“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)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. These (...) practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (shrink)

A curious feature of Aztec philosophy is that the basic metaphysical question of the “Western” tradition cannot be formulated in their language, in Nahuatl. This did not, however, prevent the Aztecs from developing an account of 'reality', or whatever it is that might exist. The article is the first of its kind to compare the work of Aristotle on ousia (being) and the Aztecs on teotl and ometeotl. Through this analysis, it suggests that both of the Nahuatl terms are fundamental (...) for expressing the basic character of reality as process for the Aztecs. It also defends their notion as prima facia reasonable, despite its wide divergence from Aristotle's view. (shrink)

This is a theoretical treatment of the term "Sha Zhu Pan" (杀猪盘) in Chinese, which translates to “Pig-Butchering” in English. The article critically examines the propagation and validation of "Pig Butchering," an animal metaphor, and its implications for the dehumanisation of victims of online fraud across various discourses. The study provides background information about this type of fraud before investigating its theoretical foundations and linking its emergence to the dehumanisation of fraud victims. The analysis highlights the disparity between academic literature, (...) subjected to rigorous peer-review processes, and sensationalised narratives prevalent in the media. While academic works subject "pig butchering" to critical scrutiny and refrain from endorsing derogatory terms to depict fraud victims, numerous media outlets employ this term uncritically, further worsening the predicament of these victims. "Pig butchering" is firmly rooted in the concept of dehumanisation, and this article underscores how language moulds perceptions of fraud and behaviour, extending to the development of preventive strategies. The role of law enforcement agencies in generating and disseminating materials is also a central theme, emphasising their responsibility as trusted sources of information. We suggest that these agencies should adopt non-victim shaming language to encourage victims to report crimes and alleviate the stigma attached to victimisation. Additionally, the article offers valuable cross-cultural insights by comparing metaphors from Chinese and Nigerian contexts. This comparative analysis enriches our comprehension of the global dimensions of online fraud and its cultural diversities, highlighting the substantial impact of language on perceptions and behaviours. We advocate for a departure from victim-blaming tendencies perpetuated by select media outlets, urging a more compassionate and accurate portrayal of those affected by online fraud. We, therefore, call for a more empathetic and accurate portrayal of individuals affected by online fraud, aligning with the broader objective of promoting understanding and support for these victims. (shrink)

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)

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)

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)

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)

This essay takes a first step in comparative ethics by looking to Aristotle and the Aztec's conceptions of the good life. It argues that the Aztec conception of a rooted life, neltiliztli, functions for ethical purposes in a way that is like Aristotle's eudaimonia. To develop this claim, it not only shows just in what their conceptions of the good consist, but also in what way the Aztecs conceived of the virtues (in qualli, in yectli).

In this paper we prove the completeness of three logical systems I LI, IL2 and IL3. IL1 deals solely with identities {a = b), and its deductions are the direct deductions constructed with the three traditional rules: (T) from a = b and b = c infer a = c, (S) from a = b infer b = a and (A) infer a = a(from anything). IL2 deals solely with identities and inidentities {a ± b) and its deductions include both (...) the direct and the indirect deductions constructed with the three traditional rules. IL3 is a hybrid of IL1 and IL2: its deductions are all direct as in IL1 but it deals with identities and inidentities as in IL2. IL1 and IL2 have a high degree of naturalness. Although the hybrid system IL3 was constructed as an artifact useful in the mathematical study of IL1 and IL2, it nevertheless has some intrinsically interesting aspects. The main motivation for describing and studying such simple systems is pedagogical. In teaching beginning logic one would like to present a system of logic which has the following properties. First, it exemplifies the main ideas of logic: implication, deduction, non-implication, counterargument(or countermodel), logical truth, self-contradiction, consistency,satisfiability, etc. Second, it exemplifies the usual general metaprinciples of logic: contraposition and transitivity of implication, cut laws, completeness,soundness, etc. Third, it is simple enough to be thoroughly grasped by beginners. Fourth, it is obvious enough so that its rules do not appear to be arbitrary or purely conventional. Fifth, it does not invite confusions which must be unlearned later. Sixth, it involves a minimum of presuppositions which are no longer accepted in mainstream contemporary logic. (shrink)

