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Applications of trees to intermediate logics

Published online by Cambridge University Press:  12 March 2014

Dov M. Gabbay
Dov M. Gabbay*
Affiliation:
Stanford University, Stanford, California 94305
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Extract

We investigate extensions of Heyting's predicate calculus (HPC). We relate geometric properties of the trees of Kripke models (see [2]) with schemas of HPC and thus obtain completeness theorems for several intermediate logics defined by schemas. Our main results are:

(a) ∼(∀x ∼ ∼ϕ(x) Λ ∼∀xϕ(x)) is characterized by all Kripke models with trees T with the property that every point is below an endpoint. (From this we shall deduce Glivenko type theorems for this extension.)

(b) The fragment of HPC without ∨ and ∃ is complete for all Kripke models with constant domains.

We assume familiarity with Kripke [2]. Our notation is different from his since we want to stress properties of the trees. A Kripke model will be denoted by (Aα, ≤ 0), α ∈ T where (T, ≤, 0) is the tree with the least member 0 ∈ T and Aα, α ∈ T, is the model standing at the node α. The truth value at α of a formula ϕ(a1 … an) under the indicated assignment at α is denoted by [ϕ(a1 … an)]α.

A Kripke model is said to be of constant domains if all the models Aα, α ∈ T, have the same domain.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 1 , March 1972 , pp. 135 - 138
Copyright
Copyright © Association for Symbolic Logic 1972

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References

REFERENCES

[1]Görnemann, S., Über eine Verschārfung der intuitionistischen Logik, Dissertation, Hannover, 1969.Google Scholar
[2]Kripke, S. A., Semantical analysis of intuitionistic logic I. Formal systems and recursive functions, Crossley, J.-Dummett, M. Editors, pp. 92–130.Google Scholar
[3]Fitting, M., Intuitionistic logic model theory and forcing, North-Holland, Amsterdam, 1969.Google Scholar