Quantifiers in nonclassical logics Rosalie Iemhoff Utrecht University Algebra and Coalgebra meets Proof Theory Vienna, 7–9 April 2016 1 / 23
The seventh edition of ALCOP On 21 July 2009 Nick Bezhanishvili and Clemens Kupke wrote to Alessandra Palmigiano and me: “Dear Alessandra, Rosalie, We recently learned that there is funding available for small scale workshops between the UK and the Netherlands. We thought this is a good opportunity to get together - ‘strengthen our already well-established contacts’ to use grant application vocabulary :)” What’s in a name: First named ACPT, soon thereafter ALCOP. 2 / 23
Quantifiers versus functions Ex If ∀ x ∃ y ϕ ( x , y ) holds, then introduce a fresh function f and add as axiom ∀ x ϕ ( x , fx ) . This is a conservative extension: ⊢ CQC ∀ x ϕ ( x , fx ) → ψ ⇔ ⊢ CQC ∀ x ∃ y ϕ ( x , y ) → ψ. 3 / 23
Skolemization Method � � Thm (Skolem 1920) ⊢ CQC ∃ x ∀ y ϕ ( x , y ) ⇔ ⊢ CQC ∃ x ϕ x , f ( x ) � � ⊢ CQC ∃ x 1 ∀ y 1 ∃ x 2 ∀ y 2 ϕ (¯ x , y 1 , y 2 ) ⇔ ⊢ CQC ∃ x 1 ∃ x 2 ϕ x , f 1 ( x 1 ) , f 2 ( x 1 , x 2 ) ¯ In general: ⊢ CQC ∃ x 1 ∀ y 1 . . . ∃ x n ∀ y n ϕ (¯ x , y 1 , . . . , y n ) ⇔ � � ⊢ CQC ∃ x 1 . . . ∃ x n ϕ x , f 1 ( x 1 ) , f 2 ( x 1 , x 2 ) , . . . , f n ( x 1 , . . . , x n ) ¯ . ⇔ � S � ⊢ CQC ∃ x 1 ∀ y 1 . . . ∃ x n ∀ y n ϕ (¯ x , ¯ y ) 4 / 23
Herbrand’s Theorem Thm (Herbrand) For every quantifier-free ϕ : k � ⊢ CQC ∃ ¯ x ϕ (¯ x ) ⇔ for some terms t 11 , . . . , t nk : ⊢ CQC ϕ ( t 1 i , . . . , t ni ) . i = 1 Cor For every quantifier-free ϕ : ⊢ CQC ∃ x 1 ∀ y 1 . . . ∃ x n ∀ y n ϕ (¯ x , ¯ y ) ⇔ � � ⊢ CQC ∃ x 1 . . . ∃ x n ϕ x , f 1 ( x 1 ) , f 2 ( x 1 , x 2 ) , . . . , f n ( x 1 , . . . , x n ) ¯ ⇔ for some terms t 11 , . . . , t nk : � k � � ⊢ CPC t i 1 , . . . , t ni , f 1 ( t i 1 ) , f 2 ( t i 1 , t i 2 ) , . . . , f n ( t i 1 , . . . , t ni ) i = 1 ϕ � k Ex ⊢ CQC ∃ x ∀ y ϕ ( x , y ) ⇔ for some t 1 , . . . , t k : ⊢ CPC i = 1 ϕ ( t i , f ( t i )) 5 / 23
Nonclassical theories Skolemization in nonclassical logics/theories? 6 / 23
Skolemization for infix formulas Dfn A quantifier occurrence in ϕ is strong when it is a positive occurrence of a universal or a negative occurrence of an existential quantifier. A quantifier occurrence in ϕ is weak if it is not strong. If put in prenex normal form, strong quantifier occurrences become universal and weak quantifier occurrences become existential quantifiers. Dfn The (infix) Skolemization, ϕ s , of ϕ is the result of replacing in ϕ all strong quantifier occurrences Qx ψ ( x , ¯ y ) by ψ ( f (¯ y ) , ¯ y ) , starting with the leftmost quantifier, where ¯ y are the variables of the weak quantifiers in the scope of which Qx ψ ( x , ¯ y ) occurs and f is a fresh function symbol in the skolem language L s ⊇ L . Thm CQC admits Skolemization: ⊢ CQC ϕ ⇔ ⊢ CQC ϕ s . 7 / 23
