The text of the treatise “The Propositional Logic of Boethius” was finished in 1939. Prof. Jan Lukasiewicz wished at that time to issue it in the second volume of “Collectanea Logica”; as a result of political events, he was not able to carry out his plan. In 1938, I published an article in “Erkenntnis” entitled “AUS- sagenlogik im Mittelalter”; this article included the contents of a paper which I read to the International Congress for the Unity of Science in (...) Cambridge, England, in 1938 (Cf. Erkenntnis, vol. 7, pp. 160-168). The subject matter of this paper touched upon that of the above-mentioned treatise. Recently an article of Mr. Rend van den Driessche, “Sur le ‘de syllogismo hypothetico’ de Boece”, was published in the journal “methodos” (vol. I, no. 3). Mr. van den Driessche referred in this article to the article on propositional logic in the middle ages, which had appeared in “Erkenntnis”. This reminded me of my yet-unpublished treatise on the propositional logic of Boethius. (shrink)

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient (...) class='Hi'>logic texts. A renaissance in ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (shrink)

ELEMENTARY LOGIC GR. C. MOISIL Institute of Mathematics, Rumanian Academy, Bucharest, Rumania 1. We shall consider a typified logic of propositions. ...

Part of rethinking philosophy today, the author believes, is to rethink our logical concepts. The author questions the ontological existence of the proposition as the content of sentential utterances—written or spoken—as it was originally proposed by John Searle. While a performative is an utterance where the speaker not only utters a sentential or illocutionary content such as a statement, but also performs the illocutionary force such as the act of stating, the author reasserts John Austin’s constative as the general (...) label (genus) of specific utterances (species) that can be rendered true or false such as a statement, assertion, description, and prediction. In the remainder of the paper, the author tries to show that it is a category mistake for someone to assert a statement or to state an assertion. (shrink)

A compilation of selected papers presented at the 1989 Charles S. Pierce International Congress Interest in Charles Sanders Peirce (1839-1914) is today worldwide. Ernest Nagel of Columbia University wrote in 1959 that "there is a fair consensus among historians of ideas that Charles Sanders Peirce remains the most original, versatile, and comprehensive philosophical mind this country has yet produced." The breadth of topics discussed in the present volume suggests that this is as true today as it was in 1959. Papers (...) concerning Peirce's philosophy of science were given at the Harvard Congress by representatives from Italy, France, Sweden, Finland, Korea, India, Denmark, Greece, Brazil, Belgium, Spain, Germany, and the United States. The Charles S. Peirce Sesquicentennial International Congress opened at Harvard University on September 5, 1989, and concluded on the 10th—Peirce's birthday. The Congress was host to approximately 450 scholars from 26 different nations. The present volume is a compilation of selected papers presented at that Congress. The philosophy of science and its logic are themes in the work of Charles Peirce that have been of greatest interest to scholars. Peirce was himself a physical scientist. He worked as an assistant at the Harvard Astronomical Observatory from 1869 to 1872 and made a series of astronomical observations there from 1872 to 1875. Solon I. Bailey says of these observations, "The first attempt at the Harvard Observatory to determine the form of the Milky Way, or the galactic system, was made by Charles S. Peirce....The investigation was of a pioneer nature, founded on scant data." Peirce also made major contributions in fields as diverse as mathematical logic and psychology. C. I. Lewis has remarked that "the head and font of mathematical logic are found in the calculus of propositional functions as developed by Peirce and Schroeder." Peirce subsequently invented, almost from whole cloth, semiotics - the science of the meaning of signs. Ogden and Richards, the British critics, say that "by far the most elaborate and determined attempt to give an account of signs and their meanings is that of the American logician C. S. Peirce, from whom William James took the idea and the term Pragmatism, and whose Algebra of Dyadic Relations was developed by Schroeder.". (shrink)

'Saṃvāda' is the live report of a dialogue between two philosophical traditions, the Indian and the western, transcribed and edited from the tapes of a week-long seminar held at Pune in 1983. The central issue whether one need postulate 'propositions' as entities to account for our understanding of sentences which are false or those whose truth and faksity is not yet known. The Indian answer is a definitive 'No.'.

