The lattice of super-Belnap logics Adam P renosil Institute of - - PowerPoint PPT Presentation

The lattice of super-Belnap logics

Adam Pˇ renosil

Institute of Computer Science, Czech Academy of Sciences Department of Logic, Faculty of Arts, Charles University in Prague

ManyVal 2015 Les Diablerets, 13 December 2015

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Introduction

The four-valued Belnap-Dunn logic B is a well-known logic for reasoning with incomplete and inconsistent information. It was introduced by Nuel Belnap in 1977 as a “useful four-valued logic”

• r a logic of “how a computer should think”.

Extensions of B will be called super-Belnap logics (following Rivieccio). Examples: strong Kleene K and the Logic of Paradox LP. Our goal is to get a better view of the landscape of super-Belnap logics.

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Truth and falsehood in B

In the logic B, truth values are computed in a perfectly classical way: 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 . . . it’s just that sentences may be both true and false or neither. In other words, the truth and falsehood values are computed separately.

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De Morgan algebras

De Morgan algebras are bounded distributive lattices with an

• rder-inverting involution −. They form a variety DMA.

DM4 DMA = SP(DM4) K3 KA = SP(K3) x ∧ −x ≤ y ∨ −y B2 BA = SP(B2) x ∧ −x ≤ y

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Logics of order

Each class of lattice-ordered algebras K naturally yields a logic of order: Γ ⊢ ϕ if Γ ≤ ϕ holds in K for some finite Γ ⊆ Γ The logic of order of DMA: B B (the Belnap-Dunn logic) The logic of order of KA: K≤ K≤ (Kleene’s logic of order) The logic of order of BA: CL CL (classical logic) Logics of order are always self-extensional: ϕ ⊣⊢ ψ ⇒ χ(ϕ) ⊣⊢ χ(ψ)

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Logics given by matrices

A matrix is an algebra A with a set of designated values D ⊆ A. A matrix (A, D) is a model of a logic L in case for each v : Fm → A: if Γ ⊢L ϕ and v[Γ] ⊆ D, then v(ϕ) ∈ D A matrix is reduced if no non-trivial congruence on A preserves D. Each matrix M has a logically equivalent reduced matrix M/θ. Mod L = class of all models of L Mod* L = class of all reduced models of L

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Handling incomplete and contradictory information: B

The Belnap-Dunn logic is given by the matrix B4: B4 The truth values are: True, False, Neither, Both. Hilbert-style axiomatization by Font (1997). Mod* B {(A, D) | A ∈ DMA, D lattice filter on A} Mod B

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Handling incomplete information: K

Consider the matrix K3: The truth values are: True, False, Neither. This logic extends B by the rule of resolution: p ∨ q, −q ∨ r ⊢ p ∨ r. This is Stephen C. Kleene’s strong three-valued logic K (1938).

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Handling contradictory information: LP

Consider the matrix LP3: The truth values are: True, False, Both. This logic extends B by the law of the excluded middle: ∅ ⊢ p ∨ −p. This is Graham Priest’s Logic of Paradox LP (1979).

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The picture so far

B K≤ LP K CL T RIV

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The picture so far

B K≤ LP K CL T RIV Until recently, these were the only known super-Belnap logics. No coincidence: these are the only well-behaved super-Belnap logics.

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Preserving exact truth: ET L

Changing the designated values of B4 yields the matrix ETL4: This logic extends B by the disjunctive syllogism: p, −p ∨ q ⊢ q. This is the Exactly True Logic introduced by Pietz and Rivieccio (2013).

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Ex contradictione quodlibet

Consider the logics ECQn extending B by the rules: (p1 ∧ −p1) ∨ . . . ∨ (pn ∧ −pn) ⊢ ∅ (ECQn) Define ET Ln = ET L ∨ ECQn (ECQ = ECQ1 and ET L = ET L1). These form an infinite increasing chain (Rivieccio 2012): ET L ET L2 . . . ET Lω

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Ex contradictione quodlibet

Consider the logics ECQn extending B by the rules: (p1 ∧ −p1) ∨ . . . ∨ (pn ∧ −pn) ⊢ ∅ (ECQn) Define ET Ln = ET L ∨ ECQn (ECQ = ECQ1 and ET L = ET L1). These form an infinite increasing chain (Rivieccio 2012): ET L ET L2 . . . ET Lω Contrary to popular opinion, p, −p ⊢ ∅ is not ex contradictione quodlibet: χ2 = (p1 ∧ −p1) ∨ (p2 ∧ −p2) is a contradiction, yet χ2 ECQ ∅.

