Consistency, Completeness, and Classicality Adam Pˇ renosil Institute of Computer Science, Academy of Sciences of the Czech Republic 18 June 2015, Hejnice Logica 2015 Adam Pˇ renosil Consistency, Completeness, and Classicality 1 / 20
Introduction This will be a talk full of trivialities. Hopefully they will add up to a somewhat sensible picture. I will try to sell you on a certain expansion of the Belnap–Dunn logic. This will illustrate a more general approach to expanding a logic by constants internalizing inconsistency and completeness in the language. By-product: a novel way of reconciling classical and intuitionistic logic. Adam Pˇ renosil Consistency, Completeness, and Classicality 2 / 20
Consistency and completeness We are in the business of asserting things about the world. Consistency and completeness are fundamental pragmatic goals here. They are inseparable: if we require only consistency, it suffices to assert nothing if we require only completeness, it suffices to assert everything Failures of consistency and completeness require different responses: if our assertions are inconsistent, we should withdraw some if our assertions are incomplete, we should assert more Adam Pˇ renosil Consistency, Completeness, and Classicality 3 / 20
Consistency and completeness We are in the business of asserting things about the world. Consistency and completeness are fundamental pragmatic goals here. They are inseparable: if we require only consistency, it suffices to assert nothing if we require only completeness, it suffices to assert everything Failures of consistency and completeness require different responses: if our assertions are inconsistent, we should withdraw some if our assertions are incomplete, we should assert more A consequence relation only gets you from the explicit to the implicit. It does not tell you whether you should assert more or assert less. Adam Pˇ renosil Consistency, Completeness, and Classicality 3 / 20
Example: minimal logic vs. the Logica logic Take Johansson’s minimal logic. That is, intuitionistic logic without ⊥ expanded by a constant f . . . . . . which is interpreted as a contradiction. In this logic, it is wrong to assert { p , p → f } . Adam Pˇ renosil Consistency, Completeness, and Classicality 4 / 20
Example: minimal logic vs. the Logica logic Take Johansson’s minimal logic. That is, intuitionistic logic without ⊥ expanded by a constant f . . . . . . which is interpreted as a contradiction. In this logic, it is wrong to assert { p , p → f } . By contrast, take the Logica logic. That is, intuitionistic logic without ⊥ expanded by a constant f . . . . . . which is interpreted as your favourite proposition . . . . . . say, as “Logica take place in Hejnice”. In this logic, it is not wrong to assert { p , p → f } . In fact, it is quite okay to assert all propositions at once. Adam Pˇ renosil Consistency, Completeness, and Classicality 4 / 20
Example: minimal logic vs. the Logica logic Take Johansson’s minimal logic. That is, intuitionistic logic without ⊥ expanded by a constant f . . . . . . which is interpreted as a contradiction. In this logic, it is wrong to assert { p , p → f } . By contrast, take the Logica logic. That is, intuitionistic logic without ⊥ expanded by a constant f . . . . . . which is interpreted as your favourite proposition . . . . . . say, as “Logica take place in Hejnice”. In this logic, it is not wrong to assert { p , p → f } . In fact, it is quite okay to assert all propositions at once. The two logics are therefore deductively equivalent but distinct. Adam Pˇ renosil Consistency, Completeness, and Classicality 4 / 20
A broader notion of logic The notion of a logic as dealing with what follows from what is too narrow. (At least for logics which deal with assertion.) It ought to be supplemented by an account of: what should not be asserted (which sets of sentences are inconsistent) what should be asserted (which sets of sentences are complete) Adam Pˇ renosil Consistency, Completeness, and Classicality 5 / 20
A broader notion of logic The notion of a logic as dealing with what follows from what is too narrow. (At least for logics which deal with assertion.) It ought to be supplemented by an account of: what should not be asserted (which sets of sentences are inconsistent) what should be asserted (which sets of sentences are complete) This can be reduced to what follows from what in an expanded language. Adam Pˇ renosil Consistency, Completeness, and Classicality 5 / 20
