Apparatus Of Logic Research Articles

Response input interface

A circuit for interfacing experimental response signals with logic apparatus that controls experimental events is described. The unit shapes and holds the signal until its presence has been detected by the controlling program.

Problem of identity transformations of nondeterministic program shcemes in algorithmic algebra systems

Ever-increasing attention to the investigation of nondeterministic processes that generalize deterministic processes because of elimination of the rigidity of execution inherent in the latter is characteristic of modern programming. Nondeterminism reflects the crux of the sampling problem in tasks allowing multivaluedness of solutions such as automation of the proofs of theorems, syntactical and semantical models of languages, distribution of resources in multiprocessor computer systems and time-sharing systems, etc. Nondeterminism is used in proving any assertions about programs by means of the apparatus of program logic that occur and are developed on the junction of mathematical logic and the theory of programming. The systems of algorithmic algebras (SAA) proposed by Glushkov (1956) and adequate conceptions of structural programming are the fundamental basis of propositional program logic. In the interests of orientating the SAA apparatus in the formalization of parallel calculations, their signature was expanded by including filtration and parallelism operations. The algebraic systems obtained in this manner were called modified SAA. Identity problems and axiomatization by using a single deduction rule, the traditional formulation, were solved by Tseitlin (1974) for s(o)-algebras modified by SAA with closed logic conditions. In this paper, the mentioned results are extended to the case of s(h)more » algebras, SAA with closed logic conditions, in whose signature the parallelism operations are replaced by nondeterministic disjunction operators. 9 references.« less

A Logical Theory of Objects

Many philosophers have attempted to offer a logical theory of objects, employing different techniques. Thus R. Carnap (1928)1 tried to reconstruct logically the world by using the modern symbolic logic (the theory of relations), while N. Goodman (1951)2 constructed the world with the help of the calculus of individuals or the logic of part-whole relations. W. V. Quine helped to steer the attention toward the question of ontological commitment (to be means to be a value of a bound variable of a system) and toward a theory of objects produced by a logical analysis of natural languages.3 Recently, there have been controversial discussions concerning the metaphysical and logical aspects of objects in actual and possible worlds (based on the acceptance or nonacceptance of modal logics) as well as new contributions to the theory of nonexistent objects, rekindling the old Russell Meinong controversy.4 It seems that even the proposed theories of events, such as that of R. M. Martin,5 maintain the great significance of the notion of an object. Then there is a plethora of the so-called intensional objects, such as the alleged objects of propositional attitudes, the story of which is tied with the names of Frege, Russell, Carnap and many others. Is it possible to give an all-embracing theory of objects which would satisfy all the parties involved? Although such possibility may be very questionable, the instrumental role of logic for the development of the metaphysical (ontological) theories of ob jects is indisputable. Two factors are, however, involved in the constitution of a logical theory of objects: (i) the basis or the given data from which the theory starts; (ii) the logical apparatus employed, i.e., the logical axioms and rules of inference, just as the relevant semantical rules which are formulated in the metalanguage (with regard to the language used). While the logical ap paratus is usually very clear, at least in the case of the formally well construed theories, the given basis of a metaphysical theory ?as we may call it?might contain many tacit assumptions which would need a lot of clarification. It seems that philosophers fight more about the presuppositions and assertions of their systems than about the logic involved.61 would like to discuss the interplay between the above two factors, starting with the realm of the given objects. As to the second factor?the logical apparatus involved? the contributions of first-order predicate logic (PL) and of the calculus of in dividuals (CI) will be considered.

Comparative Philosophy as Applied to the Concept of Natural Law

The most valuable trends in contemporary philosophy were inspired by the will to make philosophy a science. Under the general label of Analytic philosophy they succeeded in introducing new methods using the extensive development of logic. A paradigm is afforded by Russell (1905) when he ex plains inexistent objects away by analyzing definite descriptions. The first analytic programmes were ambitious enough to consider the old philosophical problems either as solvable (Russell, 1902, although his dis covery of the antinomies shattered the very foundations of his construction) or as dispelled with the confusions that originated them (Carnap's material versus formal use of language). Self-criticism and foreign objections forced changes into these concep tions. Do the consequences brought forth by Russell's versatile dogmatism differ from scepticism? As to Carnap, despite his ceaseless modification and extension of his methods, the gap was never filled between his aim and his achievement. Not to speak of some dilution of logical analysis into grammatical reconstructions of natural languages, the introduction of scientific method in philosophy led to a philosophical overturning. Ontology reappeared bound with the use of logical apparatus. Moreover different ontologies were shown to result from using different analytical tools and positivist neutrality was finally replaced by a hierarchy or even a mere multiplicity of ontological commitments.

