logic.
52 of 1538 NOTE documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| NOTE-066 v2 | Backens & Duman, graphical calculus for Spekkens’ toy bit | Take the maximal-knowledge fragment of Spekkens’ toy bit theory with post-selected measurements, viewed as relations over {1,2,3,4}ⁿ. It has a ZX-style red–green graphical calculus that is universal (§4.3), sound (§4.4) and complete (Thm 17). Completeness goes through a proof that the toy theory’s binary “check-matrix” formalism coincides with the qubit stabilizer one (valid states SᵀJS = 0, valid transformations QᵀJQ = J, Lemma 3). With it come graph-state/local-complementation results for toy states (Thms 4–5), map–state duality (§5.2) and an rGS-LO normal form (Thms 10–11). The rewrite rules differ from the stabilizer ZX-calculus only through the phase group: Z₂×Z₂ against Z₄. That difference appears as an “11-commutation” rule in place of π-commutation (§4.5). | Read |
| NOTE-069 v2 | Riehl & Shulman, type theory for synthetic ∞-categories | A three-layer type theory adds a directed interval 2 to HoTT. Its layers are cubes, topes (the coherent theory of a strict interval) and types, and simplices become shapes that can be mapped out of by extension types. One contractibility condition, “Segal” (unique fillers for the 2-1-horn), then yields associativity (Prop. 5.9) and 3-horn fillers (Prop. 5.21). It also makes every function a functor (Prop. 6.1), gives a dependent Yoneda lemma for covariant families (Thm. 9.5), and makes having a left adjoint a proposition when the adjoint’s codomain is Rezk (Thm. 11.23). The bisimplicial-sets semantics is sketched. Segal and Rezk types are shown to match Segal and complete Segal spaces (Props. A.21, A.25), and soundness is not proved in full. | Read |
| NOTE-127 v2 | Holik et al. — Quasi-set theory for a quantum ontology | The paper argues, with no new theorem, that the Lombardi–da Costa ontology of properties for quantum mechanics can be stated in Krause’s quasi-set theory Q. On that ontology a system is a bundle of I-type-properties (instances of universal type-properties, represented by the algebra of observables) with no principle of individuality. For each property instance, the indiscernible instances form a qset U([Π]). An atomic bundle is any qset x with qc(x ∩ U) = 1 for each U; the Axiom of Choice in Q guarantees one exists. The bundle’s kind is Q(B) = {x ∈ P(Q) | R(x)}, whose members are all indiscernible. N particles are a subqset N(B) ⊂ Q(B) with qc = N, a definite number with no labels (§6, pp. 19–22). | Read |
| NOTE-177 v2 | Turner — Ultrafilters as Propositional Theories | An expository piece that redefines a filter over a set W as a “propositional theory” (a set of W-propositions closed under pairwise conjunction and implication, not containing ∅), proves maximal ⇔ prime ⇔ ultra (Thm 2), the ultrafilter extension theorem as a propositional Lindenbaum’s lemma (Thm 3, assuming choice), Łoś’s theorem M_T,[A] ⊨ φ iff ⟦φ⟧_A ∈ T for ultraproducts of “intensional spaces” (Thm 9), and first-order compactness from it (Thm 12); “None of the results are novel” (p. 2). | Read |
| NOTE-179 | Brandom, inferentialism and meaning entitlement | Boghossian’s objection to substantive, entitlement-based theories of meaning (drawn from pejoratives like “Boche”) works against Dummett but not against Brandom. Brandom’s material inferences are non-monotonic and ceteris paribus, so one can grasp a concept while refusing to endorse the inferences it licenses. | Skimmed |
| NOTE-239 | The Geometry of Information Retrieval | The book recasts IR’s vector-space, probabilistic and logical models in one Hilbert-space language. Documents are state vectors. Relevance and aboutness are Hermitian observables that need not commute. Subspaces carry a non-distributive logic whose conditional is shown to be a Stalnaker conditional equal to the Sasaki hook (ch. 5). Gleason’s theorem makes tr(ρP) the probability of a subspace, which ch. 6 uses to rewrite cosine matching, cluster representatives, relevance and pseudo-relevance feedback, dynamic clustering and ostensive retrieval. It is a proposed language, not a tested model: no experiment is reported, and its “content hypothesis” that the probability of a concept is cos²θ is argued, not tested. | Read |
| NOTE-240 | The Lattice Representation Hypothesis of Large Language Models | Thresholding linear attribute directions (Fisher-LDA probes) gives a crisp object–attribute incidence, and the paper’s Theorem 1 is the standard fact that any such incidence has a complete concept lattice. The geometric meet is half-space intersection; the join is defined as the set union of two cones and “approximated by the conic hull” of their directions, which as written is not a lattice join. On five WordNet domains with GPT-4o-annotated attributes, the probes recover the incidence at F1 69.7–83.2 on three 7–8B models, and profile-based subsumption reaches F1 57.1–77.1. No lattice law is ever tested. | Read |
