Metadata of the chapter that will be visualized online Chapter Title Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism7. Copyright Year 2018 Copyright Holder Springer Nature Switzerland AG Corresponding Author Family Name Khudairi Particle Given Name Hasen Suffix Division Arché Philosophical Research Centre Organization University of St Andrews Address St. Andrews, Scotland Email hasen.khudairi@gmail.com Abstract This essay examines the philosophical significance of Ω-logic in Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between algebra and coalgebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of Ω-logical validity can then be countenanced within a coalgebraic logic, and Ωlogical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of Ω-logical validity correspond to those of second-order logical consequence, Ω-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets. Keywords (separated by "-") Modal Ω-logic Ω-logical Validity Modal Coalgebraic Automata - Neo-Logicism Set-theoretic Realism UN CO RR EC TE D PR OO F Chapter 4 1 Modal !-Logic: Automata, 2 Neo-Logicism, and Set-Theoretic Realism 3 Hasen Khudairi 4 Abstract This essay examines the philosophical significance of !-logic in 5 Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between 6 algebra and coalgebra permits Boolean-valued algebraic models of ZFC to be 7 interpreted as coalgebras. The modal profile of !-logical validity can then be 8 countenanced within a coalgebraic logic, and !-logical validity can be defined via 9 deterministic automata. I argue that the philosophical significance of the foregoing 10 is two-fold. First, because the epistemic and modal profiles of !-logical validity 11 correspond to those of second-order logical consequence, !-logical validity is 12 genuinely logical, and thus vindicates a neo-logicist conception of mathematical 13 truth in the set-theoretic multiverse. Second, the foregoing provides a modal14 computational account of the interpretation of mathematical vocabulary, adducing 15 in favor of a realist conception of the cumulative hierarchy of sets. 16 Keywords Modal !-logic * !-logical Validity * Modal Coalgebraic Automata * 17 Neo-Logicism * Set-theoretic Realism 18 4.1 Introduction 19 This essay examines the philosophical significance of the consequence relation 20 defined in the !-logic for set-theoretic languages. I argue that, as with second21 order logic, the modal profile of validity in !-Logic enables the property to be 22 epistemically tractable. Because of the dual isomorphism between algebras and 23 coalgebras, Boolean-valued models of set theory can be interpreted as coalgebras. 24 Forthcoming in the 'Proceedings of the 2016 Meeting of the International Association for Computing and Philosophy'. H. Khudairi (!) Arché Philosophical Research Centre, University of St Andrews, St. Andrews, Scotland © Springer Nature Switzerland AG 2018 D. Berkich, M. V. d'Alfonso (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence, Philosophical Studies Series 134, https://doi.org/10.1007/978-3-030-01800-9_4 UN CO RR EC TE D PR OO F H. Khudairi In Sect. 4.2, I demonstrate how the modal profile of !-logical validity can be 25 countenanced within a coalgebraic logic, and how !-logical validity can further 26 be defined via automata. In Sect. 4.3, I examine how models of epistemic modal 27 algebras to which modal coalgebraic automata are dually isomorphic are availed 28 of in the computational theory of mind. Finally, in Sect. 4.4, the philosophical 29 significance of the characterization of the modal profile of !-logical validity for the 30 philosophy of mathematics is examined. I argue (i) that it vindicates a type of neo31 logicism with regard to mathematical truth in the set-theoretic multiverse, and (ii) 32 that it provides a modal and computational account of formal grasp of the concept 33 of 'set', adducing in favor of a realist conception of the cumulative hierarchy of sets. 34 Section 4.5 provides concluding remarks. 35 4.2 Definitions 36 In this section, I define the axioms of Zermelo-Fraenkel set theory with choice. I 37 define the mathematical properties of the large cardinal axioms to which ZFC can 38 be adjoined, and I provide a detailed characterization of the properties of !-logic for 39 ZFC. Because Boolean-valued algebraic models of !-logic are dually isomorphic to 40 coalgebras, a category of coalgebraic logic is then characterized which models both 41 modal logic and deterministic automata. Modal coalgebraic models of automata are 42 then argued to provide a precise characterization of the modal and computationalAQ1 43 profiles of !-logical validity. 44 4.2.1 Axioms1 45 • Empty set: 46 ∃x∀u(u /∈ x) 47 • Extensionality: 48 x = y ⇐⇒ ∀u(u ∈ x ⇐⇒ u ∈ y) 49 • Pairing: 50 ∃x∀u(u ∈ x ⇐⇒ u = a ∨ u = b) 51 • Union: 52 ∃x∀u[u ∈ x ⇐⇒ ∃v(u ∈ v ∧ v ∈ a)] 53 • Separation: 54 ∃x∀u[u ∈ x ⇐⇒ u ∈ a ∧ φ(u)] 55 • Power Set: 56 ∃x∀u(u ∈ x ⇐⇒ u ⊆ a) 57 1For a standard presentation, see Jech (2003). For detailed, historical discussion, see Maddy (1988a). UN CO RR EC TE D PR OO F 4 Modal !-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism • Infinity: 58 ∃x∅ ∈ x ∧ ∀u(u ∈ x → {u} ∈ x) 59 • Replacement: 60 ∀u∃!vψ(u, v) → ∀x∃y(∀u ∈ x)(∃v ∈ y)ψ(u, v) 61 • Choice: 62 ∀u[u ∈ a → ∃v(v ∈ u)] ∧ ∀u, x[u ∈ a ∧ x ∈ a → ∃v(v ∈ u ⇐⇒ v ∈ 63 x) ∨ ¬v(v ∈ u ∧ v ∈ x)] → ∃x∀u[u ∈ a → ∃!v(v ∈ u ∧ u ∈ x)] 64 4.2.2 Large Cardinals 65 Borel sets of reals are subsets of ωω or R, closed under countable intersections and 66 unions.2 For all ordinals, a, such that 0 < a < ω1, and b < a, %0a denotes the open 67 subsets of ωω formed under countable unions of sets in &0b, and & 0 a denotes the 68 closed subsets of ωω formed under countable intersections of %0b . 69 Projective sets of reals are subsets of ωω, formed by complementations (ωω – u, 70 for u ⊆ ωω) and projections [p(u) = {⟨x1, . . . , xn⟩ ∈ ωω | ∃y⟨x1, . . . , xn, y⟩ ∈ u}]. 71 For all ordinals a, such that 0 < a < ω, &10 denotes closed subsets of ω ω; &1a is 72 formed by taking complements of the open subsets of ωω, %1a ; and % 1 a+1 is formed 73 by taking projections of sets in &1a . 