Abstract
Logic already spanned a great range of topics before the birth of categorical logic. Some celebrated results achieved in logic during the first half of the twentieth century are milestones in the understanding of mathematical relations between syntactic, semantic and algorithmic aspects of the structure of language and reasoning. Logical tools have been exploited in a variety of applications: from linguistics to computer science, from methodology of science to specific physical theories. The very formulation of questions and answers concerning the foundations of mathematics relies on such tools. Finally, mathematical logic has altered the face of philosophy. In view of such outcomes, it is all the more appropriate to consider the impact of categorical methods on logic, since they affect the study of proofs and models, by forging a stricter relationship between a theory and its models, and by enlarging the range of possible models beyond the universe of sets in a way that leads to a substantial refinement of the status ascribed to logic itself.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Henceforth, · will be generally omitted. As for the size of the collection of objects and maps, for simplicity’s sake the categories considered will be small, i.e., with a set of objects (rather than a proper class), or locally small, i.e., with a set of maps between any two objects.
- 2.
Some notions yet to be defined—to start with, that of a topos—are exploited in the present section to pinpoint the different components involved in the impact of categorical thinking on logic, thus affecting its use in extra-mathematical applications.
- 3.
- 4.
The logical aspects of Stone duality are located in a much wider “phenomenology” of dualities, which find unified formulation in categorical language, as results from (Johnstone, 1982). Representability is actually a constant theme of category theory, which offers the general environment to unify results in the line of Cayley and Stone (but also of Grothendieck). A reference point for this unification is the Yoneda Lemma: given the “hom-functor” \(h_{A} :\textbf{C} \longrightarrow \textbf{Set}\), with h A (−) = the set of C-maps from A to –, for a fixed object A of C (see footnote 1), and given any \(F: \textbf{C} \longrightarrow \textbf{Set}\), there is a bijection between the natural transformations \(h_{A} \rightarrow F\) and the set F(A). The Lemma also extends to contravariant functors, in which case the C-maps from – to A are considered to define h A. The embedding \(\textbf{C} \longrightarrow \textbf{Set}^{\textbf{C}^{\textbf{op}}}\) is full and faithful and thus allows one to investigate C in a wider context with no loss of information.
- 5.
That the theory of fields and in particular the theory of local fields can be expressed by coherent first-order axioms is relevant for the use of constructive reasoning in algebra, … as much as that the subtle distinctions involved in axioms of different forms for the concept of field can be confined to Boolean toposes.
- 6.
Together with various contributions by Colin McLarty on the fom- and categories-lists, the debate could have benefited more from Bell’s fair as well as suggestive remarks in (Bell, 1986).
- 7.
A topos can be shown to have power objects P(−) (represented as Ω−), in view of the unique correspondence between relation maps \(r:R \rightarrowtail B \times A\) and maps \(f_{r}: B \rightarrow P(A)\). Vice versa, by the existence of the pullback of \(f_{r}\times 1_{A} \cdot r\) along ϵ A , Ω can be obtained as P(1) up to isomorphism.
- 8.
This argument is different from relying on mutual equiconsistency relative to sets: for instance, the power of the axioms for an elementary topos being equivalent to that of Z 0, i.e., ZF with only bounded quantifiers, is significant only with reference to a universe of discrete sets of points.
- 9.
- 10.
The idea of a doctrine extends to the consideration of “monads” and 2-categories, as categories with also a further “vertical” dimension of composition. This line will not be pursued here for simplicity’s sake, but it is what allows connecting rewriting system for λ-calculi ( Seely, 1987) with proof-homotopy as remarked below.
- 11.
Already in standard topology what generally matters is whether or not connectedness, rather than total disconnectedness, holds.
- 12.
What ultimately matters in representing logic by posets is whether there is a map \(1 \rightarrow A\) or not (for completeness, whether \(\top \rightarrow \varphi\) or \(\varphi \rightarrow \bot\)) and the only map \(A \rightarrow A\) that matters is identity. Therefore, the map \(p_2 <f,g>: A \rightarrow A\), obtained by pairing \(f,g: A \rightarrow A\) to have \(<f,g>: A \rightarrow A \wedge A\) and then by composing with \(p_2: A \wedge A \rightarrow A\), is trivial, as is the composition of \(tw_1: A \wedge B \rightarrow B \wedge A\) and \(tw_2: B \wedge A \rightarrow A \wedge B\). Whereas in a general category with 1, from the existence of \(x: 1\rightarrow A\) it doesn’t follow that \(A \cong 1\), the existence of only \(id_{A}: A \rightarrow A\) collapses A onto 1, “the” absolutely true proposition (given the composition \(!_A\cdot x \cdot !_A\), necessarily \(!_A \cdot x = id_{A}\) , and if \(id_{A} = x \cdot !_A\), then \(A \cong 1\)). So, if there is more than one true proposition, there is at least one \(f: A \rightarrow A\) such that \(f \not= id_{A}\), but there is no trace of any such f when a deductive system collapses to a poset.
