Classical logic is embedded into constructive logic, through a definition of
the classical connectives and quantifiers in terms of the constructive ones.Comment: Why is this a Proof?, Festschrift for Luiz Carlos Pereira , 201

Dowek, Gilles

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International audienceClassical logic is embedded into constructive logic, through a definition of the classical connectives and quantifiers in terms of the constructive ones

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HAL Id: hal-01252221https://hal.inria.fr/hal-01252221Submitted on 7 Jan 2016HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.On the definition of the classical connectives andquantifiersGilles DowekTo cite this version:Gilles Dowek. On the definition of the classical connectives and quantifiers. Edward HermannHaeusler; Wagner de Campos Sanz; Bruno Lopes. Why is this a Proof?, Festschrift for Luiz Car-los Pereira , College Publications, 2015. ￿hal-01252221￿On the definition of the classicalconnectives and quantifiersGilles Dowek∗AbstractClassical logic is embedded into constructive logic, through a definitionof the classical connectives and quantifiers in terms of the constructiveones.The history of the notion of constructivity started with a dispute on thededuction rules that one should or should not use to prove a theorem. Dependingon the rules accepted by the ones and the others, the proposition P ∨ ¬P , forinstance, had a proof or not.A less controversial situation was reached with a classification of proofs, andit became possible to agree that this proposition had a classical proof but noconstructive proof.An alternative is to use the idea of Hilbert and Poincaré that axioms anddeduction rules define the meaning of the symbols of the language and it isthen possible to explain that some judge the proposition P ∨ ¬P true andothers do not because they do not assign the same meaning to the symbols ∨,¬, etc. The need to distinguish several meanings of a common word is usualin mathematics. For instance the proposition “there exists a number x suchthat 2x = 1” is true of false depending on whether the word “number” means“natural number” or “real number”. Even for logical connectives, the word “or”has to be disambiguated into inclusive and exclusive.Taking this idea seriously, we should not say that the proposition P ∨ ¬Phas a classical proof but no constructive proof, but we should say that theproposition P ∨c ¬cP has a proof and the proposition P ∨ ¬P does not, that iswe should introduce two symbols for each connective and quantifier, for instancea symbol ∨ for the constructive disjunction and a symbol ∨c for the classicalone, instead of introducing two judgments: “has a classical proof” and “hasa constructive proof”. We should also be able to address the question of theprovability of mixed propositions and, for instance, express that the proposition(¬(P ∧Q)) ⇒ (¬P ∨c ¬Q) has a proof.The idea that the meaning of connectives and quantifiers is expressed bythe deduction rules leads to propose a