬a is the greatest element disjoint from a. In an ortholattice this need not hold: p ∧ ◇¬p = ⊥ (Wittgenstein) but ◇¬p ≰ ¬p. This is the algebraic root of why ◇¬p ≠ ¬p. Proposition 3.7.
Pseudocomplementation fails: p ∧ ◇¬p = ⊥ but ◇¬p ≰ ¬p. x₃ witnesses ◇¬p (might not be raining) without witnessing ¬p.
Level-wise classicality: classical reasoning is safe within an epistemic level but dangerous across levels. The subortholattice B₀ = {⊥, p, ¬p, ⊤} is a four-element Boolean algebra; B₁ (generated by applying □ and ◇ to B₀) is an eight-element Boolean algebra. Distributivity only fails when mixing levels. §3.2.4.
Within-level distributivity (B₁): ◇p ∧ (◇¬p ∨ ◇p) = (◇p ∧ ◇¬p) ∨ (◇p ∧ ◇p). All operands from the same epistemic level → distributivity holds.
Cross-level failure: ◇p ∧ (p ∨ ¬p) ≠ (◇p ∧ p) ∨ (◇p ∧ ¬p). ◇p is from B₁ but p, ¬p are from B₀. At x₃ the LHS is true (◇p and excluded middle both hold) but the RHS is false (both disjuncts are empty by Wittgenstein's Law).