On Arithmetic Formulated Connexively

On Arithmetic Formulated Connexively∗ Thomas Macaulay Ferguson CUNY Graduate Center 365 Fifth Avenue New York, NY 10016 United States ABSTRACT In this paper, we reflect on some themes related to the formulation of mathematics against the backdrop of a connexive logic. From a positive perspective, we will consider some remarks of the Kneales concerning Aristotle’s position on connexive implication and suggest that common themes between the Kneales’ Aristotle and the hyperconstructive arithmetic of David Nelson may provide a philosophical basis for connexive mathematics. We will also consider some historical points, including Lukasiewicz’ argument that connexive principles may be refuted by appeal to number-theoretic intuitions. Finally, we will take more concrete steps towards the implementation of connexive mathematics by examining how weak subtheories of arithmetic fare when formulated in modest firstorder extensions of three connexive logics: Richard Angell’s PA1 and PA2 and Graham Priest’s PN . Unfortunately, we will observe that severe pathologies emerge when even extraordinarily weak subsystems of Peano arithmetic are evaluated in these logics, suggesting that Angell and Priest’s systems constitute strained, if not unserviceable, bases for arithmetic. 1. THE ALLURE OF CONNEXIVE MATHEMATICS Frequently, non-classical logics are presented as formalizations of correct deductive reasoning and mathematical reasoning, as an a priori discipline, is uniquely sensitive to the adoption or rejection of logical principles. The analysis of how various mathematical theories fare under enriched or restrained theories of inference makes up one of the most salient applications of a non-classical theory of deduction. Historically, this is most evident in the case of intuitionism, insofar as the intuitionistic standpoints with respect to deduction and mathematical practice are tightly bound together. But similar sentiments apply to mathematics investigated in other non-classical settings. For example, the analysis of mathematical principles by substructural logics has been particularly fruitful, including Robert Meyer’s investigations into the relevant arithmetic R♯ ,1 John K. Slaney, Greg Restall, and Meyer’s investigation into the linear arithmetic LL♯ in [2], and Zach Weber’s more recent investigations into set theory in weak relevant logics (e.g., [3] and [4]). The concern of this work is to begin investigating the prospects for a connexive formulation of mathematics. Connexive logics are deductive systems that contain one or more of the following as theorems: Aristotle’s Thesis Boethius’ Thesis Strawson’s Thesis ∼(ϕ → ∼ϕ) (ϕ → ψ) → ∼(ϕ → ∼ψ) ∼((ϕ → ψ) ∧ (ϕ → ∼ψ)) where “→” represents a binary conditional connective, possibly interpreted as material implication (e.g., in the classical case) or as an intensional entailment connective. The present examination of mathematics formulated with a connexive background logic is not unique, as there have been several forays into connexive mathematics in the modern era of connexive logic. Some of these investigations have been harmonious with reasonable and salient ∗ This is the author’s version of a paper accepted for publication in the IFCoLog Journal of Logics and Their Applications. Some corrections, editing, and typesetting differences will exist between this and the final version. 1 We will adhere to Meyer’s convention of, e.g., [1], according to which the theory of Peano arithmetic formulated in a logic L will be labeled L♯ . mathematical notions; some have been more negative, revealing that if the connexive principles of inference are to be retained, some revision of common mathematical practice must inevitably follow. As an example of the former, Storrs McCall’s [5] considers the thesis of classical set theory that the empty set is a subset of its complement and compares this with the intuition that a proposition should not entail its own negation. Indeed, [5] opens with a dialogue in which a mathematics student’s plausible resistance to classical set theoretic theses serves to both illustrate and motivate connexive theses. At the conclusion of the paper, McCall demonstrates that a coherent theory of classes in which no class is contained in its complement—i.e., a connexive class theory—is entirely practicable. On the other hand, J.E. Wiredu’s [6] considers what restrictions to classical set theory would have to be made in order to accommodate connexive theses. Wiredu shows that assuming connexive principles entails that even the restricted comprehension axioms of ZF are too strong. The upshot is that any reasonable connexive theory of sets must restrict comprehension even more severely than, say, Zermelo’s axiom of separation. Now, while this may demand a revision of some mathematical principles, to suggest a restriction of intuitive mathematical theses isn’t uniquely offensive; following Zermelo, mathematicians—set theorists, computability theorists, etc.