74 The full power set operation defines the cumulative hierarchy of sets, V, such thatAQ2 75 V0 = ∅; Va+1 = P(V0); and Vλ = ! a<λ Va . 76 In the inner model program (cf. Woodin 2001, 2010, 2011; Kanamori 2012a,b), 77 the definable power set operation defines the constructible universe, L(R), in the 78 universe of sets V, where the sets are transitive such that a ∈ C ⇐⇒ a ⊆ C; 79 L(R) = Vω+1; La+1(R) = Def(La(R)); and Lλ(R) = ! a<λ(La(R)). 80 Via inner models, Gödel (1940) proves the consistency of the generalized 81 continuum hypothesis, אאaa = אa+1, as well as the axiom of choice, relative to the 82 axioms of ZFC. However, for a countable transitive set of ordinals, M, in a model 83 of ZF without choice, one can define a generic set, G, such that, for all formulas, φ, 84 either φ or ¬φ is forced by a condition, f , in G. Let M[G] = !a<κ Ma[G], such that 85 M0[G] = {G}; with λ < κ , Mλ[G] = ! a<λ Ma[G]; and Ma+1[G] = Va ∩ Ma[G].3 86 G is a Cohen real over M, and comprises a set-forcing extension of M. The relation 87 of set-forcing, !, can then be defined in the ground model, M, such that the forcing 88 condition, f , is a function from a finite subset of ω into {0,1}, and f ! u ∈ G if 89 f (u) = 1 and f ! u /∈ G if f (u) = 0. The cardinalities of an open dense ground 90 model, M, and a generic extension, G, are identical, only if the countable chain 91 condition (c.c.c.) is satisfied, such that, given a chain – i.e., a linearly ordered subset 92 of a partially ordered (reflexive, antisymmetric, transitive) set – there is a countable, 93 2See Koellner (2013), for the presentation, and for further discussion, of the definitions in this and the subsequent paragraph. 3See Kanamori (2012a: 2.1; 2012b: 4.1), for further discussion. UN CO RR EC TE D PR OO F H. Khudairi maximal antichain consisting of pairwise incompatible forcing conditions. Via set94 forcing extensions, Cohen (1963, 1964) constructs a model of ZF which negates 95 the generalized continuum hypothesis, and thus proves the independence thereof 96 relative to the axioms of ZF.4 97 Gödel (1946/1990: 1–2) proposes that the value of Orey sentences such as the 98 GCH might yet be decidable, if one avails of stronger theories to which new 99 axioms of infinity – i.e., large cardinal axioms – are adjoined.5 He writes that: 100 'In set theory, e.g., the successive extensions can be represented by stronger and 101 stronger axioms of infinity. It is certainly impossible to give a combinatorial and 102 decidable characterization of what an axiom of infinity is; but there might exist, 103 e.g., a characterization of the following sort: An axiom of infinity is a proposition 104 which has a certain (decidable) formal structure and which in addition is true. 105 Such a concept of demonstrability might have the required closure property, i.e. 106 the following could be true: Any proof for a set-theoretic theorem in the next higher 107 system above set theory . . . is replaceable by a proof from such an axiom of infinity. 108 It is not impossible that for such a concept of demonstrability some completeness 109 theorem would hold which would say that every proposition expressible in set theory 110 is decidable from present axioms plus some true assertion about the largeness of the 111 universe of sets'. 112 For cardinals, x,a,C, C ⊆ a is closed unbounded in a, if it is closed [if x < C and 113! (C ∩ a) = a, then a ∈ C] and unbounded (! C = a) (Kanamori, op. cit.: 360). 114 A cardinal, S, is stationary in a, if, for any closed unbounded C ⊆ a, C ∩ S = ∅ 115 (op. cit.). An ideal is a subset of a set closed under countable unions, whereas filters 116 are subsets closed under countable intersections (361). A cardinal κ is regular if the 117 cofinality of κ – comprised of the unions of sets with cardinality less than κ – is 118 identical to κ . Uncountable regular limit cardinals are weakly inaccessible (op. cit.). 119 A strongly inaccessible cardinal is regular and has a strong limit, such that if λ < κ , 120 then 2λ < κ (op. cit.). 121 Large cardinal axioms are defined by elementary embeddings.6 Elementary 122 embeddings can be defined thus. For models A,B, and conditions φ, j: A → B, 123 φ⟨a1, . . . , an⟩ in A if and only if φ⟨j(a1), . . . , j(an)⟩ in B (363). A measurable 124 cardinal is defined as the ordinal denoted by the critical point of j, crit(j) (Koellner 125 and Woodin 2010: 7). Measurable cardinals are inaccessible (Kanamori, op. cit.). 126 Let κ be a cardinal, and η > κ an ordinal. κ is then η-strong, if there is a transitive 127 class M and an elementary embedding, j: V → M, such that crit(j) = κ, j(κ) > η, 128 and Vη ⊆ M (Koellner and Woodin, op. cit.). 129 κ is strong if and only if, for all η, it is η-strong (op. cit.). 130 4See Kanamori (2008), for further discussion. 5See Kanamori (2007), for further discussion. Kanamori (op. cit.: 154) notes that Gödel (1931/1986: fn48a) makes a similar appeal to higher-order languages, in his proofs of the incompleteness theorems. The incompleteness theorems are examined in further detail, in Sect. 4.4.2, below. 6The definitions in the remainder of this subsection follow the presentations in Koellner and Woodin (2010) and Woodin (2010, 2011). UN CO RR EC TE D PR OO F 4 Modal !-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism If A is a class, κ is η-A-strong, if there is a j : V → M, such that κ is η-strong 131 and j(A ∩ Vκ) ∩ Vη = A ∩ Vη (op. cit.). 132 κ is a Woodin cardinal, if κ is strongly inaccessible, and for all A ⊆ Vκ , there is a 133 cardinal κA < κ , such that κA is η-A-strong, for all η such that κη, η < κ (Koellner 134 and Woodin, op. cit.: 8). 135 κ is superstrong, if j : V → M, such that crit(j) = κ and Vj (κ) ⊆ M, which 136 entails that there are arbitrarily large Woodin cardinals below κ (op. cit.). 137 Large cardinal axioms can then be defined as follows. 