Trilattice Logics

Origin. Yaroslav Shramko and Heinrich Wansing, "Some Useful 16-Valued Logics: How a Computer Network Should Think" (Journal of Philosophical Logic 34(2), 2005, 121–153), the title answering Belnap's "How a Computer Should Think" (1977). Essentially the same structure appeared in Yaroslav Shramko, J. Michael Dunn, and Tatsutoshi Takenaka, "The Trilattice of Constructive Truth Values" (Journal of Logic and Computation 11, 2001, 761–788), read there as a space of constructive truth values. The generalization to networks of any depth is Shramko and Wansing, "Hypercontradictions, Generalized Truth Values, and Logics of Truth and Falsehood" (Journal of Logic, Language and Information 15, 2006, 403–424). Axiomatization: Sergei Odintsov, "On Axiomatizing Shramko-Wansing's Logic" (Studia Logica 91(3), 2009, 407–428), completed in Odintsov and Wansing, "The Logic of Generalized Truth Values and the Logic of Bilattices" (Studia Logica 103(1), 2015, 91–112). The book-length treatment is Shramko and Wansing, Truth and Falsehood: An Inquiry into Generalized Logical Values (Trends in Logic 36, Springer, 2011).

Models. The power-set construction behind Belnap's four values, iterated once. Belnap's values are already the subsets of {F, T} — what a computer has been told; a server collecting reports from several Belnap-valued computers holds a subset of Belnap's four, so its states are the sixteen members of 𝒫(4): told true-only by one source and told both by another is the value {{T}, {F, T}}, and reducing it to any of Belnap's four discards information. The structural novelty is what happens to the orders. In the bilattice FOUR₂ the logical order is a truth-and-falsity order — a rise in truth is a fall in falsity — while on the sixteen values truth and falsity come apart into two independent logical orders beside the information order, giving a trilattice where the Bilattice Logic entry's structure has two orders. Truth and falsity become separately orderable dimensions of a generalized truth value, which is the semantic thesis of the generalized-truth-value program.

Formalism.

Multilattice: An n-lattice is ⟨S, ≤₁, …, ≤ₙ⟩ with each ⟨S, ≤ᵢ⟩ a lattice and the lattices pairwise distinct. Bilattices are the n = 2 case; SIXTEEN₃ is the standard n = 3 case.

Values: The sixteen subsets of 4 = {N, F, T, B}, Belnap's values being themselves the four subsets of {F, T}. Shramko–Wansing names: 𝐍 = ∅, then N = {∅}, F = {{F}}, T = {{T}}, B = {{F, T}}, the ten intermediate combinations (NF, NT, NB, FT, FB, TB, NFT, NFB, NTB, FTB), and A = {∅, {F}, {T}, {F, T}}.

The three orders: ≤_i: set inclusion — the information order. ≤_t: defined by the presence and absence of the classical value T in the members of a generalized value — the truth order. ≤_f: likewise for F — the falsity order. SIXTEEN₃ = ⟨16, ≤_i, ≤_t, ≤_f⟩.

Connectives: Each logical order carries meet, join, and order inversion: ∧_t, ∨_t, ¬_t from ≤_t and ∧_f, ∨_f, ¬_f from ≤_f. Two parallel vocabularies, one for truth and one for falsity.

Entailment: ⊨_t and ⊨_f are order-theoretic, not designation-based: A ⊨_t B iff v(A) ≤_t v(B) under every valuation, and likewise for ≤_f.

The coincidence theorem: In the pure truth vocabulary, ⊨_t is first-degree entailment. In the pure falsity vocabulary, ⊨_f is first-degree entailment. (Shramko and Wansing 2005.)

All the way up: A server of servers runs on 𝒫(16), 65536 values, and so on; the logic of every Belnap trilattice, in either vocabulary, is still FDE (Shramko and Wansing 2006; Beall and Camrud, "A Note on FDE 'All the Way Up'", Notre Dame Journal of Formal Logic 61, 2020, 283–296).

Symbols.

Metatheory. Shramko and Wansing (2005) proved the coincidence theorem: the truth-order consequence in the truth vocabulary and the falsity-order consequence in the falsity vocabulary are each exactly first-degree entailment, and Shramko and Wansing (2006) extended this to Belnap trilattices of every degree — the value space grows without the logic moving (see also Beall and Camrud 2020). Axiomatizing the two consequence relations over the full language, with both vocabularies present, stayed open for several years. Odintsov (2009) gave the first solution, for languages extended with a truth implication and a falsity implication defined as the residua of the respective orders; the implication makes a tautology — a formula valued at the greatest element of its order under every valuation — express entailment, and the consequence relations reduce to their tautology sets. Odintsov and Wansing (2015) closed the problem by showing that the logic of SIXTEEN₃ is the logic of commutative distributive bilattices, and Shramko (2016, "Truth, Falsehood, Information and Beyond: The American Plan Generalized", in Bimbó, ed., J. Michael Dunn on Information Based Logics, Springer, 191–212) carried the result to multilattices generally. Proof theory exists in sequent form: Kamide and Wansing, "Sequent Calculi for Some Trilattice Logics" (Review of Symbolic Logic 2, 2009, 374–395), and Wansing, "The Power of Belnap: Sequent Systems for SIXTEEN₃" (Journal of Philosophical Logic 39(4), 2010, 369–393). Omori and Skurt (2019) gave two alternative semantics combining Routley's and Dunn's constructions. The consequence relations are decidable by evaluation over the sixteen-element structure.

Applies to. Reasoning in networks whose nodes are themselves Belnap-style reasoners — information integration where reports are about reports. The generalized-truth-value program: truth and falsity as independent dimensions of a semantic value. Constructive truth values, the Shramko–Dunn–Takenaka reading. Bilattice theory, through the Odintsov–Wansing identification of the trilattice logic with a bilattice class. The theory of multilattices as algebraic semantics.

Limitations. The coincidence theorem is the principal one: in each pure vocabulary the sixteen values validate exactly FDE, so the ascent from four values to sixteen refines the semantics without adding a single pure-vocabulary inference, and everything distinctive lives in the mixed language. The pure vocabularies, being FDE, have no conditional (see First-Degree Entailment), and Odintsov's axiomatization adds the residua of the orders before entailment becomes expressible inside the language. The Odintsov–Wansing theorem bounds what the third order adds: the logic of SIXTEEN₃ is the logic of commutative distributive bilattices, so as a consequence relation, rather than as a semantics, the trilattice yields nothing a bilattice class does not already determine. The values inherit FDE's silence about provenance — they record what the server was told, not by whom or how often, as the First-Degree Entailment entry records for the four-valued case. And the network reading of the higher values has been contested: the reply to Didier Dubois's criticism is Wansing and Belnap, "Generalized Truth Values: A Reply to Dubois" (Logic Journal of the IGPL 18(6), 2010, 921–935).

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