Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-16T09:45:52.992Z Has data issue: false hasContentIssue false

A pragmatic interpretation of substructural logics

from PART II - LOGIC AND COMPUTATION

Published online by Cambridge University Press:  31 March 2017

By
Gianluigi Bellin  and
Carlo Dalla Pozza
Edited by
Wilfried Sieg ,
Richard Sommer  and
Carolyn Talcott
Wilfried Sieg
Affiliation:
Carnegie Mellon University, Pennsylvania
Richard Sommer
Affiliation:
Stanford University, California
Carolyn Talcott
Affiliation:
Stanford University, California
Get access

Summary

Abstract Following work by Dalla Pozza and Garola [2, 3] on a pragmatic interpretation of intuitionistic and deontic logics, which has given evidence of their compatibility with classical semantics, we present sequent calculus system ILP formalizing the derivation of assertive judgements and obligations from mixed contexts of assertions and obligations and we prove the cut-elimination theorem for it. For the formalization of real-life normative systems it is essential to consider inferences from mixed contexts of assertions and obligations, and also of assertions justifiable relatively to a given state of information and obligations valid in a given normative system. In order to provide a formalization of the notion of causal implication and its interaction with obligations, the sequents of ILP have two areas in the antecedent, expressing the relevant and the ordinary intuitionistic consequence relation, respectively. To provide a pragmatic interpretation of reasoning with the linear consequence relation we consider the deductive properties of pragmatic schemes where the operators of illocutionary force are unknown (free logic of pragmatic force). We introduce an auxiliary system ILLP which formalizes such a logic, and has substitution rules that allow us to derive the desired mixed sequents. It is shown that in order to permit non-uniform susbtitutions and to preserve Hume's law (the underivability of obligations from truth-asserting judgements only and the potential ineffectuality of norms) ILLP must indeed be based on the linear consequence relation. It is also argued that the above uses of linear and relevant logics are perfectly compatible with a theory of pragmatics in a classical semantical setting and immune from popular confusions about the intuitive interpretations of substructural logics.

Preface Most forms of human reasoning can be formalized and those which deserve such a treatment also benefit from it. Not all philosophers may agree with such a statement, but never mind: the most distinguished opponent of the formalization of mathematical reasoning in the 20th century, Brouwer, was luckily contradicted by his followers and sympathisers, to whom we owe the beautifully insightful constructions of intuitionistic logic, from Heyting's axiomatization and informal semantics of proofs to Gentzen and Prawitz's natural deduction NJ, from Curry and Howard's correspondence between NJ and the typed lambda calculus to categorical logic, and so on and on.

Type
Chapter
Information
Reflections on the Foundations of Mathematics
Essays in Honor of Solomon Feferman
 , pp. 139 - 163
Publisher: Cambridge University Press
Print publication year: 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] N., Benton, G., Bierman, V., de Paiva, M., Hyland. A term calculus for intuitionistic linear logic. In Proceedings of the Conference on Typed Lambda Calculi and Applications. Springer LNCS 664, pp. 75-90. 1993.
[2] C., Dalla Pozza and C., Garola. A pragmatic interpretation of intuitionistic propositional logic, Erkenntnis 43. 1995, pp.81-109.Google Scholar
[3] C., Dalla Pozza. Una logica prammatica per la concezione “espressiva” delle norme, In: A., Martino, ed. Logica delle Norme, Pisa, 1997
[4] S., Feferman. Transfinite recursive progressions of axiomatic theories. The Journal of Symbolic Logic 27, 1962, pp. 259-316.Google Scholar
[5] S., Feferman. Review of Prawitz 1971. The Journal of Symbolic Logic 40, 1975, pp. 232-234.Google Scholar
[6] J-Y., Girard. Linear Logic, Theoretical Computer Science 50, 1987, pp. 1-102.Google Scholar
[7] J-Y., Girard. On the unity of logic. Annals of Pure and Applied Logic 59, 1993, pp.201-217.Google Scholar
[8] J-Y., Girard. Linear logic: its syntax and semantics. In: J-Y., Girard, Y, Lafont and L., Regnier, eds., in Advances in Linear Logic, LondonMathematical Society Lecture Note Series 222, Cambridge University Press, 1995, pp. 1-42.
[9] K., Gödel. Eine Interpretation des Intuitionistischen Aussagenkalküls, Ergebnisse eines Mathematischen Kolloquiums IV, 1933, pp. 39-40.Google Scholar
[10] B., Jacobs. Semantics of Weakening and Contraction, Annals of Pure and Applied Logic, 69, 1, May, 1992, pp.73-106.Google Scholar
[11] J.C.C., McKinsey and A., Tarski. Some theorems about the sentential calculi of Lewis and Heyting, Journal of Symbolic Logic 13, 1948, pp. 1-15.Google Scholar
[12] D., Prawitz. Ideas and Results in Proof Theory. In Proceedings of the Second Scandinavian Logic Symposium, ed. Fenstad, North-Holland, 1971.
[13] P., O'Hearn and D., Pym. The Logic of Bunched Implications. Bulletin of Symbolic Logic, Volume 5, Number 2, June 1999, 215-243.Google Scholar
[14] H., Schellinx. The noble art of linear decorating, PhD Thesis, Universiteit van Amsterdam, Faculteit Wiskunde en Informatic. ILLC dissertation series, Amsterdam 1994.
[15] R., Solovay. Provability interpretations of modal logic. Israel Journal ofMathematics, 25, 1976, pp. 287-304.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×