Absolute Decidability and Mathematical Modality Hasen Khudairi Abstract This paper aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of multi-dimensional intensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the multi-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. 1 Introduction This essay aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. I argue that mathematical modality falls under at least three types; the interpretational, the metaphysical, and the logical. The interpretational type of mathematical modality has traditionally been taken to concern the interpretation of the quantifiers (cf. Linnebo, 2009, 2010, 2013; Studd, 2013); the possible reinterpretations of the intensions of the concept of set (Uzquiano, 2015); and the possibility of reinterpreting the domain over which the quantifiers range, in order to avoid inconsistency (cf. Fine, 2005, 2006, 2007). The metaphysical type of modality concerns the ontological profile of abstracta 1 and mathematical truth. Abstracta are thus argued to have metaphysically necessary being, and mathematical truths hold of metaphysical necessity, if at all (cf. Fine, 1981; Williamson, 2016). Instances, finally, of the logical type of mathematical modality might concern the properties of consistency (cf. Field, 1989: 249-250, 257-260; Rayo, 2013: 50; Leng: 2007; 2010: 258), and can perhaps be further witnessed by the logic of provability (cf. Boolos, 1993) and the modal profile of forcing (cf. Kripke 1965; Hamkins and Löwe, 2008). The significance of the present contribution is as follows. (i) Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality.1 (ii) I argue, then, that the framework of multidimensional intensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. I target, in particular, the modal axioms that the respective interpretations of the modal operator ought to satisfy. The discrepancy between the modal systems governing the parameters in the multi-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. (iii) Finally, I examine the application of the mathematical modalities beyond the issues of unrestricted quantification and indefinite extensibility. As a test case for the 1A precedent to the current approach is Parsons (1979-1980; 1983: p. 25, chs.10-11; 2008: 176), who argues that intuition is both a species of the imagination and can be formalized by a mathematical modality. The mathematical modality is governed by S4.2, and concerns possible iterations of the successor operation in arithmetic and possible extensions of the settheoretic cumulative hierarchy. Among the differences between Parsons' approach and the one here outlined is (i) that, by contrast to the current proposal, Parsons notes that his notion of mathematical modality is not epistemic (2008: 81fn1); and (ii) that Parsons (1997: 348-351; 2008: 98-100) suggests that the intuitional mathematical modality concerning computable functions is an idealization insensitive to distinctions such as those captured by computational complexity theory, rather than being defined relative to an epistemic modal space comprising the computational theory of mind. (See Author, ms, for further discussion.) 2 multi-dimensional approach, I investigate the interaction between the epistemic and metaphysical mathematical modalities and large cardinal axioms. The multi-dimensional intensional framework permits of a formally precise means of demonstrating how the metaphysical possibility of absolute decidability and the continuum hypothesis can be accessed by their epistemic-modal-mathematical profile. The logical mathematical modalities – of consistency, provability, and forcing – provide the means for discerning whether mathematical truths are themselves epistemically possible. I argue that, in the absence of disproof, large cardinal axioms are epistemically possible, and thereby provide a sufficient guide to the metaphysical mathematical possibility of determinacy claims and the continuum hypothesis. In Section 2, I define the formal clauses and modal axioms governing the epistemic and metaphysical types of mathematical modality. In Section 3, I discuss how the properties of the epistemic mathematical modality and metaphysical mathematical modality converge and depart from previous attempts to delineate the contours of similar notions. Section 4 extends the multi-dimensional intensional framework to the issue of mathematical knowledge; in particular, to the modal profile of large cardinal axioms and to the absolute decidability of the continuum hypothesis. Section 5 provides concluding remarks. 