—have demonstrated a willingness to restrict comprehension if need be. In other words, while Wiredu shows the need to restrict some set theoretical intuitions, this does not on its face preclude a connexive mathematics. Now, connexive logics share peculiar features that undoubtedly complicate matters. The contraclassicality of connexive logics, for example, entails that the development of connexive mathematics will be more complex— and, arguably, more interesting—than intuitionistic or substructural accounts. For example, although formally undecidable sentences in classical Peano arithmetic remain independent of its intuitionistic and relevant counterparts, there exist undecidable sentences of classical arithmetic that will become decidable modulo any reasonable connexive arithmetic. In, e.g., Peano arithmetic, there exists an undecidable sentence ξ. In the classical case, that is, when the conditional → is construed as the material conditional, ξ is equivalent to the formula ∼(ξ → ∼ξ), i.e., ∼(∼ξ∨∼ξ). This entails that in classical Peano arithmetic the sentence ∼(ξ → ∼ξ) is likewise undecidable. Of course, in a connexive logic L and connexive arithmetic L♯ , one expects that L♯ should prove ∼(ξ → ∼ξ) by default, witnessing that some classically undecidable statements in number theory become decidable connexively. Although this example is extremely simple, it demonstrates that there are many subtle questions that uniquely arise in a connexive mathematics. In this paper, I wish to make a few comments on how mathematics—in particular, arithmetic—must behave if formulated connexively. We will first consider some relevant historical and philosophical topics before taking a foray into the formalization of modest subsystems of arithmetic in predicate calculi corresponding to Richard Angell’s PA1 and PA2 (described in [7] and [8], respectively) and Graham Priest’s PN (described in [9]). The more philosophical observations of this paper are guardedly optimistic. For example, we will consider a kinship between Everett Nelson’s connexive theory of self-cotenability in [10] and David Nelson’s philosophy of mathematics in [11] and suggest that Heinrich Wansing’s work on constructive connexive logic in [12] serves as a successful harmonization of the two themes. We will also diagnose problems with Jan Lukasiewicz’ argument of [13] that number theory is inconsistent with connexive principles, countering the most prominent foil to a connexive mathematics in print. In contrast, the formal results of the paper are relatively discouraging. We will see that theories of arithmetic in Angell’s logics inevitably suffer from some rather counterintuitive features and that any theory including even weak induction schema (i.e., those including induction for quantifier-free formulae) will have no consequences in Priest’s system. 2. HISTORICAL POINTS In this section, we will consider a few more philosophical points before proceeding to formalizations of fragments of arithmetic. 2.1 Anti-Zenonian Mathematics and Reductiones ad Absurdum The suggestion of a positive philosophical foundation for connexive mathematics can be discovered in an interesting remark in William and Martha Kneales’ [14]. While discussing Aristotle’s articulation of connexive principles, the Kneales suggest that it is “tempting” to infer that “what [Aristotle] attacks is in effect the positive counterpart of Zeno’s reductio ad impossibile.” [14, 97] William Kneale is even more explicit in tethering Aristotle’s Thesis to the rejection of reductiones, noting that the entailment assertions which Aristotle refuses to admit are just those required for justification of the hypothetical premisses in... the constructive counterpart of the reductio ad absurdum. [15, 66] The connection between Aristotle’s Thesis and a rejection of reductiones is also observed by Angell in [7], writing that “[t]he objection has been raised that such theorems... would eliminate reductio ad absurdum proofs.” [7, 337] The importance of reductiones to the development of mathematics suggests that there is a mathematical thesis to be squeezed out of the connexive position. While the Kneales are tentative about this suggestion, a tension between connexive principles and the technique of reductio is more also apparent in Everett Nelson’s [10]. A tension with—if not outright denial of—the legitimacy of reductiones is implicit in his connexive analysis of entailment in [10]. Nelson notes that there exists a difference between a logic in which relations [of entailment] are based on facts extrinsic to the essence of the propositions and one in which they are based on the essence itself. [10, 451] Nelson suggests that any acceptable characterization of logical entailment ought to take the latter route by analyzing the connections between the distinct essences of propositions. One of the notable assumptions distinguishing Nelson’s account of entailment can be summarized by the passage: If p entails q, then p is consistent with q. This assertion, together with p E p [i.e., the principle that p entails p], gives rise to p ◦ p [i.e., that p is consistent with p]. [10, 447] Nelson’s account of the relationship between consistency and entailment diverges from the familiar Lewisian notion of cotenability. C.I. Lewis chooses to define