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On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence pp 65–82Cite as

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

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Part of the book series: Philosophical Studies Series ((PSSP,volume 134))

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 Ω-logical 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.

Forthcoming in the ‘Proceedings of the 2016 Meeting of the International Association for Computing and Philosophy’.

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Notes

  1. 1.

    For a standard presentation, see Jech (2003). For detailed, historical discussion, see Maddy (1988a).

  2. 2.

    See Koellner (2013), for the presentation, and for further discussion, of the definitions in this and the subsequent paragraph.

  3. 3.

    See Kanamori (2012a: 2.1; 2012b: 4.1), for further discussion.

  4. 4.

    See Kanamori (2008), for further discussion.

  5. 5.

    See 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.

  6. 6.

    The definitions in the remainder of this subsection follow the presentations in Koellner and Woodin (2010) and Woodin (2010, 2011).

  7. 7.

    The definitions in this section follow the presentation in Bagaria et al. (2006).

  8. 8.

    See Lando (2015), McKinsey (1944) and Rasiowa (1963), for further details.

  9. 9.

    Note that, in cases of Boolean-valued epistemic topological algebras, models of corresponding coalgebras will be topological (cf. Takeuchi 1985 for further discussion).

  10. 10.

    See Henkin et al (op. cit.: 162–163) for the introduction of cylindric algebras, and for the axioms governing the cylindrification operators.

  11. 11.

    Cylindric frames need further to satisfy the following axioms (op. cit.: 254):

    1. p → ◇ip

    2. p → □i ◇ip

    3. ◇i ◇ip → ◇ip

    4. ◇i ◇jp → ◇j ◇ip

    5. d i,i

    6. ◇i(d i,j  ∧p) → □i(d i,j  →p)

    [Translating the diagonal element and cylindric (modal) operator into, respectively, monadic and dyadic predicates and universal quantification: ∀xyz[(Tixy ∧Ei,jy ∧Tixz ∧Ei,jz) →y = z] (op. cit.)] 7. d i,j ⇔  ◇k(d i,k], ∧ d k,j).

  12. 12.

    For an examination of the interaction between topos theory and an S4 modal axiomatization of computable functions, see Awodey et al. (2000).

  13. 13.

    The 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).

  14. 14.

    The phrase, ‘mathematical entanglement’, is owing to Koellner (2010: 2).

  15. 15.

    Cf. 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.

  16. 16.

    For 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,\(s \Vdash \phi \)(x) ⇔ M′,\(s \Vdash \phi \)(x).

  17. 17.

    For 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 single- and multi-agent systems. For an algebraic characterization of dynamic-epistemic logic, see Kurz and Palmigiano (2013). For further discussion, see Khudairi (ms). The latter proceeds by examining undecidable sentences via the epistemic interpretation of multi-dimensional 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.

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  1. Arché Philosophical Research Centre, University of St Andrews, St. Andrews, Scotland, UK

    Hasen Khudairi

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  1. Department of Humanities, Texas A&M University Corpus Christi, Corpus Christi, TX, USA

    Don Berkich

  2. Dipartimento di Studi Umanistici, Università di Ferrara, Ferrara, Italy

    Matteo Vincenzo d'Alfonso

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Khudairi, H. (2019). Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. In: Berkich, D., d'Alfonso, M. (eds) On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence. Philosophical Studies Series, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-030-01800-9_4

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