2 Mathematical Modality 2.1 Metaphysical Mathematical Modality A formula is a logical truth if and only if the formula is true in an intended model structure, M = <W, D, R, V>, where W designates a space of metaphysically possible worlds; D designates a domain of entities, constant across worlds; R 3 designates an accessibility relation on worlds; and V is an assignment function mapping elements in D to subsets of W. A formula in M is a modal truth if and only if a faithful interpretation maps the formula to a metaphysically universal proposition (cf. Williamson, 2013.: 106). A formula satisfies conditions on metaphysically universality if and only if the formula is true on its universal generalization (cf. Williamson, op. cit.: 93). 2.2 Epistemic Mathematical Modality In order to accommodate the notion of epistemic possibility, we enrich M with the following conditions: M = <C, W, D, R, V>, where C, a set of epistemically possibilities, is constrained as follows: Let JφKc ⊆ C; (φ is a formula encoding a state of information at an epistemically possible world). -pri(x) = λc.JxKc,c; (the two parameters relative to which x – a propositional variable – obtains its value are epistemically possible worlds. The function from possible formulas to values is thus an intension). -sec(x) = λc.JxKw,w (the two parameters relative to which x obtains its value are metaphysically possible worlds). Then: • Epistemic Mathematical Necessity (Apriority) JφKc,w = 1 ⇐⇒ ∀c′JφKc,c ′ = 1 (φ is true at all points in epistemic modal space). 4 • Epistemic Mathematical Possibility J⋄cφKin 6= ∅ ⇐⇒ JPrφKin 6= ∅ ∧ >.5, else 〈∅, Prin (φ | ∅)〉, where in designates an agent's state of information i in a context n. (φ might be true if and only if its value is not null and it is greater than .5). Crucially, epistemic mathematical modality is constrained by consistency, and the formal techniques of provability and forcing. A mathematical formula is false, and therefore metaphysically impossible, if it can be disproved or induces inconsistency in a model. 2.3 Interaction • Convergence ∀c∃wJφKc,w = 1 (the value of x is relative to a parameter for the space of epistemically possible worlds. The value of x relative to the first parameter determines the value of x relative to the second parameter for the space of metaphysical possibility). • Super-rigidity (2D-Intension): JφKw,c = 1 ⇐⇒ ∀w',c'JφKw ′ ,c ′ = 1 (the intension of φ is rigid in all points in metaphysical and epistemic modal space). 2.4 Modal Axioms • Metaphysical mathematical modality is governed by the modal system KTE, as augmented by the Barcan formula and its Converse (cf. Fine, 5 1981). K: [φ → ψ] → [φ → ψ] T: φ → φ E: ¬φ → ¬φ Barcan: ⋄∃xFx → ∃x⋄Fx Converse Barcan: ∃x⋄Fx → ⋄∃xFx • Epistemic mathematical modality is governed by the modal system, KT4, as augmented by the Barcan formula and the Converse Barcan formula.2 K: [φ → ψ] → [φ → ψ] T: φ → φ 4: φ → φ Barcan: ∃xFx → ∃xFx Converse Barcan: ∃xFx → ∃xFx 3 Departures from Precedent The approach to mathematical modality, according to which it yields a representation of the cumulative universe of sets, has been examined by Fine (2005; 2006) and Uzquiano (2015). Fine argues that the mathematical modality should 2Reasons adducing against including the Smiley-Gödel-Löb provability formula among the axioms of epistemic mathematical modality are examined in Section 5. GL states that '[φ → φ] → φ'. For further discussion of the properties of GL, see Löb (1955); Smiley (1963); Kripke (1965); and Boolos (1993). Löb's provability formula was formulated in response to Henkin's (1952) problem concerning whether a sentence which ascribes the property of being provable to itself is provable. (Cf. Halbach and Visser, 2014, for further discussion.) For an anticipation of the provability formula, see Wittgenstein (1933-1937/2005: 378). Wittgenstein writes: 'If we prove that a problem can be solved, the concept 'solution' must somehow occur in the proof. (There must be something in the mechanism of the proof that corresponds to this concept.) But the concept mustn't be represented by an external description; it must really be demonstrated. / The proof of the provability of a proposition is the proof of the proposition itself' (op. cit.). Wittgenstein contrasts the foregoing type of proof with 'proofs of relevance' which are akin to the mathematical, rather than empirical, propositions, discussed in Wittgenstein (2001: IV, 4-13, 30-31). 