Consequence relations old and new Workshop Proofs Paris, 1 June, - - PowerPoint PPT Presentation

Consequence relations – old and new

Workshop Proofs Paris, 1 June, 2017 Rosalie Iemhoff Utrecht University

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Logics A logic L can be given via

• semantics (algebras, Kripke models, valuations, . . . ),
• proof systems (sequent calculi, Hilbert systems, resolution, . . . ),
• or in some other way (cutting planes, . . . ).

In most cases there are many representations of the same logic. Ex The theorems of classical propositional logic CPC are the formulas

• true under all valuations,
• true in all Boolean algebras,
• derivable in the propositional fragment of natural deduction NK.

In terms of inference these representations are equal: ϕ follows from Γ in CPC iff ϕ is satisfied by all valuations that satisfy Γ iff ϕ is true in all Boolean algebras in which Γ is true iff ϕ is derivable from Γ in the propositional fragment of NK.

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Theorems versus inferences There are logics with the same set of theorems, but different inferences. Ex 1 A propositional logic with truth–values {−1, 0, 1} where x ∧ y = min{x, y} x ∨ y = max{x, y} ¬x = −x. A formula holds in L if its value is in {0, 1} for every valuation. Thus the theorems of L are the theorems of classical propositional logic CPC. L is not the same logic as CPC: p, ¬p ⊢L q p, ¬p ⊢CPC q.

1Avron, Simple Consequence Relations, 1991

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Inferences Consequence relations provide a framework to reason about inference.

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History Tarski and G¨

• del in Vienna, 1935

In 1935 Tarski spoke at the International Congress of Scientific Philosophy in Paris about logical consequence. He characterized what it means for a given sentence to follow from a given set of sentences.

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Consequence relations Dfn Given a set of expressions, a single–conclusion consequence relation (scr) is a relation ⊢ between finite sets of expressions and expressions such that reflexivity {ϕ} ⊢ ϕ weakening Γ ⊢ ϕ implies Γ, Γ′ ⊢ ϕ transitivity Γ ⊢ ϕ and Γ′, ϕ ⊢ ψ implies Γ, Γ′ ⊢ ψ structurality Γ ⊢ ϕ implies σΓ ⊢ σϕ, for all substitutions σ. Expression ϕ is a theorem if ∅ ⊢ ϕ, denoted ⊢ ϕ. The set of theorems of ⊢ is denoted by Th(⊢). The minimal scr in a given language: Γ ⊢

m ϕ iff ϕ ∈ Γ.

Ex Two scr in the language of propositional logic: Γ ⊢1 ϕ ≡df ϕ ∈ Γ or ϕ is a tautology of CPC Γ ⊢2 ϕ ≡df (

• Γ → ϕ) is a tautology of CPC.

Th(⊢1) = Th(⊢2), but not ⊢1 = ⊢2: ϕ, ψ ⊢1 ϕ ∧ ψ but ϕ, ψ ⊢2 ϕ ∧ ψ.

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Consequence relations and logics Dfn Given a logic L, Th(L) denotes the set of theorems of L. Dfn A scr ⊢ covers a logic L if Th(⊢) = Th(L). Ex The following two scr cover CPC: Γ ⊢1 ϕ ≡df ϕ ∈ Γ or ϕ is a tautology of CPC Γ ⊢2 ϕ ≡df (

• Γ → ϕ) is a tautology of CPC.

Ex For any “decent” proof system P the following is a natural scr: Γ ⊢P ϕ ≡df there is a proof of ϕ from Γ in P.

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This talk

• A full description of the consequence relations that cover a given

intermediate, modal or predicate logic.

• Complexity issues regarding consequence relations.
• The connection between consequence relations and unification.

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Rules Dfn A rule is an expression of the form Γ/ϕ or Γ ϕ, where ϕ is an expression and Γ a finite set of expressions in a certain language, which in this talk is the language of propositional, modal or predicate logic. If Γ is empty, the rule is also called an axiom. Ex Rules in Hilbert systems, sequent calculi, and resolution, respectively: ϕ(x) ∀xϕ(x) (Γ ⇒ ϕ, ∆) (Γ, ϕ ⇒ ∆) Γ ⇒ ∆ X ∪ {p} {¬p} ∪ Y X ∪ Y . Dfn Given a set of rules R and a scr ⊢, ⊢R is the smallest scr that contains ⊢ and such that Γ ⊢R ϕ for all Γ/ϕ in R.

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Derivable and admissible Dfn Given a scr ⊢ and a rule R = Γ/ϕ: R is derivable iff Γ ⊢ ϕ; R is strongly derivable iff ⊢ Γ → ϕ; R is admissible ( Γ |

∼ϕ) iff Th(⊢) = Th(⊢R).

Ex ϕ(x)/∀xϕ(x) is admissible in many theories. ⊥/ϕ is admissible in any consistent logic, but not always derivable. ϕ/✷ϕ and ✷ϕ/ϕ are admissible in many modal logics. Cut is admissible in LK − {Cut}. Note If ⊢ satisfies the deduction theorem, then derivable and strongly derivable mean the same. The minimal consequence relation ⊢ for which Th(⊢) = Th(L) is {Γ ⊢ ϕ | ϕ ∈ Γ ∪ Th(L)}.

|

∼ is the maximal consequence relation such that Th(| ∼) = Th(L).

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Aim A full description of the admissible rules of a given intermediate, modal

• r predicate logic.