| NOTE-245 | Mathematical Foundations for a Compositional Distributional Model of Meaning | The paper builds DisCoCat as a product category FVect × P of vector spaces and a free pregroup. A sentence’s meaning is f(w₁⊗…⊗wₙ), where f is the linear map got by putting vector spaces in place of the pregroup types in the reduction p₁…pₙ ≤ s. The ε maps become inner products and the η maps become Σᵢ eᵢ⊗eᵢ. So “John likes Mary” is Σ_ijk c_ijk⟨v|vᵢ⟩⟨w_k|w⟩ sⱼ ∈ S, and “not” is the 2×2 swap matrix placed on the sentence wire. There is no theorem and no experiment. The sentence space S is shown only as 1- or 2-dimensional truth values, over a toy model with one basis vector per individual, and the paper says how neither S nor the verb tensors are to be built from data. Its headline similarity numbers (3/4, 1/4, 3/8) are unnormalised inner products. Under its own Definition 5.1 they are 0.949, 0.316 and 0.6. | Read |
| NOTE-246 v2 | Reasoning about Meaning in Natural Language with Compact Closed Categories and Frobenius Algebras | The paper sets the sentence space equal to the noun space (F(s)=F(n)=W). It then uses the basis-copying map σ: v_i ↦ v_i⊗v_i of the Frobenius algebra fixed by W’s basis to lift corpus-built verb matrices Σ_i sbj_i⊗obj_i into rank-3 tensors. So a transitive sentence becomes sbj⊙(verb·obj) (“copy-subject”) or obj⊙(verbᵀ·sbj) (“copy-object”), a vector in W. On the Grefenstette–Sadrzadeh verb-disambiguation set, copy-object gets Spearman ρ=0.172, against 0.168 (Kron), 0.163 (Multp), 0.143 (copy-subject), 0.050 (additive) and an inter-annotator bound of 0.620. On a 112-term definition-matching task the best F1 is 0.21 (nouns) and 0.27 (verbs). No significance test is reported. The disambiguation numbers are identical to the authors’ COLING 2012 paper, and the definition numbers are the same data re-reported. | Read |
| NOTE-249 v2 | Measurement as Sheafification: Context, Logic, and Truth after Quantum Mechanics | A programmatic essay, with no theorem, model or computation of its own. It restates Abramsky & Brandenburger’s criterion (contextuality and non-locality as the absence of a global section of a presheaf over measurement contexts) and adds three things. It declares measurement to be “sheafification” of that presheaf, without specifying a site, a topology or any sheafified object. It asserts that the Ghose–Patra seven-valued logic is a finite Heyting algebra, without giving an order or operations. It proposes that a quantum potential scaled by 0 ≤ λ ≤ 1 (λ = 1 quantum, λ = 0 classical Hamilton–Jacobi) carries the presheaf continuously to a sheaf, which is asserted, not modelled. Its one worked formula, v2’s Čech “obstruction class” [δs], is a coboundary and so is zero by construction. | Read |
| NOTE-251 | Contextuality, Cohomology and Paradox | For a model whose outcomes live in a commutative ring R, “All-vs-Nothing” (AvN_R) means the R-linear equations satisfied by every support section of every context have no global solution. Theorem 21 proves AvN_R(S) ⇒ SC(Aff S) ⇒ CSC_R(S) ⇒ CSC_ℤ(S) ⇒ SC(S), so every AvN model is strongly contextual with a non-vanishing Čech obstruction on every section. Theorem 4 shows that every state stabilised by an “AvN triple” of n-qubit Pauli operators yields such a model. The “paradox” half proves much less: an informal identification of Liar cycles with strongly contextual support tables, one exact case (the length-4 Liar cycle is the PR box), and the cohomology of paradox deferred to future work. | Read |
| NOTE-254 | Logical and Topological Contextuality in Quantum Mechanics and Beyond | The thesis’s cohomology chapters republish the 2017 and 2018 papers with the “joint model” renamed the “line model”. The definitions, Thm IV.32 (complete on cyclic scenarios at level |M|−1), Prop IV.34, Thm IV.36 and Conjecture IV.41 carry over unchanged. So does every defect found in the preprint. Under the thesis’s own Def IV.26, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa is still detected on only 6 of 15 sections at levels 0–3 (my recomputation), and a lifted ℤ/2 family shows it is never fully detected at any level. The context-wise reading the worked examples use detects all 15 sections at level 2. The genuinely new material lies outside the cohomology programme: a probabilistic line-model construction (§IV.7), a valuation-algebra “theory of disagreement” with inference algorithms (Ch V, bound O(k^n·l^{k(n−1)+1}) on (n,k,l) scenarios), and the AvN-triple characterisation of stabiliser states (Ch VI). | Read |
| NOTE-271 | A topos perspective on the Kochen–Specker theorem: I. Quantum states as generalized valuations | Isham & Butterfield show that, for bounded self-adjoint operators with discrete spectrum, a global section of the spectral presheaf Σ(Â) = σ(Â) on the category O_d, whose arrows are functional relations B̂ = f(Â), is literally a valuation obeying the value rule and FUNC. A global section of the dual presheaf D(W) = Hom(W, {0,1}) on the poset W of Boolean subalgebras of projectors is literally a Kochen–Specker 0/1 assignment. The Kochen–Specker theorem (cited, not reproved) is therefore equivalent to “no global sections” when dim H > 2 (§§2.2–2.3). In place of the missing global valuations, they construct contextual truth values in the Heyting algebra of sieves: ν^ρ(A ∈ Δ) = {f: B̂ →  | tr(ρ Ê[B ∈ f(Δ)]) = 1}. For every density matrix these satisfy functional composition, null, monotonicity, exclusivity and unit conditions (§4.4, §5.3, Theorem 5.1). | Read |