Resident microbiota do not just shape host immunity, they can also contribute to host protection against pathogens and infectious diseases. Previous reviews of the protective roles of the microbiota have focused exclusively on colonization resistance localized within a microenvironment. This review shows that the protection against pathogens also involves the mitigation of pathogenic impact without eliminating the pathogens (i.e., “disease tolerance”) and the containment of microorganisms to prevent pathogenic spread. Protective microorganisms can have an impact beyond their niche, interfering with (...) the entry, establishment, growth, and spread of pathogenic microorganisms. More fundamentally, we propose a series of conceptual clarifications in support of the idea of a “co-immunity,” where an organism is protected by both its own immune system and components of its microbiota. -/- . (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)

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)

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)

Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08. -/- Constance Reid was an insider of the Berkeley-Stanford logic circle. Her San Francisco home was in Ashbury Heights near the homes of logicians such as Dana Scott and John Corcoran. Her sister Julia Robinson was one of the top mathematical logicians of her generation, as was Julia’s husband Raphael Robinson for whom Robinson Arithmetic was named. Julia was a Tarski PhD and, in (...) recognition of a distinguished career, was elected President of the American Mathematics Society. https://en.wikipedia.org/wiki/Julia_Robinson http://www.awm-math.org/noetherbrochure/Robinson82.html. (shrink)

Derogatory terms (racist, sexist, ethnic, and homophobic epithets) are bully words with ontological force: they serve to establish and maintain a corrupt social system fuelled by distinctions designed to justify relations of dominance and subordination. No wonder they have occasioned public outcry and legal response. The inferential role analysis developed here helps move us away from thinking of the harms as being located in connotation (representing mere speaker bias) or denotation (holding that the terms fail to refer due to inaccurate (...) descriptive content). The issue is not bad attitudes or referential misfires. An inferential role semantic analysis of derogatory terms shows exactly what is at stake between those who argue that the terms should be eliminated (Absolutists) and those who claim they can be successfully rehabilitated (Reclaimers). The Reclaimer maintains, and the Absolutist denies, that certain contexts can detach the derogatory force from deeply derogatory terms. The article looks at these claims with respect to ‘nigger’ and ‘dyke,’ setting out the inferential role of each term and examining detachability potential. Explaining detachability in terms of linguistic commitments, this article also addresses the issue of whether such terms count as political discourse, and examines the implications of that issue. (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 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)

This presentation of Aristotle's natural deduction system supplements earlier presentations and gives more historical evidence. Some fine-tunings resulted from conversations with Timothy Smiley, Charles Kahn, Josiah Gould, John Kearns,John Glanvillle, and William Parry.The criticism of Aristotle's theory of propositions found at the end of this 1974 presentation was retracted in Corcoran's 2009 HPL article "Aristotle's demonstrative logic".

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)

Chapin reviewed this 1972 ZEITSCHRIFT paper that proves the completeness theorem for the logic of variable-binding-term operators created by Corcoran and his student John Herring in the 1971 LOGIQUE ET ANALYSE paper in which the theorem was conjectured. This leveraging proof extends completeness of ordinary first-order logic to the extension with vbtos. Newton da Costa independently proved the same theorem about the same time using a Henkin-type proof. This 1972 paper builds on the 1971 “Notes on a Semantic Analysis (...) of Variable Binding Term Operators” (Co-author John Herring), Logique et Analyse 55, 646–57. MR0307874 (46 #6989). A variable binding term operator (vbto) is a non-logical constant, say v, which combines with a variable y and a formula F containing y free to form a term (vy:F) whose free variables are exact ly those of F, excluding y. Kalish-Montague 1964 proposed using vbtos to formalize definite descriptions “the x: x+x=2”, set abstracts {x: F}, minimization in recursive function theory “the least x: x+x>2”, etc. However, they gave no semantics for vbtos. Hatcher 1968 gave a semantics but one that has flaws described in the 1971 paper and admitted by Hatcher. In 1971 we give a correct semantic analysis of vbtos. We also give axioms for using them in deductions. And we conjecture strong completeness for the deductions with respect to the semantics. The conjecture, proved in this paper with Hatcher’s help, was proved independently about the same time by Newton da Costa. (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)

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)

