README⤓ .txt 2026-10-22T165042.936 000000000000784 The logics complete not for a single variety but for a family of them: the matrices with more than two truth values, where consequence is preservation of designated values. The algebras range over MV, BL, and MTL algebras, De Morgan and Kleene lattices, and bilattices.
Abelian Logic⤓ .md 2026-10-22T224914.000 000000000070976 Robert K. Meyer and John K. Slaney, "Abelian Logic (from A to Z)" (in Graham Priest, Richard Routley, and Jean Norman, eds., Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, Munich 1989, 245–288), with a sequel, "A, Still Adorable" (in Carnielli, Coniglio, and D'Ottaviano, eds., Paraconsistency: The Logical Way to the Inconsistent, 2002). Ettore Casari arrived at the same logic independently, as a logic of comparison formalizing the majority, minority, and equality constructions of natural language, ideas he traced to Aristotle: "Comparative Logics and Abelian ℓ-groups" (in Ferro, Bonotto, Valentini, and Zanardo, eds., Logic Colloquium '88, North-Holland 1989, 161–190). George Metcalfe, Nicola Olivetti, and Dov Gabbay, "Sequent and Hypersequent Calculi for Abelian and Łukasiewicz Logics" (ACM Transactions on Computational Logic 6(3), 2005, 578–613), supplied the analytic proof theory.
Basic Fuzzy Logic⤓ .md 2026-08-20T061232.000 000000000040936 Hájek (1998). The logic of all continuous t-norms, complete for BL-algebras. It unifies Łukasiewicz, Gödel, and Product logic as axiomatic extensions and founds mathematical fuzzy logic.
Belnap-Dunn Logic⤓ .md 2026-10-22T085014.000 000000000050168 Anderson, Belnap, and Dunn developed FDE as the first-degree fragment of relevance logic (Dunn 1976; Belnap 1977, "A useful four-valued logic" and "How a computer should think"). Four values—true, false, both, neither—let a reasoner handle information that is incomplete or contradictory without explosion. It is the propositional core shared by Belnap's and Belnap–Dunn's four-valued systems.
Bilattice Logic⤓ .md 2026-10-22T180154.561 000000000000848 Matthew L. Ginsberg, "Multivalued logics: a uniform approach to reasoning in artificial intelligence" (Computational Intelligence 4, 1988; conference version AAAI 1986), introduced bilattices as a single truth-value framework for first-order deduction, default logic, prioritized defaults, and assumption-based truth maintenance. Melvin Fitting made them a logic: "Bilattices and the theory of truth" (Journal of Philosophical Logic 18, 1989), "Kleene's logic, generalized" (Journal of Logic and Computation 1, 1991), and "Bilattices and the semantics of logic programming" (Journal of Logic Programming 11, 1991) located Kleene's three-valued and Belnap's four-valued logics inside bilattices. His logic-programming semantics on the knowledge order extended his "A Kripke-Kleene semantics for general logic programs" (Journal of Logic Programming 2, 1985). Ofer Arieli and Arnon Avron, "Reasoning with logical bilattices" (Journal of Logic, Language and Information 5, 1996; preliminary version LICS 1994), defined logical bilattices and gave sound and complete proof systems for their consequence relation, developed further in "The value of the four values" (Artificial Intelligence 102, 1998).
Connexive Logic⤓ .md 2026-10-22T084904.000 000000000037328 Historical roots in Aristotle's and Boethius's theses. Modern revival by McCall (1960s), Wansing (1990s-present). Non-classical logic where negation and implication interact. ¬(A → ¬A) and (A → B) → ¬(A → ¬B) are valid. Counter to material implication paradoxes.
Continuous Logic⤓ .md 2026-08-20T062834.000 000000000041232 Ben Yaacov, Berenstein, Henson, and Usvyatsov developed continuous logic (2000s). Model theory for metric structures. Truth values in [0,1] rather than {0,1}. Generalizes classical model theory to Banach spaces, probability algebras, etc. Part of continuous model theory program.
Da Costa Paraconsistent⤓ .md 2026-08-20T063134.000 000000000041144 Newton da Costa (1963). Brazilian school. Hierarchy of paraconsistent logics. Controlled contradiction. Foundation of South American paraconsistency.
