Quantifiers and Terms in Intermediate Logics Non-Classical Modal and Predicate Logics Guangzhou, December 6, 2017 Rosalie Iemhoff Utrecht University, the Netherlands 1 / 21
Quantifiers are complicated. 2 / 21
Quantifiers in intermediate logic Examples of formulas that hold in classical logic but not in intuitionistic and many other intermediate logics: ◦ ∃ x ϕ ( x ) ↔ ¬∀ x ¬ ϕ ( x ). � � ◦ ∀ x ϕ ( x ) ∨ ¬ ϕ ( x ) . � � ◦ ∃ x ϕ ( x ) → ∀ y ϕ ( y ) (the drinker’s paradox). Some weaker versions do hold in all intermediate logics: ◦ ∃ x ϕ ( x ) → ¬∀ x ¬ ϕ ( x ), � � ϕ ( x ) ∨ ¬ ϕ ( x ) . ◦ ∀ x ¬¬ 3 / 21
Constructive reading of quantifiers According to the Brouwer–Heyting–Kolmogorov interpretation : ∃ x ϕ ( x ) is given by providing a d ∈ D and a proof of ϕ ( d ). ∀ x ϕ ( x ) is a construction that transforms a proof of d ∈ D into a proof of ϕ ( d ). Thm Heyting Arithmetic HA, the constructive theory of the natural numbers, has the Existence Property : ⊢ HA ∃ x ϕ ( x ) implies ⊢ HA ϕ ( t ) for some term t . Heyting Arithmetic is consistent with Church Thesis, which states that: if ∀ x ∃ y ϕ ( x , y ), then there exists a total computable function h such that for all numbers n : ϕ ( n , hn ). Note Peano Arithmetic does not have the Existence Property and is not consistent with Church Thesis. The study of the constructive content of such quantifier combinations is pursued in contructive mathematics and proof mining . 4 / 21
Quantifiers in nonclassical logic Here: study other quantifier combinations that distinguish classical logic from (some) other intermediate logics. To better understand this differences between classical and nonclassical quantification, the relation between quantifiers and terms, as illustrated by the Skolemization method for classical logic, is explored. Dfn CQC and IQC denote classical and intuitionistic predicate logic. Given a logic L , Γ ⊢ L ϕ denotes that ϕ is derivable from Γ in the logic L . 5 / 21
Skolemization in classical logic Thm For any function symbol f not in ϕ ( fx short for f ( x )): ⊢ CQC ∃ x ∀ y ϕ ( x , y ) iff ⊢ CQC ∃ x ϕ ( x , fx ) . Prf By contraposition. For any function symbol f not in ϕ : ∃ x ∀ y ϕ ( x , y ) does not hold iff ∀ x ∃ y ¬ ϕ ( x , y ) is satisfiable iff ∀ x ¬ ϕ ( x , fx ) is satisfiable iff ∃ x ϕ ( x , fx ) does not hold. In a counter model to ∃ x ∀ y ϕ ( x , y ), f chooses, for every x , a counter witness fx such that ¬ ϕ ( x , fx ). Thm For any formula ϕ , theory T , and function symbols f 1 , f 2 not in ϕ and T , where f i has arity i ( fx 1 x 2 short for f ( x 1 , x 2 )): T ⊢ CQC ∃ x 1 ∀ y 1 ∃ x 2 ∀ y 2 ϕ ( x 1 , x 2 , y 1 , y 2 ) ⇐ ⇒ T ⊢ CQC ∃ x 1 x 2 ϕ ( x 1 , x 2 , f 1 x 1 , f 2 x 1 x 2 ). Similarly for more quantifiers. 6 / 21
Thoralf Skolem (1887–1963) 7 / 21
Skolemization for infix formulas Dfn An occurrence of a quantifier ∀ x ( ∃ x ) in a formula is strong if it occurs positively (negatively) in the formula, and weak otherwise. Strong quantifiers become universal under prenexification. Ex Quantifiers ∃ x and ∀ y occur strongly in ∃ x ϕ ( x ) → ∀ y ψ ( y ) and weakly in ∃ x ϕ ∧ ¬∀ y ψ ( y ). In the skolemization ϕ s of ϕ all strong quantifiers are replaced in the “same” way as in prenex Skolemization. Ex ∃ x ( ∃ y ϕ ( x , y ) → ∀ z ψ ( x , z )) s = ∃ x ( ϕ ( x , fx ) → ψ ( x , gx )). WLEM s WLEM ¬¬∀ x ( ϕ x ∨ ¬ ϕ x ) ¬¬ ( ϕ c ∨ ¬ ϕ c ) CD s CD ∀ x ( ϕ x ∨ ψ ) → ∀ x ϕ x ∨ ψ ∀ x ( ϕ x ∨ ψ ) → ϕ c ∨ ψ In case ϕ is in prenex normal form, this definition of Skolemization coincides with the earlier one. Fact For any formula ϕ and any theory T : T ⊢ CQC ϕ ⇔ T ⊢ CQC ϕ s . 8 / 21
Skolemization in intermediate logics Question Does there exist the same connection between terms and quantifiers in intuitionistic logic or other intermediate logics? Answer No, but . . . rest of the talk. 9 / 21
Skolemization in intuitionistic logic Question Does Skolemization hold in IQC ? For any formula ϕ , any theory T : T ⊢ IQC ϕ iff T ⊢ IQC ϕ s ? Answer No. Counterexample: � � � � ϕ ( x ) ∨ ¬ ϕ ( x ) ϕ ( c ) ∨ ¬ ϕ ( c ) �⊢ IQC ¬¬∀ x ⊢ IQC ¬¬ . 10 / 21
Nonclassical theories Dfn For a theory T , Skolemization is sound ( ⇒ ) and complete ( ⇐ ) if T ⊢ ϕ ⇔ T ⊢ ϕ s A theory admits Skolemization if Skolemization is both sound and complete. Note Many nonclassical theories (including IQC ) do not admit Skolemization: it is sound but not complete for such theories. Examples for IQC are WLEM s WLEM ¬¬∀ x ( ϕ x ∨ ¬ ϕ x ) ¬¬ ( ϕ c ∨ ¬ ϕ c ) CD s CD ∀ x ( ϕ x ∨ ψ ) → ∀ x ϕ x ∨ ψ ∀ x ( ϕ x ∨ ψ ) → ϕ c ∨ ψ From now on, ϕ x abbreviates ϕ ( x ). 11 / 21
