Using set theory in the first part of his book, and proof theory in the second, Gaisi Takeuti gives us two examples of how mathematical logic can be used to obtain results previously derived in less elegant fashion by other mathematical techniques, especially analysis. In Part One, he applies Scott- Solovay's Boolean-valued models of set theory to analysis by means of complete Boolean algebras of projections. In Part Two, he develops classical analysis including complex analysis in Peano's arithmetic, showing (...) that any arithmetical theorem proved in analytic number theory is a theorem in Peano's arithmetic. In doing so, the author applies Gentzen's cut elimination theorem. Although the results of Part One may be regarded as straightforward consequences of the spectral theorem in function analysis, the use of Boolean- valued models makes explicit and precise analogies used by analysts to lift results from ordinary analysis to operators on a Hilbert space. Essentially expository in nature, Part Two yields a general method for showing that analytic proofs of theorems in number theory can be replaced by elementary proofs. Originally published in 1978. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

The Joint International Conference on Logic Programming, sponsored by the Association for Logic Programming, is a major forum for presentations of research, applications, and implementations in this important area of computer science. Logic programming is one of the most promising steps toward declarative programming and forms the theoretical basis of the programming language Prolog and its various extensions. Logic programming is also fundamental to work in artificial intelligence, where it has been used for nonmonotonic and commonsense (...) reasoning, expert systems implementation, deductive databases, and applications such as computer-aided manufacturing. Krzysztof R. Apt is project leader at the Centre for Mathematics and Computer Science in Amsterdam and part-time Professor in Computer Science at the University of Amsterdam. Topics covered: Theory Foundations. Programming Languages. Implementation. Programming Methodologies and Tools. Applications. Deductive Databases. Artificial Intelligence. Parallelism. (shrink)

This paper analyses the notion of ‘interpretation’, which is often tied to the semantic approach to logic, where it is used when referring to truth-value assignments, for instance. There are, however, other uses of the notion that raise interesting problems. These are the cases in which interpreting a logic is closely related to its justification for a given application. The paper aims to present an understanding of interpretations that supports the model-theoretic characterization of validity to the detriment (...) of the proof-theoretic one. This is done by making use of the hierarchy of ST-related logics. Finally, a localist conception of logic is defended as the natural view stemming from the model-theoretic approach. (shrink)

We consider a paradigm of applications of Logic Engineering to illustrate the information interchange among different areas of knowledge, through the formal approach to some aspects of computing. We apply the paradigm to the area of distributed systems, taking the demand for specification formalisms, treated in three areas of knowledge: modal logics, first-order logic and algebra. In doing so, we obtain transfer of intuitions and results, establishing that, as far as input/output representation is concerned, these three formalisms are (...) equivalent. (shrink)

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)

The First International Workshop brings together researchers from the theoretical ends of the logic programming and artificial intelligence communities to discuss their mutual interests. Logic programming deals with the use of models of mathematical logic as a way of programming computers, where theoretical AI deals with abstract issues in modeling and representing human knowledge and beliefs. One common ground is nonmonotonic reasoning, a family of logics that includes room for the kinds of variations that can be found (...) in human reasoning. Topics covered: Stable Semantics. Default Logic. AutoEpistemic Logic. Truth Maintenance Systems. Implementation Issues. Diagnosis. Applications. Inheritance Reasoning. Logics of Belief. Inconsistency and Non-Monotonicity. (shrink)

This book constitutes the strictly refereed post-workshop proceedings of the 11th International Workshop on Computer Science Logic, CSL '97, held as the 1997 Annual Conference of the European Association on Computer Science Logic, EACSL, in Aarhus, Denmark, in August 1997. The volume presents 26 revised full papers selected after two rounds of refereeing from initially 92 submissions; also included are four invited papers. The book addresses all current aspects of computer science logics and its applications and thus presents (...) the state of the art in the area. (shrink)

Free logic, an alternative to traditional logic, has been seen as a useful avenue of approach to a number of philosophical issues of contemporary interest. In this collection, Karel Lambert, one of the pioneers in, and the most prominent exponent of, free logic, brings together a variety of published essays bearing on the application of free logic to philosophical topics ranging from set theory and logic to metaphysics and the philosophy of religion. The work (...) of such distinguished philosophers as Bas van Fraassen, Dana Scott, Tyler Burge, and Jaakko Hintikka is represented. Lambert provides an introductory essay placing free logic in the logical tradition beginning with Aristotle, developing it as the natural culmination of a trend begun in the Port Royal logic of the 1600s, and continuing through current predicate logic--the trend to rid logic of existence assumptions. His Introduction also provides a useful systematic overview of free logic, including both a standard syntax and some semantical options. (shrink)

The main result of this paper were announced in Kosheleva — Kreinovich [7, 8]; for other algorithmic aspects of Hilbert's Third Problem see Kosheleva [6]. The authors are greatly thankful to Alexandr D. Alexandrov , Vladimir G. Boltianskii and Patrick Suppes for valuable discussions, and to the anonymous referee for important suggestions. This work was partially supported by an NSF grant No. CDA-9015006.

