## Abstract

In standard model theory, deductions are not the things one models. But in general proof theory, in particular in categorial proof theory, one finds models of deductions, and the purpose here is to motivate a simple example of such models. This will be a model of deductions performed within an abstract context, where we do not have any particular logical constant, but something underlying all logical constants. In this context, deductions are represented by arrows in categories involved in a general *adjoint situation*. To motivate the notion of adjointness, one of the central notions of category theory, and of mathematics in general, it is first considered how some features of it occur in set-theoretical axioms and in the axioms of the lambda calculus. Next, it is explained how this notion arises in the context of deduction, where it characterizes logical constants. It is shown also how the categorial point of view suggests an analysis of propositional identity. The problem of propositional identity, i.e., the problem of identity of meaning for propositions, is no doubt a philosophical problem, but the spirit of the analysis proposed here will be rather mathematical. Finally, it is considered whether models of deductions can pretend to be a semantics. This question, which as so many questions having to do with meaning brings us to that wall that blocked linguists and philosophers during the whole of the twentieth century, is merely posed. At the very end, there is the example of a geometrical model of adjunction. Without pretending that it is a semantics, it is hoped that this model may prove illuminating and useful.

## Preview

Unable to display preview. Download preview PDF.

### References

- Buss, S. R. 1991‘The Undecidability of
*k*-Provability’Annals of Pure and Applied Logic.5375102CrossRefGoogle Scholar - Carbone, A. 1997‘Interpolants, Cut Elimination and Flow Graphs for the Propositional Calculus’Annals of Pure and Applied Logic.83249299CrossRefGoogle Scholar
- Dosen, K.: 1989, ‘Logical Constants as Punctuation Marks’,
*Notre Dame Journal of Formal Logic***30**, 362–381 (slightly amended version in D.M. Gabbay (ed.),*What is a Logical System?*, Oxford University Press, Oxford, 1994, pp. 273–296).Google Scholar - Dosen, K. 1996‘Deductive Completeness’The Bulletin of Symbolic Logic.2243283, 523CrossRefGoogle Scholar
- Dosen, K. 1998‘Deductive Systems and Categories’Publications de l’Institut Mathématique.642135Google Scholar
- Dosen, K. 1999Cut-Elimination in CategoriesKluwerDordrechtGoogle Scholar
- Dosen, K.: 2001, ‘Abstraction and Application in Adjunction’, in Z. Kadelburg (ed.),
*Proceedings of the Tenth Congress of Yugoslav Mathematicians*, Faculty of Mathematics, University of Belgrade, pp. 33–46 (available at: http://xxx.lanl.gov/math.CT.0111061). - Dummett, M. 1973Frege: Philosophy of LanguageDuckworthLondonGoogle Scholar
- Eilenberg, S., Kelly, G. M. 1966‘A Generalization of the Functorial Calculus’Journal of Algebra.3366375CrossRefGoogle Scholar
- Ishiguro, H. 1990Leibniz’s Philosophy of Logic and Language2Cambridge University PressCambridgeGoogle Scholar
- Kassel, C. 1995Quantum GroupsSpringerBerlinGoogle Scholar
- Kauffman, L. H., Lins, S. L. 1994Temperley–Lieb Recoupling Theory and Invariants of 3-ManifoldsPrinceton University PressPrincetonGoogle Scholar
- Kelly, G. M., MacLane, S. 1971‘Coherence in Closed Categories’Journal of Pure and Applied Algebra.197140, 219CrossRefGoogle Scholar
- Lambek, J. 1974‘Functional Completeness of Cartesian Categories’Annals of Mathematical Logic.6259292CrossRefGoogle Scholar
- Lambek, J., et al. 1999
# ‘Type Grammars Revisited’

Lecomte, A. eds. Logical Aspects of Computational LinguisticsSpringerBerlin127Lecture Notes in Artificial Intelligence 1582Google Scholar - Lambek, J., Scott, P. J. 1986Introduction to Higher-Order Categorical LogicCambridge University PressCambridgeGoogle Scholar
- Lawvere, F. W. 1969‘Adjointness in Foundations’Dialectica.23281296Google Scholar
- Petrić, Z.: 1997,
*Equalities of Deductions in Categorial Proof Theory*(in Serbian), Doctoral Dissertation, University of Belgrade.Google Scholar - Prawitz, D. 1965Natural Deduction: A Proof-Theoretical StudyAlmqvist and WiksellStockholmGoogle Scholar