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Rules of inference with parameters for intuitionistic logic. (English) Zbl 0788.03007
Let \(A(x_ 1,\dots,x_ n,p_ 1,\dots,p_ m)\) and \(B(x_ 1,\dots,x_ n,p_ 1, \dots,p_ m)\) be formulas of the intuitionistic propositional calculus in variables \(x_ i\), \(p_ j\). The rule \(A/B\) with parameters \(p_ j\) is admissible, if for all \(n\)-tuples of formulas \(B_ 1,\dots,B_ n\) derivability of \(A(B_ 1,\dots,B_ n,p_ 1,\dots,p_ m)\) implies derivability of \(B(B_ 1,\dots,B_ n,p_ 1,\dots,p_ m)\). An algorithm for deciding admissibility with parameters is given here by testing the rule on certain finite sets of finite Kripke models. In particular this decides existence of formulas \(B_ 1,\dots,B_ n\) such that \(A(B_ 1,\dots,B_ n,p_ 1,\dots,p_ n)\) is derivable: this is equivalent to the nonadmissibility of the rule \(A/\)false.
Reviewer: G.Mints (Stanford)

MSC:
03B20 Subsystems of classical logic (including intuitionistic logic)
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References:
[1] Doklady Akademii Nauk SSSR 287 pp 554– (1986)
[2] Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 50 pp 598– (1986)
[3] Algebra i Logika 23 pp 546– (1984)
[4] Zapiski Nauchnykh Seminarov L0M1 32 pp 85– (1972)
[5] Matematicheskie Zametki 37 pp 797– (1985)
[6] One hundred and two problems in mathematical logic 40 pp 113– (1975)
[7] Vollst√§ndige Systeme modaler und intuitionistischer Logik 42 (1968) · Zbl 0157.01602
[8] Doklady Akademii Nauk SSSR 241 pp 1288– (1978)
[9] Matematicheskiń≠ Sbornik 102 pp 314– (1977)
[10] Logical notebook: unsolved questions of mathematical logic (1986)
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