Extensions of the four-valued BelnapDunn logic Adam P renosil - - PowerPoint PPT Presentation

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

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Introduction

Classical logic is simple, elegant, well-suited for mathematics . . . . . . but not for reasoning with contradictory information.

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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.

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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.

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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.

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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.

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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.

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Preliminaries: extensions of logics

A logic L is an extension of B if Γ ⊢B ϕ ⇒ Γ ⊢L ϕ. The extensions of L

• rdered by inclusion form a lattice denoted Ext B.

In the lattice Ext B the meets and joins are computed as follows: Γ ⊢L1∩L2 ϕ ⇔ Γ ⊢L1 ϕ and Γ ⊢L2 ϕ Γ ⊢L1∨L2 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 Li is axiomatized by Ri, then L1 ∨ L2 is axiomatized by R1 ∪ R2.

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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

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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.

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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 ∨ q) ⊢B −p ∧ −q p, −p B 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.

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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: p ∨ q, −q ∨ r ⊢K p ∨ r ∅ K p ∨ −p p ∨ q, −q ∨ r LP p ∨ r ∅ ⊢LP p ∨ −p Less explored is Kleene’s logic of order K≤ = K ∩ LP.

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The lattice of super-Belnap logics so far

B K≤ LP K CL T RIV K = B + resolution LP = B + excluded middle CL = B + excluded middle + resolution K≤ = B + (p ∧ −p) ∨ r ⊢ (q ∨ −q) ∨ r

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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) protoalgebraicity: ∅ ⊢ ϕ → ϕ and ϕ, ϕ → ψ ⊢ ψ for some p → q The following super-Belnap logics have these properties: proof by cases: B, K≤, CL, K, LP contraposition: B, K≤, CL selfextensionality: B, K≤, CL protoalgebraicity: CL The only multiple-conclusion super-Belnap logics are B, K≤, K, LP, CL.

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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.

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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 2ℵ0 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 χn ∨ p, −p ∨ q ⊢ q for χn = (p1 ∧ −p1) ∨ · · · ∨ (pn ∧ −pn) for n ≥ 1.

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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 = L1 ∨ L2 with L1, L2 CL, then either LP ⊆ L1 and ET L ⊆ L2 or vice versa.

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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 LP3 for some three-valued matrix LP3. suppose that ECQ L

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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 LP3 for some three-valued matrix LP3. suppose that ECQ L, i.e. that p, −p L q

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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 LP3 for some three-valued matrix LP3. suppose that ECQ L, i.e. that p, −p L q that is, there is a model M of L and a valuation v : Fm → M such that v(p) and v(−p) are designated in M but v(q) is not

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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 LP3 for some three-valued matrix LP3. suppose that ECQ L, i.e. that p, −p L q that is, there is a model M of L and a valuation v : Fm → M such that v(p) and v(−p) are designated in M but v(q) is not in other words, there is a model M of L ⊇ B and elements a, b ∈ M with a and −a designated and b non-designated

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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 LP3 for some three-valued matrix LP3. suppose that ECQ L, i.e. that p, −p L q that is, there is a model M of L and a valuation v : Fm → M such that v(p) and v(−p) are designated in M but v(q) is not in other words, there is a model M of L ⊇ B and elements a, b ∈ M with a and −a designated and b non-designated using the closure of Mod L under suitable semantic constructions (submatrics, reductions), we show that LP3 is a model of L!

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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 LP3 for some three-valued matrix LP3. suppose that ECQ L, i.e. that p, −p L q that is, there is a model M of L and a valuation v : Fm → M such that v(p) and v(−p) are designated in M but v(q) is not in other words, there is a model M of L ⊇ B and elements a, b ∈ M with a and −a designated and b non-designated using the closure of Mod L under suitable semantic constructions (submatrics, reductions), we show that LP3 is a model of L! if LP3 is a model of L, then L ⊆ Log LP3 = LP

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Gentzen versions of super-Belnap logics

Let us take a brief look at the proof theory of super-Belnap logics. A sequent is a pair of multisets of formulas Γ ⊲ ∆. We introduce a pair of translations from sequents to formulas and back: τ: Γ ⊲ ∆ → − Γ ∨ ∆ ρ: ϕ → ∅ ⊲ ϕ To each super-Belnap logic L we assign its Gentzen version GL such that: S ⊢GL Γ ⊲ ∆ if and only if τ[S] ⊢L τ(Γ ⊲ ∆) Γ ⊢L ϕ if and only if ρ[Γ] ⊢GL ρϕ Γ ⊲ ∆ ⊣⊢GL ρτ(Γ ⊲ ∆) and ϕ ⊣⊢L τρϕ In the terminology of Raftery, GL is simply equivalent to L. We may now obtain results about super-Belnap logics (e.g. interpolation theorems) through a proof-theoretic study of their Gentzen versions.

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The sequent calculus for B

The basic sequent calculus for B will be called GCB. It: keeps the logical rules of classical logic includes their inverses, i.e. elimination rules keeps Weakening and Contraction drops Identity and Cut

(Exchange is absorbed into the multiset definition of a sequent.)

This calculus is due to Pynko (2010).

