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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Cf. Sambin (1987).
Heyting algebras are also called pseudo-Boolean algebras, e.g. in Rasiowa and Sikorski (1963).
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.
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.
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 \( \forall c.c\le a\, \& \, c\le b\rightarrow c\le a\wedge b\).
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 \(a\vee (b\wedge c)= (a\vee b)\wedge (a\vee c)\).
Cf. Dunn and Hardegree (2001), p. 293.
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).
This latter paper gives a useful summary of the results in duality theory in this line of investigation.
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).
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.
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.
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).
Observe that it is not necessary to assume a constant \(\top \) since we can use a definition such as \(\perp \supset \perp \).
References
Bezhanishvili, G. (2014). Leo Esakia on Duality in Modal and Intuitionistic Logics. Dordrecht: Springer.
Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., & Kurz, A. (2010). Bitopological duality for distributive lattices and Heyting algebras. Mathematical Structures in Computer Science, 20, 359–393.
Cederquist, J., Coquand, T., & Negri, S. (1998). The Hahn–Banach theorem in type theory. In G. Sambin & J. Smith (Eds.), Twenty-five years of constructive type theory (pp. 57–72). Oxford: Oxford University Press.
Cederquist, J., & Negri, S. (1996). A constructive proof of the heine-borel covering theorem for formal reals. In S. Berardi & M. Coppo (Eds.), Types for proofs and programs. Lecture notes in computer science (Vol. 1158, pp. 62–75). Berlin: Springer.
Celani, S., & Jansana, R. (2001). A closer look at some subintuitionistic logics. Notre Dame Journal Formal Logic, 42, 225–255.
Celani, S., & Jansana, R. (2005). Bounded distributive lattices with strict implication. Mathematical Logic Quarterly, 51, 219–246.
Chagrov, A., & Zakharyaschev, M. (1997). Modal logic. Oxford logic guides (Vol. 35). New York: The Clarendon Press.
Coquand, T., Sambin, G., Smith, J., & Valentini, S. (2003). Inductively generated formal topologies. Annals of Pure and Applied Logic, 124, 71–106.
Davey, B. A., & Priestley, H. (1990). Introduction to lattices and order. Cambridge: Cambridge University Press.
Dummett, M. A. E., & Lemmon, E. J. (1959). Modal logics between S4 and S5. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 5, 250–264.
Dunn, M., & Hardegree, G. (2001). Algebraic methods in philosophical logic. Oxford: Oxford University Press.
Dyckhoff, R., & Negri, S. (2006). Decision methods for linearly ordered Heyting algebras. Archive for Mathematical Logic, 45, 411–422.
Dyckhoff, R., & Negri, S. (2012). Proof analysis in intermediate logics. Archive for Mathematical Logic, 51, 71–92.
Dyckhoff, R., & Negri, S. (2015). Geometrization of first-order logic. The Bulletin of Symbolic Logic, 21, 123–163.
Dyckhoff, R., & Negri, S. (2016). A cut-free sequent system for Grzegorczyk logic, with an application to the Gödel–McKinsey–Tarski embedding. Journal of Logic and Computation, 26, 169–187.
Fourman, M. P., & Grayson, R. J. (1982). Formal spaces. In D. van Dalen & A. Troelstra (Eds.), L. E. J. Brouwer centenary symposium. Proceedings of the conference held in Noordwijkerhout, 8–13 June, 1981. Studies in logic and the foundations of mathematics (vol. 110, pp. 107–122). North-Holland.
Gambino, N., & Schuster, P. (2007). Spatiality for formal topologies. Mathematical Structures in Computer Science, 17, 65–80.
Grosholz, E. R. (1985). Two episodes in the unification of logic and topology. British Journal for the Philosophy of Science, 36, 147–157.
Gödel, K. (1933) Eine interpretation des intuitionistischen Aussagenkalküls. Ergebnisse eines mathematischen Kolloquiums, 4, pp. 39–40 (English tr. in Gödel’s Collected Works, vol. I, pp. 300–303, 1986).
Johnstone, P. (1982). Stone spaces. Cambridge studies in advanced mathematics (Vol. 3). Cambridge: Cambridge University Press.
Martin-Löf (1984) Intuitionistic type theory: Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980. Napoli: Bibliopolis.
Matsumoto, K. (1965). Word problems for free lattices. Kiyo of Nara Technical College, 1, 53–59. (in Japanese).
McKinsey, J. C. C., & Tarski, A. (1944). The algebra of topology. Annals of Mathematics, 45, 141–191.
McKinsey, J. C., & Tarski, A. (1948). Some theorems about the sentential calculi of Lewis and Heyting. Journal of Symbolic Logic, 13, 1–15.
