IDENTITY LOGICS

The Art of Logic by Avi Sion is a collection of recent essays on various topics in logic theory and in applied logic. The same faculty and art of logic is called for in formulating theoretical logic and in applying its findings to diverse fields. The essays here collected deal with some very important issues in logic, philosophy, and spirituality, which he had not previously treated in as much detail if at all.

The current dominant view about logic is that there is a single paradigm of logic, logic as justification, according to which logic is a means of justification and consists in a theory of deduction. The paradigm originated with Aristotle's logic and its current form is mathematical logic. The latter was created to justify mathematics by putting it on a firm foundation, it includes Aristotle's logic as a small fragment, and is adequate as a logic of justification. This paper, however, argues that this view is not valid. Aristotle's logic does not belong to the paradigm of logic as justification because it is a method of discovery, and mathematical logic is inadequate as a means of justification. Besides the paradigm of logic as justification, there is the paradigm of logic as discovery, according to which logic is a method of discovery and consists in the analytic method. The latter is adequate as a means of discovery, and hence also as a means of justification, because in the analytic method justification is part of discovery.

Dialogue, 1970

Essay on the Principles of Logic: A Defense of Logical Monism, 2023

Table of Contents for my translation of Michael Wolff's Essay on the Principles of Logic: A Defense of Logical Monism. Michael Wolff's book is an attempt to show that a non-truth-functional syllogistic logic is the only formal and general logic - in that it introduces no specific conceptual content into its formulas and principles. Further, he shows that post-Fregean logics can be reproduced within syllogistic, so long as postulates are added which are not universally valid. Yet because "classical" logics depend on postulates that are not valid in every case, they cannot earn the title of formal and general logic with the same right as the syllogistic. The book includes a full comparison of syllogistic with class-logical and function-theoretic logical languages, as well as a reconstruction of Aristotle's syllogistic within Wolff's new syllogistic language.

2017

Concise Introduction to Logic is an introduction to formal logic suitable for undergraduates taking a general education course in logic or critical thinking, and is accessible and useful to any interested in gaining a basic understanding of logic. This text takes the unique approach of teaching logic through intellectual history; the author uses examples from important and celebrated arguments in philosophy to illustrate logical principles. The text also includes a basic introduction to findings of advanced logic. As indicators of where the student could go next with logic, the book closes with an overview of advanced topics, such as the axiomatic method, set theory, Peano arithmetic, and modal logic. Throughout, the text uses brief, concise chapters that readers will find easy to read and to review.

2006

This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum algebra construction, its generalization to a Lindenbaum category construction that includes proofs, and model cardinality spectra; these are used in some examples to show logics inequivalent. Generalizations of familiar results from first order to arbitrary logics are also discussed, including Craig interpolation and Beth definability. Mathematics Subject Classification (2000). Primary 03C95; Secondary 18A15, 03G30, 18C10, 03B22.

PREPRINT 122415 : LOGIC TEACHING IN THE 21ST CENTURY-- XVIII EIDL 2015 KEYNOTE ADDRESS We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. One rather conspicuous example is that the process of refining logical terminology has been productive. Future logic students will no longer be burdened by obscure terminology and they will be able to read, think, talk, and write about logic in a more careful and more rewarding manner. Closely related is increased use and study of variable-enhanced natural language as in “Every proposition x that implies some proposition y that is false also implies some proposition z that is true”. Another welcome development is the culmination of the slow demise of logicism. No longer is the teacher blocked from using examples from arithmetic and algebra fearing that the students had been indoctrinated into thinking that every mathematical truth was a tautology and that every mathematical falsehood was a contradiction. A fifth welcome development is the separation of laws of logic from so-called logical truths, i.e., tautologies. Now we can teach the logical independence of the laws of excluded middle and non-contradiction without fear that students had been indoctrinated into thinking that every logical law was a tautology and that every falsehood of logic was a contradiction. This separation permits the logic teacher to apply logic in the clarification of laws of logic. This lecture expands the above points, which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”.