Skolemization in nonclassical logics ◦ A sufficient condition on formulas for their completeness under Skolemization in IQC (Mints). ◦ Completeness of Skolemization for the prenex fragment of first-order G¨ odel logics (Baaz, Ciabattoni, Ferm¨ uller, Zach). ◦ Completeness of Skolemization for the prenex fragment for a wide range of first-order fuzzy logics (Baaz, Metcalfe). ◦ Completeness of Skolemization for first-order � Lukasiewicz logic (Baaz, Metcalfe). ◦ Completeness of Skolemization for certain formula classes of first-order substructural logics (Cintula, Metcalfe). ◦ An alternative Skolemization method for the extension of IQC by an existence predicate (Baaz, Iemhoff). ◦ Completeness of Skolemization for a labelled version of IQC (Baaz, Iemhoff). . . . ◦ 12 / 21
Two questions about any intermediate logic ◦ For which formulas is skolemization complete? ◦ Are there useful alternative skolemization methods? The answer to the first question depends on the meaning of “alternative skolemization method”. 13 / 21
Prenex fragment Thm (Mints 1972) Sufficient condition on formulas ϕ for ( ⊢ IQC ϕ s ⇔ ⊢ IQC ϕ ). Thm (Baaz&Iemhoff 2017) Skolemization is sound and complete w.r.t. prenex formulas in any intermediate logic that is sound and complete w.r.t. a class of frames. Prf Not completely trivial. ⊣ Thm / Conj (this proof still needs to be LaTeX-iced) The prenex fragments of IQC and IQC + CD are equal. Conj The above results extend to positive prenex formulas , the class of formulas obtained by closing the class of prenex formulas under conjunction, disjunction and implication. 14 / 21
Alternative Skolemization methods Dfn An alternative Skolemization method is a computable total translation ( · ) a from formulas to formulas such that for all formulas ϕ , ϕ a does not contain strong quantifiers. A theory T admits the alternative Skolemization method if T ⊢ ϕ ⇔ T ⊢ ϕ a . (1) The method is strict if for all Kripke models K of T and all formulas ϕ : K � � ϕ a ⇒ K � � ϕ. (2) Ex Replacing quantifiers ∃ x ψ ( x , ¯ y ) and ∀ x ψ ( x , ¯ y ) by n n � � ψ ( f i (¯ y ) , ¯ y ) and ψ ( f i (¯ y ) , ¯ y ) , i =1 i =1 respectively, is an alternative Skolemization method, admitted by any intermediate logic with the finite model property (Baaz&Iemhoff 2016). Note Eskolemization is not an alternative Skolemization method. 15 / 21
� � � � Kripke models Dfn Given a class of Kripke models K , K cd denotes the set of those models in K that have constant domains. Dfn For a Kripke model K : ◦ K ↓ denotes the Kripke model that is the result of replacing every domain in K by the domain at the root of K and defining, for elements ¯ d in D : K ↓ , k � P ( ¯ d ) iff K , k � P ( ¯ d ). ⇒ K ↓ K D 1 D 2 D 0 D 0 D 0 D 0 ◦ K ↑ denotes the Kripke model that is the result of replacing every domain in K by the union of all domains in K and defining, for elements ¯ d in that union: K ↑ , k � P ( ¯ d ) if K , k � P ( ¯ d ) and ¯ d are elements in D k . 16 / 21
� � � � Kripke models Dfn Given a class of Kripke models K , K cd denotes the set of those models in K that have constant domains. Dfn For a Kripke model K : ◦ K ↓ denotes the Kripke model that is the result of replacing every domain in K by the domain at the root of K and defining, for elements ¯ d in D : K ↓ , k � P ( ¯ d ) iff K , k � P ( ¯ d ). K ↑ ⇒ � � D 1 K D 2 i D i i D i D 0 � i D i ◦ K ↑ denotes the Kripke model that is the result of replacing every domain in K by the union of all domains in K and defining, for elements ¯ d in that union: K ↑ , k � P ( ¯ d ) if K , k � P ( ¯ d ) and ¯ d are elements in D k . 17 / 21
No alternative Skolemization Dfn The strong quantifier free fragment (sqff) of a theory consists of those theorems of the theory that do not contain strong quantifiers, and likewise for weak quantifiers. Thm Let T be a theory that is sound and complete with respect to a class of Kripke models K closed under ↑ and ↓ , then the sqff of T is sound and complete with respect to K cd , and so is the wqff. Thm (Iemhoff 2017) Except for CQC , there is no intermediate logic that is sound and complete with respect to a class of frames and that admits a strict, alternative Skolemization method. Cor The intermediate logics IQC , ◦ QDn (the logic of frames of branching at most n ), ◦ QKC (the logic of frames with one maximal node), ◦ QLC (the logic of linear frames), and all tabular logics, do not admit any strict, alternative Skolemization method. 18 / 21
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