| NOTE-272 | The Theory of Representations for Boolean Algebras | The paper identifies Boolean algebras with Boolean rings with unit (xx = x), and generalized Boolean algebras with Boolean rings (Thms 1–4). Theorem 67 proves that every Boolean ring A is isomorphic to the algebra B(A) of classes 𝔈(a) = {prime ideals p : a ∉ p}, under symmetric difference and intersection. Its ideal lattice is isomorphic to the system of classes 𝔈(𝔞) under arbitrary unions and finite intersections. The existence of enough prime ideals (Thm 63) is proved by transfinite induction, and Theorem 70 shows that “every Boolean ring has an isomorphic algebra of classes” and “every ideal is the product of its prime divisors” are equivalent without choice. Theorems 68–69 classify all set-representations as restrictions of B(A) to subclasses of prime ideals. The topology, i.e. Stone spaces and duality, is explicitly deferred to “another paper”. | Read |
| NOTE-275 v2 | The logic of quantum mechanics | The paper argues heuristically that the experimental propositions of a quantum system correspond to closed linear subspaces of Hilbert space. Meet is intersection, join is closed linear span, and negation is orthogonal complement. The lattice satisfies L1–L4 and complementation L71–L73 but not the distributive law L6. The paper proposes the modular law L5 as its replacement. Under a finite-dimension assumption, it concludes that the propositional calculus of QM is an abstract projective geometry. The Appendix proves that orthocomplementations on P_{n−1}(F), n ≥ 4, are exactly the polarities of definite diagonal Hermitian forms with respect to an involutory anti-automorphism of F. | Read |
| NOTE-277 v2 | A topos foundation for theories of physics: I. Formal languages for physics | The programme paper. Its contention is that a theory of a physical system S is a representation of a typed “local language” L(S) (ground types Σ and R, function symbols A : Σ → R) in a topos. Physical quantities become arrows Σ_φ → R_φ, and propositions become sub-objects of Σ_φ, which form a Heyting algebra. Classical physics is the representation in Sets. The paper proves nothing. Its one argument is that the axiom ¬(A ε Δ) ⇔ A ε ℝ∖Δ is not validated by non-Boolean Heyting representations (§3.2.1). | Read |
| NOTE-279 | The problem of hidden variables in quantum mechanics | Theorem 1 exhibits a finite partial Boolean algebra D, generated by 117 directions in E³, that admits no homomorphism onto Z₂. §4 embeds D in the projections of the spin-1 space of orthohelium’s 2³S₁ state, via α ↦ P J_α². So no assignment of 0/1 values to those quantum propositions respects all the Boolean relations within each commeasurable set. Theorem 0 makes the target precise: a partial Boolean algebra embeds in a Boolean algebra iff its Z₂-homomorphisms separate points. That is, hidden variables in the paper’s sense need enough “global valuations”, and quantum mechanics in dimension 3 has none. The same paper builds a classical hidden-variable model for spin ½ (§6). It argues that von Neumann’s no-go needed additivity on non-commuting observables (Theorem 3). | Read |
| NOTE-285 | A topos foundation for theories of physics: IV. Categories of systems | Proposes axioms for a category Sys of physical systems, symmetric-monoidal under composite ⋄ and under disjoint sum ⊔, whose arrows are translations of local languages. Each arrow j : S₁ → S is realised by a left-exact functor τ(j) between the systems’ topoi, together with arrows φ(j) : Σ_{S₁} → τ(j)Σ_S and β(j) : τ(j)R_S → R_{S₁}. Classical physics satisfies the scheme exactly. In quantum theory the disjoint sum H₁⊕H₂ satisfies it (the pulled-back arrow of Â₁⊕Â₂ is δ̆(A₁)), but the composite H₁⊗H₂ does not: the natural pull-back of δ̆(A₁) differs from δ̆(A₁⊗1̂) off the image of V(H₁), which the authors attribute to “operator entanglement”. | Read |
| NOTE-290 | A topos foundation for theories of physics: III. The representation of physical quantities with arrows δ̆°(A): Σ̲ → ℝ̲≽ | Each bounded self-adjoint  becomes an arrow δ̆^o(A) : Σ → ℝ^⪰, where ℝ^⪰ is the presheaf of real-valued order-reversing functions on ↓V. At λ ∈ Σ_V it sends V′ ⊆ V to λ(δ^o(Â)_{V′}), and δ^o(Â)_V is the spectral-order-least operator in V above Â. The map  ↦ δ̆^o(A) is injective, but ℝ^⪰ is only an additive monoid, not the real-number object. Its Grothendieck completion k(ℝ^⪰) is an abelian-group object whose global elements are exactly the functions of bounded variation on V(H). | Read |
| NOTE-292 v2 | A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory | A projector P̂ is mapped to the clopen sub-object δ(P̂) of the spectral presheaf Σ over the poset V(H) of abelian von Neumann subalgebras, by “daseinisation” (de Groote’s V-support): δ(P̂)V = the least projector in V above P̂. The map is injective and preserves ∨, but only δ(P̂∧Q̂) ⪯ δ(P̂)∧δ(Q̂). This represents the propositional language PL(S) in the Heyting algebra Sub_cl(Σ). A pure state gives the truth object T^ψ, and “A ε Δ” gets the sieve-valued truth value {V′ ⊆ V | ⟨ψ|δ(Ê[A∈Δ]){V′}|ψ⟩ = 1}. | Read |
