Applications to mathematical logic: formalized languages of logic and arithmetic

This last chapter aims to provide a concise presentation of modal logic, the logic of necessity and possibility. A modal language is a language that contains, in addition to the symbols of a propositional or predicate language, the modal operators Open image in new window and ◊, which mean respectively ‘it is necessary that’ and ‘it is possible that’.

We now turn to the final semantics for probabilistic logic that we consider in this book: the objective Bayesian semantics. In §7.1 we provide an introduction to objective Bayesian epistemology, which holds that the strengths of an agent’s beliefs should be representable by certain probability functions. The key task is to say which probability functions are appropriate (§7.1.1). Objective Bayesian epistemology holds that an agent’s evidence should constrain her degrees of belief (§7.1.2), and that where evidence underdetermines appropriate degrees of belief they should be as equivocal as possible. §7.1.3 and §7.1.4 explain the equivocation requirement on propositional and predicate languages respectively.

The purpose of this work is to show how logic has developed in Mexican society from the middle of the last century until the present day. This process presents two characteristics: (1) works of logic were generally written by authors who, in spite of their national and international prestige, never succeeded in forming a school, but limited themselves to follow the trends outlined by others; and (2) the greater part of the work about logic was written to cover educational needs and, therefore, has been subject to national education policy changes. On the other hand, it should be taken into consideration that the authors we are going to review may be divided into two main groups: those who have used logic from the perspective of some philosophy (i.e. positivism, neo-Kantianism and Marxism) and those who have spread it within the reference framework of formal contemporary logic, whether ‘mathematical logic’ or ‘symbolic logic’. We are going to find that those writers forming the first group have been, for several reasons, opposed to the introduction of mathematical or symbolic logic.

The continuum hypothesis of G. Cantor is a starting point for much current work in set theory and mathematical logic. Most elementary texts go no further than a statement of the hypothesis, with a brief indication of its significance. The aim of this paper is to show the details of its application at an elementary level. We consider two sets; E, the set of all possible equivalence relations on the set of natural numbers N, and R, the set of real numbers, and present two proofs that they are equinumerous (E=R). The first proof uses only the basic results of transfinite arithmetic (within easy reach of first‐year undergraduates) to show that E ≤ R and E >N, and then an appeal to the continuum hypothesis immediately yields E=R. The second proof avoids the continuum hypothesis by a direct demonstration that E ≥ R, and this only involves the Schroder‐Bernstein theorem and a certain degree of care in showing that a certain map from the real interval (0, 1) to a set of subsequences of N is in fact a bijection.

In these notes, we recall some basic notions from computability, and we illustrate their use in measuring complexity of algebraic structures and classes of structures. These notes are based on a tutorial lecture given as part of the conference “Algebra and Mathematical Logic”, at Kazan Federal University in June, 2014, honoring Kazan Federal University, the group in algebra and logic there, and Marat Arslanov, who has done so much to build up the group.

For the field of manufacturing to become a science, it is necessary to develop general mathematical descriptions for the analysis and synthesis of manufacturing systems. Standard analytic models, as used extensively in the past, are ineffective for describing the general manufacturing situation due to their inability to deal with discontinuous and nonlinear phenomena. These limitations are transcended by algebraic models based on set structures. Set-theoretic and algebraic structures may be used to (1) express with precision a variety of important qualitative concepts such as hierarchies, (2) provide a uniform framework for more specialized theories such as automata theory and control theory, and (3) provide the groundwork for quantitative theories. By building on the results of other fields such as automata theory and computability theory, algebraic structures may be used as a general mathematical tool for studying the nature and limits of manufacturing systems. This paper shows how manufacturing systems may be modeled as automatons, and demonstrates the utility of this approach by discussing a number of theorems concerning the nature of manufacturing systems. In addition symbolic logic is used to formalize the Design Axioms, a set of generalized decision rules for design. The application of symbolic logic allows for the precise formulation of the Axioms and facilitates their interpretation in a logical programming language such as Prolog. Consequently, it is now possible to develop a consultive expert system for axiomatic design.