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Explosive extensions

Explosive rules are rules of the form Γ ⊢ ∅. Explosive rules are dual to axiomatic rules of the form ∅ ⊢ ϕ. Explosive extensions are extensions by explosive rules. ExpB L shall be the least explosive extension of B below L. ExpB takes L and forgets all the non-explosive rules.

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Explosive extensions

Explosive rules are rules of the form Γ ⊢ ∅. Explosive rules are dual to axiomatic rules of the form ∅ ⊢ ϕ. Explosive extensions are extensions by explosive rules. ExpB L shall be the least explosive extension of B below L. ExpB takes L and forgets all the non-explosive rules. Examples: ExpB CL = ECQω ExpET L CL = ET Lω ExpB ET Ln = ECQn

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Some completeness theorems

The operator ExpB is useful for proving completeness: Log Πi∈IAi =

• i∈I

Log Ai ∪

• i∈I

ExpB Log Ai We can now immediately compute: Log B2 × B4 = (CL ∩ B) ∪ ExpB CL ∪ ExpB B = B ∪ ECQω ∪ B = ECQω Log B2 × ETL4 = (CL ∩ ET L) ∪ ExpET L CL ∪ ExpET L ET L = ET Lω Log ETL4 × B4 = (ET L ∩ B) ∪ ExpB ET L ∪ ExpB B = ECQ

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Sidenote: paraconsistent logics

Paraconsistent logics are usually understood as logics which do not satisfy ex contradictione quodlibet, understood as p, −p ⊢ ∅. If we reject this reading of ECQ, how do we understand paraconsistency?

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Sidenote: paraconsistent logics

Paraconsistent logics are usually understood as logics which do not satisfy ex contradictione quodlibet, understood as p, −p ⊢ ∅. If we reject this reading of ECQ, how do we understand paraconsistency? Proposal: A logic is paraconsistent if it has no non-trivial explosive extension.

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Sidenote: paraconsistent logics

Paraconsistent logics are usually understood as logics which do not satisfy ex contradictione quodlibet, understood as p, −p ⊢ ∅. If we reject this reading of ECQ, how do we understand paraconsistency? Proposal: A logic is paraconsistent if it has no non-trivial explosive extension. Question: Is Lukasiewicz paraconsistent?

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Sidenote: paraconsistent logics

Paraconsistent logics are usually understood as logics which do not satisfy ex contradictione quodlibet, understood as p, −p ⊢ ∅. If we reject this reading of ECQ, how do we understand paraconsistency? Proposal: A logic is paraconsistent if it has no non-trivial explosive extension. Question: Is Lukasiewicz paraconsistent? After all, (p1 ∧ −p1) ⊕ (p2 ∧ −p2) L ∅.

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Lattices of super-Belnap logics

The lattice of (finitary) super-Belnap logics: Ext(ω) B The lattice of (finitary) explosive extensions of B: Exp Ext(ω) B ExpB is an interior operator on Extω B.

Proposition

Exp Ext(ω) L is a distributive sublattice of Ext(ω) L.

Proposition

Ext(ω) B is non-modular: (LP ∩ ET L) ∨ ECQ < (LP ∨ ECQ) ∩ ET L.

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The lattice Ext B

B LP ∩ ECQ ECQ ET L . . . . . . . . . LP ∩ ECQω ECQω ET Lω K≤ K≤ ∨ ECQ K LP LP ∨ ECQ CL

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The lattice Ext B

B LP ∩ ECQ ECQ ET L . . . . . . . . . LP ∩ ECQω ECQω ET Lω K≤ K≤ ∨ ECQ K LP LP ∨ ECQ CL L ⊇ LP ∩ ECQ or L = B

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The lattice Ext B

B LP ∩ ECQ ECQ ET L . . . . . . . . . LP ∩ ECQω ECQω ET Lω K≤ K≤ ∨ ECQ K LP LP ∨ ECQ CL L ⊇ ECQ or L ⊆ LP

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The lattice Ext B

B LP ∩ ECQ ECQ ET L . . . . . . . . . LP ∩ ECQω ECQω ET Lω K≤ K≤ ∨ ECQ K LP LP ∨ ECQ CL L ⊇ ET L or L ⊆ LP ∨ ECQ

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The lattice Ext B

B LP ∩ ECQ ECQ ET L . . . . . . . . . LP ∩ ECQω ECQω ET Lω K≤ K≤ ∨ ECQ K LP LP ∨ ECQ CL L ⊇ LP or L ⊆ K

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The lattice Ext ET L

ET L ET L2 . . . ET Lω ET L+ ET L+

2

. . . ET L+

ω

K CL ET L+

n : χn ∨ q, −q ∨ r ⊢ r

χn = (p1 ∧ −p1) ∨ . . . ∨ (pn ∧ −pn)