Corcoran’s 27 entries in the 1999 second edition of Robert Audi’s Cambridge Dictionary of Philosophy [Cambridge: Cambridge UP]. -/- ancestral, axiomatic method, borderline case, categoricity, Church (Alonzo), conditional, convention T, converse (outer and inner), corresponding conditional, degenerate case, domain, De Morgan, ellipsis, laws of thought, limiting case, logical form, logical subject, material adequacy, mathematical analysis, omega, proof by recursion, recursive function theory, scheme, scope, Tarski (Alfred), tautology, universe of discourse. -/- The entire work is available online free at more (...) than one website. Paste the whole URL. http://archive.org/stream/RobertiAudi_The.Cambridge.Dictionary.of.Philosophy/Robert.Audi_The.Cambrid ge.Dictionary.of.Philosophy -/- The 2015 third edition will be available soon. Before you think of buying it read some reviews on Amazon and read reviews of its competition: For example, my review of the 2008 Oxford Companion to Philosophy, History and Philosophy of Logic,29:3,291-292. URL: http://dx.doi.org/10.1080/01445340701300429 -/- Some of the entries have already been found to be flawed. For example, Tarski’s expression ‘materially adequate’ was misinterpreted in at least one article and it was misused in another where ‘materially correct’ should have been used. The discussion provides an opportunity to bring more flaws to light. -/- Acknowledgements: Each of these entries was presented at meetings of The Buffalo Logic Dictionary Project sponsored by The Buffalo Logic Colloquium. The members of the colloquium read drafts before the meetings and were generous with corrections, objections, and suggestions. Usually one 90-minute meeting was devoted to one entry although in some cases, for example, “axiomatic method”, took more than one meeting. Moreover, about half of the entries are rewrites of similarly named entries in the 1995 first edition. Besides the help received from people in Buffalo, help from elsewhere was received by email. We gratefully acknowledge the following: José Miguel Sagüillo, John Zeis, Stewart Shapiro, Davis Plache, Joseph Ernst, Richard Hull, Concha Martinez, Laura Arcila, James Gasser, Barry Smith, Randall Dipert, Stanley Ziewacz, Gerald Rising, Leonard Jacuzzo, George Boger, William Demopolous, David Hitchcock, John Dawson, Daniel Halpern, William Lawvere, John Kearns, Ky Herreid, Nicolas Goodman, William Parry, Charles Lambros, Harvey Friedman, George Weaver, Hughes Leblanc, James Munz, Herbert Bohnert, Robert Tragesser, David Levin, Sriram Nambiar, and others. -/- . (shrink)

Feminists have urged women to take semantic authority. This article explains what such authority is, how it depends upon community recognition, and how it differs from privilege and from authority as usually conceived under patriarchy. Understanding its natures and limits is an important part of attaining it. Understanding the role of community explains why separatism is the logical conclusion of this project, and why separatism is valuable even to those who do not separate.

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)

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)

CORCORAN REVIEWS THE 4 VOLUMES OF TARSKI’S COLLECTED PAPERS Alfred Tarski (1901--1983) is widely regarded as one of the two giants of twentieth-century logic and also as one of the four greatest logicians of all time (Aristotle, Frege and Gödel being the other three). Of the four, Tarski was the most prolific as a logician. The four volumes of his collected papers, which exclude most of his 19 monographs, span over 2500 pages. Aristotle's writings are comparable in volume, but (...) most of the Aristotelian corpus is not about logic, whereas virtually everything written by Tarski concerns logic more or less directly. There is no doubt that Tarski wrote more on logic than any other author; he started publishing on logic in 1921 at the age of 20 and continued until his death at the age of 82. (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)

This presentation includes a complete bibliography of John Corcoran’s publications devoted at least in part to Aristotle’s logic. Sections I–IV list 20 articles, 43 abstracts, 3 books, and 10 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article that antedates Corcoran’s Aristotle’s studies and the Journal of Symbolic Logic article first reporting his original results; it ends with works published in 2015. A few of the items are annotated with endnotes connecting (...) them with other work. In addition, Section V “Discussions” is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 8 items published in the 1970s, 22 in the 1980s, 39 in the 1990s, 56 in the 2000s, and 65 in the current decade. The secondary bibliography is annotated with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. As is evident from the Acknowledgements sections, all of Corcoran’s publications benefited from correspondence with other scholars, most notably Timothy Smiley, Michael Scanlan, and Kevin Tracy. All of Corcoran’s Greek translations were done in consultation with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. REQUEST: Please send errors, omissions, and suggestions. I am especially interested in citations made in non-English publications. (shrink)

Brentano's philosophy of art, contained primarily in his book, Grundzuge der Ästhetik, is the result of an original theory of intrinsic value that was derived from Brentano's philosophical psychology. In his aesthetics, Brentano endeavored to find an objective ground for the value of aesthetic contemplation through his theory of the intentional objects of emotions and desires. The lack of attention Brentano's aesthetics has received is surprising, given that two of the many students Brentano influenced, Husserl (through the development of the (...) phenomenological movement) and Ehrenfels (through the development of Gestalt psychology) have had an extraordinary influence on twentieth century perceptions of art. In this paper I will attempt to redress some of this neglect by outlining Brentano's analysis of aesthetic intentions and the relationship his aesthetics bears to his overall philosophical system. (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.

