Source
Butterfield & Isham (1998; IJTP 1999), “A Topos Perspective on the Kochen-Specker Theorem: II”, LIT-381, read in full in NOTE-329. Isham & Butterfield (1999; Found. Phys. 2000), LIT-389, read in full in NOTE-334.
What was actually shown
The general theorem (LIT-381, Thm 4.1). Let G be a presheaf of posets with 0 and 1 on a small category C, and ν a sieve-valued valuation assigning to each proposition d at A a sieve ν(A, d) on A. Then ν obeys generalized FUNC, ν(B, G(f)(d)) = f*(ν(A, d)) (4.2), iff A ↦ ν(A, ·) is a natural transformation G → Ω, that is, a subobject of G. The proof is the one-line naturality check (4.4), and it is correct.
The classical analogue (LIT-381 §3). Over the category M of measurable functions on a classical state space, with arrows B̄ = f∘Ā, the value presheaf Υ(Ā) = Ā(S) has global sections, one per microstate s, γ^s_Ā = Ā(s) (3.6–3.7). Macrostates R still give sieve-valued valuations, ν^R(A ∈ Δ) = {f | B̄(R) ⊆ f(Δ)} (3.10), and so does a single microstate (3.15). NOTE-329 checked (3.10). NOTE-329 also shows, with constants admitted as in the main text, that every false proposition gets a non-empty, non-principal sieve whose pseudo-complement is empty, so excluded middle fails at a classical microstate.
What decides the logic. A one-object base gives two truth values (LIT-381 §2.2). LIT-389 §2.2 argues that because the subobject classifier is unique up to isomorphism, the many-valued logic is not arbitrary, and §3.1 observes that a base with exactly one arrow between any two objects gives the ordinary true/false logic. LIT-389 writes out the Heyting operations on sieves for any small category (2.10)–(2.12): only S ∨ ¬S ≤ 1 holds.
What could have come out otherwise. Classical physics, having global sections, could have yielded two-valued truth values under the same construction. It does not: truth values remain sieves on its base.
What this does not say
- That the many-valued logic is non-arbitrary outright. It is fixed given the topos. The choice of base category remains free, and the series itself moved from operators (O) and Boolean subalgebras (W) to von Neumann subalgebras (V). The arbitrariness is relocated, not removed (NOTE-334, C5).
- That sieve values are forced. LIT-381’s headline, that these are among “the most natural” valuations for any presheaf of propositions, rests on Theorem 4.1 and a partial-truth argument the authors say is not a deduction (§5.2).
- That every base has a non-trivial logic, or a trivial one. LIT-389’s remark that the presheaf of Hilbert spaces over (ℝ, ≤) has “essentially trivial” logic is wrong: that base is a non-trivial poset, and its sieves form a non-Boolean Heyting algebra (NOTE-334, C8).
- Anything about probabilities. Both papers set statistical content aside; quantum probabilities enter only through probability one.
Connections
- Corrects THEORY-024, which held that Boolean logic returns when a presheaf acquires a global section or is sheafified. Its rejection already rests on Isham–Butterfield’s statement that the Heyting algebra of sieves is fixed by the base (LIT-325 §6) and on Isham 1997 (LIT-390). The classical case here is the series’ own counterexample: global sections everywhere, sieve-valued truth anyway.
- Answers in part QUESTION-055: if a communicative object is a presheaf over contexts, its truth values are sieves on the base, and their negation is the Heyting pseudo-complement the base fixes, not an orthocomplement. Which base the communicative object has is the part left open.
- The Heyting structure of propositions in the von Neumann-algebra form of the programme is THEORY-037. One Boolean context alone always has two-valued valuations, and the Kochen–Specker obstruction lies in how contexts overlap (THEORY-020).
- How a quantum state’s sieve-valued valuation is fixed by its supports on the von Neumann-algebra base is THEORY-619.