138 ∃x* is a large cardinal axiom, because: 139 (i) *x is a %2-formula; 140 (ii) if κ is a cardinal, such that V |= *(κ), then κ is strongly inaccessible; and 141 (iii) for all generic partial orders P ∈ Vκ , VP |= *(κ); INS is a non-stationary 142 ideal; AG is the canonical representation of reals in L(R), i.e. the interpretation 143 of A in M[G]; H(κ) is comprised of all of the sets whose transitive closure is 144 < κ (cf. Rittberg 2015); and L(R)Pmax |= ⟨H(ω2), ∈, INS , AG⟩ |= 'φ'. P is a 145 homogeneous partial order in L(R), such that the generic extension of L(R)P 146 inherits the generic invariance, i.e., the absoluteness, of L(R). Thus, L(R)Pmax 147 is (i) effectively complete, i.e. invariant under set-forcing extensions; and (ii) 148 maximal, i.e. satisfies all &2-sentences and is thus consistent by set-forcing 149 over ground models (Woodin (ms): 28). 150 Assume ZFC and that there is a proper class of Woodin cardinals; A ∈ P(R) ∩ 151 L(R); φ is a &2-sentence; and V(G), s.t. ⟨H(ω2), ∈, INS , AG⟩ |= 'φ': Then, it can 152 be proven that L(R)Pmax |= ⟨H(ω2), ∈, INS , AG⟩ |= 'φ', where 'φ' := ∃A ∈ 153 +∞⟨H(ω1), ∈, A⟩ |= ψ . 154 The axiom of determinacy (AD) states that every set of reals, a ⊆ ωω is 155 determined, where κ is determined if it is decidable. 156 Woodin's (1999) Axiom (*) can be thus countenanced: 157 ADL(R) and L[(Pω1)] is a Pmax-generic extension of L(R), 158 from which it can be derived that 2א0 = א2. Thus, ¬CH; and so CH is absolutely 159 decidable. 160 4.2.3 !-Logic 161 For partial orders, P, let VP = VB, where B is the regular open completion of (P).7 162 Ma = (Va)M and MBa = (VBa )M = (VM B a ). Sent denotes a set of sentences in 163 a first-order language of set theory. T ∪ {φ} is a set of sentences extending ZFC. 164 c.t.m abbreviates the notion of a countable transitive ∈-model. c.B.a. abbreviates 165 the notion of a complete Boolean algebra. 166 7The definitions in this section follow the presentation in Bagaria et al. (2006). UN CO RR EC TE D PR OO F H. Khudairi Define a c.B.a. in V, such that VB. Let VB0 = ∅; VBλ = ! b<λ V B b , with λ a limit 167 ordinal; VBa+1 = {f : X → B |X ⊆ VBa }; and VB = ! a∈On V B a . 168 φ is true in VB, if its Boolean-value is 1B, if and only if 169 VB |= φ iff !φ"B = 1B. 170 Thus, for all ordinals, a, and every c.B.a. B, VBa ≡ (Va)V B iff for all x ∈ VB, 171 ∃y ∈ VB!x = y"B = 1B iff !x ∈ VB"B = 1B. 172 Then, VBa |= φ iff VB |= 'Va |= φ'. 173 !-logical validity can then be defined as follows: 174 For T ∪ {φ} ⊆ Sent , 175 T |=! φ, if for all ordinals, a, and c.B.a. B, if VBa |= T, then VBa |= φ. 176 Supposing that there exists a proper class of Woodin cardinals and if T ∪ {φ} ⊆ 177 Sent , then for all set-forcing conditions, P: 178 T |=! φ iff VT |= 'T |=! φ', 179 where T |=! φ ≡ ∅ |= 'T |=! φ'. 180 The !-Conjecture states that V |=! φ iff VB |=! φ (Woodin ms). Thus, !181 logical validity is invariant in all set-forcing extensions of ground models in the 182 set-theoretic multiverse. 183 The soundness of !-Logic is defined by universally Baire sets of reals. For a 184 cardinal, e, let a set A be e-universally Baire, if for all partial orders P of cardinality 185 e, there exist trees, S and T on ω X λ, such that A = p[T] and if G ⊆ P is 186 generic, then p[T]G = RG – p[S]G (Koellner 2013). A is universally Baire, if it 187 is e-universally Baire for all e (op. cit.). 188 !-Logic is sound, such that V ⊢! φ → V |=! φ. However, the completeness 189 of !-Logic has yet to be resolved. 190 Finally, in category theory, a category C is comprised of a class Ob(C) of objects 191 a family of arrows for each pair of objects C(A,B) (Venema 2007: 421). A functor 192 from a category C to a category D, E: C → D, is an operation mapping objects and 193 arrows of C to objects and arrows of D (422). An endofunctor on C is a functor, E: 194 C → C (op. cit.). 195 A E-coalgebra is a pair A = (A, μ), with A an object of C referred to as the 196 carrier of A, and μ: A → E(A) is an arrow in C, referred to as the transition map of 197 A (390). 198 A = ⟨A, μ: A → E(A)⟩ is dually isomorphic to the category of algebras over the 199 functor μ (417–418). If μ is a functor on categories of sets, then Boolean-algebraic 200 models of !-logical validity are isomorphic to coalgebraic models. 201 The significance of the foregoing is that coalgebraic models may themselves be 202 availed of in order to define modal logic and automata theory. Coalgebras provide 203 therefore a setting in which the Boolean-valued models of set theory, the modal 204 profile of !-logical validity, and automata can be interdefined. In what follows, A 205 will comprise the coalgebraic model – dually isomorphic to the complete Boolean206 valued algebras defined in the !-Logic of ZFC – in which modal similarity types 207 and automata are definable. As a coalgebraic model of modal logic, A can be defined 208 as follows (407): 209 UN CO RR EC TE D PR OO F 4 Modal !-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism For a set of formulas, *, let ∇* := " " * ∧ # ⋄*, where ⋄* denotes the set 210 {⋄φ | φ ∈ * (op. cit.). Then, 211 ⋄φ ≡ ∇{φ, T}, 212 "φ ≡ ∇∅ ∨ ∇φ (op. cit.). 213 Let an E-coalgebraic modal model, A = ⟨S, λ, R[.]⟩, such that S,s ! ∇* if and 214 only if, for all (some) successors σ of s∈S, [*, σ (s) ∈ E(!A)] (op. cit.). 215 A coalgebraic model of deterministic automata can be thus defined (391). An 216 automaton is a tuple, A = ⟨A, aI , C, δ, F⟩, such that A is the state space of 217 the automaton A; aI ∈A is the automaton's initial state; C is the coding for the 218 automaton's alphabet, mapping numerals to properties of the natural numbers; δ: 219 A X C → A is a transition function, and F⊆ A is the collection of admissible 220 states, where F maps A to {1,0}, such that F: A → 1 if a∈F and A → 0 if a/∈F 221 (op. cit.). The determinacy of coalgebraic automata, the category of which is dually 222 isomorphic to the Set category satisfying !-logical consequence, is secured by the 223 existence of Woodin cardinals: Assuming ZFC, that λ is a limit of Woodin cardinals, 224 that there is a generic, set-forcing extension G ⊆ the collapse of ω < λ, and that 225 R∗ = !