Three levels of translation into many-sorted logic (original) (raw)
Related papers
Translation Methods for Non-Classical Logics: An Overview
Logic Journal of IGPL, 1993
This paper gives an overview on translation methods we have developed for nonclassical logics, in particular for modal logics. Optimized 'functional' and semi-functional translation into predicate logic is described. Using normal modal logic as an intermediate logic, other logics can be translated into predicate logic as well. As an example, the translation of modal logic of graded modalities is sketched. In the second part of the paper it is shown how to translate Hilbert axioms into properties of the semantic structure and vice versa, i.e. we can automate important parts of correspondence theory. The exact formalisms and the soundness and completeness proofs can be found in the original papers.
TOWARDS A STRONGER NOTION OF TRANSLATION BETWEEN LOGICS1
The concept of translation between logics was originally introduced in order to prove the consistency of a logic system in terms of the consistency of another logic system. The idea behind this is to interpret (or to encode) a logic into another one. In this survey we address the following question: Which logical properties a (strong) logic translation should preserve? Several approaches to the concept of translation between logics are discussed and analyzed.
Towards a stronger notion of translation between logics
The concept of translation between logics was originally introduced in order to prove the consistency of a logic system in terms of the consistency of another logic system. The idea behind this is to interpret (or to encode) a logic into another one. In this survey we address the following question: Which logical properties a (strong) logic translation should preserve? Several approaches to the concept of translation between logics are discussed and analyzed.
Two basic results on translations between logics
2016
The aim of the present paper is to show two basic results concerning translation between logics:[1] The first result establishes that given two logics S1 and S2 with languages L1 and L2, and a translation F of L1 into L2 that interprets S1 into S2, then, given any intermediate logic S3 between S1 and S2, the same translation F interprets S1 into S3.[2] The second result establishes that the translation F cannot interpret S3 into S2.
Journal of Philosophical Logic, 32, 259-285, 2003
This paper discusses the general problem of translation functions between logics, given in axiomatic form, and in particular, the problem of determining when two such logics are "synonymous" or "translationally equivalent." We discuss a proposed formal definition of translational equivalence, show why it is reasonable, and also discuss its relation to earlier definitions in the literature. We also give a simple criterion for showing that two modal logics are not translationally equivalent, and apply this to well-known examples. Some philosophical morals are drawn concerning the possibility of having two logical systems that are "empirically distinct" but are both translationally equivalent to a common logic.
Mathematical Logic and Many-Sorted Languages
2022
This short text contains a rigorous introduction to mathematical logic: Many-sorted Languages are the natural way of formulating mathematical theories. E.g. Category Theory uses two kinds (= sorts) of objects: Sets and classes. Incidence Geometry uses points and lines, and so on. We present a formalism of deduction in many-sorted languages and introduce the notion of models. The main results are the correctness and completeness theorems, the compactness theorem and the theorem of Lindenbaum, all in the many-sorted case. This introduction follows the lines of U. Felgner of the Univerisity of Tübingen, who found this extremely elegant formalism. The autor's contribution was to expand this formalism from one-sorted logic to many-sorted logic.
New dimensions on translations between logics
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.
A collection of papers from Paul Hertz to Dov Gabbay - through Tarski, Gödel, Kripke - giving a general perspective about logical systems. These papers discuss questions such as the relativity and nature of logic, present tools such as consequence operators and combinations of logics, prove theorems such as translations between logics, investigate the domain of validity and application of fundamental results such as compactness and completeness. Each of these papers is presented by a specialist explaining its context, import and influence.
2021
In order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular 'justifications' are demonstrably without epistemic value (sec. 1). Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus nonclassical logics. In this paper the question is answered positively, based on meaningpreserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic and quantum logics, can be translated into classical logic by introducing additional intensional operators into the language (sec. 2-5). Based on this result it is argued that classical logic is representationally optimal. In sec. 6 it is investigated whether non-classical logics can be likewise representationally optimal. The answer is predominantly negative but partially positive. Nevertheless the situation is not symmetric, because classical logic has important ceteris paribus advantages as a unifying metalogic.
Journal of Philosophical Logic
In order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular ‘justifications’ are demonstrably without epistemic value (sec. 1). Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus non-classical logics. In this paper the question is answered positively, based on meaning-preserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic and quantum logics, can be translated into classical logic by introducing additional intensional operators into the language (sec. 2–5). Based on this result it is argued that classical logic is representationally optimal. In sec. 6 it is investigated whether non-classical logics can be likewise representationally optimal. The answer is predominantly negative but partially positive. Never...