6 be interpretational; and thus taken to concern the reinterpretation of the domain over which the quantifiers range, in order to avoid inconsistency. Uzquiano argues similarly for an interpretational construal of mathematical modality, where the cumulative hierarchy of sets is fixed, yet what is possibly reinterpreted is the non-logical vocabulary of the language, in particular the membership relation.3 On Fine's approach, the interpretational modality is both postulational, and 'prescriptive' or imperatival. The prescriptive element consists in the rule: 'Introduction: !x.C(x)', such that one is enjoined to postulate, i.e. to 'introduce an object x conforming to the condition C(x)' (2005: 91; 2006: 38). In the setting of unrestricted quantification, suppose, e.g., that there is an interpretation for the domain over which a quantifier ranges. Fine writes that an interpretation 'I is exten[s]ible – in symbols, E(I) – if possibly some interpretation extends it, i.e. ⋄∃J(I⊂J)' (2006: 30). Then, the interpretation of the domain over which the quantifier ranges is extensible, if '∀I.E(I)'. The interpretation of the domain over which the quantifier ranges is indefinitely extensible, if '∀I.E(I)' iff '∀I⋄∃J(I⊂J)', where the reinterpretation is induced via the prescriptive imperative to postulate the existence of a new object by the foregoing 'Introduction' rule (2006: 30-31; 38). Fine clarifies that the interpretational approach is consistent with a 'realist ontology' of the set of reals. He refers to the imperative to postulate new objects, and thereby reinterpret the domain for the quantifier, as the 'mechanism' by which epistemically to track the cumulative hierarchy of sets (2007: 124-125). In accord with Fine's approach, the epistemic mathematical modality defined in the previous section was taken to have a similarly representational interpreta3Compare Gödel, 1947; Williamson, 1998; and Fine, 2005. 7 tion, and perhaps the postulational property is an optimal means of inducing a reinterpretation of the domain of the quantifier. However, the present approach avoids a potential issue with Fine's account, with regard to the the introduction of deontic modal properties of the prescriptive and imperatival rules that he mentions.4 It is sufficient that the interpretational modalities are a species of epistemic modality, i.e. possibilities that are relative to agents' spaces of states of information. Developing Fine's program, Linnebo (2013) outlines a modalized version of ZF. Similarly to the modal axioms for the epistemic mathematical modality specified in the previous section, Linnebo argues that his modal set theory ought to be governed by the system S4.2, the Converse Barcan formula, and (at least a restricted version of) the Barcan formula. However – rather than being either interpretational or epistemic – Linnebo deploys the mathematical modality in order to account for the notion of 'potential infinity', as anticipated by Aristotle.5 The mathematical modality is thereby intended to provide a formally precise answer to the inquiry into the extent of the cumulative set-theoretic hierarchy; i.e., in order to precisify the answer that the hierarchy extends 'as far as possible' (2013: 205).6 Thus, Linnebo takes the modality to be constitutive of the actual ontology of sets; and the quantifiers ranging over the actual ontology of sets are claimed to have an 'implicitly modal' profile (2010: 146; 2013: 225). He suggests, e.g., that: 'As science progresses, we formulate set theories that characterize larger 4For an analysis of the precise interaction between the semantic values of epistemic and deontic modal operators, see Author (ms). 5Cf. Aristotle, Physics, Book III, Ch. 6. 