It is shown that the Nominalism of much of Modern Logic is what has given rise to many of its problems, especially The Liar Paradox. Shifting to a Realist Logic, in which ‘that’-clauses have a central place, overcomes these problems. The move involved, from the study of mentioned sentences to the employment of ‘that’-clauses, reveals the indexicality of referring phrases, and it is that which enables an escape from The Liar. But also it is shown that no parallel (...) paradox is obtainable with the propositions that ‘that’-clauses designate. Several theoretical issues, however, then need to be faced about what ‘that’-clauses designate, primarily what has been called the problem of ‘The Unity of the Proposition’. The term ‘proposition’, it turns out, is not always appropriate for what ‘that’-clauses designate, so a richer language is introduced to handle all cases. But the problem of the unity of what is designated arises in all cases, and is solved by noting a parallel with Russell’s treatment of individuals and descriptive phrases. (shrink)

Why do logicians develop so many different philosophical logics? All their aims focus on the same question--”What is logic?” Whether they have said it is the aim question which they want answer or not when they are doing logics, this is the presumed motivation for all studies of logics. In other words, the reason for logicians to do logics is try to answer what logic is. This kind of conceptual analysis on logic is the main problem style (...) to be asked in Universal Logic, such as “What is classical propositional logic?”, “What is many‐valued logic?”, “What is paraconsistent logic?”, etc. In this paper, we discuss one of Béziau’s paradoxes, Translation Paradox in Logical Translation, due to this kind of conceptual analysis on logic. Universal Logic is not a new logic, according to Béziau, it is a general theory of logical structures analogy to Universal Algebra, even if we could see it as a new logical activity. It is not to find out a ‘universal’ logic to cover all aims of logical researches. And it is definitely not in the spirit of logical monism, that is there is only one true logic. Develop Universal Logic about 15 years, besides its own motivation and purposes, it increasingly induces to two main stream problems, one is Logical Translation and the other is Combination of Logics which are two new realms in logic research. We find it is an easier way to start with Béziau’s Translation Paradox to get involved into Translation of Logics. (shrink)

The aim of the paper is to sketch a principle of individuation that is intended to serve the Fregean notion of a proposition, a notion I take for granted. A salient feature of Fregean propositions, i.e. complexes of modes of presentation of objects, is that they are fine-grained items, so fine-grained that even synonymous sentences might express different Fregean propositions. My starting point is the principle labelled by Gareth Evans the Intuitive Criterion of Difference for Thoughts, which states that (...) it is impossible coherently to take different mental attitudes to the same proposition. As a logical truth, this is a synchronic principle, the application of which is restricted to attitudes held at a single time. I argue that such a restriction might be reasonably lifted and, on the basis of an adequate notion of attitude-retention, I propose an admissible diachronic extension of the principle. (shrink)

Explorationism (see www.bu.edu/wcp/Papers/Logi/LogiFaus.htm, WCP XX, “Conflict without Contradiction”) is a perspective concerning human knowledge: as yet, our ignorance of the Real World remains great. With this perspective, all our knowledge is so far only partial and tentative. Evidence Logic (EL) (see “The Concept of Evidence”, INTER. JOURNAL OF INTELLIGENT SYSTEMS 15 (2000), 477‐493) provides an example of a reasonable Base Logic for Explorationism:EL provides machinery for the representation and processing of gradational evidential predications. Syntactically, for any evidence level (...) e, for a proposition symbol P, Pc:e asserts that there is level e confirmatory evidence regarding P, while Pr:e asserts level e refutatory evidence (n‐ary predications, for n>0, are handled similarly). Semantically, EL has similarly enriched model spaces. The Boolean sentence algebras of the variety of EL languages, varying across stipulated families of predicate and functions symbols, have been analyzed, and EL is sound and complete. Belief Systems, we will argue, are all unnecessary: in science, as well as in all broader domains of human relations and activity, it is always sufficient to have simply commitments, which entail no assertion of Truth but rather simply entail agreed‐upon consequent actions. (Agent A believes a sentence S if A asserts S is True although A does not know (have absolute evidence) that S is True.) We will further seek to explore ways in which such a perspective may help in engendering more enhanced discourse, less absolutist and shrill advocacy and violence, and more rationality, in the Global Village of the new century. (shrink)