| NOTE-294 v2 | A theory of concepts and their combinations I: The structure of the sets of contexts and properties | Aerts & Gabora model a concept as an entity with states that change under contexts, a “state context property system” (Σ, M, L, µ, ν). They report an 81-subject e-mail study in which 7 contexts shift the rated frequency of 14 exemplars of “pet” (for example dog 0.50 under “The pet is chewing a bone” against 0.12 in the unit context) and the applicability of 14 properties. They then postulate that contexts and properties each form a complete orthocomplemented lattice, and call the structure quantum-like because a state can be an eigenstate of e ∨ f without being one of e or of f (eq. 19). No theorem is proved, the lattice is not shown to be non-distributive or orthomodular, and the paper contains no Hilbert-space model and no treatment of combination; both are deferred to Part II. | Read |
| NOTE-311 | Applications of the theory of Boolean rings to general topology | Stone topologises the set 𝔈 of prime ideals of a Boolean ring by taking the sets 𝔈(a) as a basis. 𝔈 is then a totally disconnected, locally bicompact Hausdorff space, bicompact exactly when the ring has a unit (Thm 1), and every such space arises this way (Thm 2). Theorem 4 states that “the algebraic theory of Boolean rings is mathematically equivalent to the topological theory of Boolean spaces”: isomorphism ↔ homeomorphism, ideals ↔ open sets, quotients ↔ closed subsets, unit ↔ bicompactness, and subrings ↔ continuous images (Thm 7). The same paper also contains what are now called the Stone–Čech compactification (Thms 78–79, 88), the Stone–Weierstrass theorem (Thm 82), the Banach–Stone theorem (Thm 83) and closed ideals of C(X) ↔ closed sets (Thm 85). | Read |
| NOTE-312 | Studies on the Foundation of Quantum Mechanics. I | Husimi derives Birkhoff and von Neumann’s lattice axioms from statistical premises. Implication is P̄ ≤ Q̄ in every state (and requires compatibility), negation is Q̄ = 1 − P̄, and compatible propositions obey classical logic. Meet is obtained from “Ehrenfest’s principle” (additivity of mean values) via the top eigenvalue of aP + bQ (pp. 778–779), and the modular law, equivalently the parallelogram law |P| + |Q| − |P∩Q| = |P∪Q|, is proved under a finite-chain assumption. The orthomodular identity is used but never named or stated as an axiom. It appears as the step “since we admit traditional logic to any classical part … S + P∩Q = R” for R ⊃ P∩Q (p. 784), and as the case “(Q − P) + P = Q provided Q ⊃ P” of a classical identity (p. 780). | Read |
| NOTE-326 | ’What is a Thing?’: Topos Theory in the Foundations of Physics | The Döring–Isham programme in one volume. A theory of a system is a representation of a typed language L(S) in a topos. In quantum theory, presheaves over the abelian von Neumann subalgebras V(H), daseinisation sends projectors injectively to clopen sub-objects of the spectral presheaf Σ, preserving ∨ but not ∧. A pure state gives sieve-valued truth values. Every bounded self-adjoint  becomes an arrow Σ → ℝ^↔ whose value at a point is a pair of monotone functions, an interval that widens as contexts coarsen. Most of it reprints papers I–IV. New are the inverse-image corollary linking PL(S) and L(S), values in a pseudo-state, and the algebra of the quantity-value candidates. The Heyting structure on Sub_cl(Σ) is still only sketched. | Read |
| NOTE-327 | A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category | Hamilton, Isham & Butterfield move the series’ base category from operators (O) and Boolean subalgebras (W) to the poset V of commutative von Neumann subalgebras of B(H). There they restate Kochen–Specker as “the spectral presheaf Σ(V) = Gelfand spectrum of V has no global elements” (dim H > 2, cited, §2.2). They identify the coarse-graining presheaf G(V) = L(V), G(i)(P̂) = inf{Q̂ ∈ L(V₂) | P̂ ≤ Q̂} (Def. 3.1), with the presheaf CloΣ of clopen subsets of the spectra; the isomorphism is asserted, and part IV proves it. A state ρ yields the sieve-valued valuation ν^ρ_V(P̂) = {V′ ⊆ V | ρ(G(P̂)) = 1} (Def. 3.4) and the support projectors inf{P̂ ∈ L(V) | ρ(P̂) = 1}. These form a global element of G and hence an “interval valuation”, a subobject of Σ (§§4.2–4.4). The probability-r valuations give subobjects of G but not global elements. | Read |
| NOTE-328 | A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations | Butterfield & Isham show, over the poset V of commutative von Neumann subalgebras, that sieve-valued valuations and “interval valuations” (assignments of support projectors, or of subsets of Gelfand spectra) determine each other under stated conditions. If supports s(α, V) = inf{P̂ | α_V(P̂) = true_V} form a global element of the coarse-graining presheaf G, then α_{V₁}(P̂) = {V₂ ⊆ V₁ | G(s(α, V₁)) ≤ G(P̂)} is a sieve-valued valuation obeying FUNC (Theorem 3.1). There is an analogue for “tight” subobjects of Σ (Theorem 3.2). Both are satisfied by every state’s valuation ν^ρ. The ordering ≤ (or ⊆) is shown to be the natural relation securing sievehood, null, monotonicity, exclusivity and unit (§4). The review §2.4 also supplies the proof, via state extension, that G ≅ CloΣ: restricting the clopen set of P̂ to V₂ gives the clopen set of its coarse-graining. Part III had only asserted this. | Read |
