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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

03-XX Mathematical logic and foundations
Full Text: DOI
[1] On intermediate prepositional logics 24 pp 20– (1959)
[2] Nagoya Mathematical Journal 9 pp 181– (1955) · Zbl 0066.01103 · doi:10.1017/S0027763000023412
[3] Mathematische Zeitschrift 39 pp 176– (1934)
[4] DOI: 10.2307/1969038 · Zbl 0060.06207 · doi:10.2307/1969038
[5] Ergebnisse eines mathematischen Kolloquiums pp 40– (1931)
[6] Proofs on non-deducibility in intuitionistic functional calculus 13 pp 204– (1948) · Zbl 0031.19304
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