6Precursors to the view that modal operators can be availed of in order to countenance the potential hierarchy of sets include Hodes (1984). Intensional constructions of set theory are further developed by Reinhardt (1974); Parsons (1983); Myhill (1985); Scedrov (1985); Flagg (1985); Goodman (1985); Hellman (1990); Nolan (2002); and Studd (2013). (See Shapiro (1985) for an intensional construction of arithmetic.) Chihara (2004: 171-198) argues that 'broadly logical' conceptual possibilities can be used to represent imaginary situations relevant to the construction of open-sentence tokens. The open-sentences can then be used to define the properties of natural and cardinal numbers and the axioms of Peano arithmetic. 8 and larger initial segments of the universe of sets. At any one time, precisely those sets are actual whose existence follows from our strongest, well-established set theory' (2010: 159n21). However – despite his claim that the modality is constitutive of the actual ontology of sets – Linnebo concedes that the mathematical modality at issue cannot be interpreted metaphysically, because sets exist of metaphysical necessity if at all (2010: 158; 2013: 207). In order partly to allay the tension, Linnebo remarks, then, that set theorists 'do not regard themselves as located at some particular stage of the process of forming sets' (2010: 159); and this might provide evidence that the inquiry – concerning at which stage in the process of set-individuation we happen to be, at present – can be avoided. Another distinction to note is that both Linnebo (op. cit.) and Uzquiano (op. cit.) avail of second-order plural quantification, in developing their primitivist and interpretational accounts of mathematical modality. By contrast to their approaches, the epistemic and metaphysical modalities defined in the previous section are defined with second-order singular quantification over sets. Finally, Linnebo and Uzquiano both suggest that their mathematical modalities ought to be governed by the G axiom; i.e. ⋄φ → ⋄φ. The present approach eschews, however, of the G axiom, in virtue of the following. Williamson (2009) demonstrates that – because KT4G is a sublogic of S5 – an epistemic operator which validates the conjunction of the 4 axiom of positive introspection and the E axiom of negative introspection will be inconsistent with the condition of 'recursively enumerable quasi-conservativeness'. Recursively enumerable quasi-conservativeness is a computational constraint on an epistemic agent's theorizing, according to which the intended models of the agent's theory are both maximally consistent and conservatively extended by addition of 9 the 'box'-operator, interpreted as expressing the agent's state of knowledge. As axioms of an agent's consistent, recursively axiomatizable theorizing about the theory of its own states of knowledge and belief, the conjunction of 4 and E would entail that the agent's theory is both consistent and decidable, in conflict with Gödel's (1931) second incompleteness theorem. The modal system, KT4, avoids the foregoing result. In the present setting, the circumvention is innocuous, because the undecidability – yet recursively enumerable quasi-conservativeness – of an epistemic agent's consistent theorizing about its epistemic states is consistent with the epistemic mathematical possibility that large cardinal axioms are absolutely decidable. 4 Knowledge of Absolute Decidability Williamson (2016) examines the extension of the metaphysically modal profile of mathematical truths to the question of absolute decidability. In this section, I aim to extend Williamson's analysis to the notion of epistemic mathematical modality that has been developed in the foregoing sections. The extension provides a crucial means of witnessing the signficance of the multi-dimensional intensional approach for the epistemology of mathematics. Williamson proceeds by suggesting the following line of thought. Suppose that A is a true interpreted mathematical formula which eludes present human techniques of provability; e.g. the continuum hypothesis (op. cit.). Williamson argues that mathematical truths are metaphysically necessary (op. cit.). From there, he suggests that knowledge of A satisfies the condition of safety from error, as codified via a reflexive and symmetric accessibility relation from worlds at which A is known. Thus, there is either no, or a small