In this paper, we show (among other things) that the conditional in Frege's Begriffsschrift is ambiguous

Ertola-Biraben, Rodolfo C.

Fitelson, Branden

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Univocity of Intuitionistic and Classical ConnectivesRodolfo C. Ertola-Biraben∗ and Branden Fitelson†Penultimate draft: final version to appear in the Bulletin of Symbolic Logic1 IntroductionA sentential connective ⋆ is said to be univocal, relative to a formal system F for asentential logic containing ⋆ iff any two connectives ⋆1 and ⋆2 which satisfy the same Frules (and axioms) as ⋆ are such that similar formulas involving ⋆1 and ⋆2 are inter-derivable in F .1 To be more precise, suppose ⋆ is a unary connective. Then ⋆ is univocalrelative to F iff for any ⋆1 and ⋆2 satisfying the same principles as ⋆ in F , we have⋆1α ⊢F ⋆2α. And, if ⋆ is binary, then ⋆ is univocal relative to F iff for any ⋆1 and ⋆2satisfying the same principles as ⋆ in F , we have α ⋆1 β ⊢F α ⋆2 β. In order to illustratethis definition of univocity, it is helpful to begin with a simple historical example.It is easy to see that the conditional → is univocal, relative to Gentzen’s (intuitionisticand classical) natural deduction systems LJ and LK [5]. In both of these systems, if wesuppose there are two conditionals (→1 and →2) satisfying Gentzen’s introduction andelimination rules for →, then we can quickly prove that α →1 β ⊢ α →2 β, as follows:1α α →1 β (→1 E)β (→2 I)1.α →2 βSimilar straightforward derivations may be found in order to prove that the connectives∧, ∨, ⊥ and ¬ are each univocal, relative to Gentzen’s system LJ (proofs omitted).Also, it is well known, as Gentzen himself observed in [5, §5], that LJ and LK havethe same derivations as the Hilbert systems for intuitionistic and classical sentential logic(LHJ and LHK). That is Γ ⊢LJ α iff Γ ⊢LHJ α; and, Γ ⊢LK α iff Γ ⊢LHK α.In this context, it is natural to ask whether the conditional and the other connectivesare also univocal, relative to the Hilbert systems for intuitionistic and classical sententiallogic (LHJ and LHK). This is what will be considered in what follows.We begin by reminding the reader of the axiomatic systems LHJ and LHK. Thesystem LHJ consists of modus ponens, plus the following ten axiom schemata:∗rcertola@unicamp.br, CLE, Cidade Universitária, Sérgio Buarque de Holanda, 251, PO Box 13083-859, Campinas, Brazil.†b.fitelson@northeastern.edu, Department of Philosophy and Religion, Northeastern University,360 Huntington Ave, Boston, MA 02115.1Sometimes the term “uniqueness” is used instead of “univocity.” See [1], [2] and [3] for discussion.(LHJ1) β → (α → β),(LHJ2) (γ → (α → β)) → ((γ → α) → (γ → β)),(LHJ3) α → (β → (α ∧ β),(LHJ4) (α ∧ β) → α,(LHJ5) (α ∧ β) → β,(LHJ6) (α → γ) → ((β → γ) → ((α ∨ β) → γ)),(LHJ7) α → (α ∨ β),(LHJ8) β → (α ∨ β),(LHJ9) (α → β) → ((α → ¬β) → ¬α),(LHJ10) ¬α → (α → β).One nice feature of LHJ is that the fragments of LHJ can be picked out simply byselecting those axioms which contain the connectives in question. This is not so for thesystem LHK, which has all ten axioms of LHJ plus the following eleventh axiom:2(LEM) α ∨ ¬α.Famously, this extension of LHJ is non-conservative. The implicational fragment ofLHK contains theorems not derivable in LHJ — specifically, Peirce’s Law, that is,(P ) ((α → β) → α) → α.One can easily obtain the implicational fragment of LHK by adding (P ) to the implica-tional fragment of LHJ. But, it would be preferable for our purposes to adopt a Hilbertsystem for classical sentential logic that allows us to easily extract all of the fragmentsof classical logic (just by selecting the axioms containing those connectives), as can bedone with LHJ. To that end, we will adopt the following alternative (equivalent) ax-iomatization as our classical, Hilbert system LHK. It differs in two ways from LHJ inthe axioms containing implication and negation. Specifically, our LHK replaces (LHJ9)and (LHJ10) with the following two axioms:(LHK9) (¬β → ¬α) → (α → β),(LHK10) ((α → β) → α) → α.Thus, our system LHK consists of modus ponens, plus the ten axioms LHJ1–LHJ8,and LHK9 and LHK10. Because our LHJ and LHK share these first eight axioms, wewill refer to these axioms sometimes as LHJ1–LHJ8 and sometimes as LHK1–LHK8,depending on what formal system(s) we are discussing in the context. Moreover, althoughthe resulting axiomatization is now redundant (as LHK10 follows from LHK1, LHK2,2This is Gentzen’s way of obtaining LHK from LHJ. Another equivalent option is to replace LHJ10with the double-negation axiom ¬¬α → α. This alternative is also non-conservative, of course, sincePierce’s Law is also not derivable in the implication-negation fragment of LHJ.2and LHK9), we can now simply speak about a fragment of LHK as involving only thoseaxioms of LHK that contain the connectives appearing in the fragment in question.In Section 2, we will discuss questions of univocity of connectives in (fragments of)LHJ. In Section 3, we will discuss questions of univocity of connectives in (fragmentsof) LHK. In Section 4, we will discuss questions of univocity of connectives in somealternative formal systems for classical sentential logic that use the Sheffer stroke.2 Univocity in Hilbert Systems for Intuitionism2.1 Non-Univocity of the Intuitionistic ConditionalInterestingly, the conditional is not univocal, relative to the usual Hilbert systems for theimplicational fragment of intuitionistic logic. More precisely, we have the following.Theorem 1. There exist conditionals →1 and →2 such that (a) both conditionals satisfythe laws of the implicational fragment LHJ→ of LHJ; but, (b) α →1 β ⊬ α →2 β.This result was known since at least 1962 (see [15, p. 430]) where a four-elementcounter-model is presented (see below). We recently discovered the following three-element counter-model.3→1 1 2 31 1 2 22 1 1 13 1 1 1→2 1 2 31 1 2 32 1 1 33 1 2 1So, in this model, the (sole) designated value is 1 (i.e., all and only formulas assigned thevalue 1 by the model are theorems of the logic defined by these 3-valued characteristicmatrices). Inspection of the model reveals that (a) both conditionals satisfy modus ponensand axioms LHJ1 and LHJ2; but, (b) 2 →1 3 ⊬ 2 →2 3, since 2 →1 3 is assigned thedesignated value 1, but 2 →2 3 is assigned the non-designated value 3.In fact, a stronger result was established by Smiley [15]. As it turns out, the con-ditional is not univocal, relative to LHJ as a whole. That is, even if two conditionals→1 and →2 satisfy all of the laws of LHJ, it will still not generally be the case thatα →1 β ⊢ α →2 β.Theorem 2. There exist two conditionals →1 and →2 such that (a) both conditionalssatisfy all the laws of LHJ; but, (b) α →1 β ⊬ α →2 β.3Following Smiley [15], the sole designated value in all of our matrices will be 1 (all the values greaterthan 1 will be non-designated). Moreover, all of the matrices reported in this paper will be normal inSmiley’s sense. That is, the matrices do not make a distinction between values which has no eventualeffect on the designation status of any compound in which subformulas with those values differ (in modernterminology, all of our matrices are reduced). We have made extensive use of automated theorem-provingand model-finding software (e.g., Prover9 & Mace4 [9], Otter [10], and Vampire [8]) to find and verifythe derivations and models reported in this paper. Specifically, we were able to perform exhaustivesearches with Vampire and Mace4, which is how we know that the counter-models