As it is well known, in the forties of the past century, Curry proved that in any logic S closed under Modus Ponens, uniform substitution of propositional variables and the Contraction Law, the naïve Comprehension axiom trivializes S in the sense that all propositions are derivable in S plus CA. Not less known is the fact that, ever since Curry published his proof, theses and rules weaker than W have been shown to cause the same effect as W causes. (...) Among these, the Contraction rule or the Modus Ponens axiom, for example, are to be noted. But, moreover, as Brady has proved, even the Generalized Modus Ponens axiom or the Generalized Contraction rule give rise to “Curry’s Paradox” under the same circumstances as W does. In some previous work by us, “weak relevant model structures” are defined on “weak relevant matrices” by generalizing Brady’s model structure MCL built upon Meyer’s Crystal matrix CL. We have proved that wr-ms only verify logics with the “depth relevance condition”. The aim of this paper is to show how to falsify gMPa and gRW in certain wr-ms. In particular, it will be shown that gMPa is falisfied in any wr-ms and gRW in any wr-ms verifying Routley and Meyer’s basic positive logic B+. (shrink)

There is an old and persistent tendency in the human mind to conceive of the causal relation as rationally explanatory, and therefore to assimilate it, vaguely or explicitly, to the logical relations of inclusion, implication, or equivalence. That ‘ there cannot be more in the effect than there is in the cause’ is one of the propositions that men have been readiest to accept as axiomatic; a cause, it has been supposed, does not ‘ account for ‘ its effect, unless (...) the effect is a thing which the eye of reason could somehow discern in the cause, upon a sufficiently thorough analysis. This antipathy to the notion of an absolute epigenesis has left its effect, unless the effect is a thing which the eye of reason could somehow discern in the cause, upon a sufficiently thorough analysis. This antipathy to the notion of an absolute epigenesis has left its mark deep and wide upon the history of thought; it appears, indeed, at the very outset of Western speculation in the struggles of the physiologers with the supposed difficulty of admitting qualitative change. Two of the later phases of what may be named the preformationist assumption about causality may pertinently be remembered here. (shrink)

The logical problem of evil centers on the apparent inconsistency of the following two propositions: God is omnipotent, omniscient, and wholly good, and There is evil in the world. This is the problem that Alvin Plantinga takes to task in his celebrated response to the problem of evil. Plantinga denies that and are inconsistent, arguing that J.L. Mackie's principle - that there are no limits to what an omnipotent thing can do - is false. We challenge Plantinga, and defend Mackie's (...) view. (shrink)

This article focuses on several important but obscure concepts in Tractatus Logico-Philosophicus. In order to clarify the concept of “object”, I compare it with “atom”. The analysis of the two concepts explains two important questions which have confused Wittgenstein’s reviewers for long: why is the world not the totality of things? Is object substance? “Logical space” is an important concept in Tractatus Logico-Philosophicus, although it only appears several times. If a proposition serves as the coordinates in the logical space, (...) what are the coordinate axes? Is a possible world a point in the logical space, or a set of points in it? Based on symbolic transformations, I suggest that possible worlds serve as coordinate axes in the logical space. The concept of “possible worlds” contradicts with “necessity”. In Wittgenstein’s theory, all possible worlds are “accessible” to one another. This is why Wittgenstein fails to cope with the conflict between “possibility” and “necessity”. (shrink)