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The lattice Ext ET L

ET L ET L2 . . . ET Lω ET L+ ET L+

2

. . . ET L+

ω

K CL ET L+

n : χn ∨ q, −q ∨ r ⊢ r

χn = (p1 ∧ −p1) ∨ . . . ∨ (pn ∧ −pn) L ⊇ K or L ⊆ ET L+

ω

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The lattice Ext ET L

ET L ET L2 . . . ET Lω ET L+ ET L+

2

. . . ET L+

ω

K CL ET L+

n : χn ∨ q, −q ∨ r ⊢ r

χn = (p1 ∧ −p1) ∨ . . . ∨ (pn ∧ −pn) L ⊇ K≤ or L ⊆ ET L+

ω

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Well-behaved super-Belnap logics are scarce

The following are natural properties for a logic to satisfy: proof by cases: ϕ ⊢ χ & ψ ⊢ χ ⇒ ϕ ∨ ψ ⊢ χ contraposition: ϕ ⊢ ψ ⇒ −ψ ⊢ −ϕ self-extensionality: ϕ ⊣⊢ ψ ⇒ χ(ϕ) ⊣⊢ χ(ψ) protoalgebraicity: ϕ, ϕ ⇒ ψ ⊢ ψ & ∅ ⊢ ϕ ⇒ ϕ The following super-Belnap logics have these properties: proof by cases: B, K≤, CL, K, LP contraposition: B, K≤, CL self-extensionality: B, K≤, CL protoalgebraicity: CL

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Well-behaved super-Belnap logics are scarce

The following are natural properties for a logic to satisfy: proof by cases: ϕ ⊢ χ & ψ ⊢ χ ⇒ ϕ ∨ ψ ⊢ χ contraposition: ϕ ⊢ ψ ⇒ −ψ ⊢ −ϕ self-extensionality: ϕ ⊣⊢ ψ ⇒ χ(ϕ) ⊣⊢ χ(ψ) protoalgebraicity: ϕ, ϕ ⇒ ψ ⊢ ψ & ∅ ⊢ ϕ ⇒ ϕ The following super-Belnap logics have these properties: proof by cases: B, K≤, CL, K, LP contraposition: B, K≤, CL self-extensionality: B, K≤, CL protoalgebraicity: CL Most super-Belnap logics are not very well-behaved.

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Extensions of ET L

Let us restrict our attention to Extω ET L now. Rivieccio proved: Mod* ET L {(A, {⊤}) | A ∈ DMA} Mod ET L Special quasiequation = ⊤ ≈ γ1 & . . . & ⊤ ≈ γn ⇒ ⊤ ≈ ϕ Special antiequation = ⊤ ≈ γ Finitary extensions of ET L = special quasivarieties of DMAs Finitary explosive extensions of ET L = special antivarieties of DMAs Our results about Extω ET L will have algebraic corollaries.

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From graphs to models of ET L

For us, graphs will be finite undirected graphs, possibly with loops. To each graph G, we assign the two matrices γG and µG. u v w G = (X, R) −U = X \ R[U] γG = (P(X), ∩, ∪, X, ∅, −) D = {X} u v w δu δv δw F = (W , ≤ δ) −U = W \ δ[U] µG = (P≤(W ), ∩, ∪, W , ∅, −) D = {W }

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Example: ex contradictione and graph colourings

The finite reduced models of ET L are precisely the matrices Bn

2 × µG.

The matrices γG are almost never models of B, much less ET L. . . . . . nonetheless, γG agrees with µG on Γ ⊢ ϕ for ϕ positive, Γ simple. The language of the Belnap-Dunn logic can be used to talk about graphs.

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Example: ex contradictione and graph colourings

The finite reduced models of ET L are precisely the matrices Bn

2 × µG.

The matrices γG are almost never models of B, much less ET L. . . . . . nonetheless, γG agrees with µG on Γ ⊢ ϕ for ϕ positive, Γ simple. The language of the Belnap-Dunn logic can be used to talk about graphs. Example: (p1 ∧ −p1) ∨ . . . ∨ (pn ∧ pn) µG ∅ ⇔ G is n-colourable

Proposition

The logics ET Ln (n ≥ 2) are complete w.r.t. the class of matrices µG such that G is not n-colourable.

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The homomorphism order on finite graphs

The class of finite graphs may be ordered by homomorphisms: G ≤ H if and only if there is a graph homomorphism h : G → H This yields the homomorphism order on finite graphs.