logic containing all the constructive and∗INRIA, 23 avenue d’Italie, CS 81321, 75214 Paris Cedex 13, France.gilles.dowek@inria.fr.classical connectives and quantifiers and deduction rules such that a propositioncontaining only constructive connectives and quantifiers has a proof in this logicif and only if it has a proof in constructive logic and a proposition containingonly classical connectives and quantifiers has a proof in this logic if and only ifit has a proof in classical logic. Such a logic containing classical, constructive,and also linear, connectives and quantifiers has been proposed by J.-Y. Girard[3]. This logic is a sequent calculus with unified sequents that contain a linearzone and a classical zone and rules treating differently propositions dependingon the zone they belong.Our goal in this paper is slightly different, as we want to define the meaningof a small set of primitive connectives and quantifiers with deduction rules anddefine the others explicitly, in the same way the exclusive or is explicitly definedin terms of conjunction, disjunction and negation: A⊕ B = (A ∧ ¬B) ∨ (¬A ∧B). A first step in this direction has been made by Gödel [4] who defined atranslation of constructive logic into classical logic, and Kolmogorov [7], Gödel[5], and Gentzen [2] who defined a translation of classical logic into constructivelogic. As the first translation requires a modal operator, we shall focus onthe second. This leads to consider constructive connectives and quantifiers asprimitive and search for definitions of the classical ones. Thus, we want todefine classical connectives and quantifiers ⊤c, ⊥c, ¬c, ∧c, ∨c, ⇒c, ∀c, and∃c and embed classical propositions into constructive logic with a function ‖ ‖defined as follows.Definition 1• ‖P‖ = P if P is an atomic proposition• ‖⊤‖ = ⊤c• ‖⊥‖ = ⊥c• ‖¬A‖ = ¬c‖A‖• ‖A ∧B‖ = ‖A‖ ∧c ‖B‖• ‖A ∨B‖ = ‖A‖ ∨c ‖B‖• ‖A ⇒ B‖ = ‖A‖ ⇒c ‖B‖• ‖∀x A‖ = ∀cx ‖A‖• ‖∃x A‖ = ∃cx ‖A‖If Γ = A1, ..., An is a multiset of propositions, we write ‖Γ‖ for the multiset‖A1‖, ..., ‖An‖.Kolmogorov-Gödel-Gentzen translation can be defined as follows• (P )′ = ¬¬P , if P is an atomic proposition• (⊤)′ = ¬¬⊤2• (⊥)′ = ¬¬⊥• (¬A)′ = ¬¬¬(A)′• (A ∧B)′ = ¬¬((A)′ ∧ (B)′)• (A ∨B)′ = ¬¬((A)′ ∨ (B)′)• (A ⇒ B)′ = ¬¬((A)′ ⇒ (B)′)• (∀x A)′ = ¬¬(∀x (A)′)• (∃x A)′ = ¬¬(∃x (A)′)or more succinctly as• (P )′ = ¬¬P , if P is an atomic proposition• (∗)′ = ¬¬∗, if ∗ is a zero-ary connective• (∗A)′ = ¬¬(∗(A)′), if ∗ is a unary connective• (A ∗B)′ = ¬¬((A)′ ∗ (B)′), if ∗ is a binary connective• (∗x A)′ = ¬¬(∗x (A)′), if ∗ is a quantifierFor instance(P ∨ ¬P )′ = ¬¬(¬¬P ∨ ¬¬¬¬¬P )And it is routine to prove that a proposition A has a classical proof if and onlyif the proposition (A)′ has a constructive one.But, this translation does not exactly provide a definition of the classicalconnectives and quantifiers in terms of the constructive ones, because an atomicproposition is P is translated as ¬¬P , while in a translation induced by adefinition of the classical connective and quantifiers, an atomic proposition Pmust be