entailment in terms of consistency in [16],2 so that ϕ → ψ is defined as ∼(ϕ ◦ ∼ψ), that is, ϕ entails ψ when ϕ and ∼ψ are not mutually consistent. Hence, while Nelson infers from the thesis that ϕ → ϕ is that all propositions are self-cotenable, the Lewisian account only infers from ϕ → ϕ the much tamer claim that ∼(ϕ ◦ ∼ϕ), i.e., that ϕ is inconsistent with its negation. The identification of the mutual consistency of two propositions with their cotenability brings out some of the prima facie plausibility of Nelson’s position. Under this reading, that every proposition is self-consistent is just to say that every proposition is tenable in some sense, which suggests that no proposition is so defective so as to preclude even its consideration. There is something admittedly attractive about this notion insofar as it comports with philosophical practice. Philosophers, for example, frequently engage in counterpossible reasoning, in which even inconsistent propositions can be maintained for the sake of argument. In this sense, every proposition is to some degree tenable—and one might reasonably identify self-cotenability with tenability simpliciter. The tension between Nelson’s assertion that every proposition is self-consistent and the foundation for the legitimacy of reductiones is subtle. If one considers a contradiction ϕ ∧ ∼ϕ, from this one can trivially derive a contradiction: ϕ ∧ ∼ϕ itself. But if one assumes that what has been derived is consistent with what has been assumed, to apply a reductio to disprove ϕ ∧ ∼ϕ is to reject ϕ ∧ ∼ϕ on the basis that it entails itself. In other words, that every proposition is self-consistent entails that within the essence of any proposition, one cannot discover any feature sufficiently defective to warrant its rejection solely on these grounds. Hence, 2 Lewis goes so far as to refer to the Survey System as the “Calculus of Consistencies” in [16]. if reductiones serve to reject propositions on the basis of the defectiveness of their essences, the technique of reductio is, in a sense, empty and vacuous. It is interesting to note that David Nelson’s [11] is also motivated by a resistance to reductiones ad absurdum. In the context of constructive mathematics, Nelson notes while positive formulae must always be constructibly verified in intuitionistic logic, there is no analogous requirement—indeed, no corresponding operation—for constructive disproof. The intuitionistic negation ϕ → ⊥ can be recognized as the assertion of the existence of a reductio disproof for ϕ. But while ϕ → ⊥ is a perfectly well-formed formula that imparts some information about ϕ, the formula is not identified with ∼ϕ. [J]ust as in the case of an existential proposition, we may, in the case of the negation of a generality statement ∼∀xA(x), distinguish two methods of proof. In one there is presented an effective method of constructing an n such that ∼A(n) is true, in the other there is presented a demonstration that ∀xA(x) implies an absurdity. [11, 16–17] If ∀xA(x) is judged as false, we tend to think that there is some (possibly more than one) false instance A(n) that is responsible for the falsity of ∀xA(x).3 Intuitionistically, to assert the truth of ∃xA(x) is to have a construction of a natural number n witnessing that a verifying instance A(n) is true. From a constructive standpoint, it seems just as natural to expect an effective construction of a falsifying instance of A(n) when ∀xA(x) is false. But the existence of a proof that ∀xA(x) entails an absurdity does not guarantee a procedure that produces such a witness. Hence, Nelson reasons that intuitionistic negation does not provide an adequate characterization of falsity. From this, he argues that a true commitment to constructivity in mathematics entails concern for constructive falsity as well as for constructive truth. The moral is that just as the intuitionist rejects non-constructive proof as vacuous, the intuitionist ought to reject non-constructive disproof, i.e., reductiones ad absurdum, as empty and vacuous. The proximity between the opinions concerning reductiones—the connexivist rejection on the one hand and the constructivist rejection on the other—suggests a ground for a philosophy of connexive mathematics. On both accounts, a disproof of ϕ requires more than the mere demonstration of a defect with respect to ϕ. To disprove ϕ demands something further, such as the explicit production of evidence that ϕ fails to comport with arithmetical facts. On the one hand, to derive the negation of ϕ from ϕ itself is fruitless because rather than revealing a defect with respect to ϕ, one has deduced improperly. On the other hand, it is fruitless because such a derivation fails to reveal what makes ϕ false. An interesting fact is that there already exist formalisms that apparently harmonize these two considerations. It is arguable that Wansing’s approach to connexive logic described in, e.g., [12] and [17] reflects this coincidence between connexivity and strong constructivity. Wansing’s C and related systems are presented within a framework generalizing the semantics of David Nelson’s logic of constructible falsity