Applying simple theory of types to data bases

Whereas the logical apparatus used for formulation and solution of problems connected with the data base systems is mostly a 1st order logic, here a modified version (“T-system”) of Church's simple theory of types is used. It is shown that the T-system 1. (a) is a more adequate tool for analyzing natural language, which may be important when one builds up an abstract model of a fragment of reality and when one creates a suitable philosophy of data base systems; 2. (b) has greater expressive power than the 1st order systems, which enables, e.g. to distinguish between queries according to what is the structure of the required answer, and moreover, not to be restricted by the well-known limitations of the 1st order logics; 3. (c) is based on the notion of function as on the most fundamental notion, which makes it possible to deal more directly with functional dependencies and to exploit more universally the key importance of the attribute concept; 4. (d) by using the concept of construction offers a good tool for E-C mapping and for forming query languages that are more universal than the current ones. The analysis of these properties of the T-system is accompanied by examples from the area of the data base systems theories; especially, a comparison with Codd's domain relational calculus and with Chen's and Pirotte's approaches is made. No problem of implementation is touched in this paper.

Formation of the Elements of Formal Logic in Preschool-Age Children

In contemporary psychology one of the most sophisticated theories of children's thought is, of course, that of Jean Piaget. The fundamental thesis of Piaget's ideas is that there is a close affinity between the psychological structures of intelligence and the formal constructs of mathematical logic, and that this justifies the use of the apparatus of logic as a means of analyzing the thought of a child. This logical interpretation of psychological data enables Piaget to represent the development of intelligence as an autonomous hierarchy of structures built up according to that hierarchy's own internal laws, so that no external influence (including any form of learning) can alter the sequence in which these structures appear.

Science and the Fine Arts: Reflections of Platonic Idealism and of Aristotelian Naturalism

The scientific explanation of natural phenomena by means of postulatory-deductive theoretical structures originated in the classical Greek world about 600 B. C. In the 4th century B. C. philosophical analysis of knowledge identified three essential elements for science: the idealistic view that concepts are real and that things are unreal imitations of concepts (associated with Platonic philosophy); the naturalistic view that things are real and concepts are intellectual constructs, important but not real (associated with Aristotelian philosophy); the logical apparatus essential for the intellectual manipulation of scientific knowledge (transmitted by the writings on logic of Aristotle). Only where these three elements have been available has science occurred. The visual fine arts faithfully follow later ascendancies of the two opposing views of reality: the highly idealistic early European Christian culture produced non-naturalistic art, extremely idealized; the dominance of the Aristotelian philosophy in the High Middle Ages was accompanied by a revival of naturalism in the arts; the Renaissance, in which both philosophical views were again available, resulted in highly naturalistic art used to portray idealistic conceptions. It was also during the Renaissance that, for the first time since Classical Greek Antiquity, the three essential elements for science were brought together: Platonic idealism, Aristotelian naturalism and logic. It is with this event that what is called modern science began.

Nonstandard set Theory

Infinitely small and infinitely large quantities were systematically introduced into mathematics with the invention of calculus by Newton and Leibniz. The use of such quantities, however, was accompanied by logical contradictions, which mathematicians of the seventeenth and eighteenth centuries were unable to resolve. Although the method of infinitesimals generally yielded correct results, no one ever succeeded in formulating a precise, noncontradictory set of rules governing these objects; and infinitesimal quantities were gradually displaced (at least, in pure mathematics) by the familiar e-d calculus. A mathematically sound model of infinitely small and infinitely large objects became possible only after advances in mathematical logic in the twentieth century. Nonstandard Analysis, developed by A. Robinson in 1960, not only provided foundations for the calculus of infinitesimals in the classical spirit but also enabled mathematicians to use objects in ways that could not be attempted on the basis of vague, intuitive understanding alone. Since then, interesting applications were found in various branches of mathematics, mathematical physics, and economics. Robinson's exposition in [10] and its subsequent simplifications unfortunately involve the cumbersome apparatus of mathematical logic. Our aim here is to present methods of Nonstandard Analysis at a level of formalism customary in other branches of mathematics. We view nonstandard objects as ideal, imaginary elements adjoined to the universe of the standard mathematics and formulate a few simple and reasonably intuitive principles governing their behavior. We then show, on examples selected to illustrate a variety of nonstandard constructions, how nonstandard mathematics can be developed from these principles. The basic framework for Nonstandard Analysis is presented in ?? 1-3; this system was introduced in [3], where its relative consistency with respect to the Zermelo-Fraenkel set theory is shown. We examine the real line and some concepts of general topology from our point of view in ??4-5; these results can be found in Robinson [10] and Luxemburg [7], [8]. Section 6 is devoted to nonstandard measure theory; our approach is basically that of Loeb [6] (except that we construct Loeb's extension in Theorem 3 of ?6 directly, rather than using Caratheodory's Theorem), with some ideas coming from Anderson [1]. The final ?7, part I, contains a more formal description of the logical foundations, and parts II and III discuss the relationship between our approach and the classical one based on higher-order nonstandard models, as well as some other axiomatizations of Nonstandard Analysis.