Meinander, A. (2010). A solution of the uniform word problem for ortholattices. Mathematical Structures in Computer Science, 20, 625–638.
Mormann, T. (2005). Description, construction and representation. From Russell and Carnap to stone. In G. Imaguire & B. Linsky (Eds.), On denoting 1905–2005 (pp. 333–360). München: Philosophia.
Mormann, T. (2013). Topology as an issue for history of philosophy of science. In H. Andersen, et al. (Eds.), New challenges to philosophy of sciences (pp. 423–434). Berlin: Springer.
Negri, S. (1996). Stone bases, alias the constructive content of stone representation. In A. Ursini & P. Aglianò (Eds.), Logic and algebra. Lecture notes in pure and applied mathematics (Vol. 180, pp. 617–636). New York: M. Dekker.
Negri, S. (2002). Continuous domains as formal spaces. Mathematical Structures in Computer Science, 12, 19–52.
Negri, S. (2005a). Proof analysis in modal logic. Journal Philosophical Logic, 34, 507–544.
Negri, S. (2005b). Permutability of rules for linear lattices. Journal of Universal Computer Science, 11, 1986–1995.
Negri, S. (2011). Proof theory for modal logic. Philosophy Compass, 6, 523–538.
Negri, S. (2014). Proofs and countermodels in non-classical logics. Logica Universalis, 8, 25–60.
Negri, S. (2016). Proof analysis beyond geometric theories: From rule systems to systems of rules. Journal of Logic and Computation, 26, 513–537.
Negri, S., & von Plato, J. (1998). Cut elimination in the presence of axioms. The Bulletin of Symbolic Logic, 4, 418–435.
Negri, S., & von Plato, J. (2002). Permutability of rules in lattice theory (2002). Algebra Universalis, 48, 473–477.
Negri, S., & von Plato, J. (2004). Proof systems for lattice theory. Mathematical Structures in Computer Science, 14, 507–526.
Negri, S., & von Plato, J. (2011). Proof analysis. Cambridge: Cambridge University Press.
Negri, S., von Plato, J., & Coquand, T. (2004). Proof theoretical analysis of order relations. Archive for Mathematical Logic, 43, 297–309.
Negri, S., & Soravia, D. (1999). The continuum as a formal space. Archive for Mathematical Logic, 38, 423–447.
Negri, S., & Valentini, S. (1997). Tychonoff’s theorem in the framework of formal topologies. The Journal of Symbolic Logic, 62, 1315–1332.
Priestley, H. A. (1970). Representation of distributive lattices by means of ordered Stone spaces. Bulletin of the London Mathematical Society, 2, 186–190.
Rasiowa, H., & Sikorski, R.: (1963) The Mathematics of Metamathematics. Monografie Matematyczne, Tom 41. Panstwowe Wydawnictwo Naukowe, Warsaw.
Restall, G. (1994). Subintuitionistic logics. Notre Dame Journal of Formal Logic, 35, 116–129.
Sambin, G. (1987). Intuitionistic formal spaces: A first communication. In D. Skordev (Ed.), Mathematical logic and its applications (pp. 187–204). Berlin: Plenum Press.
Schlimm, D. (2008). Bridging theories with axioms: Boole, Stone, and Tarski. In B. van Kerkhove (Ed.), New perspectives on mathematical practices (pp. 222–235). Singapore: World Scientific.
Schulte Mönting, J. (1981). Cut elimination and word problems for varieties of lattices. Algebra Universalis, 12, 290–321.
Stone, M. H. (1936). The theory of representations of Boolean algebras. Transactions of the American Mathematical Society, 40, 37–111.
Stone, M. H. (1937). Topological representation of distributive lattices and Brouwerian logics. Casopis pro pěstování matematiky a fysiky, 67, 1–25.
Tamura, S. (1988). A Gentzen formulation without the cut rule for ortholattices. Kobe Journal of Mathematics, 5, 133–150.
Troelstra, A., & Schwichtenberg, H. (1996). Basic proof theory (2nd ed.). Cambridge: Cambridge University Press.
Whitman, P. (1941). Free lattices. Annals of Mathematics, 42, 325–330.
Wolter, F., & Zakharyaschev, M. (2014) On the Blok-Esakia theorem. In Bezhanishvili, G. (Ed.), Leo Esakia on duality in modal and intuitionistic logics (pp. 99–118). Netherlands: Springer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Negri, S. The intensional side of algebraic-topological representation theorems. Synthese 198 (Suppl 5), 1121–1143 (2021). https://doi.org/10.1007/s11229-017-1331-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-017-1331-1