risk of, not believing that A, relative to a world in which A is known – although the safety condition is 10 not itself sufficient for mathematical knowledge that A. Williamson then enjoins one to consider the following scenario: It is metaphysically possible that there is a species which can prove that A. Therefore, A is absolutely provable; that is, A 'can in principle be known by a normal mathematical process' such as derivation in an axiomatizable formal system with quantification and identity. Williamson's scenario evinces one issue for the 'back-tracking' approach to modal epistemology, at least as it might be applied to the issue of possible mathematical knowledge. On the back-tracking approach, the method of modal epistemology is taken to proceed by first discerning the metaphysical modal truths – normally by natural-scientific means – and then working backward to the exigent incompleteness of an individual's epistemic states concerning such truths (cf. Stalnaker, 2003; Vetter, 2013). The issue for the back-tracking method that Williamson's scenario illuminates is that the metaphysical mathematical possibility that CH is absolutely decidable must in some way converge with the epistemic possibility thereof. However, the normal mathematical techniques that Williamson specifies – i.e. proof and forcing – fall within the remit of what is mathematically possible relative to agents' states of information; i.e. what is epistemically mathematically possible. Thus, whether CH is metaphysically necessary – and thus, as Williamson claims, metaphysically possible and absolutely decidable thereby – can only be witnessed by the epistemic means of demonstrating that its absolute decidability is not impossible. It may thus be epistemically possible that Williamson's technically advanced species, which can absolutely decide CH, exist – following Williamson (2013), they actually exist, albeit non-concretely – but the metaphysical necessity of the absolute decidability of CH needs still to be corroborated. 11 The significance of the multi-dimensional intensional framework outlined in the foregoing is that it provides an explanation of the discrepancy between metaphysical mathematical modality and epistemic mathematical modality. Further and crucially, metaphysical mathematical modality is governed by the system S5, the Barcan formula, and its Converse, whereas epistemic mathematical modality is governed by KT4, the Barcan formula, and its Converse. Thus, epistemic mathematical modality figures as the mechanism, which enables the tracking of metaphysically possible mathematical truth.7 Leitgeb (2009) endeavors similarly to argue for the convergence between the notion of informal provability – countenanced as an epistemic modal operator, K – and mathematical truth. Availing of Hilbert's (1923/1996: ¶18-42) epsilon terms for propositions, such that, for an arbitrary predicate, C(x), with x a propositional variable, the term 'ǫp.C(p)' is intuitively interpreted as stating that 'there is a proposition, x(/p), s.t. the formula, that p satisfies C, obtains' (op. cit.: 290). Leitgeb purports to demonstrate that ∀p(p → Kp), i.e. that 7A provisional definition of large cardinal axioms is as follows. ∃xΦ is a large cardinal axiom, because: (i) Φx is a Σ2-formula; (ii) if κ is a cardinal, such that V |= Φ(κ), then κ is strongly inaccessible, where a cardinal κ is regular if the cofinality of κ – comprised of the unions of sets with cardinality less than κ – is identical to κ, and a strongly inaccessible cardinal is regular and has a strong limit, such that if λ < κ, then 2λ < κ (Cf. Kanamori, 2012: 360); and (iii) for all generic partial orders P∈Vκ, VP |= Φ(κ); INS is a non-stationary ideal, where an ideal is a subset of a set closed under countable unions, whereas filters are subsets closed under countable intersections. (Cf. Kanamori, op. cit.: 361); AG is the canonical representation of reals in L(R), i.e. the interpretation of A in M[G]; H(κ) is comprised of all of the sets whose transitive closure is < κ (cf. Rittberg, 2015); and L(R)Pmax |= 〈H(ω2), ∈, INS , A G〉 |= 'φ'. P is a homogeneous partial order in L(R), such that the generic extension of L(R)P inherits the generic invariance, i.e., the absoluteness, of L(R). Thus, L(R)Pmax is (i) effectively complete, i.e. invariant under set-forcing extensions; and (ii) maximal, i.e. satisfies all Π2-sentences and is thus consistent by set-forcing over ground models (Woodin, ms: 28). Assume ZFC and that there is a proper class of Woodin cardinals; A∈P(R) ∩ L(R); φ is a Π2-sentence; and V(G), s.t. 