presented here are thesmallest possible. Input files for each of the results reported in the paper are available upon request.3Proof. In this case, four-element counter-models are required. Here is the model Smileyreports in [15, p. 430].4→1 1 2 3 41 1 2 3 42 1 1 1 43 1 1 1 44 1 1 1 1→2 1 2 3 41 1 2 3 42 1 1 3 43 1 1 1 44 1 1 1 1∧ 1 2 3 41 1 2 3 42 2 2 3 43 3 3 3 44 4 4 4 4∨ 1 2 3 41 1 1 1 12 1 2 2 23 1 2 3 34 1 2 3 41 2 3 4¬ 4 4 4 1Inspection of the model reveals that (a) both conditionals satisfy modus ponens andaxioms LHJ1–LHJ10; but, (b) 2 →1 3 ⊬ 3 →2 3, since 2 →1 3 is assigned the designatedvalue 1, but 2 →2 3 is assigned the non-designated value 3.Remark. Theorem 2 implies that the axiomatic presentations of all substructural logics(see [12, §2.3] for a survey) have non-univocal conditionals.2.2 Univocity of the Other Intuitionistic ConnectivesThe other connectives are univocal, relative to LHJ. For each connective, we will provea corresponding positive theorem to establish this.Theorem 3. If two conjunctions ∧1 and ∧2 each satisfy the laws of the implication-conjunction fragment LHJ→,∧ of LHJ, then α ∧1 β ⊢ α ∧2 β.Proof. Here is a derivation of α ∧1 β ⊢ α ∧2 β, where MP is modus ponens.1. α ∧1 β Assumption2. α 1, LHJ4, MP3. β 1, LHJ5, MP4. β → (α ∧2 β) 2, LHJ3, MP5. α ∧2 β 3, 4, MP4Strictly speaking, Smiley was interested not in the question of whether the intuitionistic conditionalis univocal (relative to LHJ); but, rather, whether any two conditionals satisfying the laws of LHJmust be synonymous, where p and q are synonymous just in case they are inter-substitutable in allcontexts (i.e., if, for any propositional function ϕ, ϕ(p) ⊢ ϕ(q)). Of course, if p and q are synonymous,then p ⊢ q (just let ϕ(x) = x). So, univocity (of, e.g., u →1 v and u →2 v) follows from synonymy. Ingeneral, the converse does not hold, i.e., there are some sentential logics in which the converse fails. Suchlogics are said to have non-congruential consequence relations [6, §3.3]. But, in intuitionistic logic LHJand classical logic LHK, univocity does entail synonymy, since both of these logics have congruentialconsequence relations (i.e., they both support general replacement theorems [7, §26]).4The strategy here is to eliminate ∧1 and then introduce ∧2. A similar strategy will workfor the other connectives as well.Theorem 4. If two disjunctions ∨1 and ∨2 each satisfy the laws of the implication-disjunction fragment LHJ→,∨ of LHJ, then α ∨1 β ⊢ α ∨2 β.Proof. Here is a derivation of α ∨1 β ⊢ α ∨2 β.1. α ∨1 β Assumption2. α → (α ∨2 β) LHJ73. β → (α ∨2 β) LHJ84. (α ∨1 β) → (α ∨2 β) 2, 3, LHJ6, MP(×2)5. α ∨2 β 1, 4, MPHere, we first introduce ∨2 and then eliminate ∨1.Theorem 5. If two negations ¬1 and ¬2 each satisfy the laws of the implication-negationfragment LHJ→,¬ of LHJ, then ¬1α ⊢ ¬2α.Proof. Here is a derivation of ¬1α ⊢ ¬2α.1. ¬1α Assumption2. α → ¬2α 1, LHJ10, MP3. α → α Theorem of LHJ→4. ¬2α 2, 3, LHJ9, MP(×2)Here, we omit the derivation of step 3, which is a well-known theorem of the implicationalfragment LHJ→ of LHJ.3 Univocity in Hilbert Systems for Classical LogicIt follows from the results in Section 2.2 that the connectives ∧,∨,¬ are univocal, relativeto (the relevant fragments of) LHK. The only remaining questions involve the univocityof the conditional in (fragments of) LHK. Here, there are two positive results and twonegative results. We begin with the negative results.Theorem 6. The conditional is non-univocal, relative to the implication-conjunctionfragment LHK→,∧ of LHK.5Proof. In this case, four-element counter-models are required. Here is one.