| NOTE-329 | A Topos Perspective on the Kochen-Specker Theorem: II. Conceptual Aspects, and Classical Analogues | Butterfield & Isham argue that the sieve-valued “generalised valuations” of part I are not a quantum peculiarity but the natural valuations for any presheaf of propositions on any small category. Three supports are offered. (1) Classical physics: the value presheaf Υ(Ā) = Ā(S) on the category M of measurable functions has global sections (one per microstate, eq. 3.7), yet a macrostate R ⊆ S yields a sieve-valued valuation ν^R(A ∈ Δ) = {f_M | B̄(R) ⊆ f(Δ)} (eq. 3.10) with the same FUNC property as the quantum ν^ρ. (2) A proof that a sieve-valued valuation on any presheaf G obeys generalised FUNC, ν(B, G(f)(d)) = f*(ν(A, d)), iff it is a natural transformation G → Ω, i.e. a subobject of G (Theorem 4.1). (3) An informal argument, explicitly “not a genuine deduction” (§5.2), that if partial truth is fixed by which weakenings of a proposition are totally true, the truth value is a sieve. | Read |
| NOTE-333 | Topos Theory and Consistent Histories: The Internal Logic of the Set of All Consistent Sets | Isham takes the poset B of all Boolean subalgebras (“windows”) of the history orthoalgebra UP, ordered by coarse-graining, and shows that the presheaf A_d(W) = {α | some d-consistent W′ ⊇ W contains α} is a subobject of the constant presheaf ΔUP, whereas the naive “realisable in W” assignment is not a presheaf at all (§4.1–4.2). The probability claim ⟨α, p⟩ then gets the sieve-valued semantic value {W ⊆ W₀ | ∃ d-consistent W′ ⊇ W with α ∈ W′} if d(α, α) = p and ∅ otherwise (4.16), a global element 1 → Ω in Set^B, so the logic of “all consistent sets at once” is the distributive but non-Boolean Heyting algebra of sieves on B. | Read |
| NOTE-334 | Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity | Isham & Butterfield argue, without proving anything new, that presheaf toposes over a base of “contexts” suit physics because the subobject classifier Ω is fixed by the base. They say this answers the objection that many-valued logics are arbitrary (§2.2), and that it keeps the logic distributive (§1.1). They then sketch seven applications. Five are speculative one-paragraph proposals (frames, observers, causal sets, TQFT, synthetic-differential-geometry time in continuous histories). One (§3.7) recaps their Kochen–Specker results: KS ⇔ the dual presheaf D(Â) = Hom(W_A, {0,1}) on the operator category O has no global sections (dim H > 2), and every state ψ gives a FUNC-obeying sieve-valued valuation ν^ψ(A ∈ Δ) = {f_O : Â → B̂ | Ê[B ∈ f(Δ)]ψ = ψ}. | Read |
| NOTE-337 | A Framework for Representing Knowledge | The frame memo. A frame is a stereotyped situation-structure whose terminals carry conditions and weakly bound defaults. Frames sharing terminals form systems whose transformations encode moves, actions and shifts of viewpoint, and mismatches are routed through a learned similarity network. Applied to vision, imagery, language, memory, search and control, with an appendix arguing that monotonic, consistency-seeking logic cannot carry commonsense knowledge. A programme statement: nothing is built or measured, and the author says what is missing. | Read |
| NOTE-361 | Jokes and the Logic of the Cognitive Unconscious | Humour as the teaching and engaging of “censors”: learned, unconscious suppressors of thought-paths known to fail (paradox, circling, false analogy, prohibited wishes). Commonsense logic is too unreliable to repair, general methods can only be corrected subtractively, so minds accumulate negative meta-knowledge, and jokes, built on unexpected frame-substitution, are how it is learned. This explains why logical nonsense is funny in the same way as Freud’s forbidden subjects. An essay of conjectures, with no data. | Read |
| NOTE-608 | The Semantics of Definite and Indefinite Noun Phrases | Argues that indefinites (and definites) are variables without quantificational force, bound by the nearest operator, which yields donkey-sentence readings; then recasts semantics as file change: each sentence updates a structured common ground of numbered cards, truth of a file supplies existential force, and the definite/indefinite contrast reduces to one felicity condition (new card versus familiar card), from which binding, presupposition projection and, with accommodation, narrow-scope definites follow. | Read |
| NOTE-617 | Defaults in Update Semantics | Sets out update semantics (meaning as change of information state), proves a dynamic system reduces to static propositions exactly when updates are total, idempotent, persistent, monotone and strengthening, and shows epistemic “might” and “presumably” are tests that fail persistence. Gives a decidable non-monotonic logic of default rules in which specificity and other priorities follow from coherence and applicability conditions, with predictions that differ from Reiter’s, Delgrande’s, Asher and Morreau’s and inheritance-net theories. | Read |
| NOTE-622 | Making It Explicit | In the core of chapter 3, asserting undertakes a commitment, licenses others to reassert it and takes on a responsibility to show entitlement if challenged. Entitlement passes by justification (across contents) and by deferral (between people), and holds by default until challenged. Each interlocutor keeps score of everyone’s commitments and entitlements, so scorekeeping is doubly perspectival. A speech act’s significance is the change it makes to the scores, determined by its content’s inferential role. | Skimmed |
