Consistency, Completeness, and Classicality Adam P renosil - - PowerPoint PPT Presentation

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.

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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.

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Road map

a logic of truth and falsehood (classical logic)

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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

Neither False True Both The matrix B4 in the signature ∧, ∨, ⊤, ⊥, − Γ ⊢B ∆ if and only if for each valuation v on B4: v(γ) ∈ D for all γ ∈ Γ ⇒ v(δ) ∈ D for some δ ∈ ∆ (We use a multiple-conclusion consequence relation.)

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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 K3 and LP3. K3 yields Kleene’s three-valued logic K. K = B + p, −p ⊢ ∅. LP3 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.

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Expanding the Belnap–Dunn logic

A proposition of B is (logically) inconsistent if it is classically inconsistent. ϕ inconsistent ⇔ ϕ ⊢B (ψ1 ∧ −ψ1) ∨ . . . ∨ (ψn ∧ ψn) for some ψ1, . . . , ψn 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 (ψ1 ∧ −ψ1) ∨ . . . ∨ (ψn ∧ ψn) for some ψ1, . . . , ψn 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, Γ ⊢ ∆, 0 ϕ, 1, Γ ⊢ ∆ 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

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Adding the inconsistency constant: semantics

Let u be a world in a de Morgan frame F = (W , ≤, δ). Then u is called inconsistent if u δ(u) . . . . . . and u is called complete if δ(u) ≤ u. u is called classical if it is both consistent and complete, i.e. if u = δ(u). The semantics of 0 and 1 is now clear: u 0 ⇔ u is inconsistent u 1 ⇔ u is complete u is inconsistent ⇔ there is a proposition p such that u p ∧ −p. u is incomplete ⇔ there is a proposition p such that u p ∨ −p. This yields precisely the logic B+.

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Basic properties of B+

Let Γ, ∆ be sets of formulas in the language {−, ∧, ∨, ⊤, ⊥}.

Theorem (intended interpretation of 0)

Γ ⊢ ∆, 0 ⇔ Γ ⊢B ∆, χ for some contradiction χ Γ, 0 ⊢ ∆ ⇔ Γ, χ ⊢B ∆ for each contradiction χ ⇔ Γ ⊢B ∆ That is, 0 behaves like ∃p (p ∧ −p). Dually, 1 behaves like ∀p (p ∨ −p).

Theorem (four-in-one conservativity)

Γ ⊢ ∆ ⇔ Γ ⊢B ∆ ⇔ Γ ∆ on all worlds Γ ⊢ ∆, 0 ⇔ Γ ⊢K ∆ ⇔ Γ ∆ on all consistent worlds 1, Γ ⊢ ∆ ⇔ Γ ⊢LP ∆ ⇔ Γ ∆ on all complete worlds 1, Γ ⊢ ∆, 0 ⇔ Γ ⊢CL ∆ ⇔ Γ ∆ on all classical worlds

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Example: expressing the inconsistency of premises

Consider the following sets of premises: Γ = −p ∨ q, p ∨ −q−p ∨ −q ∆ = −p ∨ q, p ∨ −q−p ∨ −q, p ∨ q Are these equally good sets of premises?

Adam Pˇ renosil Consistency, Completeness, and Classicality 16 / 20

Example: expressing the inconsistency of premises

Consider the following sets of premises: Γ = −p ∨ q, p ∨ −q−p ∨ −q ∆ = −p ∨ q, p ∨ −q−p ∨ −q, p ∨ q Are these equally good sets of premises? Belnap–Dunn: yes Classical logic: no Γ ⊥ but ∆ ⊢ ⊥ ∆ is equivalent to {r, −r}

Adam Pˇ renosil Consistency, Completeness, and Classicality 16 / 20

Example: expressing the inconsistency of premises

Consider the following sets of premises: Γ = −p ∨ q, p ∨ −q−p ∨ −q ∆ = −p ∨ q, p ∨ −q−p ∨ −q, p ∨ q Are these equally good sets of premises? Belnap–Dunn: yes Classical logic: no Γ ⊥ but ∆ ⊢ ⊥ ∆ is equivalent to {r, −r} Best of both worlds: Γ 0 but ∆ ⊢ 0 ∆ not equivalent to {r, −r}

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Example: disjunctive syllogism

The disjunctive syllogism is not valid in the Belnap–Dunn logic: p, −p ∨ q q In asserting p and −p ∨ q, we do not thereby assert q.

Adam Pˇ renosil Consistency, Completeness, and Classicality 17 / 20

Example: disjunctive syllogism

The disjunctive syllogism is not valid in the Belnap–Dunn logic: p, −p ∨ q q In asserting p and −p ∨ q, we do not thereby assert q. But there is something sensible about the disjunctive syllogism: p, −p ∨ q ⊢ q ∨ 0 If p and −p ∨ q are true, then so is q.

Adam Pˇ renosil Consistency, Completeness, and Classicality 17 / 20

Adding the intuitionistic implication

We can now add the intuitionistic implication → to our language. Proof-theoretically, add the standard rules for → to the sequent calculus. Semantically, add the standard clause for → to the semantics.

Theorem (conservativity over intuitionistic logic)

Let Γ, ∆ be sets of formulas in the language {∧, ∨, ⊤, ⊥, →}. Then: Γ ⊢B+ ∆ ⇔ 1, Γ ⊢B+ ∆, 0 ⇔ Γ ⊢IL ∆. Define Γ ⊢ ∆ to hold in B10

+ if and only if 1, Γ ⊢B+ ∆, 0.

B10

+ is a conservative expansion of both classical and intuitionistic logic.

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Example: reductio ad contradictionem

With the help of →, we can define a reductio ad contradictionem operator: ∼ϕ = ϕ → 0 We now have a direct, non-inferential negation (denoted −) . . . . . . as well as an indirect, inferential negation (denoted ∼). This enables us to express fine distinctions in the status of propositions. Coming back to a previous example, we now have: −p ∨ q, p ∨ −q, −p ∨ −q −p −p ∨ q, p ∨ −q, −p ∨ −q ⊢ ∼p

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Conclusion

We have expanded the Belnap–Dunn logic by a pair of constants 0 and 1. These allow us to represent inconsistency and completeness. We have seen that this brings new expressive capabilities. In particular, it enables us to naturally combine classical and intuitionistic logic within a single system in a novel way.

Adam Pˇ renosil Consistency, Completeness, and Classicality 20 / 20

Conclusion

We have expanded the Belnap–Dunn logic by a pair of constants 0 and 1. These allow us to represent inconsistency and completeness. We have seen that this brings new expressive capabilities. In particular, it enables us to naturally combine classical and intuitionistic logic within a single system in a novel way. . . . now how about doing the same for Lukasiewicz logic?

Adam Pˇ renosil Consistency, Completeness, and Classicality 20 / 20

Conclusion

We have expanded the Belnap–Dunn logic by a pair of constants 0 and 1. These allow us to represent inconsistency and completeness. We have seen that this brings new expressive capabilities. In particular, it enables us to naturally combine classical and intuitionistic logic within a single system in a novel way. . . . now how about doing the same for Lukasiewicz logic?

Thank you for your attention.

Adam Pˇ renosil Consistency, Completeness, and Classicality 20 / 20