The paper argues that the theory of Implicit Definition cannot give an account of knowledge of logical principles. According to this theory, the meanings of certain expressions are determined such that they make certain principles containing them true; this is supposed to explain our knowledge of the principles as derived from our knowledge of what the expressions mean. The paper argues that this explanation succeeds only if Implicit Definition can account for our understanding of the logical constants, and that fully (...) understanding a logical constant in turn requires the ability to apply it correctly in particular cases. It is shown, however, that Implicit Definition cannot account for this ability, even if it draws on introduction rules for the logical constants. In particular, Implicit Definition cannot account for our ability to apply negation in particular cases. Owing to constraints relating to the unique characterisation of logical constants, invoking the notion of rejection does not remedy the situation. Given its failure to explain knowledge of logic, the prospects of Implicit Definition to explain other kinds of a priori knowledge are even worse. (shrink)

I visited Professor Hintikka and discussed with him about ten times during my visit to Harvard University from 2000 to 2001. I can still vividly remember my first visit to him in his office and then Jaakko inviting me for lunch in the BU faculty club, including how I tried my best to ‘show’ my knowledge of Aristotle and Frege, with a hope that it was my first but not the last talk with him and how really excited I was (...) when he kindly said to me after lunch: “You are welcome to visit me again at the same time in two weeks”. In my view Hintikka was a brilliant philosopher and logician, and also a very kind and engaging person from and with whom one can constantly learn new ideas without limitations. (shrink)

I visited Professor Hintikka and discussed with him about ten times during my visit to Harvard University from 2000 to 2001. I can still vividly remember my first visit to him in his office and then Jaakko inviting me for lunch in the BU faculty club, including how I tried my best to ‘show’ my knowledge of Aristotle and Frege, with a hope that it was my first but not the last talk with him and how really excited I was (...) when he kindly said to me after lunch: “You are welcome to visit me again at the same time in two weeks”. In my view Hintikka was a brilliant philosopher and logician, and also a very kind and engaging person from and with whom one can constantly learn new ideas without limitations. (shrink)

This paper explores how to explain the applicability of classical mathematics to the physical world in a radically naturalistic and nominalistic philosophy of mathematics. The applicability claim is first formulated as an ordinary scientific assertion about natural regularity in a class of natural phenomena and then turned into a logical problem by some scientific simplification and abstraction. I argue that there are some genuine logical puzzles regarding applicability and no current philosophy of mathematics has resolved these puzzles. Then I introduce (...) a plan for resolving the logical puzzles of applicability. (shrink)

This text aims to convey some of the interest and charm of modal logic, and to put a reader new to the subject in a position to have an informed opinion as to its applicability to each of several areas of philosophical concern in which the merits of a modal approach' have been controversial. he main focus, for these purposes, is on normal modal logics, though some attention is given to the non-normal side of the picture.

Development of decision-support and intelligent agent systems necessitates mathematical descriptions of uncertainty and fuzziness in order to model vagueness. This paper seeks to present an outline of Peirce’s triadic logic as a practical new way to model vagueness in the context of artificial intelligence. Charles Sanders Peirce was an American scientist–philosopher and a great logician whose triadic logic is a culmination of the study of semiotics and the mathematical study of anti-Cantorean model of continuity and infinitesimals. After presenting (...) Peircean semiotics within AI perspective, a mathematical formulation of a Peircean triadic set is given in relationship with classical and fuzzy sets. Using basic logical operators, all possible respective implication operators, bi-equivalence operators, valid rules of inference, and associative, distributive and commutative logical properties are derived and verified through the truth function approach. In order to suggest practical directions, aggregation operators for Peirce’s triadic logic have been formulated. A mathematical formulation of a medical diagnostic problem and ER diagram of a library management system using Peirce’s triadic relation show potential for further applications of the proposed triadic set and triadic logic. Alongside, a classical AI game—The Wumpus World—is implemented to show practical efficacy in comparison with binary implementation. Besides giving some preliminary formulations for trichotomous set theory and definition of finite automaton, development of hybrid architectures for intelligent agents and evolutionary computations are discussed as potential practical avenues for Peirce’s triadic logic. (shrink)

In the same spirit with the theme of last issue of this SGJ journal (“Ongoing creation”), this paper shortly reviews a plausible mechanism from Aether to become ordinary matter from the perspective of Neutrosophic Logic. We also discuss two other possible applications of Neutrosophic Logic, including a resolution of conflicting paradigms in medicine. We hope that some ideas as outlined herein will be proved useful in the near future.

The modal logician's notion of possible world and the computer scientist's notion of state of a machine provide a point of commonality which can form the foundation of a logic of action. Extending ordinary modal logic with the calculus of binary relations leads to a very natural logic for describing the behavior of computer programs.

The amount of information that is produced on a daily basis in the financial markets is vast and complex; consequently, the development of systems that simplify decision-making is an essential endeavor. In this article, several intelligent systems are proposed and tested to predict the closing price of the IBEX 35 index using more than ten years of historical data and five distinct architectures for neural networks. A multi-layer perceptron was the first step, followed by a simple recurrent neural network, a (...) gated recurrent unit network and two long-short-term memory (LSTM) networks. The results of the analyses performed on these models have demonstrated a powerful capacity for prediction. Additionally, the findings of this research point to the fact that the application of intelligent systems can simplify the decision-making process in financial markets, which is a substantial advantage. Furthermore, by comparing the predicted outcome errors between the models, the LSTM presents the lowest error with a higher computational time in the training phase. The LSTM was able to accurately forecast the closing price of the day as well as the price for the following one and two days in advance. In conclusion, the empirical results demonstrated that these models could accurately predict financial data for trading purposes and that the application of intelligent systems, such as the LSTM network, represents a promising advancement in financial technology. (shrink)

This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology (...) is kept to a minimum, no background in formal set-theory is required, and the book contains proofs of all the required set theoretical results. The reader is taken on a journey starting with König's Lemma, and progressing via order relations, Zorn's Lemma, Boolean algebras, and propositional logic, to completeness and compactness of first-order logic. As applications of the work on first-order logic, two final chapters provide introductions to model theory and nonstandard analysis. (shrink)