| NOTE-644 | Semantic Unification | Puts basic DRT into sheaf form: a DRS is a section of a presheaf of consistent literals, an anaphoric resolution is a cover by variable maps, and the discourse’s meaning is the gluing, unique if it exists (Proposition 1). All the interpretive work is in choosing the cover; the only obstruction to gluing shown is inconsistency, so the paper models context dependence and ambiguity, not contextuality in the Abramsky–Brandenburger sense. Its probabilistic section ranks covers by summed corpus counts, not by gluing distributions. | Read |
| NOTE-646 | Combining contextuality and causality | Extends sheaf-theoretic contextuality to causal scenarios by an enabling relation on measurements and a game reading: Nature’s deterministic strategies, which may depend on causal history, replace the event sheaf, and the Abramsky–Brandenburger definitions then go through. Recovers flat contextuality and causal Bell scenarios, and expresses adaptive MBQC by Experimenter strategies. The strategy presheaf is not a sheaf: compatible deterministic local strategies can have no gluing or several. | Read |
| NOTE-650 | Closing Bell | Answers which maps between the empirical models of two scenarios are induced by classical, non-adaptive procedures: exactly those given by a non-contextual model of a hom scenario [S, T] whose outcomes are procedures, so deciding it is a linear program. A family of compatible local procedures glues into one global procedure exactly when that model is non-contextual; the rest are “contextual simulations”. The hom construction makes scenarios with predicates a closed category. | Read |
| NOTE-808 | Aerts — Quantum structure in cognition | Hampton’s membership weights for conjunctions and disjunctions of concepts fail a single Kolmogorovian model, item by item, in two ways: over- or underextension (∆-type) and a negative Kolmogorovian factor (k-type). A subspace lattice with meet and join can fit the k-type items, but provably never the ∆-type ones. Those need an emergent concept state in a second sector of Fock space. Negation appears only as the orthocomplement used to reach conjunction from disjunction, and is never measured. | Read |
| NOTE-817 | Sayed Ahmed — Back and forth between algebraic geometry, algebraic logic, sheaves and forcing | Distributive lattices with join-preserving idempotent operators are represented, after Comer, as continuous sections of a sheaf of quotient algebras over the prime spectrum of their fixed-point part. A small site is said to give a notion of forcing, with Kripke–Joyal clauses, and a Łukasiewicz conjunction is forced through a tensor product. Most theorems are stated without proof, and negation is never written out. | Read |
| NOTE-866 | Hansson — Logic of Belief Revision (SEP, revision of 29 July 2026) | The entry separates two ways of representing what an agent believes: a logically closed belief set, and a belief base whose elements are the beliefs “held independently of any other belief”, with everything else “merely derived”. Two bases with the same closure can respond differently to the same input (its Abe and Bob example), so the closure is not a sufficient state for revision; belief-set models recover the difference only by attaching a different selection mechanism to the same set. Bases avoid the Recovery postulate, keep distinct inconsistent states apart, and permit operations (external revision, semi-revision by consolidation) that collapse on closed sets. The entry does not discuss keeping reasons, sources or justification records: its only mention of truth maintenance systems is historical, and a base records which beliefs are basic, not why they are held. Iterated revision is treated as an open area in which most proposals carry an ordering of worlds or an entrenchment relation as the state, not a base. | Read |
| NOTE-874 | Alchourrón, Gärdenfors & Makinson — On the logic of theory change: partial meet contraction and revision | For a logically closed theory A, the contractions satisfying Gärdenfors’s six basic postulates are exactly the partial meet contractions, A ∸ x = ⋂γ(A ⊥ x), where γ picks a nonempty subset of the maximal subsets of A not implying x (Observation 2.5); adding the two supplementary postulates gives exactly the contractions whose γ is generated by a transitive relation over those subsets (Corollary 4.5). Revision is defined from contraction by the Levi identity and inherits the corresponding postulates. The result needs, besides A, a selection function or ordering that A itself does not supply: different selection functions give different contractions of the same theory, and in the finite case distinct selection functions always give distinct contractions (Observation 4.6). The setting is one-step change of a single fixed theory; nothing is said about iterated change, about where the selection function comes from, or about the reasons for which a belief is held. | Read |