translated as P .Thus, to view Kolmogorov-Gödel-Gentzen translation as a definition, wewould need to also introduce a proposition symbol P c defined by P c = ¬¬P .But this would lead us too far: we want to introduce constructive and classicalversions of the logical symbols—the connectives and the quantifiers—but not ofthe non logical ones, such as the predicate symbols.If we take the definition• ¬cA = ¬¬¬A• A ∨c B = ¬¬(A ∨B)• etc.where a double negation is put before each connective and quantifier, then theproposition P ∨c ¬cP is ¬¬(P ∨ ¬¬¬P ) where, compared to the Kolmogorov-Gödel-Gentzen translation, the double negations in front of atomic propositionsare missing. Another translation introduced by L. Allali and O. Hermant [1]leads to the definition3• ¬cA = ¬¬¬A• A ∨c B = (¬¬A) ∨ (¬¬B)• etc.where double negations are put after, and not before, each connective and quan-tifier. The proposition P ∨c ¬cP is then ¬¬P ∨ ¬¬¬¬¬P , where the doublenegation at the top of the proposition is missing. Using this translation Allaliand Hermant prove that the proposition A has a classical proof if and only ifthe proposition ¬¬‖A‖ has a constructive one and they introduce another prov-ability judgment expressing that the proposition ¬¬A has a constructive proof.This also would lead us too far: in our logic, we want a single judgment “Ahas a proof” expressing that A has a constructive proof, and not to introduce asecond judgment, whether it be “A has a classical proof” or “¬¬A has a proof”.In order to do so, we define the classical connectives and quantifiers byintroducing double negations both before and after each symbol.Definition 2 (Classical connectives and quantifiers)• ⊤c = ¬¬⊤• ⊥c = ¬¬⊥• ¬cA = ¬¬¬¬¬A• A ∧c B = ¬¬((¬¬A) ∧ (¬¬B))• A ∨c B = ¬¬((¬¬A) ∨ (¬¬B))• A ⇒c B = ¬¬((¬¬A) ⇒ (¬¬B))• ∀cx A = ¬¬(∀x (¬¬A))• ∃cx A = ¬¬(∃x (¬¬A))Notice that the propositions ⊤ ⇔ ⊤c, ⊥ ⇔ ⊥c, and ¬A ⇔ ¬cA where A ⇔ Bis defined as (A ⇒ B) ∧ (B ⇒ A), have proofs. Thus, the symbols ⊤c, ⊥c, and¬c could be just defined as ⊤, ⊥, and ¬.With this definition, neither the double negations in front of atomic propo-sitions nor those at the top of the proposition are missing. The price to pay isto have four negations instead of two in many places, but this is not harmful.Yet, there is still a problem with the translation of atomic propositions:as with any definition based translation, the atomic proposition P alone istranslated as P and not as ¬¬P . Thus, the property that a sequent Γ ⊢ Ahas a classical proof if and only if the sequent ‖Γ‖ ⊢ ‖A‖ has a constructive oneonly holds when A is not atomic. For instance, the sequent P ∧c Q ⊢ P , that is¬¬((¬¬P ) ∧ (¬¬Q)) ⊢ P , does not have a constructive proof.A solution to this problem is to decompose hypothetical provability intoabsolute provability and entailment. For absolute provability, the property that4axiomA ⊢ AΓ, A,A ⊢ ∆contr-lΓ, A ⊢ ∆Γ ⊢ A,A,∆contr-rΓ ⊢ A,∆Γ ⊢ ∆weak-lΓ, A ⊢ ∆Γ ⊢ ∆weak-rΓ ⊢ A,∆⊤-rΓ ⊢ ⊤,∆⊥-lΓ,⊥ ⊢ ∆Γ ⊢ A,∆¬-lΓ,¬A ⊢ ∆Γ, A ⊢ ∆ ¬-rΓ ⊢ ¬A,∆Γ, A,B ⊢ ∆∧-lΓ, A ∧B ⊢ ∆Γ ⊢ A,∆ Γ ⊢ B,∆∧-rΓ ⊢ A ∧B,∆Γ, A ⊢ ∆ Γ, B ⊢ ∆∨-lΓ, A ∨B ⊢ ∆Γ ⊢ A,∆∨-rΓ ⊢ A ∨B,∆Γ ⊢ B,∆∨-rΓ ⊢ A ∨B,∆Γ ⊢ A,∆ Γ, B ⊢ ∆⇒-lΓ, A ⇒ B ⊢ ∆Γ, A ⊢ B,∆ ⇒-rΓ ⊢ A ⇒ B,∆Γ, (t/x)A ⊢ ∆∀-lΓ, ∀x A ⊢ ∆Γ ⊢ A,∆∀-rΓ ⊢ ∀x A,∆Γ, A ⊢ ∆∃-lΓ, ∃x A ⊢ ∆Γ ⊢ (t/x)A,∆∃-rΓ ⊢ ∃x A,∆Figure 1: Sequent calculusa sequent ⊢ A has a classical proof if and only if the sequent ⊢ ‖A‖ has aconstructive one holds for all propositions, because atomic propositions have noproof. Thus, the sequent H1, ..., Hn ⊢ A has a classical proof if and only if thesequent ⊢ ‖H1‖ ⇒c ... ⇒c ‖Hn‖ ⇒c ‖A‖ has a constructive one. This leads toa system where we have only one notion of absolute provability, but two notionsof entailment: “A has a proof from the hypothesis H” can either be understoodas “H ⇒ A has a proof” or “H ⇒c A has a proof”.Definition 3 (Classical and constructive provability) Classical provabil-ity is defined by the cut free sequent calculus rules of Figure 1. We say that theproposition A has a classical proof if the sequent ⊢ A does.Constructive provability, our main notion of provability, is obtained by re-stricting to sequents with at most one conclusion. This requires a slight adapta-tion of the ⇒-l ruleΓ ⊢ A Γ, B ⊢ ∆⇒-lΓ, A ⇒ B ⊢ ∆We say that the proposition A has a constructive proof if the sequent ⊢ A does.Proposition 1 If the proposition ‖A‖ has a constructive proof, then the propo-sition A has a classical one.5Proof. If the proposition ‖A‖ has a constructive proof, then ‖A‖ also has aclassical proof. Hence, ‖A‖ being classically equivalent to A, A also has aclassical proof.We now want to prove the converse: that if the proposition A has a classicalproof, then ‖A‖ has a constructive proof. To do so, we first introduce anothertranslation where the top double negation is removed, when there is one.Definition 4 (Light translation)• |P | = P ,• |⊤| = ⊤,• |⊥| = ⊥,• |¬A| = ¬¬¬‖A‖,• |A ∧B| = (¬¬‖A‖) ∧ (¬¬‖B‖),• |A ∨B| = (¬¬‖A‖) ∨ (¬¬‖B‖),• |A ⇒ B| = (¬¬‖A‖) ⇒ (¬¬‖B‖),• |∀x A| = ∀x (¬¬‖A‖),• |∃x A| = ∃x (¬¬‖A‖).If Γ = A1, ..., An is a multiset of propositions, we write |Γ| for the multiset|A1|, ..., |An| and ¬|Γ| for the multiset ¬|A1|, ...,¬|An|.Proposition 2 If the proposition A is atomic, then ‖A‖ = |A|, otherwise‖A‖ = ¬¬|A|.Proof. By a case analysis on the form of the proposition A.Proposition 3 If the sequent Γ, |A| ⊢ has a constructive proof, then so doesthe sequent Γ, ‖A‖ ⊢.Proof. By Proposition 2, either ‖A‖ = |A|, or ‖A‖ = ¬¬|A|. In the first casethe result is obvious, in the second, we build a proof of Γ, ‖A‖ ⊢ with a ¬-l rule,a a ¬-r rule, and the proof of Γ, |A| ⊢.Proposition 4 If the sequent Γ ⊢ ∆ has a classical proof, then the sequent|Γ|,¬|∆| ⊢ has a constructive one.Proof. By induction on the structure of the classical proof of the sequent Γ ⊢ ∆.As all the cases are similar, we just give a few.• If the last rule is the axiom rule, then Γ = Γ′, A and ∆ = ∆′, A, and thesequent |Γ′|, |A|,¬|A|,¬|∆| ⊢, that is |Γ|,¬|∆| ⊢, has a constructive proof.6• If the last rule is the ⇒-l rule, then Γ = Γ′, A ⇒ B and by induction hy-pothesis, the sequents |Γ′|,¬|A|,¬|∆| ⊢ and |Γ′|, |B|,¬|∆| ⊢ have construc-tive proofs, thus the sequents |Γ′|,¬‖A‖,¬|∆| ⊢ and |Γ′|, ‖B‖,¬|∆| ⊢ haveconstructive proofs, thus, using Proposition 3, the sequent |Γ′|,¬¬‖A‖ ⇒¬¬‖B‖,¬|∆| ⊢, that is |Γ|,¬|∆| ⊢, has a constructive proof.