→1 1 2 3 41 1 2 2 22 1 1 1 13 1 1 1 14 1 1 1 1→2 1 2 3 41 1 2 3 42 1 1 3 33 1 2 1 24 1 1 1 15It is a corollary of Theorem 6 that → is non-univocal, relative to the implicational fragment ofLHK→ of LHK. In that case, three-element counter-models exist. In fact, the model we reported above(Theorem 1) will suffice for this purpose (as →1 and →2 in that model also satisfy Peirce’s law).5∧ 1 2 3 41 1 2 3 42 2 2 4 43 3 4 3 44 4 4 4 4Inspection of the model reveals that (a) both conditionals satisfy modus ponens andaxioms LHK1–LHK5; but, (b) 2 →1 3 ⊬ 2 →2 3, since 2 →1 3 is assigned the designatedvalue 1, but 2 →2 3 is assigned the non-designated value 3.Theorem 7. The conditional is non-univocal, relative to the implication-negation frag-ment LHK→,¬ of LHK.Proof. In this case, six-element counter-models are required. Here, we report an eight-element counter-model.6→1 1 2 3 4 5 6 7 81 1 2 3 4 5 6 7 82 1 1 1 5 5 5 1 53 1 7 1 6 5 6 7 54 1 1 1 1 1 1 1 15 1 2 3 2 1 7 7 36 1 3 3 3 1 1 1 37 1 3 3 8 5 5 1 88 1 7 1 7 1 7 7 1→2 1 2 3 4 5 6 7 81 1 2 3 4 5 6 7 82 1 1 3 5 5 7 7 33 1 2 1 6 7 6 7 24 1 1 1 1 1 1 1 15 1 2 1 2 1 2 1 26 1 1 3 3 3 1 1 37 1 2 3 8 3 2 1 88 1 1 1 7 7 7 7 11 2 3 4 5 6 7 8¬ 4 5 6 1 2 3 8 7Inspection of the model reveals that (a) both conditionals satisfy modus ponens andaxioms LHK1, LHK2, and LHK9; but, (b) 2 →1 3 ⊬ 2 →2 3, since 2 →1 3 is assignedthe designated value 1, but 2 →2 3 is assigned the non-designated value 3.Remark. Theorem 7 implies that many axiomatic presentations of classical logic —dating all the way back to Frege’s Begriffsschrift [4] — have non-univocal conditionals.Theorem 8. The conditional is univocal, relative to the implication-disjunction fragmentLHK→,∨ of LHK.Proof. Here is a derivation of α →1 β ⊢ α →2 β.1. α →1 β Assumption2. α ∨ (α →2 β) Theorem of LHK→,∨3. α →1 (β ∨ ((α →1 β) →1 (α →2 β))) Theorem of LHK→,∨4. (α →2 β) →1 (β ∨ ((α →1 β) →1 (α →2 β))) Theorem of LHK→,∨5. β ∨ ((α →1 β) →1 (α →2 β)) 2, 3, 4, LHK6, MP6We prefer this counter-model to the six-element counter-models we initially discovered for the follow-ing reason. These eight-element models allow one of the conditionals (→1) to satisfy the laws of Booleanalgebra. This is analogous to Smiley’s 4-element matrices (from Theorem 2, above), in which one of thetwo conditionals (→2) satisfies the laws of Heyting algebra.66. β →2 (α →2 β) LHK17. (α →2 β) ∨ ((α →1 β) →1 (α →2 β)) 5, 6 LHK6, MP8. (α →2 β) →1 ((α →1 β) →1 (α →2 β)) LHK19. (α →1 β) →1 (α →2 β) 7, 8, LHK6, MP10. α →2 β 1, 9, MPWe omit the derivations of steps 2, 3, 4, which are theorems of the implication-disjunctionfragment LHK→,∨ of LHK. In particular, note that steps 2 and 3 hold by Peirce’s Law.In order to derive steps 7 and 9, we have also made use of the fact that α → α is atheorem of LHK→.Theorem 9. The conditional is univocal, relative to the implication-conjunction-negationfragment LHK→,∧,¬ of LHK.Proof. This is a corollary of Theorem 8. Because we are in the implication-conjunction-negation fragment of classical logic, we can now define a disjunction connective (whichwill satisfy all of the LHK disjunction laws, relative to both →1 and →2) in terms ofnegation and conjunction, as α∨β =df ¬(¬α∧¬β). Then, we can apply the same strategythat we used above to demonstrate α →1 β ⊢ α →2 β in our proof of Theorem 8.4 Univocity in Classical Systems with Sheffer’s Stroke4.1 Univocity in Hilbert Systems Using Sheffer’s StrokeNicod [11] was the first to develop a formal system N for classical sentential logic using(only) Sheffer’s stroke (|). He presented a Hilbert system, which uses the following singlerule of inference:(D) From α | (β | γ) and α, infer γ.Various sets of axioms have been shown to be complete for classical sentential logic, inthe presence of rule (D) [14]. We will adopt Nicod’s original 23-symbol single axiom,that is,(N) (α | (β | γ)) | ((ϵ | (ϵ | ϵ)) | ((δ | β) | ((α | δ) | (α | δ)))).In other words, the system N consists of the single rule D and the single axiom N . Thequestion of univocity is a simple one in this context. And, as it happens, we have thefollowing negative result.Theorem 10. The Sheffer stroke is non-univocal, relative to Nicod’s Hilbert system N.Proof. In this case, four-element counter-models are required. Here is one.