{RG[a] |a < λ}, then R* |= the axiom of determinacy (AD) (Koellner and 226 Woodin, op. cit.: 10). 227 Finally, A = ⟨A, α:A → E(A)⟩ is dually isomorphic to the category of algebras 228 over the functor α (417–418). For a category C, object A, and endofunctor E, 229 define a new arrow, α, s.t. α:EA → A. A homomorphism, f , can further be 230 defined between algebras ⟨A, α⟩, and ⟨B, β⟩. Then, for the category of algebras, 231 the following commutative square can be defined: (i) EA → EB (Ef ); (ii) EA → 232 A (α); (iii) EB → B (β); and (iv) A → B (f ) (cf. Hughes, 2001: 7–8). The same 233 commutative square holds for the category of coalgebras, such that the latter are 234 defined by inverting the direction of the morphisms in both (ii) [A → EA (α)], and 235 (iii) [B → EB (β)] (op. cit.). 236 Thus, A is the coalgebraic category for modal, deterministic automata, dually 237 isomorphic to the complete Boolean-valued algebraic models of !-logical validity, 238 as defined in the category of sets. 239 4.3 Epistemic Modal Algebras and the Computational 240 Theory of Mind 241 Beyond the remit of Boolean-valued models of set-theoretic languages, models of 242 epistemic modal algebras are availed of by a number of paradigms in contemporary 243 empirical theorizing, including the computational theory of mind and the theory 244 of quantum computability. In Epistemic Modal Algebra, the topological boolean 245 algebra, A, can be formed by taking the powerset of the topological space, X, defined 246 above; i.e., A = P(X). The domain of A is comprised of formula-terms – eliding 247 propositions with names – assigned to elements of P(X), where the proposition248 letters are interpreted as encoding states of information. The top element of the 249 UN CO RR EC TE D PR OO F H. Khudairi algebra is denoted '1' and the bottom element is denoted '0'. We interpret modal 250 operators, f(x), – i.e., intensional functions in the algebra – as both concerning 251 topological interiority, as well as reflecting epistemic possibilities. An Epistemic 252 Modal-valued Algebraic structure has the form, F = ⟨A, DP(X), ρ⟩, where ρ is 253 a mapping from points in the topological space to elements or regions of the 254 algebraic structure; i.e., ρ : DP(X) x DP(X) → A. A model over the Epistemic255 Modal Topological Boolean Algebraic structure has the form M = ⟨F, V⟩, where 256 V(a) ≤ ρ(a) and V(a,b) ∧ ρ(a, b) ≤ V(b).8 For all xx/a,φ ,y∈A: 257 f(1) = 1; 258 f(x) ≤ x; 259 f(x ∧ y) = f(x) ∧ f(y); 260 f[f(x)] = f(x); 261 V(a, a) > 0; 262 V(a, a) = 1; 263 V(a, b) = V(b, a); 264 V(a, b) ∧ V(b, c) ≤ V(a, c); 265 V(a = a) = ρ(a, a); 266 V(a, b) ≤ f[V(a, b)]; 267 V(¬φ) = ρ(¬φ) − f(φ); 268 V(⋄φ) = ρφ − f[−V(φ)]; 269 V("φ) = f[V(φ)] (cf. Lando, op. cit.).9 270 Marcus (2001) argues that mental representations can be treated as algebraic 271 rules characterizing the computation of operations on variables, where the values 272 of a target domain for the variables are universally quantified over and the function 273 is one-one, mapping a number of inputs to an equivalent number of outputs (35–36). 274 Models of the above algebraic rules can be defined in both classical and weighted, 275 connectionist systems: Both a single and multiple nodes can serve to represent the 276 variables for a target domain (42–45). Temporal synchrony or dynamic variable277 bindings are stored in short-term working memory (56–57), while information 278 relevant to long-term variable-bindings are stored in registers (54–56). Examples 279 of the foregoing algebraic rules on variable-binding include both the syntactic 280 concatenation of morphemes and noun phrase reduplication in linguistics (37–39, 281 70–72), as well as learning algorithms (45–48). Conditions on variable-binding 282 are further examined, including treating the binding relation between variables 283 and values as tensor products – i.e., an application of a multiplicative axiom 284 for variables and their values treated as vectors (53–54, 105–106). In order to 285 account for recursively formed, complex representations, which he refers to as 286 structured propositions, Marcus argues instead that the syntax and semantics of such 287 representations can be modeled via an ordered set of registers, which he refers to as 288 'treelets' (108). 289 8See Lando (2015), McKinsey (1944) and Rasiowa (1963), for further details. 9Note that, in cases of Boolean-valued epistemic topological algebras, models of corresponding coalgebras will be topological (cf. Takeuchi 1985 for further discussion). UN CO RR EC TE D PR OO F 4 Modal !-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism A strengthened version of the algebraic rules on variable-binding can be accom290 modated in models of epistemic modal algebras, when the latter are augmented 291 by cylindrifications, i.e., operators on the algebra simulating the treatment of 292 quantification, and diagonal elements.10 By contrast to Boolean Algebras with 293 Operators, which are propositional, cylindric algebras define first-order logics. 294 Intuitively, valuation assignments for first-order variables are, in cylindric modal 295 logics, treated as possible worlds of the model, while existential and universal 296 quantifiers are replaced by, respectively, possibility and necessity operators (⋄ and 297 ") (Venema 2013: 249). For first-order variables, {vi | i < α} with α an arbitrary, 298 fixed ordinal, vi = vj is replaced by a modal constant di,j (op. cit: 250). The 299 following clauses are valid, then, for a model, M, of cylindric modal logic, with 300 Ei,j a monadic predicate and Ti for i, j < α a dyadic predicate: 301 M, w ! p ⇐⇒ w ∈ V(p); 302 M, w ! di,j ⇐⇒ w ∈ Ei,j ; 303 M, w ! ⋄iψ ⇐⇒ there is a v with wTiv and M, v ⊢ ψ (252).11 304 Finally, a cylindric modal algebra of dimension α is an algebra, A = ⟨A,+, •, 305 –, 0, 1, ⋄i , dij ⟩i,j<α , where ⋄i is a unary operator which is normal (⋄i0 = 0) and 306 additive [⋄i (x + y) = ⋄ix + ⋄iy)] (257). 