Quantifiers and Functions in Intuitionistic Logic Collegium Logicum Proof Theory: Herbrand’s Theorem revisited Vienna, 25–27 May, 2017 Rosalie Iemhoff Utrecht University, the Netherlands 1 / 28
Skolemization in classical logic Thm For any formula ϕ and function symbol f that does not occur in ϕ : ⊢ c ∃ x ∀ y ϕ ( x , y ) ⇔ ⊢ c ∃ x ϕ ( x , f ( x )) . ( ⊢ c denotes derivability in classical predicate logic CQC ) The equivalent in terms of satisfiability: Thm For any formula ϕ and function symbol f that does not occur in ϕ : ∀ x ∃ y ϕ ( x , y ) is satisfiable if and only if ∀ x ϕ ( x , f ( x )) is satisfiable. Thm For any formula ϕ and any theory T, for any function symbols f 1 , f 2 , . . . , f n not in ϕ and T: T ⊢ c ∃ x 1 ∀ y 1 . . . ∃ x n ∀ y n ϕ ( x 1 , y 1 , . . . , x n , y n ) ⇔ � � T ⊢ c ∃ x 1 . . . ∃ x n ϕ x 1 , f 1 ( x 1 ) , x 2 , f 2 ( x 1 , x 2 ) , . . . , x n , f n ( x 1 . . . x n ) . 2 / 28
Functions and quantifiers in intuitionistic logic Question Does there exist the same connection between functions and quantifiers in intuitionistic logic as in classical logic? Answer No, but . . . (content of the rest of the talk). Note Quantifiers in classical logic are different from those in intuitionistic logic. The following formulas hold in the former but not in the latter: ◦ ∃ x ϕ ( x ) ↔ ¬∀ x ¬ ϕ ( x ) , � � ◦ ∃ x ϕ ( x ) → ∀ y ϕ ( y ) , � � ◦ ¬¬∀ x ϕ ( x ) ∨ ¬ ϕ ( x ) . 3 / 28
Constructive reading of quantifiers In a constructive reading: ◦ a proof of ∀ x ϕ ( x ) consists of a construction that from a proof that d belongs to the domain produces a proof of ϕ ( d ) . ◦ a proof of ∃ x ϕ ( x ) consists of the construction of an element d in the domain and a proof of ϕ ( d ) . Thus in a constructive reading, a proof of ∀ x ∃ y ϕ ( x , y ) consists of a construction that for every d in the domain produces an element e in the domain and a proof of ϕ ( d , e ) . Heyting Arithmetic, the constructive theory of the natural numbers, is consistent with Church Thesis, which states that if ∀ x ∃ y ϕ ( x , y ) holds, then there exists a total computable function h such that ∀ x ϕ ( x , h ( x )) holds. But what about the quantifier combination ∃ x ∀ y? What is to be concluded from the derivability of ∃ x ∀ y ϕ ( x , y ) ? 4 / 28
Skolemization in intuitionistic logic Question Does Skolemization hold in IQC? For any formula ϕ and any theory T, for any function symbols f 1 , f 2 , . . . , f n not in ϕ and T: ¿ T ⊢ i ∃ x 1 ∀ y 1 . . . ∃ x n ∀ y n ϕ ( x 1 , y 1 , . . . , x n , y n ) ⇐ ⇒ � � T ⊢ i ∃ x 1 . . . ∃ x n ϕ x 1 , f 1 ( x 1 ) , x 2 , f 2 ( x 1 , x 2 ) , . . . , x n , f n ( x 1 . . . x n ) ? ⊢ i denotes derivability in intuitionistic predicate logic IQC. Answer No. Counterexample: � � � � �⊢ i ¬¬∀ x ϕ ( x ) ∨ ¬ ϕ ( x ) ⊢ i ¬¬ ϕ ( c ) ∨ ¬ ϕ ( c ) . 5 / 28
Prenex normal form Fact In intuitionistic predicate logic IQC not every formula has a prenex normal form. Dfn An occurrence of a quantifier ∀ x ( ∃ x ) in a formula is strong if it occurs positively (negatively) in the formula, and weak otherwise. Ex ∃ x and ∀ y occur strongly in ∃ x ϕ ( x ) → ∀ y ψ ( y ) and weakly in ∃ x ϕ ( x ) ∧ ¬∀ y ψ ( y ) . In ∃ x 1 ∀ y 1 . . . ∃ x n ∀ y n ϕ , the ∀ y i are strong occurrences and the ∃ x i are weak occurrences. 6 / 28
Skolemization for infix formulas Dfn An occurrence of ∀ x ( ∃ x ) in a formula is strong if it occurs positively (negatively) in the formula, and weak otherwise. ϕ s is the skolemization of ϕ if it does not contain strong quantifiers and there are formulas ϕ = ϕ 1 , . . . , ϕ n = ϕ s such that ϕ i +1 is the result of replacing the leftmost strong quantifier Qx ψ ( x , ¯ y ) in ϕ i by ψ ( f i (¯ y ) , ¯ y ) , where ¯ y are the variables of the weak quantifiers in the scope of which Qx ψ ( x , ¯ y ) occurs, and f i does not occur in any ϕ j with j ≤ i. � s = ∃ x ( ϕ ( x , fx ) → ψ ( x , gx )) . � Ex ∃ x ( ∃ y ϕ ( x , y ) → ∀ z ψ ( x , z )) From now on, fx and f ¯ x denote f ( x ) and f (¯ x ) , respectively. 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 ⊢ c ϕ ⇔ T ⊢ c ϕ s . 7 / 28