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Sequent calculus for B

Logical rules ϕ, Γ ⊲ ∆ Γ ⊲ ∆, −ϕ Γ ⊲ ∆, ϕ −ϕ, Γ ⊲ ∆ ϕ, ψ, Γ ⊲ ∆ ϕ ∧ ψ, Γ ⊲ ∆ Γ ⊲ ∆, ϕ, ψ Γ ⊲ ∆, ϕ ∨ ψ Γ ⊲ ∆, ϕ Γ ⊲ ∆, ψ Γ ⊲ ∆, ϕ ∧ ψ ϕ, Γ ⊲ ∆ ψ, Γ ⊲ ∆ ϕ ∨ ψ, Γ ⊲ ∆ Structural rules Weakening, Contraction

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Sequent calculi for super-Belnap logics

LP extends B by the axiomatic rule of Identity: ϕ ⊲ ϕ K extends B by the rule of Cut: Γ ⊲ ∆, ϕ ϕ, Γ′ ⊲ ∆′ Γ, Γ′ ⊲ ∆, ∆′ ET L and ECQ extend B by Limited Cut and Explosive Cut respectively: ∅ ⊲ ϕ ϕ, Γ ⊲ ∆ Γ ⊲ ∆ ∅ ⊲ ϕ ϕ ⊲ ∅ ∅ ⊲ ∅ More generally: each super-Belnap logic is the extension of B by a set of structural rules, i.e. rules in which no connectives occur.

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Sequent calculi for super-Belnap logics

Theorem (Normal Form Theorem)

Consider an extension of GCB by structural rules. If a sequent has a proof from a set of sequents in this extension, then it has a proof which is analytic-synthetic (in each branch elimination rules precede all intro rules) and moreover structurally atomic (structural rules are only applied to atomic sequents). The proof provides a simple procedure for the transformation. For proofs from an empty set of premises (of the empty sequent ∅ ⊲ ∅), this essentially reduces to the notion of a cut-free (identity-free) proof.

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Interpolation in super-Belnap logics

We now turn to the interpolation properties of super-Belnap logics. There are many different interpolation properties. What we consider below is a certain generalization of the Craig interpolation property (CIP). A logic L enjoys (L1, L2)-interpolation in case ϕ ⊢L χ implies that: ϕ ⊢L1 ψ and ψ ⊢L2 χ for some ψ such that Var(ψ) ⊆ Var(ϕ) ∩ Var(χ) Interpolation is well studied for superintuitionistic and modal logics. E.g. there are exactly 8 super-intuitionistic logics with CIP (Maksimova). Interpolation properties in algebraizable logics correspond to amalgamation properties of the corresponding classes of algebras (e.g. Heyting algebras). Since the link between logic and algebra is too weak for super-Belnap logics, we use proof-theoretic methods instead (Normal Form Theorem).

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Interpolation in super-Belnap logics

Interpolation for L will mean (L, L)-interpolation. Known results: CL enjoys interpolation (Craig 1957) B enjoys interpolation (Anderson and Belnap 1975) K enjoys interpolation (Bendov´ a 2005 and Milne 2016) LP enjoys interpolation (Milne 2016): ϕ ⊢LP ψ iff −ψ ⊢K −ϕ K≤ does not enjoy interpolation (Bendov´ a 2015) More interestingly to the classical logician: CL enjoys (K, LP)-interpolation (Milne 2016)

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Interpolation in super-Belnap logics

We now provide a useful sufficient condition for interpolation. A side atom of a rule is an atom which either only occurs on positively or

• nly occurs negatively in the rule.

A cut atom occurs in some premise but not in the conclusion of the rule. A generalized cut rule is a rule in which each atom is a side atom or a cut

• atom. For example, the rules p, −p ∨ q ⊢ q and p ∨ q, −q ∨ r ⊢ p ∨ r.

Theorem (Interpolation for K, LP, ECQ, ET L)

If L is an extension of B by a set of generalized cut rules, then L enjoys (L, B)-interpolation. In particular, ECQ, ET L, and K do. As a corollary, it follows that LP enjoys (B, LP)-interpolation. The proof relies on a Gentzen-style proof theory for super-Belnap logics.

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Interpolation in super-Belnap logics

These results may moreover be shown to be optimal in a sense.

Proposition

If CL enjoys (L1, L2)-interpolation, then K ⊆ L1 and LP ⊆ L2.

Proposition

If K (ET L) enjoys (L1, L2)-interpolation, then K ⊆ L1 ET L ⊆ L1.

Proposition

If LP enjoys (L1, L2)-interpolation, then LP ⊆ L2.

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Conclusion

Super-Belnap logics are a family of non-classical logics introduced by Rivieccio which has only recently been studied systematically. Some basics facts are known about the structure of the lattice Ext B. From a proof-theoretic point of view, these are obtained by weakening the rules of Cut and Identity in a calculus for CL with the elimination rules. They are also interesting from the point of view of AAL, since they provide a testing ground for some new concepts (e.g. strong versions of logics).

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Conclusion

Super-Belnap logics are a family of non-classical logics introduced by Rivieccio which has only recently been studied systematically. Some basics facts are known about the structure of the lattice Ext B. From a proof-theoretic point of view, these are obtained by weakening the rules of Cut and Identity in a calculus for CL with the elimination rules. They are also interesting from the point of view of AAL, since they provide a testing ground for some new concepts (e.g. strong versions of logics).

Thank you for your attention.

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