| NOTE-894 | Clark — Negation as failure | Clark gives a query evaluator for a data base of clauses (a Horn clause prover in which a ground negated literal ~P is taken as a lemma when the recursive evaluation of P fails on every branch) and proves it sound with respect to the completed data base: the clauses read as the if-halves of if-and-only-if definitions, plus equality axioms saying distinct names denote distinct objects. Theorem 2: if for some selection rule every branch of the evaluation tree of Q terminates with failure, then not-(exists x)Q is a theorem of the completed data base. A branch that does not terminate licenses nothing; the rule requires the search to be exhausted, which the paper calls “the exhaustive but unsuccessful search for a proof of P”. The evaluator is in general incomplete (non-ground negation, a single evaluation tree, no case analysis); it is complete for allowed queries on hierarchical data bases (Theorem 4). | Read |
| NOTE-899 | Reiter — On closed world data bases | Reiter defines the closed-world assumption (CWA) for a function-free first-order database as reasoning from DB together with the negation of every ground atom that DB does not entail, and contrasts it with the open-world assumption (OWA), under which only what DB entails is an answer. He shows that, when DB with these added negations is consistent, closed-world evaluation of any existential query reduces to ordinary (open-world) evaluation of atomic queries combined by union, intersection, difference and projection, so the negations never need to be listed. The CWA can make a consistent database inconsistent (Pa ∨ Pb yields ¬Pa and ¬Pb); consistent Horn databases, and all definite databases, stay consistent. Nothing in the paper concerns how long a search takes: non-entailment is a fixed logical relation, and the CWA’s justification is stated as an assumption of total knowledge of the domain, not proved as a soundness condition. | Read |
| NOTE-907 | Chandra & Merlin — Optimal implementation of conjunctive queries in relational data bases | Defines conjunctive queries (existentially quantified conjunctions of atoms, no equality, constants naming distinct elements) and proves that two are equivalent exactly when there are homomorphisms each way between their natural models (Lemma 13). Every conjunctive query has a minimal equivalent query, unique up to isomorphism, reached by folding (Theorem 12), and that minimal query yields an evaluation program within a constant factor of optimal in the paper’s cost model (Theorem 16). The hardness results are that evaluating a boolean conjunctive query on a database is NP-complete when the query is part of the input (Theorem 7), and that deciding whether one query is a folding of another is NP-complete (Theorem 11). The paper does not state a containment theorem or the NP-completeness of containment or equivalence as such; both are standard corollaries of Lemma 13’s proof and Theorem 11’s construction, and later papers cite them to it. | Read |
| NOTE-909 | Levy, Mendelzon, Sagiv & Srivastava — Answering queries using views | For conjunctive queries and views, a view can be used in an equivalent rewriting of a query exactly when there is a containment mapping (a homomorphism of bodies) from the view into the query, so every possible use of a stored view is found among those mappings. Without comparison predicates, a minimal rewriting needs no more literals than the query and no new variables, which puts the search in NP; deciding whether some rewriting uses a given set of views (with comparisons only in the query), whether a rewriting with fewer than k literals exists, and whether a rewriting using only the views exists are each NP-complete, and with comparisons in both query and views existence is Π2p-complete. The hardness has two independent sources, the number of mappings and the choice of which query literals to drop, and a polynomial algorithm removes the redundant literals for a given mapping, exactly under stated conditions. Only equivalent rewritings are studied; contained rewritings, the cost of the resulting plan and the size of view collections are outside it. | Read |
| NOTE-913 | Green, Karvounarakis & Tannen — Provenance semirings | Annotating each input tuple with an element of a commutative semiring, and computing union and projection with + and join with ·, gives one positive relational algebra (RA+) that is set semantics over the Booleans, bag semantics over the naturals, c-tables over positive Boolean expressions, event tables over sets of outcomes, and lineage over sets of tuple ids. The polynomials N[X] in the tuple ids are the most general such annotation: for any RA+ query and any commutative semiring K, the K-annotated answer is the provenance polynomial evaluated at the input’s K-annotations (Theorem 4.3). For datalog the paper needs ω-continuous semirings and formal power series N∞⟦X⟧, with the analogous factorization (Theorem 6.4) and a criterion for infinite coefficients (Theorem 6.5). Everything is positive: no negation, difference or aggregation, and revaluing inputs can only remove output tuples, never add them; trust and deletion propagation are not treated. | Read |