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Intermediate logics Thm All admissible rules of CPC are strongly derivable. Prf If ϕ/ψ is admissible, then for all substitutions σ to {⊤, ⊥}: if ⊢CPC σϕ, then ⊢CPC σψ. Thus ϕ → ψ is true under all valuations. Hence ⊢CPC ϕ → ψ. ⊣ Thm The Kreisel–Putnam rule, ¬ϕ → ψ ∨ χ (¬ϕ → ψ) ∨ (¬ϕ → χ) is admissible in any intermediate logic, and in Heyting Arithmetic. In any consequence relation that covers intuitionistic propositional logic IPC and satisfies the deduction theorem, the rule is nonderivable, since the corresponding implication does not hold in IPC: (¬ϕ → ψ ∨ χ) → (¬ϕ → ψ) ∨ (¬ϕ → χ).

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Admissible rules Dfn Γ/ϕ is admissible ( Γ |

∼ϕ) iff Th(⊢) = Th(⊢R).

Thm Γ |

∼ϕ iff for all substitutions σ: ⊢ σΓ implies ⊢ σϕ.

(Here appears the connection with unification.) Cor The admissible rules of any two consequence relations ⊢1 and ⊢2 that cover the same logic L are equal:

|

∼ 1 = | ∼ 2. But the derivable

rules of ⊢1 and ⊢2 are different when ⊢1 and ⊢2 are different. Dfn When given a logic L, its admissible rules are the admissible rules of any consequence relation that covers it. With every logic L, the consequence relation ⊢L is associated, where Γ ⊢L ϕ ≡df (

• Γ → ϕ) ∈ Th(L).

A rule R is derivable in L if it is derivable in ⊢L.

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Other logics Thm The Kreisel–Putnam rule, ¬ϕ → ψ ∨ χ (¬ϕ → ψ) ∨ (¬ϕ → χ) is admissible in any intermediate logic, and not derivable in IPC. Thm In Belnap’s system for relevance logic the disjunctive syllogism A ¬A ∨ B B is admissible but not derivable.

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Bases Dfn A set of rules R is a basis for the admissible rules of a logic L if

|

∼ L

is equal to ⊢R: Γ |

∼ Lϕ ⇔ Γ ⊢R ϕ.

The set of all admissible rules of L is a basis for its admissible rules. Aim: Provide minimal bases for the admissible rules of a given intermediate, modal or predicate logic. Dfn V• is the collection of rules of the form ✷Γ → ✷∆ { Γ → ϕ | ϕ ∈ ∆} V• (Γ, ∆ finite sets of formulas) (✷Γ = {✷ϕ | ϕ ∈ Γ}) Instance: ✷p → ✷q ∨ ✷r/(p → q) ∨ (p → r). Thm (Jeˇ r´ abek ’05) The set of rules V• form a basis for the admissible rules of GL.

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Decidability Thm (Chagrov 1992) There are decidable (modal) logics with an undecidable admissibility relation. Thm (Rybakov 1990s) The admissibility relation of IPC and the modal logics K4, GL, and S4 is decidable. Thm (Jeˇ r´ abek 2007) Checking admissibility is coNEXP-complete in IPC, K4, GL, and S4.

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Results about intermediate logics Dfn A logic L is structurally complete iff all admissible rules are derivable. Thm CPC is structurally complete. Thm (Rozi` ere ’95 & Iemhoff ’01, building on Ghilardi ’99) The Visser rules V are a basis for the admissible rules of IPC. Thm (Iemhoff ’05) The Visser rules are a basis for the admissible rules in all intermediate logics in which they are admissible. In particular, they are a basis in KC. Thm (Iemhoff ’05) For any intermediate logic L, if ϕ |

∼ Lψ, then ϕ ⊢V ψ.

Thm (Wro´ nski ’08) Any extension of L ⊇ LC is structurally complete. LC (ϕ → ψ) ∨ (ψ → ϕ)

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Results about modal logics Thm (Jeˇ r´ abek ’05, building on Ghilardi ’01) V• is a basis for the admissible rules in any L ⊇ GL in which it is

• admissible. Similarly for V◦ and S4.

Thm (Dzik & Wojtylak ’11) All logics L ⊇ S4.3 are structurally complete. S4.3 ✷(✷A → ✷B) ∨ ✷(✷B → ✷A)

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Results about fragments Thm (Mints ’76) In IPC, all nonderivable admissible rules contain ∨ and →. Thm (Prucnal ’83) IPC→ is structurally complete, as is IPC→,∧. Thm (Cintula & Metcalfe ’10) The Wro´ nski rules are a basis for the admissible rules of IPC→,¬.

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Results about predicate theories Thm (Pogorzelski & Prucnal ’75) Classical predicate logic is structurally complete. (under the correct definition of substitution.) Thm (Iemhoff & Visser) The Visser rules form a basis for the propositional admissible rules of Heyting Arithmetic.

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Unification Key ingredients is many of the results on admissibility are results about unification in logic. They concern the substitutions under which a formula becomes derivable. This aspect of admissibility connects it to computer science, and in particular to areas such as Description Logic.

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Complexity Can the addition of admissible rules reduce the size of proofs in a logic? Sometimes yes: Thm There is a nonelementary increase in proof size between LK and LK–Cut. Sometimes no: Dfn A Hilbert system (Frege system) for IPC consists of a finite number

• f formulas that hold in IPC and a finite number of rules admissible in

IPC such that any formula ϕ that holds in IPC can be derived using these axioms and rules. Thm (Mints & Kojevnikov ’04) Any two Hilbert systems F1 and F2 for IPC polynomially simulate each

• ther: There is a polynomial p(x) such that any formula with a proof in

F1 of size n, there exists a proof of the formula of size ≤ p(n) in F2.

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To study a logic, play by the rules.

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What I would like to do

• Predicate logic;
• Nontransitive modal logics;
• Substructural logics.

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Finis

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