Internalizing inconsistency and completeness We shall expand the language by a pair of constants: the weakest contradiction 0 the strongest tautology 1 Consistency says that: if Γ ⊢ 0, then you should not assert all γ ∈ Γ Completeness says that: if 1 ⊢ ∆, then you should assert some δ ∈ ∆ Meaning: the less you violate these, the better. Adam Pˇ renosil Consistency, Completeness, and Classicality 6 / 20
Road map a logic of truth and falsehood (classical logic) Adam Pˇ renosil Consistency, Completeness, and Classicality 7 / 20
Road map a logic of truth and falsehood (classical logic) ⇓ a logic of assertion and denial (the Belnap–Dunn logic) Adam Pˇ renosil Consistency, Completeness, and Classicality 7 / 20
Road map a logic of truth and falsehood (classical logic) ⇓ a logic of assertion and denial (the Belnap–Dunn logic) ⇓ a logic of both truth and assertion Adam Pˇ renosil Consistency, Completeness, and Classicality 7 / 20
The Belnap–Dunn logic B True Neither Both False The matrix B 4 in the signature ∧ , ∨ , ⊤ , ⊥ , − Γ ⊢ B ∆ if and only if for each valuation v on B 4 : v ( γ ) ∈ D for all γ ∈ Γ ⇒ v ( δ ) ∈ D for some δ ∈ ∆ (We use a multiple-conclusion consequence relation.) Adam Pˇ renosil Consistency, Completeness, and Classicality 8 / 20
The motivation behind B Suppose we have a geographical database into which different users can enter data. Suppose that one user enters the proposition p : Crimea is part of Ukraine. Now suppose another user has a different view and enters − p : Crimea is not part of Ukraine. He also fills in a gap in the database and enters the proposition q : The capital of Crimea of Simferopol. We now want an affirmative answer to a query about q ∨ p but not to a query about − q ∧ p . Adam Pˇ renosil Consistency, Completeness, and Classicality 9 / 20
Kleene’s logic K and Priest’s logic LP The matrices K 3 and LP 3 . K 3 yields Kleene’s three-valued logic K . K = B + p , − p ⊢ ∅ . LP 3 yields Priest’s Logic of Paradox LP . LP = B + ∅ ⊢ p , − p . Adam Pˇ renosil Consistency, Completeness, and Classicality 10 / 20
Relational semantics for B A de Morgan frame is a poset ( W , ≤ ) with an order-inverting involution δ . Atomic propositions interpreted by upsets. For complex propositions: u � ϕ ∧ ψ ⇔ u � ϕ and v � ϕ u � ϕ ∨ ψ ⇔ u � ϕ or v � ϕ u � ⊤ , u � ⊥ u � − ϕ ⇔ δ ( u ) � ϕ Consequence is defined as truth-preservation in all worlds. Γ � F ∆ if for each u ∈ F and each valuation � on F : u � γ for each γ ∈ Γ ⇒ u � δ for some δ ∈ ∆ Γ � ∆ if Γ � F for each de Morgan frame F . This yields precisely the logic B . Adam Pˇ renosil Consistency, Completeness, and Classicality 11 / 20
Expanding the Belnap–Dunn logic A proposition of B is (logically) inconsistent if it is classically inconsistent. ϕ inconsistent ⇔ ϕ ⊢ B ( ψ 1 ∧ − ψ 1 ) ∨ . . . ∨ ( ψ n ∧ ψ n ) for some ψ 1 , . . . , ψ n ϕ inconsistent ⇔ ϕ ⊢ B ∃ p ( p ∧ − p ) We thus want to expand B by 0 ≡ ∃ p ( p ∧ − p ) and 1 ≡ ∀ p ( p ∨ − p ). Adam Pˇ renosil Consistency, Completeness, and Classicality 12 / 20
Expanding the Belnap–Dunn logic A proposition of B is (logically) inconsistent if it is classically inconsistent. ϕ inconsistent ⇔ ϕ ⊢ B ( ψ 1 ∧ − ψ 1 ) ∨ . . . ∨ ( ψ n ∧ ψ n ) for some ψ 1 , . . . , ψ n ϕ inconsistent ⇔ ϕ ⊢ B ∃ p ( p ∧ − p ) We thus want to expand B by 0 ≡ ∃ p ( p ∧ − p ) and 1 ≡ ∀ p ( p ∨ − p ). Note: a different notion of inconsistency than in paraconsistent logic. There, ϕ is usually called inconsistent if ∃ ψ ϕ ⊢ ψ and ϕ ⊢ − ψ . Accordingly, ex contradictione quodlibet interpreted as p , − p ⊢ q . This is incompatible with the above interpretation of inconsistency. Adam Pˇ renosil Consistency, Completeness, and Classicality 12 / 20
Adding the inconsistency constant: proof theory Consider now the expansion B + of B by constants 0 and 1 such that p , − p ⊢ 0 1 ⊢ p , − p and 0 ⊣⊢ − 1. These are equivalent to the rules Γ ⊢ ∆ , 0 , ϕ ϕ, 1 , Γ ⊢ ∆ − ϕ, Γ ⊢ ∆ , 0 1 , Γ ⊢ ∆ , − ϕ The de Morgan negation now behaves classically in the context 1 , Γ ⊢ ∆ , 0: 1 , Γ ⊢ ∆ , 0 , ϕ ϕ, 1 , Γ ⊢ ∆ , 0 − ϕ, 1 , Γ ⊢ ∆ , 0 1 , Γ ⊢ ∆ , 0 , − ϕ Adam Pˇ renosil Consistency, Completeness, and Classicality 13 / 20
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