An algebraic approach to intuitionistic connectives

Journal of Symbolic Logic 66 (4):1620-1636 (2001)
  Copy   BIBTEX

Abstract

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed" conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 91,271

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Analytics

Added to PP
2009-01-28

Downloads
99 (#170,904)

6 months
30 (#103,283)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

What is a Relevant Connective?Shawn Standefer - 2022 - Journal of Philosophical Logic 51 (4):919-950.
Implicit connectives of algebraizable logics.Xavier Caicedo - 2004 - Studia Logica 78 (1-2):155 - 170.
Implicit connectives of algebraizable logics.Xavier Caicedo - 2004 - Studia Logica 78 (1-2):155-170.

View all 18 citations / Add more citations

References found in this work

Intuitionistic Logic Model Theory and Forcing.F. R. Drake - 1971 - Journal of Symbolic Logic 36 (1):166-167.
Distributive Lattices.Raymond Balbes & Philip Dwinger - 1977 - Journal of Symbolic Logic 42 (4):587-588.
Bi-Heyting algebras, toposes and modalities.Gonzalo E. Reyes & Houman Zolfaghari - 1996 - Journal of Philosophical Logic 25 (1):25 - 43.
Finite limitations on Dummet's LC.Ivo Thomas - 1962 - Notre Dame Journal of Formal Logic 3 (3):170-174.
The Mathematics of Metamathematics.Donald Monk - 1963 - Journal of Symbolic Logic 32 (2):274-275.

View all 10 references / Add more references