In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of the system is deducible (...) by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)

For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...) is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

Argumentations are at the heart of the deductive and the hypothetico-deductive methods, which are involved in attempts to reduce currently open problems to problems already solved. These two methods span the entire spectrum of problem-oriented reasoning from the simplest and most practical to the most complex and most theoretical, thereby uniting all objective thought whether ancient or contemporary, whether humanistic or scientific, whether normative or descriptive, whether concrete or abstract. Analysis, synthesis, evaluation, and function of argumentations are described. Perennial philosophic (...) problems, epistemic and ontic, related to argumentations are put in perspective. So much of what has been regarded as logic is seen to be involved in the study of argumentations that logic may be usefully defined as the systematic study of argumentations, which is virtually identical to the quest of objective understanding of objectivity. (shrink)

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...) involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss (...) many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (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 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)

Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's two-volume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning showing (...) by deductively evident steps that its conclusion is a consequence of its premises. In particular, a demonstration is a deduction whose premises are known to be true. Aristotle's general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of intermediate immediately evident steps that shows its final conclusion to follow logically from its premises. To illustrate his general theory of deduction, he presented an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

Information-theoretic approaches to formal logic analyse the "common intuitive" concept of propositional implication (or argumental validity) in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; an argument is valid if the conclusion contains no information beyond that of the premise-set. This paper locates information-theoretic approaches historically, philosophically and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves and (...) by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyse validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. (shrink)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. Several (...) attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)

Stephen Law developed a challenge to theism, known as the evil-god challenge (Law (2010) ). The evil-god challenge to theism is to explain why the theist’s responses to the problem of evil are any better than the diabolist’s – who believes in a supremely evil god – rejoinders to the problem of good, when all the theist’s ploys (theodicy, sceptical theism, etc.) can be parodied by the diabolist. In the first part of this article, I extend the evil-god challenge by (...) showing that additional theist replies to the problem of evil (more theodicies, the privation view of evil, and others) also may be appropriated, with just as much plausibility, in support of the diabolist position. In the second part of the article, I defend the evil-god challenge against several objections. (shrink)

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s Induction-Axiom (...) Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

Thirteen meanings of 'implication' are described and compared. Among them are relations that have been called: logical implication, material implication,deductive implication, formal implication, enthymemic implication, and factual implication. In a given context, implication is the homogeneous two-place relation expressed by the relation verb 'implies'. For heuristic and expository reasons this article skirts many crucial issues including use-mention, the nature of the entities that imply and are implied, and the processes by which knowledge of these relations are achieved. This paper is (...) better thought of as an early stage of a dialogue than as a definitive treatise. (shrink)

One innovation in this paper is its identification, analysis, and description of a troubling ambiguity in the word ‘argument’. In one sense ‘argument’ denotes a premise-conclusion argument: a two-part system composed of a set of sentences—the premises—and a single sentence—the conclusion. In another sense it denotes a premise-conclusion-mediation argument—later called an argumentation: a three-part system composed of a set of sentences—the premises—a single sentence—the conclusion—and complex of sentences—the mediation. The latter is often intended to show that the conclusion follows from (...) the premises. The complementarity and interrelation of premise-conclusion arguments and premise-conclusion-mediation arguments resonate throughout the rest of the paper which articulates the conceptual structure found in logic from Aristotle to Tarski. This 1972 paper can be seen as anticipating Corcoran’s signature work: the more widely read 1989 paper, Argumentations and Logic, Argumentation 3, 17–43. MR91b:03006. The 1972 paper was translated into Portuguese. The 1989 paper was translated into Spanish, Portuguese, and Persian. (shrink)

This study concerns logical systems considered as theories. By searching for the problems which the traditionally given systems may reasonably be intended to solve, we clarify the rationales for the adequacy criteria commonly applied to logical systems. From this point of view there appear to be three basic types of logical systems: those concerned with logical truth; those concerned with logical truth and with logical consequence; and those concerned with deduction per se as well as with logical truth and logical (...) consequence. Adequacy criteria for systems of the first two types include: effectiveness, soundness, completeness, Post completeness, "strong soundness" and strong completeness. Consideration of a logical system as a theory of deduction leads us to attempt to formulate two adequacy criteria for systems of proofs. The first deals with the concept of rigor or "gaplessness" in proofs. The second is a completeness condition for a system of proofs. An historical note at the end of the paper suggests a remarkable parallel between the above hierarchy of systems and the actual historical development of this area of logic. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