|1 1 2 3 41 2 1 4 32 1 1 1 13 4 1 4 14 3 1 1 3|2 1 2 3 41 4 3 2 12 3 3 1 13 2 1 2 14 1 1 1 17Inspection of the model reveals that (a) both strokes satisfy D and N ; but, (b) 1|12 ⊬ 1|22,since 1 |1 2 is assigned the designated value 1, but 1 |2 2 is assigned the non-designatedvalue 3.4.2 Univocity in Gentzen Systems Using Sheffer’s StrokeWe know of two Gentzen systems for classical sentential logic including Sheffer’s stroke(|). One is due to Read [13], and the other is due to Zach [16]. In both systems, theSheffer stroke is univocal, and this is our final theorem.Theorem 11. The Sheffer stroke is univocal, relative to both Read’s Gentzen system SC,and Zach’s Gentzen system NS.Proof. Here is a derivation of α |1 β ⊢SC α |2 β.α |1 β1α1β (|1E)α |2 β (|2Ic)1α |2 β ∨ ⊥2α |2 β2⊥ (⊥E)α |2 β (∨E)2α |2 βAnd, here is a derivation of α |1 β ⊢NS α |2 β.α |1 β1α1β (|1E)⊥ (|2I)1α |2 βRemark. It may seem surprising that the ⊥-rule is not used in our derivation in Zach’ssystem (of α |1β ⊢NS α |2β). But, something completely analogous happens when provingthe univocity of negation, relative to Gentzen’s original systems. To wit:1α ¬1α (¬1E)⊥ (¬2I)1¬2αAcknowledgements. Rodolfo C. Ertola-Biraben is grateful to Xavier Caicedo for usefuldiscussions. Both authors are indebted to an anonymous referee of this journal, whoprovided very detailed and helpful comments on an earlier version of the paper.8References[1] Belnap, N. (1962). Tonk, Plonk and Plink. Analysis, 22(6), 130–4.[2] Caicedo, X. and Cignoli, R. (2001). An algebraic approach to intuitionistic connec-tives. The Journal of Symbolic Logic, 66(4), 1620–1636.[3] Ertola-Biraben, R. (2009). On univocal connectives. Logic and Logical Philosophy,19(1), 5–13.[4] Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprachedes reinen Denkens. Halle: Nebert.[5] Gentzen, G. (1969). Investigations into logical deduction. The collected papers ofGerhard Gentzen, North Holland, pages 68–131.[6] L. Humberstone. (2011). The connectives. MIT Press.[7] S. Kleene. (1952). Introduction to metamathematics, North-Holland.[8] Kovács, L. and A. Voronkov. First-Order Theorem Proving and Vampire. In N. Shary-gina and H. Veith (Eds.), Computer Aided Verification, Springer LNCS 8044, 1–35.[9] McCune, W. Prover9 and Mace4. URL: http://www.cs.unm.edu/~mccune/prover9/.[10] McCune, W. Otter 3.3 Reference Manual. Technical Report ANL/MSC-TM-263,Argonne National Laboratory, Argonne, USA, 2003.[11] Nicod, J. (1917). A reduction in the number of primitive propositions of logic. InProceedings of the Cambridge Philosophical Society, 19(1917), 32–41.[12] F. Paoli. (2002). Substructural Logics: A Primer. Springer.[13] Read, S. (1999). Sheffer’s stroke: a study in proof-theoretic harmony. Danish Year-book of Philosophy, 34(1), 7–23.[14] Scharle, T. (1965). Axiomatization of propositional calculus with Sheffer functors.Notre Dame Journal of Formal Logic, 6(3), 209–217.[15] Smiley, T. (1962). The Independence of Connectives. The Journal of Symbolic Logic,27(4), 426–436.[16] Zach, R. (2016). Natural deduction for the Sheffer stroke and Peirce’s arrow (andany other truth-functional connective). Journal of Philosophical Logic, 45(2), 183–197.9

Univocity of Intuitionistic and Classical Connectives

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Univocity of Intuitionistic and Classical Connectives

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