Theorem

Each countable pre-order embeds into the hom order on finite graphs. (Usually formulated in terms of partial orders instead of pre-orders.) Restricting to loopless graphs = dropping the top equivalence class Restricting to graphs with edges = dropping the bottom equivalence class

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Extω ET L and Exp Extω ET L

Theorem

The interval [ET L, ET Lω] ⊆ Extω ET L is dually isomorphic to the lattice

• f classes of graphs closed under finite disjoint unions, surjective graph

homomorphisms, and vertex reductions. Vertex reduction means contracting all the outgoing edges of a vertex.

Theorem

The lattice Exp Extω ET L is dually isomorphic to the lattice of classes of graphs (with a non-empty set of edges) closed under homomorphisms.

Corollary

There is a continuum of finitary explosive extensions of ET L (and of B). The lattice Extω B contains infinite increasing and decreasing chains.

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Explosive extensions of B

The lattice Exp Extω B is essentially the same as Exp Extω ET L. Exp Extω ET L = Exp Extω ECQ = LP ∩ Exp Extω ECQ. via L → ExpB L and L → ET L ∨ L via L → LP ∩ L and L → ECQ ∨ L Exp Extω B = {B} ∪ Exp Extω ECQ = {B} ∪ Exp Extω ET L. Each of the intervals [B, LP], [ECQ, LP ∨ ECQ] and Extω ET L thus contains a distributive sublattice of the cardinality of the continuum.

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Example: from graph theory to super-Belnap logics

The girth of a (loopless) graph is the length of its shortest cycle.

Theorem (Erd˝

• s, 1959)

The class of non-n-colourable (loopless) graphs has unbounded girth.

Corollary

The logics ET Ln are not complete w.r.t. a finite set of finite matrices.

Proof.

If it were, then the class of non-n-colourable graphs would be the closure of a finite set of graphs under disjoint unions, surjective homomorphisms, and vertex reductions. But these operations all preserve or decrease girth.

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Example: a non-finitary super-Belnap logic

For each graph H (with some edges), there is a formula ϕH such that: µG satisfies ϕH ⊢ ∅ ⇔ G H. Let LK extend ET L by the rule {ϕH | H ∈ K} ⊢ ∅. Then: µG / ∈ Mod LK ⇔ G ≤ H for all H ∈ K.

Theorem

There is a non-finitary super-Belnap logic.

Proof.

Consider a free countably generated meet-semilattice embedded into the hom order on finite graphs. Let K be its antichain of generators. If LK is finitary, then LK = LK for some finite K ⊆ K. Thus Mod LK = Mod LK, hence the finite meet of K would have to be below each H ∈ K.

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Example: completeness of ET L+

ω

Recall that ET L+

n extends B by the rule χn ∨ q, −q ∨ r ⊢ r.

G is weakly n-colourable if there is a partial n-colouring which leaves some vertices U uncoloured such that not every vertex of G is a neighbour of U. We may observe: µG ∈ Mod ET L+

n ⇔ G not weakly n-colourable.

Let G2 = and let ETL8 be the 8-element matrix µG2.

Proposition

ET L+

ω = Log ETL8.

Proof.

Each graph which is not weakly n-colourable for any n is a surjective homomorphic image of a disjoint union of copies of G2.

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Conclusion

To sum up, Extω B is a non-modular lattice of cardinality 2ℵ0 which splits into three parts, namely [B, LP], [ECQ, LP ∨ ECQ], and Extω ET L.

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Conclusion

To sum up, Extω B is a non-modular lattice of cardinality 2ℵ0 which splits into three parts, namely [B, LP], [ECQ, LP ∨ ECQ], and Extω ET L. The notion of an explosive extension and the explosive part operator ExtB are useful in the study of super-Belnap logics.

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Conclusion

To sum up, Extω B is a non-modular lattice of cardinality 2ℵ0 which splits into three parts, namely [B, LP], [ECQ, LP ∨ ECQ], and Extω ET L. The notion of an explosive extension and the explosive part operator ExtB are useful in the study of super-Belnap logics. Perhaps somewhat surprisingly, graph-theoretic methods may be used to

• btain non-trivial results about super-Belnap logics.

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Conclusion

To sum up, Extω B is a non-modular lattice of cardinality 2ℵ0 which splits into three parts, namely [B, LP], [ECQ, LP ∨ ECQ], and Extω ET L. The notion of an explosive extension and the explosive part operator ExtB are useful in the study of super-Belnap logics. Perhaps somewhat surprisingly, graph-theoretic methods may be used to

• btain non-trivial results about super-Belnap logics.

Thank you for your attention.

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