Contrary to common misconceptions, today's logic is not devoid of existential import: the universalized conditional ∀ x [S→ P] implies its corresponding existentialized conjunction ∃ x [S & P], not in all cases, but in some. We characterize the proexamples by proving the Existential-Import Equivalence: The antecedent S of the universalized conditional alone determines whether the universalized conditional has existential import, i.e. whether it implies its corresponding existentialized conjunction.A predicate is an open formula having only x free. An existential-import predicate (...) Q is one whose existentialization, ∃ x Q, is logically true; otherwise, Q is existential-import-free or simply import-free.How abundant or widespread is existential import? How abundant or widespread are existential-import predicates in themselves or in comparison to import-free predicates? We show that existential-import predicates are quite abundant, and no less so than import-free predicates. Existential.. (shrink)

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our traditional intuitive logical framework and (...) that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving radical repudiation of a part of our scientific tradition, a tradition that is defended by classical logicians. But it is also viewed as reactionary: as being analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been. (shrink)

This interesting and imaginative monograph is based on the author’s PhD dissertation supervised by Saul Kripke. It is dedicated to Timothy Smiley, whose interpretation of PRIOR ANALYTICS informs its approach. As suggested by its title, this short work demonstrates conclusively that Aristotle’s syllogistic is a suitable vehicle for fruitful discussion of contemporary issues in logical theory. Aristotle’s syllogistic is represented by Corcoran’s 1972 reconstruction. The review studies Lear’s treatment of Aristotle’s logic, his appreciation of the Corcoran-Smiley paradigm, and his understanding (...) of modern logical theory. In the process Corcoran and Scanlan present new, previously unpublished results. Corcoran regards this review as an important contribution to contemporary study of PRIOR ANALYTICS: both the book and the review deserve to be better known. (shrink)

The five English words—sentence, proposition, judgment, statement, and fact—are central to coherent discussion in logic. However, each is ambiguous in that logicians use each with multiple normal meanings. Several of their meanings are vague in the sense of admitting borderline cases. In the course of displaying and describing the phenomena discussed using these words, this paper juxtaposes, distinguishes, and analyzes several senses of these and related words, focusing on a constellation of recommended senses. One of the purposes of this paper (...) is to demonstrate that ordinary English properly used has the resources for intricate and philosophically sound investigation of rather deep issues in logic and philosophy of language. No mathematical, logical, or linguistic symbols are used. Meanings need to be identified and clarified before being expressed in symbols. We hope to establish that clarity is served by deferring the extensive use of formalized or logically perfect languages until a solid “informal” foundation has been established. Questions of “ontological status”—e.g., whether propositions or sentences, or for that matter characters, numbers, truth-values, or instants, are “real entities”, are “idealizations”, or are “theoretical constructs”—plays no role in this paper. As is suggested by the title, this paper is written to be read aloud. -/- I hope that reading this aloud in groups will unite people in the enjoyment of the humanistic spirit of analytic philosophy. (shrink)

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. This lecture expands points which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

RÉSUMÉ : Dans son récent recueil d’articles Language, Truth and History, Donald Davidson semble pencher en faveur d’une philosophie du langage mettant l’accent sur la notion de l’intention communicative du sujet parlant; en quoi il se rapproche du point de vue de Paul Grice. Si cela est juste, la pensée de Davidson se serait dégagée de l’approche sémantique formelle qu’il soutenait dans ses Inquiries into Truth and Interpretation. Dans cet article, je soutiens que, bien qu’il y ait beaucoup de similitudes (...) entre ces deux penseurs, Davidson n’a pas abandonné ses précédentes vues sur le langage. (shrink)

This essay takes logic and ethics in broad senses: logic as the science of evidence; ethics as the science justice. One of its main conclusions is that neither science can be fruitfully pursued without the virtues fostered by the other: logic is pointless without fairness and compassion; ethics is pointless without rigor and objectivity. The logician urging us to be dispassionate is in resonance and harmony with the ethicist urging us to be compassionate.

This is an encyclopedia article about epistemic closure principles. The article explains what they are, their various philosophical uses, how they are argued for or against, and provides an overview of the related literature.