and, indeed, the discussion of falsification in [18] is consistent with the above considerations on reductiones ad absurdum. A particular feature of Wansing’s C that might be thought to exhibit the convergence of the two notions is the theoremhood of formulae ∼(A → ⊥) is a theorem of C, i.e., the theoremhood of ∼¬A where ¬ denotes intuitionistic negation. On the one hand, when this feature is read as the thesis that all intuitionistically negated formulae are false in C, this might be thought to be counterintuitive. But providing a natural Brouwer-HeytingKolmogorov-type reading, the theoremhood of ∼(A → ⊥) is interpreted as the statement that for an arbitrary formula A, it is false that there exists a procedure to convert a proof of A into a proof of ⊥,4 an interpretation that might be thought to capture the shared ground between Everett and David Nelson. 3 As an example, it is natural to assert that “all prime numbers are odd” is false because “2 is a prime number and is odd” is false. 4 But n.b. that C is inherently negation inconsistent, a point that problematizes a BHK-style interpretation. Because both ⊥ → ⊥ and ∼(⊥ → ⊥) are theorems, for example, our naive BHK interpretation entails both the existence and non-existence of a procedure turning proofs of ⊥ into proofs of ⊥. 2.2 Lukasiewicz’ Counterexample to Aristotle’s Thesis We have encountered one apparently negative result concerning connexive mathematics in Wiredu’s remarks of [6] that connexive principles might require further restriction to the comprehension axioms of set theory. A more dangerous specter for connexive mathematics appears in a remark of Jan Lukasiewicz in [13], in which he aims to demonstrate the inconsistency of Aristotle’s Thesis with respect to a fundamental number-theoretic principle: [Euclid] states first that ‘If the product of two integers, a and b, is divisible by a prime number n, then if a is not divisible by n, b should be divisible by n.’ Let us now suppose that a = b and the product a × a (a2 ) is divisible by n. It results from this supposition that ‘If a is not divisible by n, then a is divisible by n.’ Here we have an example of a true implication the antecedent of which is the negation of the consequent. [13, 50–51] Let us make this argument somewhat more perspicuous by formalizing the sequence of reasoning. Where “a | b” symbolizes the relation that b is divisible by a, we provide the following scheme with n prime: 1. 2. 3. 4. ∀a, b[n | (a × b) → [n ∤ a → n | b]] n | n2 n | n2 → [n ∤ n → n | n] n∤n→n|n Number-Theoretic Truth Number-Theoretic Truth Universal Instantiation, 1. Modus Ponens, 2,3. Hence, it appears that, granted very weak logical assumption, instances of Euclid’s lemma serve as counterexamples to Aristotle’s Thesis. Furthermore, it is not clear that Euclid’s lemma can be restricted as naturally or as readily as naive comprehension.5 Despite Euclid’s employing a conditional in the statement of the lemma, to state that if a prime n divides a composite number a × b then either n divides a or n divides b is an equally good (and perhaps superior) formulation of Euclid’s lemma. If so, Lukasiewicz’ argument requires an enthymematic assumption of the validity of disjunctive syllogism. Formally, the initial steps would be: 1. 2. 3. 4. ∀a, b[n | (a × b) → [n | a ∨ n | b]] n | n2 n | n2 → [n | n ∨ n | n] n| n∨n|n Number-Theoretic Truth Number-Theoretic Truth Universal Instantiation, 1. Modus Ponens, 2,3. If we state Euclid’s lemma in this way, the most natural way to generate the counterexample to Aristotle’s Thesis is by inferring n ∤ n → n | n from n | n ∨ n | n, e.g., by producing a conditional proof making an explicit appeal to the validity of disjunctive syllogism. But this sort of argument is much less compelling. Lukasiewicz intends to argue against Aristotle’s Thesis on purely mathematical grounds, that is, Aristotle’s Thesis is to be rejected not due to its inconsistency with competing logical principles, but due to its inconsistency with mathematical intuitions. Moreover, if one follows Richard Sylvan in maintaining that connexivism “coincides with the broad requirement of relevance” [19, 393], then it is virtually obligatory that one rejects disjunctive syllogism as an archetypal fallacy of relevance. Hence, the force of Lukasiewicz’ counterexample to Aristotle’s Thesis is not generated by features of number theory, but by assumptions concerning logic.6 5 A referee has noted that such restrictions to Euclid’s lemma—by, e.g., stipulating that a and b must be either nonequal or relatively prime—would in fact stave off counterexamples to Aristotle’s Thesis. It is worth mentioning that this reply shares an affinity with the approach to connexive logic described in Graham Priest’s [9], in which counterexamples to Aristotle’s Thesis are avoided by filtering out cases in which an antecedent is contradictory. 6 A referee has noted that in any logic in which Weakening is admissible, n ∤ n → n | n will follow from the truth of n | n. This suggests that Lukasiewicz could have formulated a similar argument against Aristotle’s Thesis by appeal to Weakening. Such an argument, of course, would also have to make an appeal to explicitly logical principles, something Lukasiewicz appears to be attempting to avoid in [13]. 