Dialogue, 1973

Starting from the idea that an abstract notion of logic is inevitably interlaced to the mathematical instruments necessary to its development, we are describing four forms of logic (related, respectively, to naïve set theory of concrete mathematics, second-order arithmetic, basic and advanced set theory) and studying their mathematic relationships. We aim at pointing out the conceptual naturalness and the technical efficiency of the form of logic founded on the advanced Set Theory.

One of the central logical ideas in Wittgenstein's Tractatus logico-philoso-phicus is the elimination of the identity sign in favor of the so-called "exclusive interpre-tation" of names and quantifiers requiring different names to refer to different objects and (roughly) different variables to take different values. In this paper, we examine a recent development of these ideas in papers by Kai Wehmeier. We diagnose two main problems of Wehmeier's account, the first concerning the treatment of individual constants, the second concerning so-called "pseudo-propositions" (Scheinsätze) of classical logic such as a = a or a = b ∧ b = c → a = c. We argue that overcoming these problems requires two fairly drastic departures from Wehmeier's account: (1) Not every formula of classical first-order logic will be translatable into a single formula of Wittgenstein's exclusive notation. Instead, there will often be a multiplicity of possible translations, revealing the original "inclusive" formulas to be ambiguous. (2) Certain formulas of first-order logic such as a = a will not be translatable into Wittgenstein's notation at all, being thereby revealed as nonsensical pseudo-propositions which should be excluded from a "correct" conceptual notation. We provide translation procedures from inclusive quantifier-free logic into the exclusive notation that take these modifications into account and define a notion of logical equivalence suitable for assessing these translations.

Classical and Nonclassical Logics, 2020

Typically, a logic consists of a formal or informal language together with a deductive system and/or a modeltheoretic semantics. The language has components that correspond to a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record arguments that are valid for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions for at least part of the language. The following sections provide the basics of a typical logic, sometimes called "classical elementary logic" or "classical first-order logic". Section 2 develops a formal language, with a rigorous syntax and grammar. The formal language is a recursively defined collection of strings on a fixed alphabet. As such, it has no meaning, or perhaps better, the meaning of its formulas is given by the deductive system and the semantics. Some of the symbols have counterparts in ordinary language. We define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion. The other sentences (if any) in an argument are its premises. Section 3 sets up a deductive system for the language, in the spirit of natural deduction. An argument is derivable if there is a deduction from some or all of its premises to its conclusion. Section 4 provides a model-theoretic semantics. An argument is valid if there is no interpretation (in the semantics) in which its premises are all true and its conclusion false. This reflects the longstanding view that a valid argument is truth-preserving. In Section 5, we turn to relationships between the deductive system and the semantics, and in particular, the relationship between derivability and validity. We show that an argument is derivable only if it is valid. This pleasant feature, called soundness, entails that no deduction takes one from true premises to a false conclusion. Thus, deductions preserve truth. Then we establish a converse, called completeness, that an argument is valid only if it is derivable. This establishes that the deductive system is rich enough to provide a deduction for every valid argument. So there are enough deductions: all and only valid arguments are derivable. We briefly indicate other features of the logic, some of which are corollaries to soundness and completeness.

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations.

Essay on the Principles of Logic: A Defense of Logical Monism, 2023

This is a sample of my translation of Michael Wolff's Essay on the Principles of Logic: A Defense of Logical Monism. In the book, Wolff sets out an elementary language of syllogistic logic, and shows it to be sufficient to express all universally valid principles of general and formal logic. In the excerpt, Wolff begins by distinguishing a syllogistic language as one that expresses only relations between concepts, rather than (with Frege) asymmetrical object-concept relations. Key features of the elementary language are also: the basic form of negation is non-truth-functional, and expressions for modality are necessary for the elementary language itself. This excerpt gives readers a first picture of the elementary language. The heart of the book's argument lies in Wolff's comparison of this language with class-logical and function-theoretic attempts to express general or formal logic.