This accessible essay treats knowledge and belief in a usable and applicable way. Many of its basic ideas have been developed recently in Corcoran-Hamid 2014: Investigating knowledge and opinion. The Road to Universal Logic. Vol. I. Arthur Buchsbaum and Arnold Koslow, Editors. Springer. Pp. 95-126. http://www.springer.com/birkhauser/mathematics/book/978-3-319-10192-7 .

This paper raises obvious questions undermining any residual confidence in Mates work and revealing our embarrassing ignorance of true nature of Stoic deduction. It was inspired by the challenging exploratory work of JOSIAH GOULD.

Misinformation can have significant societal consequences. For example, misinformation about climate change has confused the public and stalled support for mitigation policies. When people lack the expertise and skill to evaluate the science behind a claim, they typically rely on heuristics such as substituting judgment about something complex (i.e. climate science) with judgment about something simple (i.e. the character of people who speak about climate science) and are therefore vulnerable to misleading information. Inoculation theory offers one approach to effectively neutralize (...) the influence of misinformation. Typically, inoculations convey resistance by providing people with information that counters misinformation. In contrast, we propose inoculating against misinformation by explaining the fallacious reasoning within misleading denialist claims. We offer a strategy based on critical thinking methods to analyse and detect poor reasoning within denialist claims. This strategy includes detailing argument structure, determining the truth of the premises, and checking for validity, hidden premises, or ambiguous language. Focusing on argument structure also facilitates the identification of reasoning fallacies by locating them in the reasoning process. Because this reason-based form of inoculation is based on general critical thinking methods, it offers the distinct advantage of being accessible to those who lack expertise in climate science. We applied this approach to 42 common denialist claims and find that they all demonstrate fallacious reasoning and fail to refute the scientific consensus regarding anthropogenic global warming. This comprehensive deconstruction and refutation of the most common denialist claims about climate change is designed to act as a resource for communicators and educators who teach climate science and/or critical thinking. (shrink)

This paper questions the idea that theism can function as an explanatory hypothesis to account for the nature and origins of the cosmos. Invoking God cannot dissolve the mystery of existence, and the characteristic religious response here is one of awe and humility. I then address David E. Cooper’s challenge of showing how a ”doctrine of mystery’ can have any discursible content. It is argued that certain aspects of our human experience afford us glimpses of the divine nature -- intimations (...) of the transcendent, which shine through from the ineffable source of our being to the human world we inhabit. (shrink)

This elementary 4-page paper is a preliminary survey of some of the most important uses of ‘condition’ and ‘consequence’ in American Philosophy. A more comprehensive treatment is being written. Your suggestions, questions, and objections are welcome. A statement of a conditional need not be a conditional statement and conditional statement need not be a statement of a conditional.

An information recovery problem is the problem of constructing a proposition containing the information dropped in going from a given premise to a given conclusion that folIows. The proposition(s) to beconstructed can be required to satisfy other conditions as well, e.g. being independent of the conclusion, or being “informationally unconnected” with the conclusion, or some other condition dictated by the context. This paper discusses various types of such problems, it presents techniques and principles useful in solving them, and it develops (...) algorithmic methods for certain classes of such problems. The results are then applied to classical number theory, in particular, to questions concerning possible refinements of the 1931 Gödel Axiom Set, e.g. whether any of its axioms can be analyzed into “informational atoms”. Two propositions are “informationally unconnected” [with each other] if no informative (nontautological) consequence of one also follows from the other. A proposition is an “informational atom” if it is informative but no information can be dropped from it without rendering it uninformative (tautological). Presentation, employment, and investigation of these two new concepts are prominent features of this paper. (shrink)

A Inseparabilidade entre Lógica e a Ética. Philósophos. 18 (2013) 245–259. Portuguese translation by Décio Krause and Pedro Merlussi: The Inseparability of Logic and Ethics, Free Inquiry, Spring 1989, 37–40. This essay takes logic and ethics in broad senses: logic as the science of evidence; ethics as the science of justice. One of its main conclusions is that neither science can be fruitfully pursued without the virtues fostered by the other: logic is pointless without fairness and compassion; ethics is pointless (...) without rigor and objectivity. The logician’s advice to be dispassionate is in resonance and harmony with the ethicist’s advice to be compassionate. (shrink)

We begin with an introductory overview of contributions made by more than twenty scholars associated with the Philosophy Department at the University of Buffalo during the last half-century to our understanding and evaluation of Aristotle's logic. More well-known developments are merely mentioned in..

