Skip to main content Accessibility help
Home

# A new version of Beth semantics for intuitionistic logic

## Extract

We use the notation of Kripke's paper [1]. Let M = (G, K, R) be a tree structure and D a domain and η a Beth model on M. The truth conditions of the Beth semantics for ∨ and ∃ are (see [1]):

(a) η (AB, H) = T iff for some BK, B bars H and for each H′ ∈ B, either η(A, H′) = T or η(B, H′) = T.

(b) η(∃xA(x), H) = T iff for some BK, B bars H and for each H′ ∈ B there exists an aD such that η(A (a), H′) = T.

Suppose we change the truth definition η to η* by replacing condition (b) by the condition (b*) (well known from the Kripke interpretation):

Call this type of interpretation the new version of Beth semantics. We prove

Theorem 1. Intuitionistic predicate logic is complete for the new version of the Beth semantics.

Since Beth structures are of constant domains, and in the new version of Beth semantics the truth conditions for ∧, →, ∃, ∀, ¬ are the same as for the Kripke interpretation, we get:

Corollary 2. The fragment without disjunction of the logic CD of constant domains (i.e. with the additional schemax(AB(x))→ A ∨ ∀xB(x), x not free in A) equals the fragment without disjunction of intuitionistice logic.

## References

Hide All
[1]Kripke, S., Semantical analysis for intuitionistic logic. I, Formal systems and recursive functions (Crossley, J. and Dummett, M., Editors), North-Holland, Amsterdam, 1965.
[2]Fitting, M., Intuitionistic logic model theory and forcing, North-Holland, Amsterdam, 1969.
[3]Gabbay, D., Applications of trees to intermediate logics, this Journal, vol. 37 (1972), pp. 135138.
[4]Görnemann, S., A logic stronger than intuitionism, this Journal, vol. 36 (1971), pp. 249261.
[5]Gabbay, D., Semantical investigations in Heyting's predicate logic, D. Reidel (in press).

# A new version of Beth semantics for intuitionistic logic

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

### Abstract viewsAbstract views reflect the number of visits to the article landing page.

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.