The intensional side of algebraic-topological representation theorems

Abstract

Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic methods will also be used to establish the embedding of subintuitionistic logics into the corresponding modal logics. Finally, proof-theoretic embeddings will be interpreted as a reduction of classes of word problems.

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Notes

  1. 1.

    Cf. Sambin (1987).

  2. 2.

    Heyting algebras are also called pseudo-Boolean algebras, e.g. in Rasiowa and Sikorski (1963).

  3. 3.

    The condition of having enough points, or being spatial, means that if \(a\ \nleqslant \ b\) in the algebra, there is a formal point that contains a but not b, that is, the (formal) points are enough to witness the order relation in the algebra. See Johnstone (1982) for the condition of spatiality in locale theory and Gambino and Schuster (2007) for a survey on spatiality in formal topology.

  4. 4.

    We recall that a relation \(\le \) on a set P is a partial order if it is reflexive, transitive, and antisymmetric, i.e. if it satisfies \(\forall x. x\le x\), \( \forall xyz. x\le y \, \& \, y\le z\rightarrow x\le z\), and \( \forall xy. x\le y \, \& \, y\le x\rightarrow x= y\). Since a partial order is an antisymmetric preorder and antisymmetry holds by definition if equality is defined by the \(\le \) relation in the two direction, preorders are often preferred to partial orders as more basic structures.

  5. 5.

    We recall that a partially ordered set L is a meet-semilattice if for any two elements a, b of L the greatest lower bound of a and b exists, i.e. there is a binary operation \(\wedge \) on elements of L with the properties \(a\wedge b\le a\), \(a\wedge b\le b\), and \( \forall c.c\le a\, \& \, c\le b\rightarrow c\le a\wedge b\).

  6. 6.

    A lattice is distributive if the operations of meet and join distribute over each other, i.e. the condition \(a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)\) holds for arbitrary elements of the lattice. By a general result in lattice theory, the condition is equivalent to the dual condition \(a\vee (b\wedge c)= (a\vee b)\wedge (a\vee c)\).

  7. 7.

    Cf. Dunn and Hardegree (2001), p. 293.

  8. 8.

    We recall that an Heyting algebra is a distributive lattice endowed with an operation \(\rightarrow \) that satisfies \(a\wedge b\le c\) if and only if \(a\le b\rightarrow c\). A complete Heyting algebra is an Heyting algebra which is complete as a lattice, i.e. one in which the supremum (least upper bound) and infimum (greatest lower bound) exist for arbitrary subsets of elements. By a known result (cf. e.g. I.4.3 of Johnstone (1982) it is enough to require closure under arbitrary meets (resp. joins) to obtain closure under arbitrary joins (resp. meets), i.e. a complete semilattice is also a complete lattice. The two notions are however distinct when morphisms are considered because a map that preserves arbitrary joins (resp. meets) need not preserve arbitrary meets (resp. joins).

  9. 9.

    This latter paper gives a useful summary of the results in duality theory in this line of investigation.

  10. 10.

    Inductive generation of formal covers has been the key property in the presentation of formal reals, formal intervals and formal linear functionals for establishing results such as the constructive version of the Tychonoff, the Heine–Borel, and the Hahn–Banach theorem and the representation of continuous domains (cf. Negri and Soravia 1999; Negri et al. 1997; Cederquist and Negri 1996; Cederquist et al. 1998; Negri 2002). For a survey on inductive generation of formal topologies in the wider context of inductive definitions in type theory and examples of formal topologies that cannot be inductively generated cf. Coquand et al. (2003). For examples of inductively generated formal topologies see also Sect. 4 of Gambino and Schuster (2007).

  11. 11.

    See the self-contained survey by Wolter and Zakharyaschev (2014), whose notation we have followed and which includes an extension of the Blok–Esakia theorem to intuitionistic modal logics.

  12. 12.

    As discussed in Sect. 8 of Dyckhoff and Negri (2012), analyticity is guaranteed by the possibility of restricting the above rule to labels x and y found in its conclusion.

  13. 13.

    Indeed, by the replacement of rules with systems of rules, the method can be further extended to frame classes expressed by generalized geometric implications, which are first-order properties with an arbitrary number of quantifier alternations (cf. Negri 2016), and even to arbitrary first-order frame conditions with the method detailed in Dyckhoff and Negri (2015).

  14. 14.

    Observe that it is not necessary to assume a constant \(\top \) since we can use a definition such as \(\perp \supset \perp \).

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Negri, S. The intensional side of algebraic-topological representation theorems. Synthese (2017). https://doi.org/10.1007/s11229-017-1331-1

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Keywords

  • Stone representation
  • Logical embeddings
  • Philosophical logics
  • Intermediate logics
  • Subintuitionistic logics
  • Modal companions
  • Word problems
  • Cut elimination
  • Pointfree topology