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”. (shrink)

C. I. Lewis (I883-I964) was the first major figure in history and philosophy of logic—-a field that has come to be recognized as a separate specialty after years of work by Ivor Grattan-Guinness and others (Dawson 2003, 257).Lewis was among the earliest to accept the challenges offered by this field; he was the first who had the philosophical and mathematical talent, the philosophical, logical, and historical background, and the patience and dedication to objectivity needed to excel. He was blessed with (...) many fortunate circumstances, not least of which was entering the field when mathematical logic, after only six decades of toil, had just reaped one of its most important harvests with publication of the monumental Principia Mathematica. It was a time of joyful optimism which demanded an historical account and a sober philosophical critique. Lewis was one of the first to apply to mathematical logic the Aristotelian dictum that we do not understand a living institution until we see it growing from its birth. (shrink)

The logical form of a discourse—such as a proposition, a set of propositions, an argument, or an argumentation—is obtained by abstracting from the subject-matter of its content terms or by regarding the content terms as mere place-holders or blanks in a form. In a logically perfect language the logical form of a proposition, a set of propositions, an argument, or an argumentation is determined by the grammatical form of the sentence, the set of sentences, the argument-text, or the argumentation-text expressing (...) it. Two such discourse-texts are said to have the same grammatical form, in this sense, if a uniform one-one substitution of content words transforms the one exactly into the other. The sentence ‘Abe properly respects every agent who respects himself’ may be regarded as having the same grammatical form as the sentence ‘Ben generously assists every patient who assists himself’. (shrink)

This work treats the correlative concepts knowledge and opinion, in various senses. In all senses of ‘knowledge’ and ‘opinion’, a belief known to be true is knowledge; a belief not known to be true is opinion. In this sense of ‘belief’, a belief is a proposition thought to be true—perhaps, but not necessarily, known to be true. All knowledge is truth. Some but not all opinion is truth. Every proposition known to be true is believed to be true. Some but (...) not every proposition believed to be true is known to be true. Our focus is thus on propositional belief (“belief-that”): the combination of propositional knowledge (“knowledge-that”) and propositional opinion (“opinion-that”). Each of a person’s beliefs, whether knowledge or opinion, is the end result of a particular thought process that continued during a particular time interval and ended at a particular time with a conclusive act—a judgment that something is the case. This work is mainly about beliefs in substantive informative propositions—not empty tautologies. We also treat objectual knowledge (knowledge of objects in the broadest sense, or “knowledge-of”), operational knowledge (abilities and skills, “knowledge-how-to”, or “know-how”), and expert knowledge (expertise). Most points made in this work have been made by previous writers, but to the best of our knowledge, they have never before been collected into a coherent work accessible to a wide audience. -/- Key words: belief, knowledge/opinion, propositional, operational, objectual, cognition, . (shrink)

ABSTRACT: In its strongest unqualified form, the principle of wholistic reference is that in any given discourse, each proposition refers to the whole universe of that discourse, regardless of how limited the referents of its non-logical or content terms. According to this principle every proposition of number theory, even an equation such as "5 + 7 = 12", refers not only to the individual numbers that it happens to mention but to the whole universe of numbers. This principle, its history, (...) and its relevance to some of Oswaldo Chateaubriand's work are discussed in my 2004 paper "The Principle of Wholistic Reference" in Essays on Chateaubriand's "Logical Forms". In Chateaubriand's réplica (reply), which is printed with my paper, he raised several important additional issues including the three I focus on in this tréplica (reply to his reply): truth-values, universes of discourse, and formal ontology. This paper is self-contained: it is not necessary to have read the above-mentioned works. The principle of wholistic reference (PWR) was first put forth by George Boole in 1847 when he espoused a monistic fixed-universe viewpoint similar to the one Frege and Russell espoused throughout their careers. Later, Boole elaborated PWR in 1854 from the pluralistic multiple-universes perspective. (shrink)