- 13.
This notion of theory-equivalence requires some care, as Johnstone explains in (Johnstone, 2002, vol. 2).
- 14.
- 15.
In most applications of set-theoretic semantics, one argues as if \(\{ a \} \not= \{ b \}\) for \(a \not= b\), but in Set the two singletons are isomorphic and that’s all. Hence either urelements are given or some content foreign to explicit semantics is used, to confirm the claim that meaning is in the eye of the beholder: in other words, one supposes to know more than what is allowed by the theory. On the other hand, the very notion of singleton is far from trivial in categories corresponding to constructive reasoning, see (Fourman et al., 1979).
- 16.
In this regard, the equiconsistency of the elementary topos axioms with Z 0, and the internalization of a topos into a local set theory, can be both misleading. What is achieved by means of categories is a theory of variable and cohesive sets. In particular classical sets appear as the limiting case of vanishing variation and cohesion. Languages qua special mathematical structures have no ontological priority.
Bibliography
Artin, M., Grothendieck, A., and Verdier, J. L. (1972). Séminaire de Géometrie Algébrique, IV: Théorie des Topos et Cohomologie Étale des Schemas (1964). Lecture Notes in Mathematics volumes 269, 270, 305. Springer, Berlin.
Asperti, A. and Longo, G. (1991). Categories, Types and Structures. MIT Press, Cambridge, MA.
Bell, J. L. (1986). From absolute to local mathematics. Synthese, 69:409–426.
Bell, J. L. (1988). Toposes and Local Set Theories: An Introduction. Clarendon Press, Oxford.
Bell, J. L. (1993). Hilbert’s ε-operator and classical logic. Journal of Philosophical Logic, 22:1–18.
Bell, J. L. (1996). Logical reflections on the Kochen-Specker theorem. In Clifton, R., editor, Perspectives on Quantum Reality, pages 227–235. Kluwer, Amsterdam.
Bell, J. L. (1998). A Primer of Infinitesimal Analysis. Cambridge University Press, Cambdrige.
Bell, J. L. (2005). The development of categorical logic. In Gabbay, D. and Guenthner, F., editors, Handbook of Philosophical Logic, volume 12, pages 279–361. Springer, New York, NY.
Bell, J. L. (2006). Cover schemes, frame-valued sets and their potential uses in space-time physics. In Reimer, A., editor, Spacetime Physics Research Trends. Horizons in World Physics, volume 248. Nova Science, New York, NY.
Benabou, J. (1985). Fibred categories and the foundations of naive category theory. Journal of Symbolic Logic, 50:10–37.
Boileau, A. and Joyal, A. (1981). La logique de topos. Journal of Symbolic Logic, 46:6–16.
Borceux, F. (1994). Handbook of Categorical Algebra. Cambridge University Press, Cambdrige.
Cartier, P. (2001). A mad day’s work: From Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry. Bulletin of the American Mathematical Society (New Series), 38(4):389–408.
Dummett, M. (1975). The philosophical basis of intuitionistic logic. In Rose, H. and Sheperdson, J., editors, Logic Colloquium ’73, pages 5–40. North-Holland, Amsterdam. Reprinted in Truth and Other Enigmas, Duckworth, 1978, pp. 215–247.
Eilenberg, S. and Mac Lane, S. (1945). General theory of natural equivalences. Transactions of the American Mathematical Society, 58:231–294. Reprinted in S. Eilenberg and S. Mac Lane, Collected Works. Academic Press, New York, NY, 1986.
Fourman, M., Mulvey, C., and Scott, D., editors (1979). Applications of Sheaves. Proceedings of the London Mathematical Society Durham Symposium 1977, volume 753 of Lecture Notes in Mathematics. Springer, Berlin.
Fourman, M. and Scott, D. (1979). Sheaves and logic. In Fourman, M., Mulvey, C., and Scott, D., editors, Applications of Sheaves, pages 302–401. Springer, Berlin.
Fourman, M. (1977). The logic of topoi. In Barwise, J., editor, Handbook of Mathematical Logic, pages 1053–1090. North-Holland, Amsterdam.
