Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of ∀<i>p</i>and ∃<i>p</i>

Kremer, Philip

doi:10.2307/2275341

Cited by 21 publications

(18 citation statements)

References 8 publications

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“…This was proved independently by Fine and Kripke shortly after Fine's paper [8] was published, as Kremer remarked in [11]. Also Kremer's strategy from [12] can be extended to prove the same result, as he claims in [11]. In particular it means that these systems are undecidable while their propositional counterparts are decidable.…”

Section: Propositional Quantifiersmentioning

confidence: 76%

Uniform Interpolation and Propositional Quantifiers in Modal Logics

Bílková

2007

Stud Logica

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We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view.Our approach is adopted from Pitts' proof of uniform interpolation in intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.We shall present such a proof of the uniform interpolation theorem for normal modal logics K and T. It provides an explicit algorithm constructing the interpolants.

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Section: Propositional Quantifiersmentioning

confidence: 76%

Uniform Interpolation and Propositional Quantifiers in Modal Logics

Bílková

2007

Stud Logica

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“…that there are formulas valid in all full RP-model structures that are not RP-theorems. This was confirmed by Kremer in [7] by proving that the set of all formulas valid in all full RP-model structures is not recursively axiomatisable, and indeed is recursively isomorphic to full second-order logic. This shows that the use we have made of models with a restricted set of admissible propositions is essential for providing a complete relational semantics for RP.…”

Section: Incompletenessmentioning

confidence: 77%

“…This was by analogy with Henkin's primary interpretations of higher-order logic [6], given that this interpretation of ∀ was second-order in nature. Kremer [7] eventually proved their conjecture by showing that the set of formulas validated by the Routley-Meyer primary semantics for RP is not recursively axiomatisable.…”

Section: Introductionmentioning

confidence: 99%

An Admissible Semantics for Propositionally Quantified Relevant Logics

Goldblatt

Kane

2009

J Philos Logic

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The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀pA true when there is some true admissible proposition that entails all p-instantiations of A.It is also shown that without the admissibility qualification many of the systems considered are semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap's system E of entailment, extended by the mingle axiom and the Ackermann constant t. The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras.

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“…These results were later strengthened and unified by Kaminski and Tiomkin (1996), who show for every class of frames containing all frames validating S4.2, i.e, all reflexive, transitive and convergent frames, that the set of sentences of second-order predicate logic valid on all standard models (henceforth SOL) can be recursively embedded in the propositionally quantified logic of this class of frames. It follows by routine considerations that the two sets are recursively isomorphic; see Kremer (1993) for a careful presentation of the relevant complexity-theoretic matters. The fact that adding propositional quantifiers to propositional modal logic often gives rise to such a complex logic may be one reason why the study of propositional quantifiers has been comparatively marginal within modal logic.…”

Section: Introductionmentioning

confidence: 99%

Propositional Quantification in Bimodal $$\mathbf {S5}$$S5

Fritz

2018

Erkenn

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Propositional quantifiers are added to a propositional modal language with two modal operators. The resulting language is interpreted over socalled products of Kripke frames whose accessibility relations are equivalence relations, letting propositional quantifiers range over the powerset of the set of worlds of the frame. It is first shown that full second-order logic can be recursively embedded in the resulting logic, which entails that the two logics are recursively isomorphic. The embedding is then extended to all sublogics containing the logic of so-called fusions of frames with equivalence relations. This generalizes a result due to Antonelli and Thomason, who construct such an embedding for the logic of such fusions.

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Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of ∀pand ∃p

Cited by 21 publications

References 8 publications

Uniform Interpolation and Propositional Quantifiers in Modal Logics

Uniform Interpolation and Propositional Quantifiers in Modal Logics

An Admissible Semantics for Propositionally Quantified Relevant Logics

Propositional Quantification in Bimodal $$\mathbf {S5}$$S5

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