An arithmetic circuit is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper we investigate the algebraic structure of complex algebras of natural numbers and make some observations regarding the complexity of various theories of such algebras.

is sometimes called the foundation of mathematics: the logician studies the kinds of reasoning used in the individual steps of a proof. Alonzo Church was a pioneer in the field of mathematical logic, whose contributions to number theory and the theories of algorithms and computability laid the theoretical foundations of computer science. His first Princeton book, The Calculi of Lambda-Conversion (1941), established an invaluable tool that computer scientists still use today. Even beyond the accomplishment of that book, however, his second Princeton book, Introduction to Mathematical Logic, defined its subject for a generation. Originally published in Princeton's Annals of Mathematics Studies series, this book was revised in 1956 and reprinted a third time, in 1996, in the Princeton Landmarks in Mathematics series. Although new results in mathematical logic have been developed and other textbooks have been published, it remains, sixty years later, a basic source for understanding formal logic. Church was one of the principal founders of the Association for Symbolic Logic; he founded the Journal of Symbolic Logic in 1936 and remained an editor until 1979 At his death in 1995, Church was still regarded as the greatest mathematical logician in the world.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsSolomon FefermanSOLOMON FEFERMAN is professor of Mathematics and Philosophy, and Suppes Family Professor in Humanities and Sciences at Stanford University. He is past President of the Association for Symbolic Logic and is a Fellow of the American Academy of Arts and Sciences. He received his Ph.D. at U.C. Berkeley in 1957 under the direction of Alfred Tarski. He is Editor-in-Chief of the Collected Works of Kurt Gödel, and the author of numerous articles on mathematical logic, the foundations of mathematics, and the history of modern logic.

This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven equivalences (namely: geometric, universal geometric, quasi-equational, universal, elementary, and combinations thereof) in specific classes of algebraic structures (equationally Noetherian, qω-compact, uω-compact, equational domains, equational co-domains, etc.). The main questions are the following: (1) Which equivalences coincide inside a given class K, which do not? (2) With respect to which equivalences a given class K is invariant, with respect to which it is not?

Donald Davidson (1984g) famously proposed that what we want from a theory of meaning we may get from a theory of truth. What we want from a theory of meaning for a natural language, that being the case at issue, includes at least an account of the phenomena that give rise to the principle of compositionality: We want an explanation of how the meaning of a complex expression, such as a sentence, depends upon and is determined by the meanings of its syntactic parts. One central advantage of Davidson’s proposal, as he saw it, is that we know from the work of Tarski (1958) how to construct theories of truth for languages with certain sorts of structures, and such a theory does help us to understand how at least some of the semantic properties of complex expressions are determined by those of their parts. Davidson’s practice suggests he may have hoped that all sentences of natural language could be assigned logical forms of the variety familiar from mathematical logic. If so, then semantics would all but reduce to syntax: The only outstanding problem would be to uncover the ‘logical forms’ of the sentences of, say, English. This hope does not now seem reasonable. But the methods available for investigating logical form—that is, for investigating syntax—have become so powerful that it is not at all unreasonable to suppose that the syntactic part of the problem is solvable, at least in principle, and Tarski’s work can be adapted to many of the resulting structures. The work that is left for the semanticist proper then becomes to solve the problems syntax and Tarski do not solve: These include, for example, discovering what sorts of semantic values need to be assigned to novel semantic primitives, such as mass terms, and explaining the semantic significance of novel modes of composition, such as [V PV [CPS]]. It is unfortunate that a proposal with such great advantages faces such grave difficulties. Indeed, it is probably fair to say that most philosophers of language, and perhaps even most working semanticists, regard truththeoretic semantics as a non-starter, for the simple reason that it seems to have the wrong subject-matter. We will not get what we want from a theory of meaning from any theory that does not specify meanings: No theory that does not specify what the sentence ‘snow is white’ means, for example, can possibly explain how this sentence’s meaning what it does is determined by its parts’ meaning what they do. But a theory of truth for English does not specify what even one sentence of English means, since any such theory is consistent with infinitely many different suppositions concerning what any sentence of English means. For example, a theory of truth that proves this T-sentence2 for ‘snow is white’:

The work of George Boole (1805–64) is often considered a first attempt at mathe-matization of logic. It is often opposed to that of Gottlob Frege (1848–1925), who is regarded as the true founder of mathematical logic. This way of thinking may be considered as the appraisal of the twentieth-century philosopher, logician or mathematician who is well informed about historical developments in this field since the last century. It lays great stress on the formalization of logic but tends to neglect what could now be called the semantic question, despite its importance for Boole with the problem of the interpretation of symbols, and for Frege with the problem of representation by which he tries to understand the relation between propositions and reality.

