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Syntactical truth predicates for formulas with atomic negation

Published online by Cambridge University Press:  26 February 2002

LOIC COLSON
LOIC COLSON
Affiliation:
University of Metz - L.I.T.A. Département d’Informatique, UFR MIM, Ile du Saulcy 57000 Metz - France. Email: colson@lita.univ-metz.fr
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Abstract

We define a language for second-order arithmetic using the (positive) connectives ∧ and ∨, the (positive) quantifiers ∀ and ∃, and with negation ‘pushed to the atoms’ under the form of atomic formula t[esdot ]u, t≠u, t∈X, t∈X where t and u stand for first-order terms and X for a second-order variable.

We define and study the translation from this language to the more usual implicational language containing the entailment connective →. We also study the converse translation. Next we define the appropriate notion of syntactical truth predicate for this language (such a notion has been introduced for the implicational language in Colson and Grigorieff (1998)). We establish using the previous translations that the existence of such a predicate for this language is equivalent to the the existence of such a predicate for the implicational language. This result is established in a predicative formal system. We conclude by discussing some elementary attempts to construct such a truth predicate by a fixed point technique.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 12 , Issue 1 , February 2002 , pp. 1 - 18
Copyright
2002 Cambridge University Press

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