Freyd, P. and Scedrov, A. (1990). Categories, Allegories. North-Holland, Amsterdam.
Freyd, P. (1972). Aspects of topoi. Bulletin of the Australian Mathematical Society, 7:1–76, 467–480.
Girard, J.-Y. (2004). Logique à Rome. Università di Roma Tre, Roma.
Goldblatt, R. (1979). Topoi: The Categorial Analysis of Logic. North-Holland, Amsterdam. Revised edition, 1984.
Grothendieck, A. and Dieudonné, J. (1971). Éléments de géometrie algébrique I, volume 166 of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Springer, Berlin.
Grothendieck, A. (1970). Catégories fibrés et descente (1960/61), volume 224 of Lecture Notes in Mathematics. Springer, Berlin.
Hatcher, W. (1968). Foundations of Mathematics. WB Saunders, Philadelphia, PA.
Hyland, M. (1982). The effective topos. In Troelstra, A. and van Dalen, D., editors, The L.E.J. Brouwer Centenary Symposium, pages 162–217. North-Holland, Amsterdam.
Jacobs, B. (1999). Categorical Logic and Type Theory. Elsevier, Amsterdam.
Johnstone, P. (1977). Topos Theory, volume 10 of London Mathematical Society Monographs. Academic Press, New York, NY.
Johnstone, P. (1982). Stone Spaces. Cambridge University Press, Cambridge.
Johnstone, P. (2002). Sketches of an Elephant. A Topos Theory Compendium. Clarendon Press, Oxford.
Kan, D. (1958). Adjoint functors. Transactions of the American Mathematical Society, 87: 294–329.
Kock, A. and Reyes, G. (1977). Doctrines in categorical logic. In Barwise, J., editor, Handbook of Mathematical Logic, pages 284–313. North-Holland, Amsterdam.
Lambek, J. and Scott, P. (1986). Introduction to Higher-Order Categorical Logic. Cambridge University Press, Cambridge.
Lambek, J. (1968). Deductive systems and categories I. Journal of Mathematical Systems Theory, 2:278–318.
Lawvere, F. W., Maurer, C., and eds., G. W. (1975). Model Theory and Topoi, volume 445 of Lecture Notes in Mathematics. Springer, Berlin.
Lawvere, F. W. and Rosebrugh, R. (2003). Sets for Mathematics. Cambridge University Press, Cambridge.
Lawvere, F. W. (1963). Functorial semantics of algebraic theories. PhD thesis, Columbia University, New York, NY.
Lawvere, F. W. (1964). An elementary theory of the category of sets. Proceedings of the National Academy of Sciences USA, 52:1506–1511.
Lawvere, F. W. (1966). The category of categories as a foundation for mathematics. In Eilenberg, S., Harrison, D., Mac Lane, S., and Röhrl, H., editors, Proceedings of the Conference on Categorical Algebra, La Jolla, pages 1–21. Springer, Berlin.
Lawvere, F. W. (1968). Diagonal arguments and cartesian closed categories. In Category Theory, Homology Theory and their Applications, II, volume 92 of Lecture Notes in Mathematics, pages 134–145. Reprinted in Theory and Applications of Category Theory, No.15 (2006), pp. 1–13. Springer, Berlin.
Lawvere, F. W. (1969). Adjointness in foundations. Dialectica, 23:281–295.
Lawvere, F. W. (1971). Quantifiers and sheaves. In Actes du Congrès International des Mathématiciens, volume 1, pages 329–334. Gauthier-Villars, Paris.
Lawvere, F. W., editor (1972). Toposes, Algebraic Geometry and Logic, volume 274 of Lecture Notes in Mathematics. Springer, Berlin.
Lawvere, F. W. (1973). Teoria delle Categorie sopra un Topos di Base. Universitá di Perugia, Perugia.
Lawvere, F. W. (1975). Continuously variable sets: algebraic geometry = geometric logic. In Rose, H. and Sheperdson, J., editors, Logic Colloquium ’73, pages 135– 153. North Holland, Amsterdam.
MacIntyre, A. (1973). Model-completeness for sheaves of structures. Fundamenta Mathematicae, 81:73–89.
Macnamara, J. and Reyes, G., editors (1994). The Logical Foundations of Cognition. Oxford University Pres, New York, NY.
Mac Lane, S. and Moerdijk, I. (1992). Sheaves in Geometry and Logic: a First Introduction to Topos Theory. Springer, New York, NY.