• If the last rule is the ⇒-r rule, then ∆ = ∆′, A ⇒ B and by inductionhypothesis, the sequent |Γ|, |A|,¬|B|,¬|∆′| ⊢ has a constructive proof,thus, using Proposition 3, the sequent |Γ|, ‖A‖,¬‖B‖,¬|∆′| ⊢ has a con-structive proof, thus the sequent |Γ|,¬(¬¬‖A‖ ⇒ ¬¬‖B‖),¬|∆′| ⊢, thatis |Γ|,¬|∆| ⊢, has a constructive proof.Proposition 5 If the sequent Γ ⊢ A has a classical proof and A is not anatomic proposition, then the sequent ‖Γ‖ ⊢ ‖A‖ has a constructive one.Proof. By Proposition 4, as the sequent Γ ⊢ A has a classical proof, the se-quent |Γ|,¬|A| ⊢ has a constructive one. Thus, by Proposition 3, the sequent‖Γ‖,¬|A| ⊢ has a constructive proof, and the sequent ‖Γ‖ ⊢ ¬¬|A| also. ByProposition 2, as A is not atomic, ‖A‖ = ¬¬|A|. Thus, the sequent ‖Γ‖ ⊢ ‖A‖has a constructive proof.Theorem 1 The proposition A has a classical proof if and only if the proposi-tion ‖A‖ has a constructive one.Proof. By Proposition 1, if the proposition ‖A‖ has a constructive proof, thenthe proposition A has a classical one. Conversely, we prove that if the propo-sition A has a classical proof, then the proposition ‖A‖ has a constructive one.If A is atomic, the proposition A does not have a classical proof, otherwise, byProposition 5, the proposition ‖A‖ has a constructive proof.Corollary 1 The sequent H1, ..., Hn ⊢ A has a classical proof if and only if thesequent ⊢ ‖H1‖ ⇒c ...‖Hn‖ ⇒c ‖A‖ has a constructive one.Proof. The sequent H1, ..., Hn ⊢ A has a classical proof if and only if thesequent ⊢ H1 ⇒ ...Hn ⇒ A has one and, by Theorem 1, if and only if thesequent ⊢ ‖H1‖ ⇒c ...‖Hn‖ ⇒c ‖A‖ has a constructive proof.There is no equivalent of Theorem 1 if we add double negations after theconnectors only. For instance, the proposition P ∨¬P has a classical proof, butthe proposition ¬¬P ∨¬¬¬¬¬P has no constructive proof. O. Hermant [6] hasproved that there is also no equivalent of Theorem 1 if we add double negationsbefore the connectors only. For instance, the proposition (∀x (P (x) ∧ Q)) ⇒(∀x P (x)) has a classical proof, but the proposition ¬¬((¬¬∀x ¬¬(P (x)∧Q)) ⇒(¬¬∀x P (x))) has no constructive proof.Let H1, ..., Hn be an axiomatization of mathematics with a finite number ofaxioms, H = H1 ∧ ... ∧Hn be their conjunction, and A be a proposition. If theproposition H ⇒ A has a classical proof, then, by Theorem 1, the proposition‖H‖ ⇒c ‖A‖ has a constructive one. Thus, in general, not only the proposition7A must be formulated with classical connectives and quantifiers, but the axiomsof the theory and the entailment relation also.Using Proposition 5, if A is not an atomic proposition, then the proposition‖H‖ ⇒ ‖A‖ has a constructive proof. In this case, the axioms of the theory mustbe formulated with classical connectives and quantifiers, but the entailmentrelation does not.In many cases, however, even the proposition H ⇒ ‖A‖ has a constructiveproof. For instance, consider the theory formed with the axiom H “The unionof two finite sets is finite”∀x∀y (F (x) ⇒ F (y) ⇒ F (x ∪ y))—or, as the cut rule is admissible in sequent calculus, any theory where thisproposition has a proof—and let A be the proposition “If the union of two setsis infinite then one of them is”∀a∀b ((¬F (a ∪ b)) ⇒ (¬F (a) ∨ ¬F (b)))which is, for instance, at the heart of the proof of Bolzano-Weierstrass theorem,then the proposition H ⇒ ‖A‖ has a constructive proofaxiomF (a), F (b) ⊢ F (a)axiomF (a), F (b) ⊢ F (b)axiomF (a ∪ b), F (a), F (b) ⊢ F (a ∪ b)⇒-lF (b) ⇒ F (a ∪ b), F (a), F (b) ⊢ F (a ∪ b)⇒-lF (a) ⇒ F (b) ⇒ F (a ∪ b), F (a), F (b) ⊢ F (a ∪ b)∀-l, ∀-lH,F (a), F (b) ⊢ F (a ∪ b)¬-l, ¬-r, ¬-l, ¬-r, ¬-l, ¬-r, ¬-lH,¬¬(¬cF (a ∪ b)), F (a), F (b) ⊢¬-r, ¬-l, ¬-r, ¬-l, ¬-rH,¬¬(¬cF (a ∪ b)), F (a) ⊢ ¬cF (b)¬-l, ¬-r, ∨-r, ¬-r, ¬-lH,¬¬(¬cF (a ∪ b)), F (a) ⊢ ¬cF (a) ∨c ¬cF (b)¬-lH,¬¬(¬cF (a ∪ b)),¬(¬cF (a) ∨c ¬cF (b)), F (a) ⊢¬-r, ¬-l, ¬-r, ¬-l, ¬-rH,¬¬(¬cF (a ∪ b)),¬(¬cF (a) ∨c ¬cF (b)) ⊢ ¬cF (a)¬-r, ¬-l, ∨-r, ¬-r, ¬-lH,¬¬(¬cF (a ∪ b)),¬(¬cF (a) ∨c ¬cF (b)) ⊢ ¬cF (a) ∨c ¬cF (b)¬-lH,¬¬(¬cF (a ∪ b)),¬(¬cF (a) ∨c ¬cF (b)),¬(¬cF (a) ∨c ¬cF (b)) ⊢contr-lH,¬¬(¬cF (a ∪ b)),¬(¬cF (a) ∨c ¬cF (b)) ⊢¬-rH,¬¬(¬cF (a ∪ b)) ⊢ ¬¬(¬cF (a) ∨c ¬cF (b))¬-r, ¬-l, ⇒-rH ⊢ (¬cF (a ∪ b)) ⇒c (¬cF (a) ∨c ¬cF (b))(¬-r, ¬-l, ∀-r, ¬-r, ¬-l)2H ⊢ ∀ca∀cb ((¬cF (a ∪ b)) ⇒c (¬cF (a) ∨c ¬cF (b)))⇒-r⊢ H ⇒ ∀ca∀cb ((¬cF (a ∪ b)) ⇒c (¬cF (a) ∨c ¬cF (b)))In this case, even the propositionH ⇒ ∀a∀b ((¬F (a ∪ b)) ⇒ (¬F (a) ∨c ¬F (b)))where the only classical connective is the disjunction, has a constructive proof.Which mathematical results have a classical formulation that can be provedfrom the axioms of constructive set theory or constructive type theory andwhich require a classical formulation of these axioms and a classical notion ofentailment remains to be investigated.8AcknowledgementsThe author wishes to thank Olivier Hermant and Sara Negri for many helpfulcomments on a previous version of this paper.References[1] L. Allali and O. Hermant. Cut Elimination in Classical Sequent CalculusModulo a Super-consistent Theory, manuscript, 2010.[2] G. Gentzen. Über das Verhältnis zwischen intuitionistischer and klassischerLogik, 1933, in M.E. Szabo (Ed.) The Collected Works of Gerhard Gentzen,53-67, 1969.[3] J.-Y. Girard. On the Unity of Logic. Annals of Pure and Applied Logic,59(3):201-217, 1993.[4] K. Gödel. Zum intuitionistischen Aussagenkalkül, Anzeiger der Akademieder Wissenschaften in Wien, 69:65-66, 1932.[5] K. Gödel. Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisseeinses mathematischen Kolloquiums, 4:34-38, 1933.[6] O. Hermant. Personal communication, 2012.[7] A.N. Kolmogorov. On the principle of the excluded middle” (russian),Matematicheskij Sbornik, 32:646-667, 1925.9

On the definition of the classical connectives and quantifiers

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On the definition of the classical connectives and quantifiers

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