〈H(ω2), ∈, INS , A G〉 |= 'φ': Then, it can be proven that L(R)Pmax |= 〈H(ω2), ∈, INS , A G〉 |= 'φ', where 'φ' := ∃A∈Γ∞〈H(ω1), ∈, A〉 |= ψ. The axiom of determinacy (AD) states that every set of reals, a⊆ωω is determined, where κ is determined if it is decidable. Woodin's (1999) Axiom (*) can be thus countenanced: ADL(R) and L[(Pω1)] is a Pmax-generic extension of L(R), from which it can be derived that 2א0 = א2. Thus, ¬CH; and so CH is absolutely decidable. 12 informal provability is absolute; i.e. truth and provability are co-extensive.8 He argues as follows. Let A(p) abbreviate the formula 'p ∧ ¬K(p)', i.e., that the proposition, p, is true while yet being unprovable. Let K be the informal provability operator reflecting knowability or epistemic necessity, with 〈K〉 its dual.9 Then: 1. ∃p(p ∧ ¬Kp) ⇐⇒ ǫp.A(p) ∧ ¬Kǫp.A(p). By necessitation, 2. K[∃p(p ∧ ¬Kp)] ⇐⇒ K[ǫp.A(p) ∧ ¬Kǫp.A(p)]. Applying modal axioms, KT, to (1), however, 3. ¬K[ǫp.A(p) ∧ ¬Kǫp.A(p)]. Thus, 4. ¬K∃p(p ∧ ¬Kp). Leitgeb suggests that (4) be rewritten 5. 〈K〉∀p(p → Kp). Abbreviate (5) by B. By existential introduction and modal axiom K, both 6. B → ∃p[K(p → B) ∨ K(p → ¬B) ∧ p], and 7. ¬B → ∃p[K(p → B) ∨ K(p → ¬B) ∧ p]. Thus, 8. ∃p[K(p → B) ∨ K(p → ¬B) ∧ p]. Abbreviate (8) by C(p). Introducing epsilon notation, 9. [K(ǫp.C(p) → B) ∨ K(ǫp.C(p) → ¬B)] ∧ ǫp.C(p). By K, 10. [K(ǫp.C(p) → KB) ∨ K(ǫp.C(p) → K¬B)]. From (9) and necessitation, one can further derive 8The formula is referred to as the Principle of Knowability, and discussed in further detail in Section 5, below. 9See Section 5, for further discussion of the duality of knowledge, and its relation to doxastic operators. 13 11. Kǫp.C(p). By (10) and (11), 12. KB ∨ K¬B. From (5), (12), and K, Leitgeb derives 13. KB. By, then, the T axiom, 14. ∀p(p → Kp) (291-292). Rather than accounting for the coextensiveness of epistemic provability and truth, Leitgeb interprets the foregoing result as cause for pessimism with regard to whether the formulas countenanced in epistemic logic and via epsilon terms are genuinely logical truths if true at all (292). In response to the attending pressure on the status of epistemic logic as concerning truths of logic, one can challenge the derivation, in the above proof, from lines (12) to (13). The inference depends on line (5), i.e., the epistemic possibility of completeness: 〈K〉∀p(p → Kp). One can question how, from (4), i.e. the unprovability of the unprovability of a proposition [¬K∃p(p ∧ ¬Kp)], one can derive (5), i.e. that it is epistemically possible that all propositions are informally provable. Assume, however, that line (5) is valid. Then, the validity of the inference from (12) to (13) can be challenged by the restriction on the quantifier on worlds in the Knowability Principle expressed by (5). The epistemic operator in lines (12) and (13) records, by contrast, the epistemic necessity, rather than the possibility, of the truth of the formulas and subformulas therein. Thus, from (12) either the provability of the provability of propositions or the provability of the unprovability of propositions, one cannot derive (13) the provability of the provability of propositions, because – by (5) – it is only epistemically possible that all true propositions are provable. 14 5 Concluding Remarks In this paper, I have endeavored to delineate the types of mathematical modality, and to argue that the epistemic interpretation of multi-dimensional intensional semantics can be applied in order to explain, in part, the epistemic status of large cardinal axioms and the decidability of Orey sentences. The formal constraints on mathematical conceivability adumbrated in the foregoing can therefore be considered a guide to our possible knowledge of unknown mathematical truth. 15 References Aristotle. 1987. Physics, tr. E. Hussey (Clarendon Aristotle Series, 1983), text: W.D. Ross (Oxford Classical Texts, 1950). In J.L. Ackill (ed.), A New Aristotle Reader. 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