Mac Lane, S. (1935). A logical analysis of mathematical structures. The Monist, 45:118–130.
Mac Lane, S. (1986). Mathematics: Form and Function. Springer, Berlin.
Mac Lane, S. (1997). Despite physicists, proof is essential to mathematics. Synthese, 111:147–154.
Makkai, M. and Paré, R. (1989). Accessible Categories: the Foundations of Categorical Model Theory, volume 104 of Contemporary Mathematics. American Mathematical Society, Providence, RI.
Makkai, M. and Reyes, G. (1977). First Order Categorical Logic. Springer, Berlin.
Makkai, M. (1987). Stone duality for first order logic. Advances in Mathematics, 65:97–170.
Mangione, C. (1976). Logica e teoria delle categorie. In Geymonat, L., editor, Storia del Pensiero Filosofico e Scientifico, volume 7, pages 519–653. Garzanti, Milano.
Mayberry, J. (1994). What is required of a foundation for mathematics? Philosophia Mathematica, 2:16–35.
McLarty, C. (1992). Elementary Categories, Elementary Toposes. Clarendon Press, Oxford.
McLarty, C. (2005). Saunders Mac Lane (1909–2005). His mathematical life and philosophical works. Philosophia Mathematica, 13:237–251.
Mitchell, W. (1972). Boolean topoi and the theory of sets. Journal of Pure and Applied Algebra, 2:261–274.
Moore, E. (1908). On a form of general analysis with application to linear differential and integral equations. In Atti del IV Congresso Internazionale dei Matematici, pages 98–114. Accademia dei Lincei, Roma.
Osius, G. (1974). The internal and external aspects of logic and set theory in elementary topoi. Cahiers de Topologie et Géométrie Différentielle, 15:157–180.
Peruzzi, A. (1991a). Categories and logic. In Usberti, G., editor, Problemi Fondazionali nella Teoria del Significato, pages 137–211. Olschki, Firenze.
Peruzzi, A. (1991b). Meaning and truth: the ILEG project. In Wolf, E., Nencioni, G., and Tscherdanzeva, H., editors, Semantics and Translation, pages 53–59. Moscow Academy of Sciences, Moscow.
Peruzzi, A. (1994). On the logical meaning of precategories. http://www.philos.unifi.it/persone/peruzzi.htm.
Peruzzi, A. (2000a). Anterior future. Rendiconti del Circolo Matematico di Palermo, 64:227–248.
Peruzzi, A. (2000b). The geometric roots of semantics. In Albertazzi, L., editor, Meaning and Cognition, pages 169–201. John Benjamins, Amsterdam.
Pitts, A. (2001). Categorical logic. In Abramsky, S., Gabbay, D. M., and Maibaum, T. S. E., editors, Handbook of Logic in Computer Science, volume 5, Logic and algebraic methods, pages 39–123. Oxford University Press, Oxford.
Rasiowa, H. and Sikorski, R. (1963). The Mathematics of Metamathematics. Polish Scientific Publishers, Warsaw.
Reyes, G. (1974). From sheaves to logic. In Daigneault, A., editor, Studies in Algebraic Logic, volume 9 of MAA Studies in Mathematics, pages 143–204. The Mathematical Association of America, Providence, RI.
Seely, R. (1984). Locally cartesian closed categories and type theory. Mathematical Proceedings of the Cambridge Philosophical Society, 95:33–48.
Seely, R. (1987). Modelling computations: A 2–categorical framework. In Proceedings of the Second Symposium on Logic in Computer Science, pages 61–71. IEEE Computer Science Press, Trier.
Troelstra, A. S. and van Dalen, D. (1988). Constructivism in Mathematics: An Introduction, volume 1–2. North-Holland, Amsterdam.
Vickers, S. (1988). Topology via Logic. Cambridge University Press, Cambridge.
Yanofsky, N. (2003). A universal approach to self-referential paradoxes, incompleteness and fixed points. arXiv:math.LO/0305282 v1. May 19.
Zangwill, J. (1977). Local set theory and topoi. MSc thesis, Bristol University, Bristol.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Peruzzi, A. (2011). Logic in Category Theory. In: DeVidi, D., Hallett, M., Clarke, P. (eds) Logic, Mathematics, Philosophy, Vintage Enthusiasms. The Western Ontario Series in Philosophy of Science, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0214-1_15
Download citation
DOI: https://doi.org/10.1007/978-94-007-0214-1_15
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0213-4
Online ISBN: 978-94-007-0214-1
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)