In the eleventh of his Antiquarian Letters, Gotthold Ephraim Lessing discusses a phrase from Lucian's description of the painting by Zeuxis called A Family of Centaurs: ‘at the top of the painting a centaur is leaning down as if from an observation point, smiling’. ‘This as if from an observation point, Lessing notes, obviously implies that Lucian himself was uncertain whether this figure was positioned further back, or was at the same time on higher ground. We need to recognize the (...) logic of ancient bas-reliefs where figures further to the back look over those at the front, not because they are actually positioned above them but because they are meant to appear as if standing behind.'1 · 1. The passage is in George Lessing, Briefe, antiquarischen Inhalts, p. 81. Panofsky's discussion does not note that the original text of Lucian makes clear that what is described is a copy of the original painting. This means that some of the issues of misunderstanding situated by Lessing and Panofsky in Lucian's court may in principle be attributable to the copyist. This makes no difference to the conceptual thrust of Panofsky's case.—Trans. (shrink)

How much violence can a society expect its members to accept? A comparison between the language theories of Julia Kristeva and Jacques Lacan is the starting point for answering this question. A look at the early stages of language acquisition exposes the sacrificial logic of patriarchal society. Are those forces that restrict the individual to be conceived in a martial imagery of castration or is it possible that an existing society critically questions those points of socialization that leave their members (...) in a state of homelessness? The following considerations should help to distinguish between unavoidable and avoidable forms of violence. (shrink)

With special reference to the Tupi-speaking Aweti people, this article reconsiders the nature of Xinguan pacifism in an analysis of sorcery and its relation to war in the Upper Xingu region of Brazil. It is argued that the mechanism that keeps violence there under control is probably less the result of an applied pacifist ideology—that is, rejection of war as the socius’s generative matrix—than the effect of a specific conception of knowledge. It is through the Xinguans’ refusal of the idea (...) of singular truth, rather than through their rejection of war, that their logic is “good to think” through the question of peace. This article divides roughly into two parts, the first concentrating on Xinguan sorcery, and the second on their knowledge politics. (shrink)

A translation of formulas in a language L1 to formulas in a language L2 is a mapping which preserves the parameters and commutes with the substitution prefix, the propositional connectives and the quantifiers. Every translation generates a corresponding transformation of theories in L1 to theories in L2. We formulate the equiconsistency problem for such transformations and propose a variant of its solution. First, for a transformation F we find the least theory A in L1 such that its inclusion (...) in a theory T guarantees equiconsistency of F and F, then we propose axiomatizations of A for some F's. (shrink)

It is shown that there exists no grammatical translation into classical (propositional) logic of the modal logics, nor of intuitionistic logic and of the relatedness and dependence logics, as defined in Richard L. Epstein's bookThe Semantic foundations of logic. In the book the result is proved for translations without parameters.Classical propositional logicPC can be translated into other logics. Usually the grammatical structure of propositions is preserved, in the sense of the following definition.

In this article we discuss Zhang Shizhao’s famous essay “Lun Fanyi Mingyi〈論翻譯名義〉” (On the Meanings of Names in Translation), which played a key role in establishing what is now the standard translation of ‘logic’ into Chinese, sketching the historical context and analyzing and evaluating the argument he gives for providing a phonemic rather than semantic translation.

Despite its importance, the work is rather secondary in the sense that it relies, for the most part, on the Buddhist logical legacy. The first extant commentary is the Nyayavatara-vivrti of Siddharsigani.

Following a proposal by Kooi and Tamminga, we introduce a conservative translation manual for every four-valued truth-functional propositional logic into a modal logic. However, the application of this translation does not preserve the intuitive reading of the truth-values for every four-valued logic. In order to solve this problem, we modify the translation manual and prove its conservativity by exploiting the method of generalized truth-values.

In the second half of Book One of the Prior Analytics, Aristotle reflects on the application of the formalized logic has developed in the first half, focusing particularly on the non-modal or assertoric syllogistic developed in the first seven chapters. These reflections lead Alexander of Aphrodisias, who was a great exponent of Aristotelianism in the late second century, to explain and sometimes argue against subsequent developments of Aristotle's logic and alternatives and objections to it, ideas associated mainly with his colleague (...) Theophrastus and with the Stoics. The other main topic of this part of the Prior Analytics is the specification of a method for discovering true premises needed to prove a given proposition. Aristotle's presentation is sometimes difficult to follow, and Alexander's discussion is extremely helpful to the uninitiated reader. In his commentary on the final chapter translated in this volume, Alexander provides an insightful account of Aristotle's criticism of Plato's method of division. (shrink)

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. 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. 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. (shrink)

In this paper we prove a bounded translation of intuitionistic propositional logic into basic propositional logic. Our new theorem, compared with the translation theorem in [1], has the advantage that it gives an effective bound on the translation, depending on the complexity of formulas.