Complexity of Skolemization Thm (Baaz & Leitsch 1994) Prefix Skolemization may result in a nonelementary increase of Herbrand complexity (the minimal number of disjuncts in a Herbrand disjunction). Thm (Avigad 2003) If the underlying theory allows for a modicum of coding, Skolem functions can be eliminated in polynomial time. Thm (Baaz & Hetzl & Weller 2012) The complexity of deSkolemization is the complexity of the function that given a proof of the Skolemization of a formula gives the length of the shortest proof of the formula. For cut-free proofs this complexity is exponential. Related work on Herbrand expansions in classical logic by Baaz & Hetzl & Straßburger appeared recently. 8 / 23
Skolemization in nonclassical logics Note Not all intermediate logics admit Skolemization. Ex In intuitionistic predicate logic IQC : �⊢ IQC ¬¬∃ x ϕ ( x ) → ∃ x ¬¬ ϕ ( x ) ⊢ IQC ¬¬ ϕ ( c ) → ∃ x ¬¬ ϕ ( x ) DNS �⊢ IQC ∀ x ¬¬ ϕ ( x ) → ¬¬∀ x ϕ ( x ) ⊢ IQC ∀ x ¬¬ ϕ ( x ) → ¬¬ ϕ ( c ) CD �⊢ IQC ∀ x ( ϕ ( x ) ∨ ψ ) → ∀ x ϕ ( x ) ∨ ψ ⊢ IQC ∀ x ( ϕ ( x ) ∨ ψ ) → ϕ ( c ) ∨ ψ. 9 / 23
Two directions I Develop alternative methods that a given theory admits. II Establish which theories admit a given alternative method. Question: What is an alternative Skolemization method? Dfn Necessary condition for an alternative Skolemization method: It is a (computable) translation ( · ) a of formulas in L to formulas in L a ⊇ L such that ϕ a does not contain strong quantifiers. A logic L admits the alternative method if ⊢ L ϕ ⇔ ⊢ L ϕ a . The method is strict if for all ϕ and all models K for L a : K � ϕ a , K ↾ L � ϕ ⇔ where K ↾ L the restriction of K to L . 10 / 23
No strict alternatives Thm (Iemhoff ’16) No intermediate logic that is sound and complete with respect to a class of frames, admits a strict, alternative Skolemization method. This holds for IQC is particular. Dfn Given a logic L the fragment consisting of formulas without strong (weak) quantifiers is denoted by L w ( L s ) . The theorem is a corollary of the following theorem. Thm For every intermediate logic that is sound and complete with respect to a class F of frames, the fragments L w ( L s ) are sound and complete with respect to the class of constant domain models on frames in F . 11 / 23
� � � No strict alternatives Thm For every intermediate logic that is sound and complete with respect to a class F of frames, the fragments L w ( L s ) are sound and complete with respect to the class of constant domain models on frames in F . Dfn Given a rooted Kripke model K , K ↓ is the model obtained by replacing every domain by the domain at the root, and K ↑ is the model obtained by replacing every domain by the union of all domains in the model. K ↓ : K ↑ : K : 1 R 1 N 1 R 0 N 0 N 0 R ∀ r �∈ N � � P ( r ) Lem For all formulas ψ without weak quantifiers: K ↓ , k � ψ K ↑ , k � � ψ. K , k � ψ ⇒ K , k � � ψ ⇒ For all formulas ψ without strong quantifiers: K ↑ , k � ψ K ↓ , k � � ψ. K , k � ψ ⇒ K , k � � ψ ⇒ Cor �⊢ L w ϕ implies K ↓ � � ϕ , and �⊢ L s ϕ implies K ↑ � � ϕ . 12 / 23