3. FRAGMENTS OF ARITHMETIC IN CONNEXIVE LOGICS Lukasiewicz, I have suggested, failed to give a definitive number-theoretic refutation of Aristotle’s Thesis. We have also considered some apparent connections between the connexive account of entailment and David Nelson’s philosophy of mathematics. This is not to say, however, that existent connexive logics are compatible with our standard convictions concerning arithmetic. In this section, we will examine three connexive propositional logics and modest extensions thereof, revealing that any reasonable extensions of these systems are either straightforwardly incompatible with arithmetical principles or lead to severely pathological formulations of arithmetic. 3.1 Three Connexive Logics Richard Angell’s propositional logic PA1 was introduced in [7] as an attempt to capture a notion of a subjunctive conditional, in which Boethius’ Thesis is presented as the principle of subjunctive contrariety. A further connexive logic PA2 was introduced by Angell in work first appearing as the abstract [20] and subsequently appearing in Italian as [8]. PA2 was intended to rectify certain shortcomings of PA1 and McCall’s system CC1 of [21], one of which shall make an appearance in the sequel. To define Angell’s PA1 and PA2, we will follow the presentation of many-valued logics in [22]. ∼ ∧ → Definition 1. The semantic matrix for PA1 is hVPA1 , DPA1 , fPA1 , fPA1 , fPA1 i where • VPA1 = {0, 1, 2, 3} • DPA1 = {0, 1} and the truth functions are defined by the following matrices: ∼ 3 2 1 0 ϕ 0 1 2 3 ϕ∧ψ 0 1 2 3 0 1 0 3 2 1 0 1 2 3 2 3 2 3 2 3 2 3 2 3 ϕ→ψ 0 1 2 3 0 1 2 1 2 1 2 1 2 1 2 3 2 1 2 3 2 3 2 1 T ∼ ∧ → Definition 2. The semantic matrix for PA2 is hVPA2 , DPA2 , fPA2 , fPA2 , fPA2 , fPA2 i where VPA2 = VPA1 and DPA2 = DPA1 and the truth functions are defined by the following matrices: T 1 1 3 3 ϕ 0 1 2 3 ∼ 3 2 1 0 ϕ 0 1 2 3 ϕ∧ψ 0 1 2 3 0 0 1 2 3 1 1 1 2 3 2 2 2 2 3 3 3 3 3 3 ϕ→ψ 0 1 2 3 0 1 3 1 3 1 3 1 3 1 2 3 3 1 3 3 3 3 3 1 Semantic validity in both PA1 and PA2 is defined in the standard way, that is, as preservation of designated values from premises to conclusion. Angell fails to provide any intuitive reading for these matrices. The nearest thing to a natural interpretation the matrices for PA1 might be derived from Routley and Montgomery’s interpretation of McCall’s CC1. With notation adjusted to reflect Angell’s matrices, Routley and Montgomery write: CC1, for instance, can be given a semantics by associating the matrix value [0] with logical necessity, value [3] with logical impossibility, value [1] with contingent truth, and value [2] with contingent falsehood. [23, 95] However, Routley and Montgomery concede that such an interpretation is given to anomalies—e.g., the conjunction of two necessary truths is a contingent truth—and these anomalies follow when such an interpretation is given to PA1. It is common to treat these matrices merely as a theoretical tool. Wansing, for example, states that the semantics for CC1 “appears to be a purely formal method with little explanatory power.” [12, 370] Despite the apparent artificiality of the semantics for PA1 and PA2, Graham Priest has introduced a pair of connexive logics in [9] whose semantics are much more philosophically salient. Priest’s semantics are motivated by a theme of negation as “cancellation,” so that a formula ∼ϕ cancels a formula ϕ. One of the formal features of this account of negation is that the cancellation of ϕ by ∼ϕ entails that ϕ ∧ ∼ϕ has no content and therefore entails nothing. Priest traces the provenance of this notion of negation-as-cancellation through Western philosophy, describing appearances in the work of not only Aristotle and Boethius, but also Abelard and Berkeley. In this paper, we will focus only on the “non-symmetrized” version PN .7 Definition 3. A model for PN is a 3-tuple M = hW, g, vi, where W is a nonempty set of points with g ∈ W and v is a function mapping At to subsets of W . • M, w p iff w ∈ v(p) for p ∈ At • M, w ∼ϕ iff M, w 1 ϕ • M, w ϕ ∧ ψ iff M, w • M, w ϕ ∨ ψ iff M, w ϕ or M, w ψ ( ∃w′ ∈ W such that M, w′ ϕ, and ϕ → ψ iff ∀w′ ∈ W , if M, w′ ϕ then M, w′ ψ • M, w ϕ and M, w ψ Validity is defined by appealing to the designated world g. Definition 4. Validity is defined so that: Γ PN ϕ if ( there exists an M such that for all ψ ∈ Γ, M, g ψ for all M such that for all ψ ∈ Γ, M, g ψ, also M, g ϕ The case of validity in PN is atypical in that it is not Tarskian. Notably, the presumption of reflexivity (i.e., self-entailment) fails, which can be clearly illustrated. Consider the set {p ∧ ∼p}. Then there exists no model M such that M, g p ∧ ∼p, whence p ∧ ∼p 2PN p ∧ ∼p may be inferred. 