A counterfactual is a conditional statement in the subjunctive mood. For example: If Suzy hadn’t thrown the rock, then the bottle wouldn’t have shattered. The philosophical importance of counterfactuals stems from the fact that they seem to be closely connected to the concept of causation. Thus it seems that the truth of the above conditional is just what is required for Suzy’s throw to count as a cause of the bottle’s shattering. If philosophers were reluctant to exploit this idea prior (...) to 1970, it was because of a widespread feeling that the truth-conditions of the counterfactual conditional were not sufficiently well understood. The development of a formal semantics for counterfactuals by Robert Stalnaker [1968] and David Lewis [1973b] stands as a major recent achievement in philosophical logic. (shrink)

This paper corrects a mistake I saw students make but I have yet to see in print. The mistake is thinking that logically equivalent propositions have the same counterexamples—always. Of course, it is often the case that logically equivalent propositions have the same counterexamples: “every number that is prime is odd” has the same counterexamples as “every number that is not odd is not prime”. The set of numbers satisfying “prime but not odd” is the same as the set of (...) numbers satisfying “not odd but not not-prime”. The mistake is thinking that every two logically-equivalent false universal propositions have the same counterexamples. Only false universal propositions have counterexamples. A counterexample for “every two logically-equivalent false universal propositions have the same counterexamples” is two logically-equivalent false universal propositions not having the same counterexamples. The following counterexample arose naturally in my sophomore deductive logic course in a discussion of inner and outer converses. “Every even number precedes every odd number” is counterexemplified only by even numbers, whereas its equivalent “Every odd number is preceded by every even number” is counterexemplified only by odd numbers. Please let me know if you see this mistake in print. Also let me know if you have seen these points discussed before. I learned them in my own course: talk about learning by teaching! (shrink)

Critical thinking involves deliberate application of tests and standards to beliefs per se and to methods used to arrive at beliefs. Pedagogical license is authorization accorded to teachers permitting them to use otherwise illicit means in order to achieve pedagogical goals. Pedagogical license is thus analogous to poetic license or, more generally, to artistic license. Pedagogical license will be found to be pervasive in college teaching. This presentation suggests that critical thinking courses emphasize two topics: first, the nature and usefulness (...) of critical thinking; second, the nature and pervasiveness of pedagogical license. Awareness of pedagogical license alerts the student to the need for critical thinking. Indoctrination is done to students; education is done by students. (shrink)

Information-theoretic approaches to formal logic analyze the "common intuitive" concepts of implication, consequence, and validity in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; one given proposition is a consequence of a second if the latter contains all of the information contained by the former; an argument is valid if the conclusion contains no information beyond that of the premise-set. This (...) paper locates information-theoretic approaches historically, philosophically, and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves and by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyze validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. (shrink)

Tarski’s Convention T—presenting his notion of adequate definition of truth (sic)—contains two conditions: alpha and beta. Alpha requires that all instances of a certain T Schema be provable. Beta requires in effect the provability of ‘every truth is a sentence’. Beta formally recognizes the fact, repeatedly emphasized by Tarski, that sentences (devoid of free variable occurrences)—as opposed to pre-sentences (having free occurrences of variables)—exhaust the range of significance of is true. In Tarski’s preferred usage, it is part of the meaning (...) of true that attribution of being true to a given thing presupposes the thing is a sentence. Beta’s importance is further highlighted by the fact that alpha can be satisfied using the recursively definable concept of being satisfied by every infinite sequence, which Tarski explicitly rejects. Moreover, in Definition 23, the famous truth-definition, Tarski supplements “being satisfied by every infinite sequence” by adding the condition “being a sentence”. Even where truth is undefinable and treated by Tarski axiomatically, he adds as an explicit axiom a sentence to the effect that every truth is a sentence. Surprisingly, the sentence just before the presentation of Convention T seems to imply that alpha alone might be sufficient. Even more surprising is the sentence just after Convention T saying beta “is not essential”. Why include a condition if it is not essential? Tarski says nothing about this dissonance. Considering the broader context, the Polish original, the German translation from which the English was derived, and other sources, we attempt to determine what Tarski might have intended by the two troubling sentences which, as they stand, are contrary to the spirit, if not the letter, of several other passages in Tarski’s corpus. (shrink)