307 The philosophical interest of cylindric modal algebras to Marcus' cognitive 308 models of algebraic variable-binding is that variable substitution is treated in the 309 modal algebras as a modal relation, while universal quantification is interpreted as 310 necessitation. The interest of translating universal generalization into operations of 311 epistemic necessitation is, finally, that – by identifying epistemic necessity with 312 apriority – both the algebraic rules for variable-binding and the recursive formation 313 of structured propositions can be seen as operations, the implicit knowledge of 314 which is apriori. 315 In quantum information theory, let a constructor be a computation defined over 316 physical systems. Constructors entrain nomologically possible transformations from 317 admissible input states to output states (cf. Deutsch 2013). On this approach, 318 10See Henkin et al (op. cit.: 162–163) for the introduction of cylindric algebras, and for the axioms governing the cylindrification operators. 11Cylindric frames need further to satisfy the following axioms (op. cit.: 254): 1. p → ⋄ip 2. p → "i ⋄i p 3. ⋄i ⋄i p → ⋄ip 4. ⋄i ⋄j p → ⋄j ⋄i p 5. di,i 6. ⋄i (di,j ∧ p) → "i (di,j → p) [Translating the diagonal element and cylindric (modal) operator into, respectively, monadic and dyadic predicates and universal quantification: ∀xyz[(Tixy∧Ei,j y∧Tixz∧Ei,j z) → y = z] (op. cit.)] 7. di,j ⇐⇒ ⋄k(di,k],∧ dk,j ). UN CO RR EC TE D PR OO F H. Khudairi information is defined in terms of constructors, i.e., intensional computational 319 properties. The foregoing transformations, as induced by constructors, are referred 320 to as tasks. Because constructors encode the counterfactual to the effect that, were 321 an initial state to be computed over, then the output state would result, modal 322 notions are thus constitutive of the definition of the tasks at issue. There are, further, 323 both topological and algebraic aspects of the foregoing modal approach to quantum 324 computation.12 The composition of tasks is formed by taking their union, where the 325 union of tasks can be satisfiable while its component tasks might not be. Suppose, 326 e.g., that the information states at issue concern the spin of a particle. A spin-state 327 vector will be the sum of the probabilities that the particle is spinning either upward 328 or downward. Suppose that there are two particles which can be spinning either 329 upward or downward. Both particles can be spinning upward; spinning downward; 330 particle-1 can be spinning upward while particle-2 spins downward; and vice versa. 331 The state vector, V which records the foregoing possibilities – i.e., the superposition 332 of the states – will be equal to the product of the spin-state of particle-1 and the 333 spin-state of particle-2. If the particles are both spinning upward or both spinning 334 downward, then V will be .5. However – relative to the value of each particle 335 vector, referred to as its eigenvalue – the probability that particle-1 will be spinning 336 upward is .5 and the probability that particle-2 will be spinning downward is .5, 337 such that the probability that both will be spinning upward or downward = .5 × .5 338 = .25. Considered as the superposition of the two states, V will thus be unequal to 339 the product of their eigenvalues, and is said to be entangled. If the indeterminacy 340 evinced by entangled states is interpreted as inconsistency, then the computational 341 properties at issue might further have to be defined on a distribution of epistemic 342 possibilities which permit of hyperintensional distinctions.13 343 4.4 Modal Coalgebraic Automata and the Philosophy of 344 Mathematics 345 This section examines the philosophical significance of the Boolean-valued models 346 of set-theoretic languages and the modal coalgebraic automata to which they are 347 dually isomorphic. I argue that, similarly to second-order logical consequence, 348 (i) the 'mathematical entanglement' of !-logical validity does not undermine its 349 status as a relation of pure logic; and (ii) both the modal profile and model350 theoretic characterization of !-logical consequence provide a guide to its epistemic 351 12For an examination of the interaction between topos theory and an S4 modal axiomatization of computable functions, see Awodey et al. (2000). 13The nature of the indeterminacy in question is examined in Saunders and Wallace (2008), Deutsch (2010), Hawthorne (2010), Wilson (2011), Wallace (2012: 287–289), Lewis (2016: 277– 278), and Khudairi (ms). For a thorough examination of approaches to the ontology of quantum mechanics, see Arntzenius (2012: ch. 3). UN CO RR EC TE D PR OO F 4 Modal !-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism tractability.14 I argue, then, that there are several considerations adducing in favor of 352 the claim that the interpretation of the concept of set constitutively involves modal 353 notions. The role of the category of modal coalegebraic deterministic automata 354 in (i) characterizing the modal profile of !-logical consequence, and (ii) being 355 constitutive of the formal understanding-conditions for the concept of set, provides, 356 then, support for a realist conception of the cumulative hierarchy. 357 4.4.1 Neo-Logicism 358 Frege's (1884/1980; 1893/2013) proposal – that cardinal numbers can be explained 359 by specifying an equivalence relation, expressible in the signature of second-order 360 logic and identity, on lower-order representatives for higher-order entities – is the 361 first attempt to provide a foundation for mathematics on the basis of logical axioms 362 rather than rational or empirical intuition. In Frege (1884/1980. cit.: 68) and Wright 363 (1983: 104–105), the number of the concept, A, is argued to be identical to the 364 number of the concept, B, if and only if there is a one-to-one correspondence 365 between A and B, i.e., there is a bijective mapping, R, from A to B. With Nx: a 366 numerical term-forming operator, 367 • ∀A∀B∃R[[Nx: A = Nx: B ≡ ∃R[∀x[Ax → ∃y(By ∧ Rxy ∧∀z(Bz ∧ Rxz → y 368 = z))] ∧∀y[By → ∃x(Ax ∧ Rxy ∧∀z(Az ∧ Rzy → x = z))]]]. 