3.2 Formal Languages Of course, it is impossible to sufficiently express the richness of arithmetical theses in a purely propositional language. For the purposes of this section—showing that PA1, PA2, and PN are questionable bases for arithmetic— we need not develop the full language of arithmetic. Rather, we appeal to weak subsystems of arithmetic, simplifying matters by considering only a modest fragment of the full language of arithmetic, i.e., the language with a constant 0, a binary symbol = ˙ representing identity, and a successor function (·)′ . We will define very weak languages that one may expect to be included in any sufficiently expressive language of arithmetic and then proceed to describe general schema for extending the semantics of PA1, PA2, and PN to a framework rich enough to accommodate these languages. Now, let us define the formal languages with which we will work. Suppose that we have a denumerable set Var of variables {x0 , x1 , ..., y0 , ...}; then the set of terms Tm♯ is ′ {τ |{z} ... ′ | n ∈ ω and τ ∈ Var ∪ {0}} n times 7 Priest’s [9] only refers to this system as the “plain connexive logic.” The nomenclature PN was introduced in [24]. The set of closed terms Tm♯C is the subset comprising instances of the symbol 0 followed by finitely (possibly zero) many applications of (·)′ . Definition 5. At♯ is defined as the set {s=t ˙ | s, t ∈ Tm♯ } and the set of closed atoms At♯C is defined as the ♯ set {s=t ˙ | s, t ∈ TmC }. Given the distinct logical connectives of PA1 and PA2, we must define two languages of arithmetic. Definition 6. If ϕ is a formula and s, t ∈ Tm♯ , then ϕ[s ::= t] is the formula generated by replacing each instance of s with an instance of t. Definition 7. L ♯ is defined • If ϕ ∈ At♯ then ϕ ∈ L ♯ • If ϕ ∈ L ♯ then ∼ϕ ∈ L ♯ • If ϕ, ψ ∈ L ♯ then ϕ ∧ ψ ∈ L ♯ • If ϕ, ψ ∈ L ♯ then ϕ → ψ ∈ L ♯ • If ϕ ∈ L ♯ then ∀x(ϕ[t ::= x]) ∈ L ♯ for x ∈ Var Definition 8. LT♯ is defined in a similar fashion, appending the recursive clause: • If ϕ ∈ LT♯ then Tϕ ∈ LT♯ 3.3 Protoarithmetical Theories in Angell’s PA1 and PA2 In this section, we will consider some of the idiosyncrasies that will meet arithmetic if its axioms are formulated in appropriately rich extensions of PA1 and PA2. Definition 9. A universal-identity (UI) extension of PA1 is any deductive system extending PA1 rich enough to ensure that: • there exist valuations v : At♯C → VPA1 • the valuations respect the truth functions of PA1 • for any valuation v there exists a recursive method of evaluating each formula ∀xϕ[t ::= x] so that v(∀xϕ[t ::= x]) ∈ VPA1 A UI extension of PA2 is defined analogously. We will denote these systems generically by “PA1+ ” and “PA2+ ,” defining validity in the expected way: Definition 10. We say that Γ PA1+ ϕ (respectively, Γ PA2+ ϕ) when in every PA1+ model (respectively, PA2+ model) such that v(ψ) ∈ DPA1 for all ψ ∈ Γ, also v(ϕ) ∈ DPA1 (respectively, DPA2 ). A further logical definition is required before turning to look at arithmetic-specific considerations. We will define a theory semantically as a set of sentences closed under semantic consequence: Definition 11. In a deductive system L, a theory T is a collection of sentences closed under semantic consequence, that is, T is a set such that ϕ ∈ T iff T L ϕ. Note an important aspect of the foregoing definition. In classical logic, the standard definition of a theory is only that T L ϕ entails that ϕ ∈ T . While the converse holds classically by the definition of consequence— guaranteeing the felicity of the above definition—the converse must be explicitly expressed in cases in which consequence is non-Tarskian. Now, suppose we also employ a convention defining an extension of L ♯ to include formulae with open variables x, y, etc. Then the standard definition of a bounded universal quantifier is expressible in the theories of L ♯ . Definition 12. For a natural number n, define a bounded universal quantifier so that ′ (∀x ≤ n)ϕ(x) =df ϕ(x)[x := 0] ∧ ϕ(x)[x := 0′ ] ∧ ... ∧ ϕ(x)[x := 0|{z} ... ′ ] n times A very reasonable expectation concerning bounded universal quantifiers in arithmetic is that if all natural numbers less than n have a property ϕ, then when m < n, all natural numbers less than m have the property ϕ. Despite this expectation, we observe the following idiosyncrasy with respect to arithmetic in any UI extension of PA1: Observation 1. There exists a formula ψ such that for any theory T in a UI extension PA1+ and natural numbers m and n, T PA1+ (∀x ≤ n + 2m)ψ(x) → (∀x ≤ n)ψ(x) although T 2PA1+ (∀x ≤ n + 2m − 1)ψ(x) → (∀x ≤ n)ψ(x) ˙ y denote the formula ∼(x=y) Proof. Let x6= ˙ and let ψ(x) denote the formula ˙ ˙ x)). Then it can easily be confirmed that for any natural number n, the ∼((x=x ˙ → (x6=x)) ∧ (x=x ˙ → x6= value assigned to ψ(x)[x := pnq] is 0. Then the value assigned to (∀x ≤ n)ψ(x) is determined entirely by the parity of n, that is, for any valuation v, we have the following: v((∀x ≤ n)ψ(x)) = ( 0 1 if n is odd if n is even Hence, the formula (∀x ≤ n + 2m)ψ(x) → (∀x ≤ n)ψ(x) will take a value of 1 in any model of T while (∀ ≤ n + 2m − 1)ψ(x) → (∀x ≤ n)ψ(x) will take a value of 2 in any model of T . This means that for all n, T PA1+ (∀x ≤ n + 2m)ψ(x) → (∀x ≤ n)ψ(x) although T 2PA1+ (∀x ≤ n + 2m − 1)ψ(x) → (∀x ≤ n)ψ(x). Note that this is common to all PA1+ theories, not merely those including some fragment of arithmetic. In other words, this pathology is intimately related to John Woods’ diagnosis of the “defects” of McCall’s connexive