We study here equiformity, the standard identity criterion for sentences. This notion was put forward by Lesniewski, mentioned by Tarski and defined explicitly by Presburger. At the practical level this criterion seems workable but if the notion of sentence is taken as a fundamental basis for logic and mathematics, it seems that this principle cannot be maintained without vicious circle. It seems also that equiformity has some semantical features ; maybe this is not so clear for individual signs but (...) sentences are often considered as meaningful combinations of signs. If meaning has to play a role, we are thus maybe in no better position than when dealing with identity criterion for propositions. In formal logic, one speaks rather about well-formed formulas, but closed formulas are called sentences because they are meaningful in the sense that they can be true or false. Formulas look better like mathematical objects than material inscriptions and equiformity does not seem to apply to them. Various congruencies can be considered as identities between formulas and in particular "to have the same logical form". One can say that the objects of study of logic are rather logical forms than sentences conceived as material inscriptions. (shrink)

The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valuation in quantum mechanics as exemplified, in particular, by Kochen-Specker’s theorem. In the present study, this is realized by representing categorically the global structure of a quantum algebra of events in terms of sheaves of local Boolean frames forming Boolean localization functors. The category of sheaves is a topos providing the possibility of applying the powerful logical classification methodology of topos theory with (...) reference to the quantum world. In particular, we show that the topos-theoretic representation scheme of quantum event algebras by means of Boolean localization functors incorporates an object of truth values, which constitutes the appropriate tool for the definition of quantum truth-value assignments to propositions describing the behavior of quantum systems. Effectively, this scheme induces a contextualist account of truth in the quantum domain of discourse. The philosophical implications of the resulting account are analyzed. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to state the totality of facts of nature. (shrink)

First, ‘Is Justified True Belief Knowledge’ is imprecise but Gettier is explicit that ‘know’ is analysed as the definiendum is ‘S knows that P’. Second, Gettier does not misrepresent as Plato’s definition as the expressions used are ‘Plato considers’ and ‘seems to accept’. Third, Gettier is not mistaken to apply Plato’s definition to propositions since propositional knowledge is a species of Plato’s definition. Fourth, for Plato true belief temporally precedes an account. ‘Jones owns a Ford’ is never a true opinion, (...) hence no account for it can be given. The counterexample is reconstructed with temporality built into it. Fifth, Gettier does not fail to establish the equivalence of ‘believe’, ‘accepts’ and ‘sure’ in the three versions as this is implicitly established in the shifts made in the paper. Sixth, ‘entails’ logically is used only when the entailing proposition is true, but in the counterexamples a false proposition is taken to imply a true one. ‘Entail’ is to be taken in the ordinary sense of implies. Seventh, in Case I, the implication is preserved with the proper representation: → [Gy ↔ ], which implies { → [Gy ↔ ]}’. Eighth, the counterexample is reworked to avoid the objection that justification for p and justification for q may not be sufficient justification for ‘p & q’. (shrink)

Wittgenstein’s early philosophy of language is shaped by his attention to Parmenides’ paradox of false propositions and the problem of the unity of the proposition. Wittgenstein (dis)solves these two (pseudo)problems through his discussion of the “internal pictorial relation” between propositions and states of affairs, which is an artifact of language and the world being “constructed according to a common logical pattern” (TLP 4.014). After examining these issues, I argue that this treatment points to a further problem, namely, the question (...) of the agency responsible for this construction, and in the paper I briefly explore this byconsidering a parallel set of problems as they arise for Kant. For Kant, the question is what guarantees that nature conforms to our concepts and their logic, which he answers through his account of the normative authority we exercise in our cognitive activities. Wittgenstein, I argue, faces a parallel question, which he fails to face in his early work, and only begins to address in his later work on rule-following. (shrink)