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On formulas of one variable in intuitionistic propositional calculus. (English) Zbl 0108.00302
The author gives a very clear description of the lattice of equivalence classes of formulae of intuitionistic propositional logic (IPC) in one variable, by choosing one representative of each class. He also introduces the curious notion of an \(LJ\)-closed formula \(P(X)\) in the single free variable \(X\), if the corresponding class of formulae \(\{A: A\) is a formula of IPC and \(\vdash_{\text{IPC}} P(A)\}\) is closed under the propositional connectives. (This notion seems to have nothing to do with adding the schema \(P(X)\) to IPC.) The only classes of formulae so obtained are (i) all propositional formulae, (ii) none, (iii) \(\{A\colon \vdash A\) or \( \vdash \rightharpoondown A\}\), and (iv) \(\{A\colon \vdash \rightharpoondown A\) or \( \vdash \rightharpoondown\rightharpoondown A\}\).
Reviewer: G. Kreisel

MSC:
03-XX Mathematical logic and foundations
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References:
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[2] Nagoya Mathematical Journal 9 pp 181– (1955) · Zbl 0066.01103 · doi:10.1017/S0027763000023412
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