In this paper we present an approach to conditional reasoning tasks based on two main ideas. The first idea is that, in contrast with what is usually assumed, an ‘if… then…’ sentence is not an adequate translation in natural language of a logical formula containing a material implication as its principal operator. The second idea is that when subjects are required to check the validity of a sentence in a task, their inferences are not driven uniquely by the content (...) of the sentence, but also by other information embedded in the task scenario and, eventually, by their knowledge about the topic (i.e. information stored in memory). Data from 30 subjects tested on six different tasks are reported as evidence for our approach. The results show that conditional tasks are significantly more complex only when they are presented with ‘if… then…’ sentences and when the subject cannot rely on any extra information, such as contextual information or knowledge about the topic. (shrink)

The thirteenth-century logician Lambert of Auxerre was well known for his Summa Lamberti, or simply Logica, written in the mid-1250s, which became an authoritative textbook on logic in the Western tradition. Our knowledge of medieval logic comes in great part from Lambert's Logica and three other texts: William of Sherwood's Introductiones in logicam, Peter of Spain's Tractatus, and Roger Bacon's Summulae dialectics. Of the four, Lambert's work is the best example of question-summas that proceed principally by asking and answering questions (...) on the subject matter. Thomas S. Maloney's translation of Logica, the only complete translation of this work in any language, is a milestone in the study of medieval logic. More than simply a translation, Maloney's project is a critical, comprehensive study of Lambert's logic situated in the context of his contemporaries and predecessors. As such, it offers a wealth of annotation and commentary. The lengthy introduction and extensive notes to the text explain the origin, theoretical context, and intricacies of the text and its doctrines. Maloney also addresses the disputed issues of authorship, date, and place of publication of the Summa Lamberti and makes available to the English-only audience the French, German, and Italian secondary sources--all translated--that are needed to enter the discussion. "Thomas S. Maloney fully commands the primary and secondary sources necessary to elucidate Lambert's Logica. An expert on Roger Bacon's philosophy, he demonstrates a rare proficiency in medieval Latin and scholastic logic. His references to sources from the ancient (Aristotle and Boethius) and medieval worlds are apposite, perspicuous, and useful. The volume's presentation with an appropriate introduction and commentary in the endnotes will no doubt establish it as an indispensable resource for scholars in the twenty-first century." --Alan Perreiah, University of Kentucky. (shrink)

This paper deals with the translation of first order formulas to predicate S5 formulas. This translation does not bring the first order formula itself to a modal system, but modal interpretation of the first order formula can be given by the translation. Every formula can be translated, and the additional condition such as formula's having only one variable, or having both world domain and individual domain is not required. I introduce an indexical predicate 'E' for the (...) class='Hi'>translation. The meaning that 'E(a)' is true is 'this world is 'a' '. Because of this meaning, I call 'E' an indexical predicate. 'E' plays an important role for the translation. In addition that the modal formulas can be translated into first order formulas, we can conclude that the first order logic and modal predicate logic isintertranslatable. (shrink)

The main objective o f this descriptive paper is to present the general notion of translation between logical systems as studied by the GTAL research group, as well as its main results, questions, problems and indagations. Logical systems here are defined in the most general sense, as sets endowed with consequence relations; translations between logical systems are characterized as maps which preserve consequence relations (that is, as continuous functions between those sets). In this sense, logics together with translations (...) form a bicomplete category of which topological spaces with topological continuous functions constitute a full subcategory. We also describe other uses of translations in providing new semantics for non-classical logics and in investigating duality between them. An important subclass of translations, the conservative translations, which strongly preserve consequence relations, is introduced and studied. Some specific new examples of translations involving modal logics, many-valued logics, para- consistent logics, intuitionistic and classical logics are also described. (shrink)

In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a (...) corresponding sorted, residuated (multi) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness and the Löwenheim-Skolem property and it opens up new directions for deducing properties of substructural logic systems by establishing appropriate transfer theorems. (shrink)

This book offers the first English translation of the _Quodlibetal Questions _of William of Ockham —reflections on a variety of topics in logic, ontology, natural philosophy, philosophical psychology, moral theory, and theology by one of the preeminent thinkers of the Middle Ages. It is based on the recent critical edition of Ockham’s theological and philosophical works.

Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.

This volume contains selections of Ockham's philosophical writings which give a balanced introductory view of his work in logic, metaphysics, and ethics. This edition includes textual markings referring readers to appendices containing changes in the Latin text and alterations found in the English translation that have been made necessary by the critical edition of Ockham’s work published after Boehner prepared the original text. The updated bibliography includes the most important scholarship produced since publication of the original edition.

For nearly four centuries Peter of Spain's influential Summaries of Logic was the basis for teaching logic; few university texts were read by more people. This new translation presents the Latin and English on facing pages, and comes with an extensive introduction, chapter-by-chapter analysis, notes, and a full bibliography.

We show that the polynomial translation of the classical propositional normal modal logic S4 into the intuitionistic propositional logic Int from Fernández is incorrect. We give a modified translation and prove its correctness, and provide implementations of both translations to allow others to test our results.

This is the first of a three-volume anthology intended as a companion to The Cambridge History of Later Medieval Philosophy. Volume 1 is concerned with the logic and the philosophy of language, and comprises fifteen important texts on questions of meaning and inference that formed the basis of Medieval philosophy. As far as is practicable, complete works or topically complete segments of larger works have been selected. The editors have provided a full introduction to the volume and detailed introductory headnotes (...) to each text; the volume is also indexed comprehensively. (shrink)

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.

This article employs an internal logics approach, developed in a recent work by Sher Ali Tareen, to the study of the Barelwi school in South Africa. This approach ties metaphysics to practice. Specifically, the article addresses some works of the school’s founder, Ahmad Raza Khan, as translated by South African Islamic scholar Abdul Hadi Qadiri, in the light of this approach. It then extrapolates the insights of this approach to a recent article by Sepetla Molapo, which highlights the importance (...) of appreciating the metaphysical role of ancestors in any academic approach to understanding traditional African worldviews and African self-concept. Taken together, the article suggests that the internal logics approach is helpful in bringing to the surface the crucial, but often obscured, metaphysical presuppositions concerning the nature of time-presuppositions that inform not only the worldview of the object being studied, but, equally, the often different ones that shape the perspective of the researcher. Contribution: This article wishes to underscore the importance of metaphysical considerations in the study of religion, advocating an approach that highlights such considerations and examining some of its broader academic implications. While it specifically focuses on a theological contestation in Islam, it extends the implications of this contestation to the academic study of traditional African worldviews and to world religions more generally. (shrink)

We describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities is shown (...) to be equivalent to the finite identity basis problem for the finite members of a finitely based variety with definable principal congruences. (shrink)

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a (...) large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management. (shrink)

Early in the 1240s the University of Paris hired a recent graduate from Oxford, Roger Bacon by name, to teach the arts and introduce Aristotle to its curriculum. Along with eight sets of questions on Aristotle's natural works and the Metaphysics he claims to have authored another eight books before he returned to Oxford around 1247. Within the prodigious output of this period we find a treatise on logic titled Summulae dialectices, and it is this that is here annotated and (...) presented in translation. The book is unique in several respects. First, there is the breadth of its sources. Not only do we find explicit reference to the usual authors such as Aristotle, Plato, Boethius, Porphyry, Cicero, and Priscian, we also find unexpected reference to Augustine, Bernardus Silvestris, Donatus, Terence, and Themistius, along with mention of the Muslim philosophers Algazel and Ibn Rushd. Second, it is clear that Bacon is drawing on or reacting to an extraordinarily wide variety of medieval sources: Garland the Computist, Hugh of St. Victor, Master Hugo, Hugutius of Pisa, Isidore of Seville, Nicholas of Damas, Nicholas of Paris, Richard of Cornwall, Robert Kilwardby, Robert of Lincoln, and Robert the Englishman. Third, it unexpectedly presents a full-blown treatment of Aristotle's theory of demonstration. And finally, Bacon reveals a highly unorthodox view of the signification of common terms. Bacon, here, takes his students and us deeper into medieval sources and controversy than any of his rivals do. (shrink)