The logico-linguistic evidence underlying Montague's language descriptions

Richard Montague's analyses of certain fragments of English have deservedly received much attention as the first sophisticated attempts to use model theoretic semantics for natural language description. They are especially interesting in that they exploit for linguistic purposes the apparatus of intensional logic developed in the 1960's by Kripke and others. In view of the influence they have had it is natural to ask: How accurate are these language analyses, regarded as empirical descriptions of actual English? And in particular, how successful is their use of the intensional apparatus? Montague presents detailed syntactic and semantic rules for three fragments of English. The first, which I shall call Fragment I, is described in his (1970a); the second, or Fragment II, in his (1970b); and the third, Fragment III, in his (1973). I wish to examine here the descriptive accuracy of his semantical rules for these fragments, those for Fragment I in some detail and those for Fragments II and III more briefly. Some prior acquaintance with the three relevant papers will be assumed; they are conveniently available in Thomason (1974), to which my pagination will refer. I should warn the reader that I plan to make the examination of Montague's work a vehicle for illustrating my own ideas about how evidence may be brought to bear in testing linguistic hypotheses.

Evaluating Global and Local Theories of Induction

The construction of inductive logics is an attempt to explicate the justification for the acceptance, or rational preference, of a given hypothesis on the basis of empirical evidence. As such it is a main part of epistemology. This paper explores this connection by focusing on the epistemic distinction between the grounds that are relevant for justification in normal knowledge-claim contexts (‘local’) and those that are relevant in philosophical knowledge-claim contexts (‘global’).In Gambling with Truth, Issac Levi introduces the global/local distinction with respect to scepticism and justification. Levi writes,Supplying this [local] justification requires an appeal to evidence, which will include observation reports and theoretical assumptions, as well as much of the apparatus of logic and mathematics.

The old and the new ?Erkenntnis?

In this first issue of the new Erkenntnis, it seems fitting to recall at least briefly the character and the main achievements of its distinguished namesake and predecessor. The old Erkenntnis came into existence when Hans Reichenbach and Rudolf Carnap assumed the editorship of the Annalen der Philosophie and gave the journal its new title and its characteristic orientation; the first issue appeared in 1930. The journal was backed by the Gesellschaft f?r Empirische Philosophie in Berlin, in which Reichenbach, Walter Dubislav, and Kurt Grelling were the leading figures, and by the Verein Ernst Mach in Vienna, whose philosophical position was strongly in? fluenced by that of the Vienna Circle; a brief account of these groups, and of several kindred schools and trends of scientific and philosophical thinking, was given by Otto Neurath in his 'Historische Anmerkungen' (Vol. 1, pp. 311-314). As Reichenbach noted in his introduction to the first issue, the editors of Erkenntnis were concerned to carry on philosophical inquiry in close consideration of the procedures and results of the various scientific disciplines: analysis of scientific research and its presuppositions was expected to yield insight into the character of all human knowledge, while at the same time, the objectivity and the progressive character of science inspired the convection that philosophy need not remain an array of con? flicting 'systems', but could attain to the status of objective knowledge. As a student in Berlin and Vienna during those years, I experienced vividly the exhilarating sense, shared by those close to those two philo? sophical groups, of being jointly engaged in a novel and challenging intellectual enterprise in which philosophical issues were dealt with 'scientifically' and philosophical claims were amenable to support or criticism by logically rigorous arguments. The 'logical analyses' and 'rational reconstructions' set forth by ad? herents of this program often made extensive use of the concepts, methods, and symbolic apparatus of contemporary symbolic logic, whose impor

