Extensions of the four-valued Belnap–Dunn logic Adam Pˇ renosil Institute of Computer Science, Czech Academy of Sciences Department of Logic, Faculty of Arts, Charles University PhDs in Logic IX Bochum, 4 May 2017 1 / 25
Introduction Classical logic is simple, elegant, well-suited for mathematics . . . . . . but not for reasoning with contradictory information. 2 / 25
Introduction Classical logic is simple, elegant, well-suited for mathematics . . . . . . but not for reasoning with contradictory information. Various non-classical logics have been proposed for this purpose. One of the best known is the four-valued Belnap–Dunn logic. 2 / 25
Introduction Classical logic is simple, elegant, well-suited for mathematics . . . . . . but not for reasoning with contradictory information. Various non-classical logics have been proposed for this purpose. One of the best known is the four-valued Belnap–Dunn logic. This logic has attracted a good deal of attention from logicians. Much less attention was paid to its extensions, super-Belnap logics. 2 / 25
Introduction Classical logic is simple, elegant, well-suited for mathematics . . . . . . but not for reasoning with contradictory information. Various non-classical logics have been proposed for this purpose. One of the best known is the four-valued Belnap–Dunn logic. This logic has attracted a good deal of attention from logicians. Much less attention was paid to its extensions, super-Belnap logics. In this talk we will give a very brief introduction to this family of logics. 2 / 25
Preliminaries: logics A rule is a pair Γ ⊢ ϕ of a set of formulas and a formula. A matrix is a algebra A with a filter F ⊆ A . A rule is valid (holds) in a matrix if for each valuation v : Fm → A : if v [Γ] ⊆ F , then v ( ϕ ) ∈ F This yields a Galois connection between syntax and semantics: K �→ Log K = { Γ ⊢ ϕ | Γ ⊢ ϕ holds in each matrix in K } R �→ Mod R = {� A , F � | each rule in R holds in � A , F �} The Galois closed sets of rules are logics. The Galois closed classes of matrices are equality-free quasivarieties. 3 / 25
Preliminaries: logics These notions admit the following intrinsic characterizations. Theorem (cf. Birkhoff’s completeness theorem for equational logic) Logics are precisely the sets of rules which satisfy: ϕ ⊢ ϕ (reflexivity) if Γ ⊢ ϕ , then Γ , ∆ ⊢ ϕ (monotonicity) if Γ ⊢ ϕ , then σ [Γ] ⊢ σϕ (structurality) if Γ ⊢ π for each π ∈ Π and Π , ∆ ⊢ ϕ , then Γ , ∆ ⊢ ϕ (cut) Theorem (cf. Birkhoff’s HSP theorem) Equality-free quasivarieties are precisely the classes of matrices which reflect Leibniz reductions and are closed under Leibniz reductions, submatrices, and reduced products of matrices. 4 / 25
Preliminaries: extensions of logics A logic L is an extension of B if Γ ⊢ B ϕ ⇒ Γ ⊢ L ϕ . The extensions of L ordered by inclusion form a lattice denoted Ext B . In the lattice Ext B the meets and joins are computed as follows: Γ ⊢ L 1 ∩L 2 ϕ ⇔ Γ ⊢ L 1 ϕ and Γ ⊢ L 2 ϕ Γ ⊢ L 1 ∨L 2 ϕ ⇔ ϕ provable from Γ using the rules of both logics L is axiomatized by a set of rules R (relative to B ) if it is the smallest logic (extending B ) which validates each rule of R , or equivalently if validity in L coincides with provability using instances of rules in R . Thus if L i is axiomatized by R i , then L 1 ∨ L 2 is axiomatized by R 1 ∪ R 2 . 5 / 25
Preliminaries: classical logic Let us recall how classical logic CL is defined. A truth valuation assigns to each formula one of two values: true and false (t and f). The truth values of atomic formulas are arbitrary. The truth values of complex formulas are determined by the following truth conditions: ϕ ∧ ψ is true ⇔ ϕ is true and ψ is true ϕ ∧ ψ is false ⇔ ϕ is false or ψ is false ϕ ∨ ψ is true ⇔ ϕ is true or ψ is true ϕ ∨ ψ is false ⇔ ϕ is false and ψ is false − ϕ is true ⇔ ϕ is false − ϕ is false ⇔ ϕ is true 6 / 25
Introducing the Belnap–Dunn logic The Belnap–Dunn logic B is defined similarly, except we abandon: completeness: the assumption that each formula is either true or false consistency: the assumption that no formula is both true and false A truth relation (between formulas and the truth values t and f) says of each formula whether it is true and whether it is false. The truth relation on atomic formulas are arbitrary. The truth relation on complex formulas are determined by the classical truth conditions: ϕ ∧ ψ is true ⇔ ϕ is true and ψ is true ϕ ∧ ψ is false ⇔ ϕ is false or ψ is false . . . it’s just that formulas may be both true and false or neither. 7 / 25