In Sect. 1 it is argued that systems of logic are exceptional, but not a priori necessary. Logics are exceptional because they can neither be demonstrated as valid nor be confirmed by observation without entering a circle, and their motivation based on intuition is unreliable. On the other hand, logics do not express a priori necessities of thinking because alternative non-classical logics have been developed. Section 2 reflects the controversies about four major kinds of non-classical logics—multi-valued, (...) intuitionistic, paraconsistent and quantum logics. Its purpose is to show that there is no particular domain or reason that demands the use of a non-classical logic; the particular reasons given for the non-classical logic can also be handled within classical logic. The result of Sect. 2 is substantiated in Sect. 3, where it is shown (referring to other work) that all four kinds of non-classical logics can be translated into classical logic in a meaning-preserving way. Based on this fact a justification of classical logic is developed in Sect. 4 that is based on its representational optimality. It is pointed out that not many but a few non-classical logics can be likewise representationally optimal. However, the situation is not symmetric: classical logic has ceteris paribus advantages as a unifying metalogic, while non-classical logics can have local simplicity advantages. (shrink)

1 Meaning and Truth Objection to propositions Propositions as information Diffuseness of empirical meaning Propositions dismissed Truth and semantic ascent Tokens and eternal sentences 2 Grammar Grammar by recursion Categories Immanence and transcendence Grammarian's goal reexamined Logical grammar Redundant devices Names and functors Lexicon, particle, and name Criterion of lexicon Time, events, adverbs Attitudes and modality 3 Truth Truth and satisfaction Satisfaction by sequences Tarski's definition of truth Paradox in the object language Resolution in set theory 4 Logical Truth In (...) terms of structure In terms of substitution In terms of models Adequacy of substituteon In terms of proof In terms of grammar 5 The Scope of Logic Affinities of identity Identity reduced Set theory Set theory in sheep's clothing Logic in wolf's clothing Scope of the virtual theory Simulated class quantification Other simulated quantification Annexes 6 Deviant Login Change of logic, change of subject Logic in translation Law of excluded middle Debate about the dichotomy Intuitionism Branched quantifiers Substitutional quantification Its strength 7 The Ground of Logical Truth The semblance of a theory An untenable dualism The place of logic For Further Reading Index. (shrink)

Epistles 10-14 of the Brethren of Purity focus on the foundations of logic, offering a tenth-century interpretation of the classic Aristotelian texts by scholars in Islamic civilization. This is the first critical edition, presenting the original Arabic text with a fully annotated English translation.

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 a logic into another one. In this survey we address the following question: Which logical properties a logic translation should preserve? Several approaches to the concept of translation between logics are discussed and analyzed.

The Ikhwan al-Safa (Brethren of Purity), the anonymous adepts of a tenth-century esoteric fraternity based in Basra and Baghdad, hold an eminent position in the history of science and philosophy in Islam due to the wide reception and assimilation of their monumental encyclopaedia, the Rasa'il Ikhwan al-Safa ( Epistles of the Brethren of Purity ). This compendium contains fifty-two epistles offering synoptic accounts of the classical sciences and philosophies of the age; divided into four classificatory parts, it treats themes in (...) mathematics, logic, natural philosophy, psychology, metaphysics, and theology, in addition to didactic fables. The Rasa'il constitutes a paradigmatic legacy in the canonization of philosophy and the sciences in mediaeval Islamic civilization, as well as having shown a permeating influence in Western culture. The present volume is the second of this definitive series, consisting of the very first critical edition of the Rasa'il in its original Arabic, complete with the first fully annotated English translation. Prepared by Professor Carmela Baffioni, Epistles 10-14 comprise the foundations of logic, which remained a fundamental component in pedagogy until the twentieth century. The Ikhwan treat the Isagoge and the larger part of the Organon , both of which were circulating through the Islamic world at that time, as they set about detailing the ten categories of existents, the five predicables, and other such commonplaces of Aristotelian logic, including his seminal method of syllogistic inference. With the claim that logic is the noblest of man's arts, and man the noblest of creatures, the Ikhwan cast Aristotelian tropes in a spiritual light. (shrink)

Based on the notion of time translation, we develop a formalism to deal with the logic of quantum properties at different times. In our formalism it is possible to enlarge the usual notion of context to include composed properties involving properties at different times. We compare our results with the theory of consistent histories.