Psychophysics of Time

If any single abstraction has claim to be central to the rather ill-matched concerns of artist, philosopher and scientist, it is surely the concept of time [1]. That time is inescapable may be a platitude, yet it is one that demands a certain resolution, one in which we sense uncomfortably the need either to accommodate or succumb. The particulars of our response to the experience of time are fascinating, puzzling and magnificently varied-but they are nothing if not inseparable from the element that gives rise to them. Time, the ineffable paradox, both sustains and defeats us, at once the unyielding matrix of our works and aspirations, the fluid, pervasive and mobile that informs and transfigures our achievement in the very act of bearing it away. Yet, in keeping with this general ambivalence, our very intimacy with time confuses and alienateswhatever form of logical apparatus we bring to bear, the reference remains elusive; we seem as much at a loss to focus as to grasp. Likewise the critic, seeking a common truth at the exposed margins of art and science, will find many clues, though hardly a ready synthesis in timebearing schemata. The interplay of temporal elements in languages is itself complex and notoriously deceptive. The rigour cultivated by physicist and mathematician is fugitive outside their own territory, being in any case haunted by a whole literature of logical paradoxes that, ironically enough, have themselves evolved to match the increasing subtlety and precision of the timeconcept. The insidious truism of time and our sense of

Axiomatic Theory, Informative Value of Propositions, and "Derivation Rules of Ordinary Language"

We present a logical analysis of Hans Zetterberg's strategy of axiomatic construction, especially his derivation of ordinary from theoretical propositions. Using formal criteria for assessing the relative logical or empirical content of propositions, we show that not only is Zetterberg's distinction vague but his derivation rules are logically faulty, and his assessment of the relative informative value of propositions is inconsistent. Thus we argue that Zetterberg failed to produce a coherent strategy for theory construction. Finally, we suggest that for the critical analysis of strategies of theory construction in sociology in general, the logical apparatus of modern symbolic logic and nondeductive inference be used.

Basic particulars and the Identity Thesis

This paper begins with a discussion of the logical apparatus of Frege, where his use ofSinn suggests a modification of Leibniz's Principle of the Identity of Indiscernibles. Then, it turns to Strawson's “basic particulars” with its essentially Kantian orientation. This brings forward the logical ground upon which the Identity Thesis rests. Finally, following Frege with some modifications, the paper suggests that an “ontological list” where concepts can be treated as objective (materially dependent) subsistent entities would be necessary in order to avoid errors of J. J. C. Smart and other analytic philosophers who hold the Identity Thesis.

