Abstract
∈ T -logic was first designed by Werner Sträter as a first-order propositional logic with quantification, reference, and predicates for true and false. It is motivated by reconstruction of natural language semantics and allows, as a logic with self-reference and impredicativity, among others the treatment of the liar paradox despite the totality of its truth predicates. Its intensional models form a theory of propositions for which a correct and complete calculus is given.
∈ T -logic was picked up by Philip Zeitz to study the extension of abstract logics by the concepts of truth, reference and classical negation, thereby rebuilding the meta-level of judgements in a formal level of propositional logic. His parameterized ∈ T -logic allows formulas from a parameter logic to become the constants in his ∈ T -logic. Parameter-passing of logics with correct and complete calculus also admits, under certain conditions, the entailment of a calculus which is correct and complete for the extended logic.
Since in parameterized ∈ T -logic Tarski Biconditionals not only apply for the truth of ∈ T -logic sentences, but also for the meta-level truth of the parameter logic it is natural to view ∈ T -logic as a theory of judgements whose propositions are expressed in the parameter logic.
We add a new interpretation to ∈ T -logic as a theory of truth and judgements, and introduce ∈ T -logic as a means for the integration of logics. Based on a particular choice of uniform view and treatment of logics we define ∈ T -logics and ∈ T -extensions as the foundation for ∈ T -integration of logics and models.
Studies in ∈ T -logic, which have started to deal with the difficulties of truth in natural language semantics, have evolved into a concept of logic integration where application oriented logics can be plugged in as parameters. This paper very much relies on the work of Philip Zeitz, but opens it for the new perspective of integration.
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Mahr, B., Bab, S. (2005). ∈ T -Integration of Logics. In: Kreowski, HJ., Montanari, U., Orejas, F., Rozenberg, G., Taentzer, G. (eds) Formal Methods in Software and Systems Modeling. Lecture Notes in Computer Science, vol 3393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31847-7_12
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