CC1. Although CC1 is distinct from PA1, the similarity of the two entails that Woods’ objection applies equally to both systems: The upshot would appear to be that p connexively implies only odd-numbered conjunctions of occurrences of itself, and never even-numbered ones. [25, 474] This feature is arguably more troubling in the context of arithmetic, as the foregoing observation lifts the pathology from matters of logical form to the behavior of bounded quantification over natural numbers. Although the revisions to conjunction central to Angell’s PA2 seem to solve these apparently counterintuitive features of bounded quantification—PA2 is introduced precisely to repair the pathology concerning conjunction in PA1—there remain some peculiarities facing PA2 theories of very weak subsystems of arithmetic. Let us define conditions for PA2+ theories to be protoarithmetical. Definition 13. Call a theory T protoarithmetical if • For all t ∈ Tm♯C , t=t ˙ ∈T • For all s, t ∈ Tm♯C , s′ =t ˙ ′ → s=t ˙ ∈T Essentially, that T is protoarithmetical is to say that it provides a natural interpretation of identity and that each instance of the successor axiom holds. Lemma 1. Let v be a model of a protoarithmetical theory in a UI extension of PA2. Then • for all t ∈ Tm♯C , v(t=t) ˙ = 0 or ˙ = 1. • for all t ∈ Tm♯C , v(t=t) ˙ = 0 although v(t=t) ˙ = 1. For a term Proof. Suppose not and that there exist s, t ∈ Tm♯C such that v(s=s) ♯ ′′′ t ∈ TmC , let κ(t) denote the number of primes occurring in t so that, e.g., κ(0 ) = 3. Suppose without loss of generality that κ(s) < κ(t). Then there exists a term u ∈ Tm♯C (possibly s) such that κ(s) ≤ κ(u) < κ(u′ ) ≤ κ(t) where v(u=u) ˙ = 0 and v(u′ =u ˙ ′ ) = 1. Simple calculation entails that ′ ′ v(u =u ˙ → u=u) ˙ = 3, which is not designated. This contradicts the assumed protoarithmeticity of the theory of v. First, consider the following definitions. Let PA2+ be a UI extension of PA2: Definition 14. A protoarithmetical PA2+ theory T is literal if for every term t ∈ At♯C , T PA2+ (t=t) ˙ ↔ T(t=t). ˙ Definition 15. A protoarithmetical PA2+ theory T is illiterate if for every term t ∈ At♯C , T PA2+ ∼((t=t) ˙ ↔ T(t=t)). ˙ Given the interpretation given to T in [8], a theory is literal if all statements of self-identity are literally true and a theory is illiterate if self-identity is always meant hypothetically, that is, formulae of the form t=t ˙ are considered absent any assumption that t=t ˙ is true. If we use “completeness” of a theory in the model-theoretic sense, i.e., that T is complete when it is the theory of some model, these are the only types of complete protoarithmetical PA2+ theory—and a fortiori, the only two types of complete theories of PA2+ arithmetic. Consider the following observation: Observation 2. For each UI extension of PA2, every complete protoarithmetical theory is either literal or illiterate. Proof. By model-theoretic completeness, T is the theory of a model v; from Lemma 1, either for all formulae t=t ˙ ∈ At♯C , v(t=t) ˙ = 0 or for all t=t ˙ ∈ At♯C , v(t=t) ˙ = 1. By examining the truth tables for PA2—which PA2+ must respect—we draw the inference that v will either uniformly assign formulae (t=t) ˙ ↔ T(t=t) ˙ the value of 3 (in the former case) or uniformly assign such formulae the value 1. As T is the theory of v, it follows that either T is literal or illiterate. 3.4 Numerically Inductive Theories in Priest’s PN Just as we considered a very general scheme for extending PA1 and PA2, we can give a similar definition to suitable extensions of PN . Definition 16. A universal extension of PN is any extension enriching PN sufficiently so that: • models have valuations v : At♯C → ℘(W ) • the forcing conditions for the connectives are identical to those in PN • for any model there is a recursive method of evaluating a formula ∀xϕ[t ::= x] governing when w x] ∀xϕ[t ::= Recall the earlier qualification made with respect to the definition of a theory. The peculiarities of PN —and any first-order extensions of PN —have important consequences for this notion. Standardly, any set of sentences has a deductive closure modulo most deductive systems. But the notion of logical consequence in PN deviates from the standard Tarskian account and not all sets of formulae can serve as the kernel of a deductive closure. For example, inasmuch as p ∧ ∼p 0PN p ∧ ∼p, the set {p ∧ ∼p} may not have a deductive closure modulo ⊢PN , i.e., {p ∧ ∼p}⊢PN = ∅. This follows intuitively from Priest’s formalization of negation-as-cancellation. The set {p ∧ ∼p} has no content and therefore has no deductive closure. We are still concerned with showing problems with formalizing even weak arithmetics in connexive logic but will now consider theories besides protoarithmetical theories. It is likely that we would wish to retain some aspect of induction in our theories of arithmetic. Let us define the following notion: Definition 17. Let T be a theory in a UI extension of PN whose language extends L ♯ . Then T is numerically inductive if the signature of T extends the signature of arithmetic and for every formula ϕ(x), the formula [(ϕ(0) ∧ ∀x(ϕ(x) → ϕ(x′ ))) → ∀xϕ(x)] ∈ T . Observation 