369 Frege's Theorem states that the Dedekind-Peano axioms for the language of 370 arithmetic can be derived from the foregoing abstraction principle, as augmented 371 to the signature of second-order logic and identity.15 Thus, if second-order logic 372 may be counted as pure logic, despite that domains of second-order models are 373 definable via power set operations, then one aspect of the philosophical significance 374 of the abstractionist program consists in its provision of a foundation for classical 375 mathematics on the basis of pure logic as augmented with non-logical implicit 376 definitions expressed by abstraction principles. 377 There are at least three reasons for which a logic defined in ZFC might not 378 undermine the status of its consequence relation as being logical. The first reason for 379 which the mathematical entanglement of !-logical validity might be innocuous is 380 that, as Shapiro (1991: 5.1.4) notes, many mathematical properties cannot be defined 381 within first-order logic, and instead require the expressive resources of second-order 382 logic. For example, the notion of well-foundedness cannot be expressed in a first383 order framework, as evinced by considerations of compactness. Let E be a binary 384 relation. Let m be a well-founded model, if there is no infinite sequence, a0, . . . , 385 14The phrase, 'mathematical entanglement', is owing to Koellner (2010: 2). 15Cf. Dedekend (1888/1963) and Peano (1889/1967). See Wright (1983: 154–169) for a proof sketch of Frege's theorem; Boolos (1987) for the formal proof thereof; and Parsons (1964) for an incipient conjecture of the theorem's validity. UN CO RR EC TE D PR OO F H. Khudairi ai , such that Ea0, . . . , Eai+1 are all true. If m is well-founded, then there are no 386 infinite-descending E-chains. Suppose that T is a first-order theory containing m, 387 and that, for all natural numbers, n, there is a T with n + 1 elements, a0, . . . , an, 388 such that ⟨a0, a1⟩, . . . , ⟨an, an−1⟩ are in the extension of E. By compactness, there 389 is an infinite sequence such that a0 . . . ai , s.t. Ea0, . . . , Eai+1 are all true. So, m is 390 not well-founded. 391 By contrast, however, well-foundedness can be expressed in a second-order 392 framework: 393 ∀X[∃xXx → ∃x[Xx ∧ ∀y(Xy → ¬Eyx)]], such that m is well-founded iff 394 every non-empty subset X has an element x, s.t. nothing in X bears E to x. 395 One aspect of the philosophical significance of well-foundedness is that it 396 provides a distinctively second-order constraint on when the membership relation in 397 a given model is intended. This contrasts with Putnam's (1980) claim, that first-order 398 models mod can be intended, if every set s of reals in mod is such that an ω-model in 399 mod contains s and is constructible, such that – given the Downward Lowenheim400 Skolem theorem16 – if mod is non-constructible but has a submodel satisfying 's 401 is constructible', then the model is non-well-founded and yet must be intended. 402 The claim depends on the assumption that general understanding-conditions and 403 conditions on intendedness must be co-extensive, to which I will return in Sect. 4.4.2 404 A second reason for which !-logic's mathematical entanglement might not 405 be pernicious, such that the consequence relation specified in the !-logic might 406 be genuinely logical, may again be appreciated by its comparison with second407 order logic. Shapiro (1998) defines the model-theoretic characterization of logical 408 consequence as follows: 409 '(10) * is a logical consequence of [a model] + if * holds in all possibilities 410 under every interpretation of the nonlogical terminology which holds in +' (148). 411 A condition on the foregoing is referred to as the 'isomorphism property', 412 according to which 'if two models M, M' are isomorphic vis-a-vis the nonlogical 413 items in a formula *, then M satisfies * if and only if M' satisfies *' (151). 414 Shapiro argues, then, that the consequence relation specified using second-order 415 resources is logical, because of its modal and epistemic profiles. The epistemic 416 tractability of second-order validity consists in 'typical soundness theorems, where 417 one shows that a given deductive system is 'truth-preserving' (154). He writes that: 418 '[I]f we know that a model is a good mathematical model of logical consequence 419 (10), then we know that we won't go wrong using a sound deductive system. Also, 420 we can know that an argument is a logical consequence . . . via a set-theoretic proof 421 in the metatheory' (154–155). 422 The modal profile of second-order validity provides a second means of account423 ing for the property's epistemic tractability. Shapiro argues, e.g., that: 'If the 424 isomorphism property holds, then in evaluating sentences and arguments, the only 425 'possibility' we need to 'vary' is the size of the universe. If enough sizes are 426 16For any first-order model M , M has a submodel M ′ whose domain is at most denumerably infinite, s.t. for all assignments s on, and formulas φ(x) in, M ′, M ,s ! φ(x) ⇐⇒ M ′,s ! φ(x). UN CO RR EC TE D PR OO F 4 Modal !-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism represented in the universe of models, then the modal nature of logical consequence 427 will be registered . . . [T]he only 'modality' we keep is 'possible size', which is 428 relegated to the set-theoretic metatheory' (152). 429 Shapiro's remarks about the considerations adducing in favor of the logicality 430 of non-effective, second-order validity generalize to !-logical validity. In the 431 previous section, the modal profile of !-logical validity was codified by the dual 432 isomorphism between complete Boolean-valued algebraic models of !-logic and 433 the category, A, of coalgebraic modal logics. As with Shapiro's definition of logical 434 consequence, where * holds in all possibilities in the universe of models and 435 the possibilities concern the 'possible size' in the set-theoretic metatheory, the !436 Conjecture states that V |=! φ iff VB |=! φ, such that !