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Switch to: References
Citations of:
Conservative translations
Annals of Pure and Applied Logic 108 (1-3):205-227 (2001)
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We provide sufficient conditions for the existence of a conservative translation from a consequence system to another one. We analyze the problem in many settings, namely when the consequence systems are generated by a deductive calculus or by a logic system including both proof-theoretic and model-theoretic components. We also discuss reflection of several metaproperties with the objective of showing that conservative translations provide an alternative to proving such properties from scratch. We discuss soundness and completeness, disjunction property and metatheorem of (...) |
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In this paper we treat metasequents—objects which stand to sequents as sequents stand to formulas—as first class logical citizens. To this end we provide a metasequent calculus, a sequent calculus which allows us to directly manipulate metasequents. We show that the various metasequent calculi we consider are sound and complete w.r.t. appropriate classes of tetravaluations where validity is understood locally. Finally we use our metasequent calculus to give direct syntactic proofs of various collapse results, closing a problem left open in (...) |
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Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question whether there are (...) |
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We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof- theoretic and model-theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quantifiers, although not present in the definition of logic and logic translation, can be recovered by their abstract (...) |
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In this paper we study interpretations and equivalences of propositional deductive systems by using a quantale-theoretic approach introduced by Galatos and Tsinakis. Our aim is to provide a general order-theoretic framework which is able to describe and characterize both strong and weak forms of interpretations among propositional deductive systems also in the cases where the systems have different underlying languages. |
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What does it take for two logics to be mere notational variants? The present paper proposes a variety of different ways of cashing out notational variance, in particular isolating a constraint on any reasonable account of notational variance which makes plausible that the only kinds of translations which can witness notational variance are what are sometimes called definitional translations. |
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After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: ( conservative ) translations , transfers and contextual translations . Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another. |
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A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads (...) |
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In this paper we treat metasequents—objects which stand to sequents as sequents stand to formulas—as first class logical citizens. To this end we provide a metasequent calculus, a sequent calculus which allows us to directly manipulate metasequents. We show that the various metasequent calculi we consider are sound and complete w.r.t. appropriate classes of tetravaluations where validity is understood locally. Finally we use our metasequent calculus to give direct syntactic proofs of various collapse results, closing a problem left open in (...) |
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This work presents the concepts of translation and conservative translation between logics. By using algebraic semantics we introduce several conservative translations involving the classical propositional calculus and the many-valued calculi of Post and Lukasiewicz. |
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We consider an example of four valued semantics partially inspired by quantum computations and negation-like operations occurred therein. In particular we consider a representation of so called square root of negation within this four valued semantics as an operation which acts like a cycling negation. We define two variants of logical matrices performing different orders over the set of truth values. Purely formal logical result of our study consists in axiomatizing the logics of defined matrices as the systems of binary (...) |
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