Discussive Logic⤓ .md 2026-10-22T180154.561 000000000000792 Stanisław Jaśkowski, "Propositional calculus for contradictory deductive systems" (Studia Societatis Scientiarum Torunensis, 1948; English 1969) — the first paraconsistent logic, published the year Priest was born and fifteen years before da Costa. Jaśkowski was answering Łukasiewicz's 1910 challenge to construct a system in which the law of non-contradiction fails without triviality. A second paper (1949) added the discussive conjunction, which the first had done without. Kotas (1974) gave the first axiomatization; da Costa and Dubikajtis (1977) worked it into Jaśkowski's own discussive language.
Dual-Intuitionistic Logic⤓ .md 2026-10-22T090631.000 000000000044280 C. Rauszer, "A formalization of the propositional calculus of H-B logic" (1974) and the series through 1980, which gave Heyting–Brouwer logic with both implication and co-implication; the dual fragment alone is Goodman's (1981) and Urbas's (1996). McKinsey and Tarski's closure algebras (1946) are the algebraic ancestor. Crolard (2001) gave the type-theoretic reading; Pinto and Uustalu (2009) the sequent calculus.
First-Degree Entailment⤓ .md 2026-10-22T085014.000 000000000050168 Anderson, Belnap, and Dunn developed FDE as the first-degree fragment of relevance logic (Dunn 1976; Belnap 1977, "A useful four-valued logic" and "How a computer should think"). Four values—true, false, both, neither—let a reasoner handle information that is incomplete or contradictory without explosion. It is the propositional core shared by Belnap's and Belnap–Dunn's four-valued systems.
Full Lambek Calculus⤓ .md 2026-10-22T180154.561 000000000000888 Lambek's syntactic calculus (1958) supplies the residuated core; the additives and constants were added and the family systematized by Ono and Komori (1985) and Ono (1993, 2003). Galatos, Jipsen, Kowalski, and Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics (2007), is the standard reference and the reason FL is the base of the field rather than one system in it.
Fuzzy Logic⤓ .md 2026-08-20T065009.000 000000000044248 Lotfi Zadeh introduced fuzzy sets (1965) and fuzzy logic for reasoning under vagueness. Petr Hájek provided mathematical foundations (Metamathematics of Fuzzy Logic, 1998). Practical applications in control systems, AI, and decision making. Distinct from probability: fuzziness is about vague boundaries, not uncertain events.
Godel Logic⤓ .md 2026-08-20T070143.000 000000000039512 Gödel introduced Gödel logics (1932) studying intuitionistic logic. Infinite-valued with min/max operations. Also called Gödel-Dummett logic. Characterized by linearity axiom: (A→B)∨(B→A). Intermediate between intuitionistic and classical.
Inconsistent Mathematics⤓ .md 2026-10-22T090631.000 000000000049440 Robert K. Meyer's relevant arithmetic R# (1976) and his conjecture that it proves its own non-triviality; Ross Brady's non-triviality proof for naive set theory (1971, 1989); Chris Mortensen's Inconsistent Mathematics (1995), which named the programme; Graham Priest, "Inconsistent models of arithmetic" (1997); Zach Weber's Paradoxes and Inconsistent Mathematics (2021) for the current state.
Jaina Saptabhangi⤓ .md 2026-10-22T180154.561 000000000000864 Jaina logicians of classical India. The sevenfold predication (saptabhaṅgī) under the qualifier syāt, the logical arm of anekāntavāda (non-one-sidedness) and of the naya theory of standpoints. First full exposition: Samantabhadra's Āptamīmāṃsā (also Devāgamastotra), 114 verses — traditionally dated to the 2nd century CE, placed in the 6th by most modern scholarship, with proposals from the 4th to the 8th; commented by Akalaṅka's Aṣṭaśatī. Mature treatments in Prabhācandra and in Vādideva Sūri (1086–1169), whose Pramāṇanayatattvālokālaṃkāra is the text the modern formalizations anchor on; Mallisena's Syādvādamañjarī (1292) is the late dedicated treatise. Modern formalizations: Jonardon Ganeri, "Jaina Logic and the Philosophical Basis of Pluralism" (History and Philosophy of Logic 2002); Graham Priest, "Jaina Logic: A Contemporary Perspective" (History and Philosophy of Logic 2008); Fabien Schang, "A One-Valued Logic for Non-One-Sidedness" (International Journal of Jaina Studies 2013).