3. In any UI extension of PN there are no numerically inductive theories. Proof. Let ψ be an arbitrary formula of the language (e.g., 0=0), ˙ let ϕ(x) = (ψ → ∼ψ), and let P+ N be an arbitrary UI extension of PN . Then for no w in any model M will M, w ϕ(0). Thus for arbitrary formulae ξ and ζ, the formula (ψ(0) ∧ ξ) → ζ will be logically false. Hence, no matter how quantification is handled in ′ ′ P+ N —i.e., irrespective of the points at which ∀x(ϕ(x) → ϕ(x )) and ∀x(ϕ(x)) are true—ϕ(0) ∧ ∀x(ϕ(x) → ϕ(x )) will hold at no w in the model by the forcing conditions for connexive implication. It follows the formula (ϕ(0) ∧ ∀x(ϕ(x) → ϕ(x′ ))) → ∀xϕ(x) will hold at no w and a fortiori at no designated point g. Moreover, if (ϕ(0) ∧ ∀x(ϕ(x) → ϕ(x′ ))) → ∀xϕ(x) ∈ T then the deductive closure of T is ∅, that is, there are no theories including all instances of the arithmetical induction schema. We noted that the scheme (ϕ(0) ∧ ξ) → ζ will have logically false instances for all ξ and ζ in any UI extension of PN . Thus, even significant restrictions to induction—e.g., induction on quantifier-free formulae—cannot obtain. However, it is worth noting that this pathology does not at first blush conflict with Robinson’s Q, as Q lacks any type of induction axiom scheme. Hence, advocates of very weak subsystems of Peano arithmetic may not be discouraged by this observation. Furthermore, it is worth mentioning that this constitutes a trivial decidability result concerning the Peano + ♯ ♯ axioms in P+ N , i.e., (PN ) . Consider an arbitrary formula ϕ in a language extending L and ask: Is ϕ a logical + ♯ ♯ consequence of the theory (PN ) ? Of course, we have an answer: No. By Observation 3, (P+ N ) has no logical consequences and is thus trivially decidable. On its face, Observation 3 might seem to entail that first-order extensions of PN are not suitable bases for the formulation of arithmetic. Despite this, it is important to note that the fact that Peano arithmetic cannot be formulated in this system does not entail that arithmetic cannot be so formulated. All this shows is that systems of arithmetic that include even meager species of induction are not practicable. But not all theories of arithmetic posit induction. Observation 3 fails to rule out that Robinson’s Q—the axioms of Peano arithmetic without the induction schema—can be formulated in Priest’s system. Moreover, several philosophical standpoints anticipate that inclusion of the induction axioms with the other Peano axioms should yield a trivial result. From the perspective of strict finitism—a position with which Priest himself has flirted in [26]—the true pathology is found not in the failure of induction but in the supposition that it holds. That an arithmetic with induction is empty in the sense of Observation 3, after all, is a side of the same medal as e.g., Edward Nelson’s attempts to show that arithmetic with induction is inconsistent (see, e.g., [27] for Edward Nelson’s criticism of the inherent impredicativity of the induction schema). 4. CONCLUSION In this paper, we have surveyed a number of topics concerning the implementation of a connexive arithmetic. The foregoing observations have ranged from the encouraging (e.g., the rebuttal to Lukasiewicz’ argument against Aristotle’s Thesis) to the discouraging (e.g., the pathologies of PA1). However, especially in the cases of UI extensions of PA2 and PN , what has been uncovered is that conjoining connexive and arithmetical concerns yields a landscape that is not insuperable, although its terrain may appear quite alien from the perspective of classical mathematics. One can justifiably interpret this as an invitation to further study. Supposing that there exist models of, say, Q in UI extensions of PA2 and PN , there are many natural questions that emerge: Is there a robust and natural way of interpreting the distinction between literal and illiterate models in PA2+ ? Is there any recursive restriction of numerical induction (just as we restrict comprehension) that would permit induction in P+ N arithmetic by, e.g., considering only induction for negation-free formulae? Given the pathologies that greet arithmetic formulated in these systems, it is fair to say that the supposition that there exist a model of even Q in these systems is a much stronger assumption than the existence of a model for classical PA. In this regard, Wansing’s C provides a bit of a ray of light with respect to the pursuit of connexive arithmetic. Given the faithful embedding of quantified C into positive intuitionistic logic described in [12], it seems very likely that C♯ is Post consistent relative to Heyting arithmetic J♯ , that is: Conjecture 1. If J♯ has a model then there exists a model of C♯ . This conjecture is very plausible and suggests that in the case of C, the presumption of the existence of a model of C♯ will be a corollary of mathematical orthodoxy. As a field, modern studies in connexive logic have generally struggled with reconciling the prima facie plausibility of connexive theses with the pathologies that emerge during their formalization. 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