This paper has as its topic two recent philosophical disputes. One of these disputes is internal to the project known as decision theory, and while by now familiar to many, may well seem to be of pressing concern only to specialists. It has been carried on over the last twenty years or so, but by now the two opposing camps are pretty well entrenched in their respective positions, and the situation appears to many observers (as well as to some of (...) the parties involved) to have reached a sort of stalemate. The second of these two disputes is, on the other hand, very much alive. While it has been framed in decision theoretic terms, it is definitely not a dispute internal to that enterprise. It is, rather, a debate about the very coherence of the notion of objective value, and as such touches on issues of central importance to, for example, meta–ethics and moral psychology. (shrink)

I argue that the dispute between two leading theories of interpretation of legal texts, textual originalism and textual evolutionism, depends on the false presupposition that changes in the way a word is used necessarily require a change in the word’s meaning. Semantic externalism goes a long way towards reconciling these views by showing how a word’s semantic properties can be stable over time, even through vicissitudes of usage. I argue that temporal externalism can account for even more semantic stability, however. (...) Temporal externalism is the theory that the content of an utterance at time t may be determined by developments in linguistic usage subsequent to t. If this semantic theory is correct, then the originalist and evolutionist positions effectively collapse. Originalism is correct in that the original meaning of the text is the meaning that is binding on jurists, but evolutionism is vindicated, as it is the current practices and standards that determine the meaning the text now has, and has always had. Objections to temporal externalism, and to its application to the interpretation of legal texts, are considered and addressed. (shrink)

Our modern egalitarian and individualistic age is suspicious of authority, and in recent times there have been almost daily reports in the press of cases where trust in various authorities, including financial, governmental, political and religious, has been found to have been abused or misplaced. Such disappointments seem to bolster the case for withholding trust in external authority and falling back on one’s own resources. But if the lessons from Linda Zagzebski’s groundbreaking work are accepted, 1 self- reliance turns out (...) to be a confused and probably incoherent ideal ; and the rational and self-reflective person is committed to believing and acting on authority. In the second half of this short discussion paper I shall raise some possible concerns about Zagzebski’s positive case for reliance on authority, focussing on the moral and religious spheres. First however, let me say something about the negative part of Zagzebski’s work, her critique of self-reliance. Since I find this wholly convincing, I shall confine myself to some supplementary observations, mainly to do with the historical context in which her critique is located. (shrink)

I am saying farewell after more than forty happy years of teaching logic at the University of Buffalo. But this is only a partial farewell. I will no longer be at UB to teach classroom courses or seminars. But nothing else will change. I will continue to be available for independent study. I will continue to write abstracts and articles with people who have taken courses or seminars with me. And I will continue to honor the LogicLifetimeGuarantee™, which is earned (...) by taking one of my logic courses or seminars. As you know, according to the terms of the LogicLifetimeGuarantee™, I stand behind everything I teach. If you find anything to be unsatisfactory, I am committed to fixing it. If you forget anything, I will remind you. If you have questions, I will answer them or ask more questions. And if you need more detail on any topic we discussed, I will help you to broaden and deepen your knowledge—and maybe write an abstract or article. Stay in touch. (shrink)

This review concludes that if the authors know what mathematical logic is they have not shared their knowledge with the readers. This highly praised book is replete with errors and incoherency.

With the growth of epistemology, an important debate in philosophy of religion has arisen: can mystical encounters—purported feelings of intense unity with the divine—serve as epistemic warrants? In this paper, I examine two of the most prominent and promising standards by which to determine the veridicality of such encounters—those of William Alston and Richard Swinburne—and demonstrate their respective strengths and shortcomings. Considering these shortcomings, I compose and defend my own set of criteria to use in evaluating the veridicality of putative (...) mystical experiences which draws upon the subject’s religious tradition, rationality, and affectivity. (shrink)

Newcomb’s problem is a decision puzzle whose difficulty and interest stem from the fact that the possible outcomes are probabilistically dependent on, yet causally independent of, the agent’s options. The problem is named for its inventor, the physicist William Newcomb, but first appeared in print in a 1969 paper by Robert Nozick [12]. Closely related to, though less well-known than, the Prisoners’ Dilemma, it has been the subject of intense debate in the philosophical literature. After three decades, the issues remain (...) unresolved. Newcomb’s problem is of genuine importance because it poses a challenge to the theoretical adequacy of orthodox Bayesian decision theory. It has led both to the development of causal decision theory and to efforts aimed at defending the adequacy of the orthodox theory. (shrink)