-logical validity is invariant 437 in all set-forcing extensions of ground models in the set-theoretic multiverse. 438 Finally, the epistemic tractability of !-logical validity is secured, both – as on 439 Shapiro's account of second-order logical consequence – by its soundness, but also 440 by its isomorphism to the coalgebraic category of deterministic automata, where the 441 determinacy thereof is again secured by the existence of Woodin cardinals. 442 4.4.2 Set-Theoretic Realism 443 In this section, I argue, finally, that the modal profile of !-logic can be availed of 444 in order to account for the understanding-conditions of the concept of set, and thus 445 crucially serve as part of the argument for set-theoretic realism. 446 Putnam (op. cit.: 473–474) argues that defining models of first-order theories is 447 sufficient for both understanding and specifying an intended interpretation of the 448 latter. Wright (1985: 124–125) argues, by contrast, that understanding-conditions 449 for mathematical concepts cannot be exhausted by the axioms for the theories 450 thereof, even on the intended interpretations of the theories. He suggests, e.g., that: 451 '[I]f there really were uncountable sets, their existence would surely have to flow 452 from the concept of set, as intuitively satisfactorily explained. Here, there is, as it 453 seems to me, no assumption that the content of the ZF-axioms cannot exceed what is 454 invariant under all their classical models. [Benacerraf] writes, e.g., that: 'It is granted 455 that they are to have their 'intended interpretation': 'e' is to mean set-membership. 456 Even so, and conceived as encoding the intuitive concept of set, they fail to entail 457 the existence of uncountable sets. So how can it be true that there are such sets? 458 Benacerraf's reply is that the ZF-axioms are indeed faithful to the relevant informal 459 notions only if, in addition to ensuring that 'E' means set-membership, we interpret 460 them so as to observe the constraint that 'the universal quantifier has to mean all or at 461 least all sets' (p. 103). It follows, of course, that if the concept of set does determine 462 a background against which Cantor's theorem, under its intended interpretation, is 463 sound, there is more to the concept of set that can be explained by communication of 464 the intended sense of 'e' and the stipulation that the ZF-axioms are to hold. And the 465 residue is contained, presumably, in the informal explanations to which, Benacerraf 466 reminds us, Zermelo intended his formalization to answer. At least, this must be so if 467 UN CO RR EC TE D PR OO F H. Khudairi the 'intuitive concept of set' is capable of being explained at all. Yet it is notable that 468 Benacerraf nowhere ventures to supply the missing informal explanation – the story 469 which will pack enough into the extension of 'all sets' to yield Cantor's theorem, 470 under its intended interpretation, as a highly non-trivial corollary'(op. cit). 471 In order to provide the foregoing explanation in virtue of which the concept of set 472 can be shown to be associated with a realistic notion of the cumulative hierarchy, I 473 will argue that there are several points in the model theory and epistemology of set474 theoretic languages at which the interpretation of the concept of set constitutively 475 involves modal notions. The aim of the section will thus be to provide a modal 476 foundation for mathematical platonism. 477 One point is in the coding of the signature of the theory, T, in which Gödel's 478 incompleteness theorems are proved (cf. Halbach and Visser 2014). Relative to, 479 (i) a choice of coding for an ω-complete, recursively axiomatizable language, L, 480 of T – i.e. a mapping between properties of numbers and properties of terms 481 and formulas in L; 482 (ii) a predicate, phi; and 483 (iii) a fixed-point construction: 484 Let phi express the property of 'being provable', and define (iii) such that, 485 for all consistent theories T of L, there are sentences, pphi , corresponding to 486 each formula, phi(x), in T, s.t. for 'm' := pphi , 487 |–T pphi iff phi(m). 488 One can then construct a sentence, 'm' := ¬phi(m), such that L is incom489 plete (the first incompleteness theorem). 490 Moreover, L cannot prove its own consistency: 491 If: 492 |–T 'm' iff ¬phi(m), 493 Then: 494 |–T C → m. 495 Thus, L is consistent only if L is inconsistent (the second incompleteness 496 theorem). 497 In the foregoing, the choice of coding bridges the numerals in the language 498 with the properties of the target numbers. The choice of coding is therefore 499 intensional, and has been marshalled in order to argue that the very notion of 500 syntactic computability – via the equivalence class of partial recursive functions, 501 λ-definable terms, and the transition functions of discrete-state automata such as 502 Turing machines – is constitutively semantic (cf. Rescorla 2015). Further points 503 at which intensionality can be witnessed in the phenomenon of self-reference in 504 arithmetic are introduced by Reinhardt (1986). Reinhardt (op. cit.: 470–472) argues 505 that the provability predicate can be defined relative to the minds of particular agents 506 – similarly to Quine's (1968) and Lewis' (1979) suggestion that possible worlds can 507 be centered by defining them relative to parameters ranging over tuples of spacetime 508 coordinates or agents and locations – and that a theoretical identity statement can be 509 established for the concept of the foregoing minds and the concept of a computable 510 system. 511 UN CO RR EC TE D PR OO F 4 Modal !-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism In the previous section, intensional computational properties were defined via 512 modal coalgebraic deterministic automata, where the coalgebraic categories are 513 dually isomorphic to the category of sets in which !-logical validity was defined. 