Logic of Paradox⤓ .md 2026-08-20T062447.000 000000000042024 Priest (1979). Three-valued paraconsistent logic underwriting dialetheism—the view that some contradictions are true. Same tables as Strong Kleene/K3, but with the glut value designated.
Logical Matrix⤓ .md 2026-10-22T085300.000 000000000046952 Łukasiewicz and Tarski, "Investigations into the sentential calculus" (1930), where the matrix method is set out; Jerzy Łoś and Roman Suszko, "Remarks on sentential logics" (1958), which gave the general theory and the notion of a structural consequence relation. Wójcicki's Theory of Logical Calculi (1988) is the reference; Font and Jansana's abstract algebraic logic is where it now lives.
Logics of Formal Inconsistency⤓ .md 2026-10-22T180154.561 000000000000968 Walter Carnielli and João Marcos, "A Taxonomy of C-systems", in Paraconsistency: The Logical Way to the Inconsistent, the proceedings of the Second World Congress on Paraconsistency (Marcel Dekker, 2002), defined the class and mapped it; Carnielli, Marcelo Coniglio and Marcos, "Logics of Formal Inconsistency" (Handbook of Philosophical Logic, 2nd edition, volume 14, Springer, 2007, pp. 1–93), is the standard treatment, and Carnielli and Coniglio's Paraconsistent Logic: Consistency, Contradiction and Negation (Springer, 2016) the book-length one. The idea descends from Newton da Costa's Cₙ calculi (1963), where consistency enters as the defined formula ¬(A ∧ ¬A); the LFIs take the operator as primitive and make da Costa's hierarchy one corner of a far larger family.
Lukasiewicz Logic⤓ .md 2026-08-20T061121.000 000000000037016 Jan Łukasiewicz introduced three-valued logic (1920), then infinitely-valued (1930). Foundation for fuzzy logic. MV-algebras as algebraic semantics (Chang, 1958). One of the main many-valued logics. Motivated by future contingents.
Many-Valued Logic⤓ .md 2026-10-22T165042.936 000000000000864 Jan Łukasiewicz introduced three-valued logic (1920) for future contingents ("there will be a sea battle tomorrow"). Emil Post explored n-valued and infinite-valued logics (1921). Stephen Kleene developed three-valued logic for partial recursive functions (1938). Lotfi Zadeh introduced fuzzy logic (1965). Multiple traditions with different motivations.
Monoidal T-norm Logic⤓ .md 2026-08-20T061417.000 000000000035720 Esteva and Godo (2001). Logic of left-continuous t-norms. Weaker than BL (basic logic). Subsumed by Łukasiewicz, Gödel, Product. Foundation for fuzzy logic hierarchy.
Nelson Logic⤓ .md 2026-10-22T165042.936 000000000000792 David Nelson (1949). Constructive logic with strong negation. Two negations: ¬ (weak) and ~ (strong). N3 and N4 variants. Foundation for constructive falsity.
Nilpotent Minimum Logic⤓ .md 2026-08-20T061504.000 000000000035936 Esteva and Godo (2001). Fuzzy logic with nilpotent minimum t-norm. Left-continuous but not continuous. Strict negation properties. Foundation for certain fuzzy reasoning.
Paraconsistent Logic⤓ .md 2026-08-20T064854.000 000000000043352 Stanisław Jaśkowski (1948) and Newton da Costa (1963) independently developed logics tolerating contradiction. Motivated by: inconsistent but useful theories (early calculus, naive set theory), real-world databases with conflicts, dialethism (some contradictions are true). Graham Priest developed dialetheism and Logic of Paradox (LP).
