Evidence Logic

Origin. Johan van Benthem and Eric Pacuit, "Dynamic Logics of Evidence-Based Beliefs" (Studia Logica 99(1–3), 2011, 61–92), interpreted neighborhood models as records of evidence and derived belief from them. Van Benthem, Fernández-Duque and Pacuit, "Evidence and plausibility in neighborhood structures" (Annals of Pure and Applied Logic 165(1), 2014, 106–133; conference version in Advances in Modal Logic 9, 2011), supplied the static axiomatization the 2011 paper left open. Baltag, Bezhanishvili, Özgün and Smets, "Justified Belief and the Topology of Evidence" (WoLLIC 2016, LNCS 9803, 83–103), with the journal version "Justified belief, knowledge, and the topology of evidence" (Synthese 200, article 512, 2022), made the models topological; Özgün's dissertation Evidence in Epistemic Logic: A Topological Perspective (ILLC Dissertation Series, 2017) develops the program.

Models. Epistemic logic with evidence as the primitive. An evidence model assigns each world a family of sets of worlds — the propositions the agent's evidence directly supports — and the family is deliberately left unclosed: not under supersets, since the paper rejects monotonicity as a property of evidence itself, and not under intersections, since two pieces of evidence may be jointly inconsistent while each remains usable. The modality □φ, "the agent has evidence for φ", is then a monotone neighborhood modality (see Neighborhood Semantics), and belief is not primitive but computed: the agent believes what is supported by every maximal consistent way of combining the evidence. That inverts the order of Plausibility Logic, where belief is read off a primitive ordering of worlds — here the combinatorics of evidence does the ordering's work, and the 2014 paper connects the two model classes. The topological turn reads the same family as the subbasis of a topology: combined evidence is the generated basis, having combined factive evidence for φ is being in the interior of its truth set, and belief and knowledge become density conditions — believed is what some dense open set supports, known at a world is what some dense open neighborhood of it supports.

Formalism.

Evidence model (van Benthem–Pacuit 2011): M = (W, E, V) with E ⊆ W × ℘(W); E(w) = {X : wEX}. Cons: ∅ ∉ E(w). Triv: W ∈ E(w). No closure under supersets or intersections is imposed. The 2011 paper works with finite models, mostly uniform ones (E constant across worlds).

Evidence modality: M, w ⊨ □φ iff there is X ∈ E(w) with X ⊆ ‖φ‖. Monotone: from a theorem φ → ψ infer □φ → □ψ. Aggregation fails: (□φ ∧ □ψ) → □(φ ∧ ψ) is not valid. A is the universal modality: Aφ iff φ holds at every world.

Derived belief: A body of evidence: a subfamily of E(w) every finite subfamily of which has nonempty intersection — on the 2011 paper's finite models, a family X with ⋂X ≠ ∅. M, w ⊨ Bφ iff ⋂X ⊆ ‖φ‖ for every maximal such family X ⊆ E(w).

Evidence dynamics (2011): [!φ] public announcement: restrict W to ‖φ‖ and each evidence set to its nonempty intersection with ‖φ‖. [+φ] addition: E(w) becomes E(w) ∪ {‖φ‖}; precondition Eφ (φ holds somewhere). [−φ] removal: delete every X ⊆ ‖φ‖; precondition ¬Aφ. [⇑φ] upgrade: E(w) becomes {X ∪ ‖φ‖ : X ∈ E(w)} ∪ {‖φ‖}. [#] combination: close E(w) under nonempty intersection — stated validities only, no completeness.

Topological evidence models (Baltag–Bezhanishvili–Özgün–Smets): (X, E₀, τ, V): basic evidence E₀ with X ∈ E₀ and ∅ ∉ E₀; τ the topology with E₀ as subbasis. Combined evidence: the nonempty finite intersections of basic evidence — a basis of τ. □P = Int(P): factive combined evidence at the point. A justification for P: a dense open subset of P. BP iff some dense open U ⊆ P — equivalently, Int(P) is dense. KP at x iff P includes a dense open neighborhood of x — equivalently □P ∧ BP: factively justified belief. [∀]: the global modality, infallible knowledge.

Symbols.

Metatheory. The 2011 paper's Fact 3.1 locates the three modalities: A satisfies S5, derived belief satisfies KD on the paper's finite models, and □ satisfies only the minimal monotone modal logic, since aggregation fails by design; on uniform models every formula of the base language reduces to modal depth one. The dynamic operators are axiomatized by recursion axioms relative to the static base — public announcement (Theorem 4.2), addition (Theorem 4.7, requiring conditional-evidence operators in the static language), removal (Theorem 4.15) — and the full static logic of the neighborhood models is axiomatized in van Benthem, Fernández-Duque and Pacuit (2014). On topological evidence models the Synthese paper proves: the knowledge fragment is exactly S4.2 (their Theorem 1), the belief fragment is exactly KD45 and has the finite model property (Theorem 3), Stalnaker's combined knowledge–belief system is sound and complete, with Bφ → BKφ holding (Theorem 4), and the logic of factive basic and combined evidence with the global modality is strongly complete and decidable with the strong finite model property (Theorem 5). The comparison between the two belief definitions is itself a theorem of the topological paper: on the infinite model whose evidence sets are the tails [n, ∞) of ℕ, the single maximal body has empty intersection and van Benthem–Pacuit belief proves ⊥, while dense-open belief is consistent on every model, no finite body having empty intersection; the two coincide on maximally compact models, where every body of evidence is equivalent to a finite one, which covers every model with finite E₀. The dense-open reading obtains KD45 on all topological evidence models, where the closure-of-interior reading of belief requires extremally disconnected spaces (see Topological Epistemology); and the van Benthem–Pacuit doxastic logic lacks the finite model property (van Benthem, Fernández-Duque and Pacuit 2014).

Applies to. Formal epistemology: the gap between justified and merely true belief, and knowledge resistant to misleading evidence. Belief revision — the dynamics act below AGM, changing the evidence family from which belief is computed rather than the belief set itself, so revision behaviour is derived where AGM postulates it (see AGM Belief Revision). Neighborhood semantics, as its standing epistemic interpretation. Topological epistemology, where the evidential topology supplies the spaces. Multi-agent extensions of evidence-based belief in dynamic epistemic logic.

Limitations. The 2011 belief definition does not survive infinite models: maximal bodies of evidence can have empty intersection, making belief inconsistent — the 2011 paper works on finite models, where belief is KD, and the topological reformulation's density condition is consistent on every model; the van Benthem–Pacuit doxastic logic also lacks the finite model property. The evidence modality is weak by construction — aggregation fails, so nothing follows from □φ and □ψ jointly — and the strengthened combined-evidence modality restores intersection only piece by finite piece. Logical omniscience is untouched: evidence sets are propositions, so logically equivalent contents are the same evidence, and derived belief is a normal modality closed under consequence — the coarseness is inherited from neighborhood semantics itself (see Neighborhood Semantics). In the uniform and topological models that carry the completeness theorems, the evidence family is the same at every world, so belief holds everywhere or nowhere and world-dependent belief lies outside the analyzed fragment. Evidence carries no provenance, weight or reliability — all pieces count equally, and what Plausibility Logic grades by an ordering must here be encoded in the family's combinatorics. And the dynamics are regimented: removal requires ¬Aφ, addition requires φ to hold somewhere, and evidence combination is left without a completeness theorem.

© 2026 Lingenic LLC