One of the epistemological results emerging from this initial study is that the different forms of co-relational inference, known in the Islamic jurisprudence as qiyās, represent an innovative and sophisticated form of reasoning that not only provides new epistemological insights into legal reasoning in general but also furnishes a fine-grained pattern for parallel reasoning which can be deployed in a wide range of problem-solving contexts and does not seem to reduce to the standard forms of analogical argumentation studied in contemporary (...) philosophy of science. However, in the present paper we will only discuss the case of so-called co-relational inferences of the occasioning factor and only in the context of Islamic jurisprudence. (shrink)

RésuméCette première étude permet notamment de dégager ce résultat épistémologique: les différentes formes d’“inférence co-relationnelle” connues dans la jurisprudence islamique sous le nom de qiyās représentent une forme innovante et sophistiquée de raisonnement qui permet non seulement d'avoir une conception épistémologique plus claire du raisonnement légal en général, mais aussi de produire une mécanique bien huilée pour le “raisonnement parallèle”; cette mécanique du “raisonnement parallèle” peut être déployée selon un large spectre dans différents cadres de résolution de problèmes et ne (...) semble pas se réduire aux formes standard de l'argumentation analogique étudiées en philosophie des sciences contemporaine. Nous n'aborderons cependant dans la présente étude que le cas de la dénommée “inférence co-relationnelle du facteur occasionnel”, et ce seulement dans le contexte de la jurisprudence islamique. (shrink)

In its first meaning, a logic is a collection of closely related artificial languages. There are certain languages called first‐order languages, and together they form first‐order logic. In the same spirit, there are several closely related languages called modal languages, and together they form modal logic. Likewise second‐order logic, deontic logic and so forth.

Though deceptively simple and plausible on the face of it, Craig's interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, (...) especially of many-sorted interpolation theorems. Attention is also paid to methodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for "reasonable" logics extending first-order logic within the framework of abstract model theory. (shrink)

Though deceptively simple and plausible on the face of it, Craig's interpolation theorem has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted interpolation (...) theorems. Attention is also paid to methodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for "reasonable" logics extending first-order logic within the framework of abstract model theory. (shrink)

Given some links between Lyndon’s Interpolation Theorem, term distribution, and Sommers and Englebretsen’s logic, in this contribution we attempt to capture a sense of interpolation for Sommers and Englebretsen’s Term Functor Logic. In order to reach this goal we first expound the basics of Term Functor Logic, together with a sense of term distribution, and then we offer a proof of our main contribution.

This paper is a brief history of natural logic at the interface of logic, linguistics, and nowadays also other disciplines. It merely summarizes some facts that deserve to be common knowledge.

Syllogistics reduces to only two rules of inference: monotonicity and symmetry, plus a third if one wants to take existential import into account. We give an implementation that uses only the monotonicity and symmetry rules, with an addendum for the treatment of existential import. Soundness follows from the monotonicity properties and symmetry properties of the Aristotelean quantifiers, while completeness for syllogistic theory is proved by direct inspection of the valid syllogisms. Next, the valid syllogisms are decomposed in terms of the (...) rules they involve. The implementation uses Haskell [8], and is given in ‘literate programming’ style [9]. (shrink)

We implement the extension of the logical consequence relation to a partial order ≤ on arbitary types built from e (entities) and t (Booleans) that was given in [1], and the definition of monotonicity preserving and monotonicity reversing functions in terms of ≤. Next, we present a new algorithm for polarity marking, and implement this for a particular fragment of syntax. Finally, we list the reseach agenda that these definitions and this algorithm suggest. The implementations use Haskell [8], and are (...) given in ‘literate programming’ style [9]. (shrink)