On an interpretation of second order quantification in first order intuitionistic propositional logic

Journal of Symbolic Logic 57 (1):33-52 (1992)
Abstract
We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, φ, built up from propositional variables (p,q,r,...) and falsity $(\perp)$ using conjunction $(\wedge)$ , disjunction (∨) and implication (→). Write $\vdash\phi$ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula φ there exists a formula Apφ (effectively computable from φ), containing only variables not equal to p which occur in φ, and such that for all formulas ψ not involving $p, \vdash \psi \rightarrow A_p\phi$ if and only if $\vdash \psi \rightarrow \phi$ . Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions. An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/2275175
Options
 Save to my reading list
Follow the author(s)
Edit this record
My bibliography
Export citation
Find it on Scholar
Mark as duplicate
Request removal from index
Revision history
Download options
Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 30,248
Through your library
References found in this work BETA

No references found.

Add more references

Citations of this work BETA
The Many Faces of Interpolation.van Benthem Johan - 2008 - Synthese 164 (3):451-460.

View all 18 citations / Add more citations

Similar books and articles
Added to PP index
2009-01-28

Total downloads
25 ( #208,391 of 2,192,221 )

Recent downloads (6 months)
1 ( #289,802 of 2,192,221 )

How can I increase my downloads?

Monthly downloads
My notes
Sign in to use this feature