Introducing the Belnap–Dunn logic Consequence in B is again defined in terms of truth preservation: Γ ⊢ B ϕ ⇔ ϕ is true in each truth relation where each γ ∈ Γ is true. Equivalently, it may be defined in terms of backward falsity preservation: Γ ⊢ B ϕ ⇔ some γ ∈ Γ is false in each truth relation where ϕ is false. Some examples and non-examples of consequence in B : p ∧ − p ⊢ B − p ∨ q ∅ � B p ∨ − p p , − p � B q − ( p ∨ q ) ⊢ B − p ∧ − q p , q ∨ r ⊢ B ( p ∧ q ) ∨ ( p ∧ r ) p , − p ∨ q � B q Each truth valuation yields a truth relation, hence Γ ⊢ B ϕ ⇒ Γ ⊢ CL ϕ . 8 / 25
Introducing the Belnap–Dunn logic Imposing consistency and completeness on truth relations yields CL . Imposing only consistency yields the so-called strong Kleene logic K . The semantics of K first considered by Kleene (1938). Originally introduced for dealing with partial functions. Later used in Kripke’s Outline of a Theory of Truth (1975). Imposing only completeness yields the so-called Logic of Paradox LP . Introduced by Priest (1979) to deal with semantic paradoxes. Popular among dialetheists, who believe some contradictions are true. K adds resolution to B , while LP adds the excluded middle to B : ∅ � K p ∨ − p p ∨ q , − q ∨ r ⊢ K p ∨ r p ∨ q , − q ∨ r � LP p ∨ r ∅ ⊢ LP p ∨ − p Less explored is Kleene’s logic of order K ≤ = K ∩ LP . 9 / 25
The lattice of super-Belnap logics so far T RIV CL LP K K ≤ B K = B + resolution LP = B + excluded middle CL = B + excluded middle + resolution K ≤ = B + ( p ∧ − p ) ∨ r ⊢ ( q ∨ − q ) ∨ r 10 / 25
The lattice of super-Belnap logics These are from many perspectives the only well-behaved extensions of B . The following are natural properties for a logic to satisfy: proof by cases: if ϕ ⊢ χ and ψ ⊢ χ , then ϕ ∨ ψ ⊢ χ contraposition: if ϕ ⊢ ψ , then − ψ ⊢ − ϕ selfextensionality: if ϕ ⊣⊢ ψ , then χ ( ϕ ) ⊣⊢ χ ( ψ ) ∅ ⊢ ϕ → ϕ and ϕ, ϕ → ψ ⊢ ψ for some p → q protoalgebraicity: The following super-Belnap logics have these properties: B , K ≤ , CL , K , LP proof by cases: B , K ≤ , CL contraposition: B , K ≤ , CL selfextensionality: protoalgebraicity: CL The only multiple-conclusion super-Belnap logics are B , K ≤ , K , LP , CL . 11 / 25
Other super-Belnap logics More recently, the Exactly True Logic ET L was introduced by Marcos (2011) and, under this name, independently by Pietz & Rivieccio (2011). This logic is defined by preserving truth-and-non-falsity. That is: Γ ⊢ ET L ϕ if and only if ϕ is true and not false in each truth relation in which each γ ∈ Γ is true and not false. Lemma (Pietz & Rivieccio) ϕ ⊢ ET L ψ if and only if ϕ ⊢ B − ϕ ∨ ψ . Proposition (Pietz & Rivieccio) ET L is axiomatized rel. to B by the disjunctive syllogism: p , − p ∨ q ⊢ q. 12 / 25
The lattice of super-Belnap logics The logic ECQ is axiomatized by the rule p , − p ⊢ q . The logic LP ∩ ECQ is axiomatized by the rule p , − p ⊢ q ∨ − q . Theorem Each non-trivial proper extension of B belongs to one of the three disjoint intervals [ LP ∩ ECQ , LP ] , [ ECQ , LP ∨ ECQ ] , and [ ET L , CL ] . Moreover, each of these three intervals contains 2 ℵ 0 finitary logics. Theorem (Rivieccio) ET L has a smallest proper extension, namely by ( p ∧ − p ) ∨ ( q ∧ − q ) ⊢ ⊥ . Theorem There is a largest proper extensions of ET L below K , axiomatized by the rules χ n ∨ p , − p ∨ q ⊢ q for χ n = ( p 1 ∧ − p 1 ) ∨ · · · ∨ ( p n ∧ − p n ) for n ≥ 1 . 13 / 25
The lattice of super-Belnap logics Theorem Each super-Belnap logic L either has the same theorems as B , in which case L ⊆ K , or the same theorems as CL , in which case LP ⊆ L . Theorem Each super-Belnap logic L either has the same antitheorems as B , in which case L ⊆ LP , or ECQ ⊆ L , in which case p ∧ − p is an antitheorem of L . Corollary (Canonical decomposition of CL ) CL = LP ∨ ET L . Moreover, if CL = L 1 ∨ L 2 with L 1 , L 2 � CL , then either LP ⊆ L 1 and ET L ⊆ L 2 or vice versa. 14 / 25
The lattice of super-Belnap logics How do we prove these kinds of results? Proposition For each super-Belnap logic L either ECQ ⊆ L or L ⊆ LP . Proof. We know that LP = Log LP 3 for some three-valued matrix LP 3 . suppose that ECQ � L 15 / 25
The lattice of super-Belnap logics How do we prove these kinds of results? Proposition For each super-Belnap logic L either ECQ ⊆ L or L ⊆ LP . Proof. We know that LP = Log LP 3 for some three-valued matrix LP 3 . suppose that ECQ � L , i.e. that p , − p � L q 15 / 25
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