Ancient Interpretations of Aristotle's Doctrine of Homonyma

Ancient Interpretations of Aristotle's Doctrine of Homonyma JOHN P. ANTON Tins MAIN PURPOSE OF TinS PAPER is to offeran exposition and a criticalexamination of the ancient interpretationsof Aristotle'sdoctrine of homonym. A circumlocution of what Aristotle means by homonym things is given in Categories, Ch. I Ia. The ancient interpretationswith which we are concerned in this paper are to be found in the extant commentaries on this treatise.Evidently, more commentaries had been written on the Categoriesthan the vicissitudesof time allowed to survive, but we have only those of the following writers: Porphyrius (c: 233-303), Dexippus (ft.c. 350), Ammonius (ft.C. 485), Philoponus (c.490-530), Olympiodorus (ft.c. 535), $implicius (ft.c. 533), Elias (ft.c. 550). One might add here the relevant writings of John Damascene (675-749), Photius (820-891), and Michael Psellus (1018-1079), which are useful paraphrases rather than fullcommentaries; for that reason the interpretationsthey support are not discussed in thispaper. The main body of this paper is given to a discussion of the interpretations which the ancient commentators offered and to an analysis of the assumptions which underlie them. It can be stated here in anticipation of what follows that the commentators often attached to Aristotle'smeaning of homonyma aspects that were quite foreign to his views and that by doing so these commentators were taking extensive libertieswith the text at hand. As we hope to show, the commentators brought into their discussions of this particular portion of the Categoriesissues and views that were far more relevant to theirown ontologies and logical theories than to Aristotle'sdoctrines. In order to show how this is the case, we must first give a summary of what we believe our textpermits us to say about the meaning of homonym, as given in the opening chapter of the Categories. ~ Suffice it to add This is the second of a two-part article. This part is based on an invited paper presented at the annual meeting of the Society for Ancient Greek Philosophy in New York City, Dec. 27, 1965. a For a full discussion of the doctrine of homonyma and the background against which it can be understood, see my "The Aristotelian Doctrine of Homonyma in the Categories and its Platonic Antecedents," in the last number of this journal (VI, 4 [Oct., 1968], 315-326). I tried to show in that paper how Aristotle's formulation is indebted linguistically to Plato but was worked out not in response to Plato's usage but to his own philosophical position and as part of his logical apparatus to attack the Speusippean wing of the Academy. On the whole, the textual evidence from the Platonic writings indicates clearly that Plato had not proposed a technical use of homonyma. Speusippus and Aristotle, however, offered two different and opposing technical formulations intended to fit their respective ontologies. [H 2 HISTORY OF PHILOSOPHY at this point that the interpretations of the doctrine of homonyma with which we are concerned here are only those that are discussed exclusively in the relevant commentaries on this work. Textual tradition has preserved two versions, referred to here as Vl and 3/2, of Aristotle's formulation of homonyma. The difference between them is that while V1 preserves the key expression ),d,yJ~.~ o6~(~, the other, V2, omits it. V1, as established by modern textual criticism, reads: 'Ol~6wtzce My~'~cct~o~~vollQc,~6vo~xow~, 6 ~ xc~Sc~o~vo~ccM'l,o~~ o6~(c~;~'~po;. Not only did we defend the inclusion of the expression logos t~s ousias (hereafter abbreviated as L of O) as indispensable for distinguishing Aristofle's formulation from the one propounded by Speusippus, but we also found it necessary to advance the view that Aristotle used the expression L of O in a definitely technical sense. We showed why the deletion of the term ousia would make the passage open to misinterpretation, since by omitting it one could argue that Aristofle meant to include in the class of homonyma such things as accidental properties and individuals with proper names. We also argued in favor of a view that delimits the meaning of ousia to definable substance, i.e., in the...

The Failure of Nowell-Smith's ‘New Logical Apparatus’

When Hume observed that there is no apparent logical relationship between ‘is'-type and ‘ought'-type statements he of course posed one of the fundamental problems in modern philosophy; and the various approaches to his challenge serve in large part to distinguish various ‘camps’ of contemporary moral philosophers. In this paper I shall be concerned only with one such approach, that of Nowell-Smith in his Ethics in which he tries to make the point that what he terms “Hume's gap” is not really there at all. (pp. 82-83) In his attempt to eliminate “Hume's gap” Nowell-Smith introduces a “new logical apparatus” (p. 61) the special features of which are the related doctrines of “logical oddness” and “contextual implication”. I shall argue that there are serious short-comings in this ‘new logical apparatus', and that, in fact, he fails in his attempt to eliminate ‘Hume's gap'.

Note sur un essai d’approche probabiliste à la théorie économique

A reformulation of the theory of marginal utility derined from the author’s inquiries twenty years ago is reproduced as a specimen of the dynamic theory advanced by him at that lime and as an introduction to the present essay. Dynamics was associated by the author with an individual’ s imperfect knowledge and not with variations. Statics was defined as the limiting case of dynamics. The bridge between the two, from the point of view of the logical apparatus of analysis, was the consideration of homogeneous doses, dated at different instants of time, of the same good as different goods. In this framework the author suggested a reformulation of the concepts of uncertainty and risk. The individual approaches them on the basis of the principle that his decisions regarding the future depend upon a judgement representing an application of statistical methods, albeit a rough one. Consequently he acknowledged a relation between subjective probabilities and statistical method. The author regrets that a formal dynamic theory organized around anticipations has not yet been completly realized. The problem of individuals’ expectations still remains the basic point to define. In this framework the author suggests considering Shackle’s theory as a limiting case. Finally the general problem of the measure of the degree of probability of every theory is dealt with.

The British Institute of Radiology: Annual Congress and Exhibition

THE eighth Annual Congress of the British Insti-JL tute of Radiology, and an exhibition of radio logical apparatus organised by the X-ray industry were held at the Central Hall, Westminster, on December 5-7. The morning of December 6 was devoted to the reading of physical and technical papers, contributions being made by Prof. F. L. Hopwood, Dr. A. Miiller, Dr. K. E. Clay and Mr. A. G. Warren.