Gödel's incompleteness theorem states that there is no system of axioms and rules of inference such that the totality of all assertions deducible from the axioms is the same as the totality of all true assertions in arithmetic (indeed, for every consistent system one can construct effectively a true but unprovable assertion). The present article is devoted to a proof of this theorem, based on the concepts and methods of the theory of algorithms; the necessary information from the theory of algorithms is provided. The paper does not require specialized knowledge of any kind (in particular, none from mathematical logic), but assumes only a familiarity with elementary mathematical terminology and symbolism.

This article deals with general problem of connection between legal reasoning, legal argumentation and, on the other hand, logic. Although this connection seems obvious since ancient times, its nature is still the subject of non-trivial discussions. The purpose of my study is to explore one important aspect of this connection, namely, is general, or formal, logic necessary and sufficient to determine relevant form of legal reasoning and argumentation? To reach this purpose, method of comparative analysis is employed both synchronically and diachronically. First, in order to prove importance of the issue, a diachronic comparison of several basic approaches is carried out (G.W. von Leibniz, O.W. Holmes, Jr., etc.). Then a comparative analysis of the views of some contemporary experts is carried out, especially Ilmar Tammelo and Stephen Toulmin. It is argued, in the field of law there is a special legal validity that differs significantly from the formal-logical validity. The legal validity is determined directly and mainly by the value of the right not by the value of formal or material truth. It is not the subject of formal logic, but of a special legal logic and, more broadly, informal logic. Therefore, formal logic does not determine relevant form of legal reasoning and argumentation, since, first of all, there is the essential difference between the formal-logical validity and the legal validity. It is argued, formal logic is neither sufficient nor necessary to think like a jurist. In this regard, it is shown that the formal-logical standard of proof is not sufficient to determine the relevant form of valid legal proof. There are at least three standards of legal proof – proof without reasonable doubt, proof by clear and convincing evidence, and proof by preponderance of the evidence. Each of them differs significantly from the formal-logical standard. The conclusion of this research is this: it cannot be said that general logic does not work at all in the field of law. However, in order to reason and argue like a successful jurist, one must to grasp and use, without any exceptions, the special legal logic that belongs to the domain of contemporary informal logic.

Willard Van Orman Quine (b. 1908–d. 2000) was the main agenda setter in post–World War II philosophy of language and related fields. He was also a highly active logician; if his views on formal and mathematical logic did not shape the field in quite the same way as his work on language, epistemology, and metaphysics, the latter largely sprang from his work on logic and the related domain of the foundations of mathematics. His doctoral thesis (“The Logic of Sequences,” 1933) attempted to clean up and generalize yet boil down the great Principia Mathematica of Bertrand Russell and Alfred North Whitehead (Cambridge, UK: Cambridge University Press, 1910, 1927). Quine’s subsequent works—including A System of Logistic (Cambridge, MA: Harvard University Press, 1934), Mathematical Logic (1940, 1951; Quine 1951, cited under Quine’s Monographs), “New Foundations for Mathematical Logic” (1937, 1953, 1961; Quine 1961, cited under Logic and Set Theory), and Set Theory and Its Logic (1963, 1969; Quine 1969, cited under Quine’s Textbooks)—developed more refined and original ideas on largely the same topics. In philosophy he is known most generally for his commitment to extensionalism, which, for scientific purposes, rules out as inadmissible that two expressions could have the same truth-value or extension or refer to the same object yet not be interchangeable in containing sentences without affecting the truth-values of the containing sentences. Closely related are his celebrated criticisms of the unexplicated reliance on the concepts “meaning,” the “a priori,” and “analyticity” among empiricist philosophers; pressed harder, Quine believed, those notions cannot support the edifices empiricists had envisaged. A scientifically rigorous attitude toward language shows that the notion of the meaning of a sentence, in the intuitive sense that sustains the idea that two correct translations of a given foreign sentence must be in some clear sense be equivalent, must be discarded; meaning in that sense is “indeterminate.” In metaphysics, specifically ontology, he is celebrated for formulating a clear and seemingly rigorous test for ontological commitment in his “On What There Is” (1948; Quine 1961, cited under Ontology): existence is equated with being the value of variable in the first-order predicate calculus. Less well known but equally interesting are his later views concerning the question of what exists; he held that certain considerations undermine the idea that there is a single answer to the question “What is there?” Finally, the notion that epistemology should be “naturalized” and more generally that philosophy itself should be thought of as taking its place within natural science has caused considerable commentary and indeed controversy.