514 Coalgebraic modal logic was shown to elucidate the modal profile of !-logical con515 sequence in the Boolean-valued algebraic models of set theory. The intensionality 516 witnessed by the choice of coding may therefore be further witnessed by the modal 517 automata specified in the foregoing coalgebraic logic. 518 A second point at which understanding-conditions may be shown to be con519 stitutively modal can be witnessed by the conditions on the epistemic entitlement 520 to assume that the language in which Gödel's second incompleteness theorem is 521 proved is consistent (cf. Dummett 1963/1978; Wright 1985). Wright (op. cit.: 91, 522 fn.9) suggests that '[T]o treat [a] proof as establishing consistency is implicitly 523 to exclude any doubt . . . about the consistency of first-order number theory'. 524 Wright's elaboration of the notion of epistemic entitlement, appeals to a notion 525 of rational 'trust', which he argues is recorded by the calculation of 'expected 526 epistemic utility' in the setting of decision theory (2004; 2014: 226, 241). Wright 527 notes that the rational trust subserving epistemic entitlement will be pragmatic, 528 and makes the intriguing point that 'pragmatic reasons are not a special genre of 529 reason, to be contrasted with e.g. epistemic, prudential, and moral reasons' (2012: 530 484). Crucially, however, the very idea of expected epistemic utility in the setting 531 of decision theory makes implicit appeal to the notion of possible worlds, where the 532 latter can again be determined by the coalgebraic logic for modal automata. 533 A third consideration adducing in favor of the thought that grasp of the concept 534 of set might constitutively possess a modal profile is that the concept can be defined 535 as an intension – i.e., a function from possible worlds to extensions. The modal 536 similarity types in the coalgebraic modal logic may then be interpreted as dynamic537 interpretational modalities, where the dynamic-interpretational modal operator has 538 been argued to entrain the possible reinterpretations both of the domains of the 539 theory's quantifiers (cf. Fine 2005, 2006), as well as of the intensions of non-logical 540 concepts, such as the membership relation (cf. Uzquiano 2015).17 541 The fourth consideration avails directly of the modal profile of !-logical 542 consequence. While the above dynamic-interpretational modality will suffice for 543 17For an examination of the philosophical significance of modal coalgebraic automata beyond the philosophy of mathematics, see Baltag (2003). Baltag (op. cit.) proffers a colagebraic semantics for dynamic-epistemic logic, where coalgebraic functors are intended to record the informational dynamics of singleand multi-agent systems. For an algebraic characterization of dynamicepistemic logic, see Kurz and Palmigiano (2013). For further discussion, see Khudairi (ms). The latter proceeds by examining undecidable sentences via the epistemic interpretation of multidimensional intensional semantics. See Reinhardt (1974), for a similar epistemic interpretation of set-theoretic languages, in order to examine the reduction of the incompleteness of undecidable sentences on the counterfactual supposition that the language is augmented by stronger axioms of infinity; and Maddy (1988b), for critical discussion. Chihara (2004) argues, as well, that conceptual possibilities can be treated as imaginary situations with regard to the construction of open-sentence tokens, where the latter can then be availed of in order to define nominalistically adequate arithmetic properties. UN CO RR EC TE D PR OO F H. Khudairi possible reinterpretations of mathematical terms, the absoluteness and generic 544 invariance of the consequence relation is such that, if the !-conjecture is true, then 545 !-logical validity is invariant in all possible set-forcing extensions of ground models 546 in the set-theoretic multiverse. The truth of the !-conjecture would thereby place 547 an indefeasible necessary condition on a formal understanding of the intension for 548 the concept of set. 549 4.5 Concluding Remarks 550 In this essay I have examined the philosophical significance of the isomorphism 551 between Boolean-valued algebraic models of modal !-logic and modal coalgebraic 552 models of automata. I argued that – as with the property of validity in second553 order logic – !-logical validity is genuinely logical, and thus entails a type 554 of neo-logicism in the foundations of mathematics. I argued, then, that modal 555 coalegebraic deterministic automata, which characterize the modal profile of !556 logical consequence, are constitutive of the interpretation of mathematical concepts 557 such as the membership relation. The philosophical significance of modal !-logic 558 is thus that it can be availed of to vindicate both a neo-logicist foundation for set 559 theory and a realist interpretation of the cumulative hierarchy of sets. 560 References 561 Arntzenius, F. 2012. Space, Time, and Stuff. Oxford: Oxford University Press. 562 Awodey, S., L. Birkedal, and D. Scott. 2000. Local Realizability Toposes and a Modal Logic for 563 Computability. Technical Report No. CMU-PHIL-99. 564 Bagaria, J., N. Castells, and P. Larson. 2006. An !-logic Primer. Trends in Mathematics: Set 565 Theory. Basel: Birkhäuser Verlag. 566 Baltag, A. 2003. A coalgebraic semantics for epistemic programs. Electronic Notes in Theoretical 567 Computer Science 82: 1. 568 Boolos, G. 1987. 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Please provide details of "Woodin (2001), Gödel (1940), Cohen (1963, 1964), Hughes (2001) and Parsons (1964)" in reference list. AQ3. Please provide publisher location for "Boolos (1987), Reinhardt (1974), Frege (1884/1980), Gödel (1931/1986), and Gödel (1946/1990)". AQ4. Please cite "Henkin et al. (1971), Wright (2004), Wright (2012), and Wright (2014)" in text. AQ5. Please provide publisher details for "Koellner (2013) and Peano (1889/1967)". AQ6. Please provide conference location for "Woodin (2010)".