Partial Logic⤓ .md 2026-08-20T065531.000 000000000053904 Kleene's partial recursive functions and three-valued tables (1938, 1952) gave the computational motive: a predicate defined by an unbounded search has no value where the search does not halt. Bas van Fraassen, "Singular terms, truth-value gaps, and free logic" (1966), gave the semantic motive and the supervaluational treatment of empty descriptions. Dana Scott's domain theory supplied the information ordering with ⊥ at the bottom. Blamey's survey "Partial logic" (Handbook of Philosophical Logic, 1986) named and organized the field; Langholm, Partiality, Truth and Persistence (1988), and Muskens (1995) for the type-theoretic version, are the standard developments.
Post Logic⤓ .md 2026-10-22T090153.000 000000000032960 Emil Post (1921). First systematic many-valued logics. n-valued truth tables. Generalized negation cycles. Functional completeness results. Foundation for multi-valued computation.
Product Logic⤓ .md 2026-10-22T085501.000 000000000035256 Hájek, Godo, Esteva developed Product Logic (1990s). Many-valued logic with multiplication as conjunction. One of three basic fuzzy logics (with Łukasiewicz, Gödel). Algebraically: product algebras. Part of monoidal t-norm logic family.
Rational Pavelka Logic⤓ .md 2026-10-22T090154.000 000000000036416 Pavelka (1979), Hájek (1998). Graded provability. Truth constants for rationals. Complete degree calculus. Foundation for graded deduction.
RM3⤓ .md 2026-08-20T065209.000 000000000045592 The three-element Sugihara matrix, from Sugihara's work on mingle (1955); the logic R-mingle (RM) is Anderson and Belnap's (Entailment I, 1975, §29), and RM3 is its three-valued characteristic matrix, due to Dunn (1970), who proved RM's completeness for the Sugihara algebras and RM3's role among them.
Strict-Tolerant Logic⤓ .md 2026-10-22T165042.936 000000000000896 Pablo Cobreros, Paul Égré, David Ripley, and Robert van Rooij, "Tolerant, classical, strict" (Journal of Philosophical Logic, 2012), introduced for vagueness; Ripley, "Paradoxes and failures of cut" (2013) and "Conservatively extending classical logic with transparent truth" (2012), turned it on the semantic paradoxes. The claim that made it notorious: ST is classical logic, and it has a transparent truth predicate.
Strong Kleene Logic⤓ .md 2026-10-22T165042.936 000000000000880 Kleene's strong three-valued logic (1938, 1952). Third value: unknown/undefined. Some operations short-circuit: F∧U=F, T∨U=T. Foundation for logic programming semantics. Connection to fixed-point semantics and partial predicates.
Trilattice Logics⤓ .md 2026-10-22T224935.000 000000000067456 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).
Vasiliev Imaginary Logic⤓ .md 2026-08-20T062209.000 000000000048928 Nikolai Aleksandrovich Vasiliev (1880–1940) taught at Kazan University, where Lobachevsky had built non-Euclidean geometry, and took that construction as his explicit model: as geometry had been freed from the parallel postulate, logic was to be freed from the law of non-contradiction. Three papers carry the system — "On particular judgments, the triangle of opposition, and the law of the excluded fourth" (1910), "Imaginary (non-Aristotelian) logic" (1912), and "Logic and metalogic" (1913). They predate Łukasiewicz's three-valued tables (1920) and Post (1921) and were arrived at independently. Written in Russian and unnoticed abroad, the work was recovered by V. A. Smirnov in the 1960s and brought into the paraconsistent literature by Ayda Arruda (1977) and later by Puga and da Costa (1988). Bazhanov's studies supply the biography and the textual history.
Weak Kleene Logic⤓ .md 2026-10-22T085716.000 000000000042904 Kleene's weak three-valued logic (1952) coincides with Bochvar's B₃ (1938): a third value that infects every connective, modelling meaninglessness or nonsense. Contrast Strong Kleene, where some operations ignore the undefined value.
CRITERIA⤓ .txt 2026-10-22T165042.936 000000000000800 Not sufficient: Two-valued semantics (Boolean). Completeness for a Heyting variety (Heyting). Failure of distribution alone (Orthomodular). A supertruth predicate defined by quantifying over classical precisifications, whose connectives are not truth-functional and which therefore has no matrix—that is a modal construction over a bivalent base and belongs in Modal. A degree read as probability rather than as truth (Applications/Probabilistic).