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 1 / 29

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” or 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. 2 / 29

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. 3 / 29

De Morgan algebras De Morgan algebras are bounded distributive lattices with an order-inverting involution − . They form a variety DMA. DM 4 K 3 B 2 DMA = SP ( DM 4 ) KA = SP ( K 3 ) BA = SP ( B 2 ) x ∧ − x ≤ y ∨ − y x ∧ − x ≤ y 4 / 29

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 B B The logic of order of DMA: (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 ) 5 / 29

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 6 / 29

Handling incomplete and contradictory information: B The Belnap-Dunn logic is given by the matrix B 4 : B 4 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 7 / 29

Handling incomplete information: K Consider the matrix K 3 : 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). 8 / 29

Handling contradictory information: LP Consider the matrix LP 3 : 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). 9 / 29

The picture so far T RIV CL LP K K ≤ B 10 / 29

The picture so far T RIV CL LP K K ≤ B Until recently, these were the only known super-Belnap logics. No coincidence: these are the only well-behaved super-Belnap logics. 10 / 29

Preserving exact truth: ET L Changing the designated values of B 4 yields the matrix ETL 4 : This logic extends B by the disjunctive syllogism: p , − p ∨ q ⊢ q . This is the Exactly True Logic introduced by Pietz and Rivieccio (2013). 11 / 29

Ex contradictione quodlibet Consider the logics ECQ n extending B by the rules: ( p 1 ∧ − p 1 ) ∨ . . . ∨ ( p n ∧ − p n ) ⊢ ∅ ( ECQ n ) Define ET L n = ET L ∨ ECQ n ( ECQ = ECQ 1 and ET L = ET L 1 ). These form an infinite increasing chain (Rivieccio 2012): ET L � ET L 2 � . . . � ET L ω 12 / 29

Ex contradictione quodlibet Consider the logics ECQ n extending B by the rules: ( p 1 ∧ − p 1 ) ∨ . . . ∨ ( p n ∧ − p n ) ⊢ ∅ ( ECQ n ) Define ET L n = ET L ∨ ECQ n ( ECQ = ECQ 1 and ET L = ET L 1 ). These form an infinite increasing chain (Rivieccio 2012): ET L � ET L 2 � . . . � ET L ω Contrary to popular opinion, p , − p ⊢ ∅ is not ex contradictione quodlibet : χ 2 = ( p 1 ∧ − p 1 ) ∨ ( p 2 ∧ − p 2 ) is a contradiction, yet χ 2 � ECQ ∅ . 12 / 29

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. Exp B L shall be the least explosive extension of B below L . Exp B takes L and forgets all the non-explosive rules. 13 / 29

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. Exp B L shall be the least explosive extension of B below L . Exp B takes L and forgets all the non-explosive rules. Examples : Exp B CL = ECQ ω Exp ET L CL = ET L ω Exp B ET L n = ECQ n 13 / 29

Some completeness theorems The operator Exp B is useful for proving completeness: � � Log Π i ∈ I A i = Log A i ∪ Exp B Log A i i ∈ I i ∈ I We can now immediately compute: Log B 2 × B 4 = ( CL ∩ B ) ∪ Exp B CL ∪ Exp B B = B ∪ ECQ ω ∪ B = ECQ ω Log B 2 × ETL 4 = ( CL ∩ ET L ) ∪ Exp ET L CL ∪ Exp ET L ET L = ET L ω Log ETL 4 × B 4 = ( ET L ∩ B ) ∪ Exp B ET L ∪ Exp B B = ECQ 14 / 29

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? 15 / 29

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. 15 / 29

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? 15 / 29

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, ( p 1 ∧ − p 1 ) ⊕ ( p 2 ∧ − p 2 ) � � L ∅ . 15 / 29

Lattices of super-Belnap logics Ext ( ω ) B The lattice of (finitary) super-Belnap logics: The lattice of (finitary) explosive extensions of B : Exp Ext ( ω ) B Exp B is an interior operator on Ext ω 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 . 16 / 29

The lattice Ext B CL LP ∨ ECQ K K ≤ ∨ ECQ ET L ω LP . . . K ≤ ECQ ω . . . LP ∩ ECQ ω ET L . . . ECQ LP ∩ ECQ B 17 / 29

The lattice Ext B CL LP ∨ ECQ K K ≤ ∨ ECQ ET L ω LP . . . K ≤ ECQ ω . . . LP ∩ ECQ ω ET L . . . ECQ L ⊇ LP ∩ ECQ or L = B LP ∩ ECQ B 17 / 29

The lattice Ext B CL LP ∨ ECQ K K ≤ ∨ ECQ ET L ω LP . . . K ≤ ECQ ω . . . LP ∩ ECQ ω ET L . . . ECQ L ⊇ ECQ or L ⊆ LP LP ∩ ECQ B 17 / 29

The lattice Ext B CL LP ∨ ECQ K K ≤ ∨ ECQ ET L ω LP . . . K ≤ ECQ ω . . . LP ∩ ECQ ω ET L . . . ECQ L ⊇ ET L or L ⊆ LP ∨ ECQ LP ∩ ECQ B 17 / 29

The lattice Ext B CL LP ∨ ECQ K K ≤ ∨ ECQ ET L ω LP . . . K ≤ ECQ ω . . . LP ∩ ECQ ω ET L . . . ECQ L ⊇ LP or L ⊆ K LP ∩ ECQ B 17 / 29

The lattice Ext ET L CL K ET L + ω . . . ET L ω . . . ET L + 2 ET L + ET L + n : χ n ∨ q , − q ∨ r ⊢ r ET L 2 χ n = ( p 1 ∧ − p 1 ) ∨ . . . ∨ ( p n ∧ − p n ) ET L 18 / 29

The lattice Ext ET L L ⊇ K or L ⊆ ET L + CL ω K ET L + ω . . . ET L ω . . . ET L + 2 ET L + ET L + n : χ n ∨ q , − q ∨ r ⊢ r ET L 2 χ n = ( p 1 ∧ − p 1 ) ∨ . . . ∨ ( p n ∧ − p n ) ET L 18 / 29

The lattice Ext ET L L ⊇ K ≤ or L ⊆ ET L + CL ω K ET L + ω . . . ET L ω . . . ET L + 2 ET L + ET L + n : χ n ∨ q , − q ∨ r ⊢ r ET L 2 χ n = ( p 1 ∧ − p 1 ) ∨ . . . ∨ ( p n ∧ − p n ) ET L 18 / 29

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 B , K ≤ , CL contraposition: self-extensionality: B , K ≤ , CL CL protoalgebraicity: 19 / 29