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An intuitionistically plausible interpretation of intuitionistic logic

Published online by Cambridge University Press:  12 March 2014

H. C. M. de Swart
H. C. M. de Swart*
Affiliation:
Department of Philosophy, Erasmuslaan 40, Nijmegen, Holland
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Extract

Let IPC be the intuitionistic first-order predicate calculus. From the definition of derivability in IPC the following is clear:

(1) If A is derivable in IPC, denoted by “⊦IPCA”, then A is intuitively true, that means, true according to the intuitionistic interpretation of the logical symbols. To be able to settle the converse question: “if A is intuitively true, then ⊦IPCA”, one should make the notion of intuitionistic truth more easily amenable to mathematical treatment. So we have to look then for a definition of “A is valid”, denoted by “⊨A”, such that the following holds:

(2) If A is intuitively true, then ⊨ A.

Then one might hope to be able to prove

(3) If ⊨ A, then ⊦IPCA.

If one would succeed in finding a notion of “⊨ A”, such that all the conditions (1), (2) and (3) are satisfied, then the chain would be closed, i.e. all the arrows in the scheme below would hold.

Several suggestions for ⊨ A have been made in the past: Topological and algebraic interpretations, see Rasiowa and Sikorski [1]; the intuitionistic models of Beth, see [2] and [3]; the interpretation of Grzegorczyk, see [4] and [5]; the models of Kripke, see [6] and [7]. In Thirty years of foundational studies, A. Mostowski [8] gives a review of the interpretations, proposed for intuitionistic logic, on pp. 90–98.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 4 , December 1977 , pp. 564 - 578
Copyright
Copyright © Association for Symbolic Logic 1977

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References

REFERENCES

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