Nonclassical theories Dfn Skolemization is sound and complete for a theory T if T ⊢ ϕ s T ⊢ ϕ ⇒ (sound) T ⊢ ϕ s T ⊢ ϕ ⇐ (complete) A theory admits Skolemization if Skolemization is both sound and complete. Many nonclassical theories (including IQC) do not admit Skolemization: it is sound but not complete for such theories. For infix formulas it is in general not complete. Examples are formula skolemization DLEM ¬¬∀ x ( ϕ x ∨ ¬ ϕ x ) ¬¬ ( ϕ c ∨ ¬ ϕ c ) CD ∀ x ( ϕ x ∨ ψ ) → ∀ x ϕ x ∨ ψ ∀ x ( ϕ x ∨ ψ ) → ϕ c ∨ ψ EDNS ¬¬∃ x ϕ x → ∃ x ¬¬ ϕ x ¬¬ ϕ c → ∃ x ¬¬ ϕ x . From now on, ϕ x abbreviates ϕ ( x ) . 8 / 28
Existence predicate Extend IQC with an existence predicate E: Et is interpreted as t exists. Dfn (Scott 1977) The logic IQCE has quantifier rules: [ ϕ y , Ey ] Ey . . . . . . . . ϕ t ∧ Et ∃ x ϕ x ψ ϕ y ∀ x ϕ x ∧ Et . ∃ x ϕ x ∀ x ϕ x ϕ t ψ Note IQCE is conservative over IQC. IQCE has a well–behaved sequent calculus and Kripke semantics. A Kripke model K for IQCE is a regular Kripke model with constant domain D , where E is interpreted as a unary predicate E on D and the forcing conditions for the qfs are: K , k � ∃ x ϕ ( x ) ≡ df ∃ d ∈ D K , k � E ( d ) ∧ ϕ ( d ) K , k � ∀ x ϕ ( x ) ≡ df ∀ d ∈ D∀ l � k K , l � E ( d ) → ϕ ( d ) . 9 / 28
Eskolemization Dfn The eskolemization of ϕ is a formula ϕ e without strong quantifiers such that there are formulas ϕ = ϕ 1 , . . . , ϕ n = ϕ e such that ϕ i +1 is the result of replacing the leftmost strong quantifier Qx ψ ( x , ¯ y ) in ϕ i by � E ( f ¯ y ) → ψ ( f ¯ y , ¯ y ) if Q = ∀ E ( f ¯ 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 i does not occur in any ϕ j with j ≤ i. If only existential qfs are replaced, the result is denoted by ϕ E . Ex �⊢ IQCE ¬¬∃ x ϕ x → ∃ x ¬¬ ϕ x �⊢ IQCE ¬¬ ( Ec ∧ ϕ c ) → ∃ x ¬¬ ϕ x. � � �⊢ IQCE ∀ x ( ϕ x ∨ ψ ) → ∀ x ϕ x ∨ ψ �⊢ IQCE ∀ x ( ϕ x ∨ ψ ) → ( Ec → ϕ c ) ∨ ψ . 10 / 28
Soundness and completeness of eskolemization for existential quantifiers Thm (Baaz&Iemhoff 2011) For theories T not containing the existence predicate: T ⊢ IQCE ϕ ⇔ T ⊢ IQCE ϕ E . Thus for ϕ not containing the existence predicate: T ⊢ i ϕ ⇔ T ⊢ IQCE ϕ E . 11 / 28
Three questions about Skolemization ◦ For which (intermediate) logics is eskolemization complete? ◦ For which Skolem functions is skolemization sound and complete? ◦ Are there useful alternative skolemization methods? 12 / 28
Soundness and completeness of eskolemization Thm (Baaz&Iemhoff 2016) For all theories T with the witness property: T ⊢ ϕ ⇔ T ⊢ ϕ e . Cor Eskolemization is sound and complete for all theories complete w.r.t. a class of well-founded and conversely well-founded models. This holds in particular for theories with the finite model property. The previous results have the following theorems, proved by Craig Smory´ nski in the 1970s, as a corollary. Thm (Smory´ nski) The constructive theory of decidable equality is decidable. Thm (Smory´ nski) The constructive theory of decidable monadic predicates is decidable. 13 / 28
� � � The witness property Dfn A model has the witness property if for all nodes k refuting a formula ∀ x ϕ x there is a l � k and d ∈ D l such that l � � ϕ d and l � ϕ d ↔ ∀ x ϕ x. D = N • � ∀ x ϕ ( x ) Note Every conversely well-founded model has the witness property. . . . Dfn A theory has the witness property if it is sound and complete w.r.t. a class of well-founded models that all have D = N • � ϕ (2) the witness property. D = N • � ϕ (1) D = N • � ϕ (0) model without the witness property 14 / 28
Two remaining questions ◦ For which (intermediate) logics is eskolemization complete? At least for logics with the witness property. ◦ For which Skolem functions is skolemization sound and complete? ◦ Are there useful alternative skolemization methods? 15 / 28
Question For which Skolem functions is skolemization sound and complete? Aim Extend IQCE in a minimal way to a theory, say IQCO, that admits a translation similar to Skolemization. Such an extension, IQCO, has been developed some years ago. A related but “lighter” extension, IQCT, is currently being developed. 16 / 28
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