| NOTE-915 | Gupta, Mumick & Subrahmanian — Maintaining views incrementally | Two algorithms compute the changes to a stored derived relation (a view) from changes to the base relations, without recomputing it. Counting stores, for each derived tuple, the number of its derivations and propagates signed counts through delta rules stratum by stratum; it is stated correct for nonrecursive Datalog or SQL views with stratified negation and aggregation, under set or duplicate semantics, and is argued to cost little beyond evaluating the view. DRed (delete and rederive), for recursive views under set semantics only, deletes an overestimate of what a deletion invalidates, puts back those tuples that still have a derivation from what remains, then adds what insertions derive; it is stated to yield exactly the tuples derivable from the new base. The paper is an extended abstract: the theorems are stated without proof (proofs are referred to a 1992 AT&T report), cost claims are argued, not measured, and there are no experiments. | Read |
| NOTE-920 | Buneman, Khanna & Tan — Why and where: a characterization of data provenance | The paper separates two meanings of the provenance of a piece of output data. Why-provenance is the witness basis: for each valuation of the query that produces the value, the part of the source the query’s patterns matched, one witness per derivation. Where-provenance is the location in the source a value was copied from. For a hierarchical data model and a query language that covers positive (SPJU) relational queries, the witness basis is preserved by the paper’s query rewriting, and the minimal witness basis is the same for equivalent queries with equality conditions, while where-provenance is not invariant over equivalent queries in general and is preserved by rewriting only for a restricted class it calls traceable. Proofs are omitted, and nothing covers negation, aggregation, updates or additions to the source. | Read |
| NOTE-921 | Halevy — Answering queries using views: a survey | A survey that separates two problems of answering queries from stored views: finding an equivalent rewriting, which query optimization needs, and finding a maximally-contained rewriting or all certain answers, which data integration needs when sources are incomplete. Its theoretical findings, mostly reported from other papers: an equivalent conjunctive rewriting exists only if one exists no longer than the query, so finding one is NP-complete in the size of the query and view definitions, and checking a candidate is itself NP-complete; the inverse-rules algorithm builds the maximally-contained rewriting, as a possibly recursive datalog program, in time polynomial in the query and views; and finding all certain answers is polynomial in the size of the stored data in many open-world cases, but co-NP-hard in that size under the closed-world assumption even for conjunctive views and queries, or open-world with union in the views or ≠ in the query. Choosing which views to store is left as the next problem; cost-based choice among rewritings is described for practical optimizers, without guarantees. | Read |
| NOTE-1231 | Clarke, Grumberg, Jha, Lu & Veith — Counterexample-guided abstraction refinement | A model checker runs on an existential abstraction that merges concrete states, so a property true of the abstraction is true of the program (Theorem 1). An abstract counterexample is checked against the concrete system by image computation; if it has no concrete counterpart, the first abstract state where it breaks is split to separate the concrete states reachable along it from those with the transition onward, and checking runs again. The coarsest such split is stated to be NP-hard (proof omitted), so a polynomial, non-optimal split is used. For finite-state programs and ACTL* properties whose counterexamples are paths or loops, the loop ends in a real counterexample or a proof. The property is fixed throughout. | Read |
| NOTE-1348 | Goguen & Burstall — Institutions | An institution is a category of signatures with a sentence functor, a contravariant model functor and a satisfaction relation per signature, subject to one axiom, the Satisfaction Condition: a translated sentence holds in a model exactly when the original holds in the model’s reduct. From that axiom alone theories and model classes form a Galois connection, theories can be glued by colimits whenever signatures can (Theorem 11), constraints that force standard or free interpretations can be added as sentences without losing this, and an institution morphism with surjective model translation lets a prover for one logic decide entailment for another (Proposition 35). The Satisfaction Condition is a criterion of correct translation relative to a given model translation; it says nothing about where that model translation comes from. | Read |
| NOTE-1381 | Halpern and Moses — Knowledge and common knowledge in a distributed environment | Defines distributed, someone-, everyone- and common knowledge for processors in a system of runs, and shows that a guaranteed coordinated attack makes the attack common knowledge (Proposition 4), while over a channel that may lose any message nothing becomes common knowledge that would not be common knowledge with no messages delivered (Theorem 5), so no correct protocol ever attacks (Corollary 6). Temporal imprecision alone, uncertain start times and delivery times, makes common knowledge unattainable (Theorem 8). Weaker fixed-point variants (ε-, eventual and timestamped common knowledge) are attainable in many systems. | Read |