README⤓ .txt 2026-07-17T121634.146 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.
Basic Fuzzy Logic⤓ .md 2026-07-15T203122.000 000000000022312 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.
Connexive Logic⤓ .md 2026-07-15T211228.000 000000000018848 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 2026-07-15T060051.000 000000000020952 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 2026-07-17T120407.600 000000000000912 Newton da Costa (1963). Brazilian school. Hierarchy of paraconsistent logics. Controlled contradiction. Foundation of South American paraconsistency.
Discussive Logic⤓ .md 2026-07-16T004541.000 000000000034608 Stanisław Jaśkowski, "Propositional calculus for contradictory deductive systems" (Studia Societatis Scientiarum Torunensis, 1948; English 1969) — the first paraconsistent logic, four years before Priest was born and fifteen before da Costa. Jaśkowski was answering Łukasiewicz's 1910 challenge to construct a system in which the law of non-contradiction fails without triviality. Da Costa and Dubikajtis (1977) axiomatized it.
Dual-Intuitionistic Logic⤓ .md 2026-07-16T004541.000 000000000034544 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 2026-07-15T203033.000 000000000029584 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 2026-07-15T235854.000 000000000035992 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 2026-07-15T055108.000 000000000026192 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 2026-07-15T061745.000 000000000018008 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 2026-07-16T001414.000 000000000036608 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.
Logic of Paradox⤓ .md 2026-07-15T203105.000 000000000020400 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 2026-07-17T120407.600 000000000000856 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.
Lukasiewicz Logic⤓ .md 2026-07-15T061333.000 000000000019096 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 2026-07-15T053335.000 000000000026280 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 2026-07-15T070812.000 000000000013808 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 2026-07-15T070816.000 000000000013648 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 2026-07-15T065322.000 000000000014456 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 2026-07-15T053337.000 000000000026792 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 2026-07-15T235628.000 000000000015776 Kleene's partial recursive functions and three-valued tables (1938, 1952); Blamey's survey "Partial logic" (Handbook of Philosophical Logic, 1986); Langholm, Partiality, Truth and Persistence (1988); Muskens (1995) for the type-theoretic version. Partial functions. Undefined terms. Gaps. Foundation of partiality semantics.
Post Logic⤓ .md 2026-07-15T064410.000 000000000014400 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 2026-07-15T061747.000 000000000017456 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 2026-07-15T073802.000 000000000013808 Pavelka (1979), Hájek (1998). Graded provability. Truth constants for rationals. Complete degree calculus. Foundation for graded deduction.
RM3⤓ .md 2026-07-16T004722.000 000000000030744 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 2026-07-15T233755.000 000000000035520 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 2026-07-15T061934.000 000000000018000 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.
Vasiliev Imaginary Logic⤓ .md 2026-07-15T204207.000 000000000015672 Nikolai Vasiliev (1910). Russian logician. Non-Aristotelian logic. Restriction of the law of non-contradiction (excluded middle is retained, extended to the law of the excluded fourth). Foundation of paraconsistent thought.
Weak Kleene Logic⤓ .md 2026-07-15T203049.000 000000000023848 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 2026-07-17T121634.146 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).