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**1 - 3**of**3**### The ubiquity of conservative translations Emil Jeˇrábek ∗

, 2011

"... We study the notion of conservative translation between logics introduced by Feitosa and D’Ottaviano [7]. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. Th ..."

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We study the notion of conservative translation between logics introduced by Feitosa and D’Ottaviano [7]. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal. 1

### Universal Logic as a Science of Patterns

"... Abstract. This article addresses Béziau’s [11] vision that universal logic should be capable of helping other fields of knowledge to build the right logic for the right situation, and that for some disciplines mathematical abstract concep-tualization is more appropriate than symbolic formalization. ..."

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Abstract. This article addresses Béziau’s [11] vision that universal logic should be capable of helping other fields of knowledge to build the right logic for the right situation, and that for some disciplines mathematical abstract concep-tualization is more appropriate than symbolic formalization. Hertz’s [67] dia-grams of logical inference patterns are formalized and extended to present the universal logic conceptual framework as a comprehensible science of patterns. This facilitates those in other disciplines to develop, visualize and apply logical representation and inference structures that emerge from their problématique. A family of protologics is developed by resemantifying the sign for deduction,→, with inference patterns common to many logics, and specifying possible constraints on its use to represent the structural connectives and defeasible reasoning. Proof-theoretic, truth-theoretic, intensional and extensional proto-semantics are derived that supervene on the inference patterns. Examples are given of applications problem areas in a range of other disciplines, including the representation of states of affairs, individuals and relations.

### The ubiquity of conservative translations Emil Jeˇrábek ∗

, 2012

"... We study the notion of conservative translation between logics introduced by Feitosa and D’Ottaviano [7]. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. Th ..."

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We study the notion of conservative translation between logics introduced by Feitosa and D’Ottaviano [7]. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal. 1