Alternatives that are not strict Dfn IQCE is a conservative extension of IQC with an existence predicate E such that Et denotes that t exists. Dfn The eskolemization, ϕ e , of ϕ is the result of replacing all strong quantifier occurrences Qx ψ ( x , ¯ y ) in ϕ (starting with the leftmost) by � Ef (¯ y ) ∧ ψ ( f (¯ y ) , ¯ y ) if Q = ∃ Ef (¯ y ) → ψ ( f (¯ y ) , ¯ y ) if Q = ∀ , where ¯ y are the variables of the weak quantifiers in the scope of which Qx ψ ( x , ¯ y ) occurs and f is a fresh function symbol. Thm (Baaz & Iemhoff ’11) The logic IQCE , which is a conservative extension of IQC with an existence predicate E , admits eskolemization for existential quantifiers. 13 / 23
Alternatives that are not strict Thm (Baaz & Iemhoff ’08) The logic IQCO , which is a logic with labelled formulas that is a conservative extension of IQC with an existence predicate and partial order, admits an alternative Skolemization method called orderization. Question: In intuitionistic logic Skolem functions are partial? Related work: Skolemization in fuzzy logics (Baaz, Ciabattoni, Cintula, Diaconescu, Metcalfe, . . . ) 14 / 23
Two directions I Develop alternative methods that a given logic/theory admits. II Establish which logics/theories admit the (alternative) Skolemization method. Mix: Develop an alternative method and establish which logics/theories admit the alternative method. 15 / 23
Parallel Skolemization Dfn (Baaz & Iemhoff) The parallel Skolemization (pskolemization), ϕ p , of ϕ is the result of replacing, starting with the leftmost, strong quantifier occurrences Qx ψ ( x , ¯ y ) in ϕ by � � n i = 1 ψ ( f i (¯ y ) , ¯ y ) if Q = ∃ � n i = 1 ψ ( f i (¯ y ) , ¯ y ) if Q = ∀ , where ¯ y are the variables of the weak quantifiers in the scope of which Qx ψ ( x , ¯ y ) occurs, n ∈ N , and the f i are fresh function symbols. Thm (Baaz & Iemhoff ’14) Every intermediate logic that is sound and complete with respect to a class of constant domain finite width Kripke models with the witness property, admits parallel Skolemization. Cor Every intermediate logic with the constant domain finite model property admits parallel Skolemization. Thm (Cintula & Diaconescu & Metcalfe ’15) Every substructural logic with a semantics that has the n -witnessed model property, admits parallel Skolemization left and right of degree n . 16 / 23
� � � � Parallel Skolemization Thm (Baaz & Iemhoff ’14) Every intermediate logic that is sound and complete with respect to a class of constant domain finite width Kripke models with the witness property, admits parallel Skolemization. Prf Sufficient: If K � � ϕ , then there exists a model K p such that K p � � ϕ p . Simple case: ϕ = ϕ [ ∃ x ψ ( x )] and ϕ p = ϕ [ ψ ( c 1 ) ∨ ψ ( c 2 )] . To show: K , k � ϕ [ ∃ x ψ ( x )] ⇔ K p , k � ϕ [ ψ ( c 1 ) ∨ ψ ( c 2 )] . K p k 1 � ψ ( 1 ) k 2 � ψ ( 2 ) k 1 � ψ ( c 1 ) k 2 � ψ ( c 2 ) K k 0 k 0 If c 1 is interpreted as 1 and c 2 as 2, then for i = 0 , 1 , 2 : K , k i � ∃ x ψ ( x ) ⇔ K p , k i � ψ ( c 1 ) ∨ ψ ( c 2 ) . The witness property garantees the existence of elements in the domain that can be the interpretation of the c i , or the f i , in the general case. ⊣ 17 / 23
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