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Quantifying over propositions in relevance logic: Nonaxiomatisability of primary interpretations of \({\forall}p\) and \({\exists}p\). (English) Zbl 0786.03016
This nicely written paper presents “a quite general method for proving that systems resulting from an algebraically-motivated primary interpretation of propositional quantification are recursively isomorphic to full second-order classical logic” (p. 346). It turns out that both the universal quantifier associated with Urquhart’s semilattice semantics and \(\forall p\) and \(\exists p\) associated with Routley and Meyer’s relational semantics are not axiomatizable.
Reviewer: M.Urchs (Leipzig)

MSC:
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
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References:
[1] A completeness theorem in modal logic 24 pp 1– (1959) · Zbl 0091.00902
[2] Formal systems and recursive functions. Proceedings of the eighth logic colloquium pp 92– (1963)
[3] Semantics for relevant logics 37 pp 159– (1972)
[4] S5 with quantifiable propositional variables 35 pp 355– (1970)
[5] An introduction to modal logic (1968) · Zbl 0205.00503
[6] Recursion-Theoretic hierarchies (1978) · Zbl 0371.02017
[7] Completeness in the theory of types 15 pp 81–
[8] Models for entailment pp 347– (1974)
[9] Theoria 36 pp 331– (1970)
[10] Introduction to mathematical logic (1956)
[11] Entailment, the logic of relevance and necessity 2 (1992)
[12] Entailment, the logic of relevance and necessity 1 (1975) · Zbl 0345.02013
[13] Relevant logics and their rivals 1. The basic philosophical and semantical theory (1982) · Zbl 0579.03011
[14] Truth, syntax and modality. Proceedings of the Temple University conference on alternative semantics pp 199– (1973)
[15] DOI: 10.1007/BF00650498 · Zbl 0317.02019
[16] DOI: 10.1007/BF00649991 · Zbl 0317.02018
[17] DOI: 10.1090/S0002-9904-1944-08111-1 · Zbl 0063.06328
[18] Classical recursion theory: The theory of functions and sets of natural numbers (1989)
[19] The Kleene symposium pp 181– (1980)
[20] DOI: 10.1002/malq.19550010205 · Zbl 0065.00105
[21] Entailment, the logic of relevance and necessity 2 (1992)
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