It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in each possible world, (3) (...) some entities existing in the actual world do not exist in every possible world. Since these principles seem to be true, the paper is a refutation of the Frege-Russell definition. The paper does more. It shows that the contradictory of the Frege-Russell definition follows even when principles 2 and 3 are replaced by one considerably weaker principle. The ideas contained in the paper are related to two earlier objections to the definition. The first, sometimes attributed to the mathematician, C. S. Keyser, is that existence of the numbers as defined implies the existence of infinitely many particulars in each possible world. The second is, in effect, an idea which is said to have led Whitehead to reject the definition of number to which he had subscribed in Principia Mathematica. Whitehead is supposed to have said that he could not believe that the number two changes every “time twins are born”. The mathematician H. Jeffreys expressed similar ideas [Philos. of Sci. 5 (1938), 434–451]. One of the merits of the author’s work is that it refutes the Frege-Russell definition without the need to take sides on controversial points presupposed by the Keyser and Whitehead objections. The objections made by the author are therefore not to be identified with the Keyser and Whitehead objections. Even if the author’s work is to be regarded as a refinement and integration of previous ideas, it is nevertheless a contribution—not only because the basic points are well worth repeating but also because the refinements are logically significant improvements and because the author has stated them clearly and concisely in the idiom of contemporary philosophy. (shrink)

PUTNAM has made highly regarded contributions to mathematics, to philosophy of logic and to philosophy of science, and in this book he brings his ideas in these three areas to bear on the traditional philosophic problem of materialism versus (objective) idealism. The book assumes that contemporary science (mathematical and physical) is largely correct as far as it goes, or at least that it is rational to believe in it. The main thesis of the book is that consistent acceptance of contemporary (...) science requires the acceptance of some sort of Platonistic idealism affirming the existence of abstract, non-temporal, non-material, non-mental entities (numbers, scientific laws, mathematical formulas, etc.). The author is thus in direct opposition to the extreme materialism which had dominated philosophy of science in the first three quarters of this century. The book can be recommended to the scientifically literate, general reader whose acquaintance with these areas is limited to the literature of the 1950’s and before, when it had been assumed that empiricistic materialism was the only philosophy compatible with a scientific outlook. To this group the book presents an eye-opening challenge fulfilling the author’s intention of “shaking up preconceptions and stimulating further discussion”. QUINE’S book is not easy to read, partly because the level of sophistication fluctuates at high frequency between remote extremes and partly because of convoluted English prose style and devilish terminology. Almost all of the minor but troublesome technical errata in the first printing have been corrected [see reviews, e.g., the reviewer, Philos. Sci. 39 (1972), no. 1, 97–99]. In the opinion of the reviewer the book is not suitable for undergraduate instruction, and without external motivation few mathematicians are likely to have the patience to appreciate it. Nevertheless, a careful study of the book will more than repay the effort and one should expect to find frequent references to this book in coming years. (shrink)

This presentation includes a complete bibliography of John Corcoran’s publications relevant on Aristotle’s logic. The Sections I, II, III, and IV list respectively 23 articles, 44 abstracts, 3 books, and 11 reviews. Section I starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article—from Corcoran’s Philadelphia period that antedates his discovery of Aristotle’s natural deduction system—and the Journal of Symbolic Logic article—from his Buffalo period first reporting his original results. It ends with works published (...) in 2015. Some items are annotated as listed or with endnotes connecting them with other work and pointing out passages that, in retrospect, are seen to be misleading and in a few places erroneous. In addition, Section V, “Discussions”, is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 10 items published in the 1970s, 24 in the 1980s, 42 in the 1990s, 60 in the 2000s, and 70 in the current decade. The secondary bibliography is also annotated as listed or with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. Section VI, “Alternatives”, lists recent works on Aristotle’s logic oblivious of Corcoran’s research and, more generally in some cases, even of the Łukasiewicz-initiated tradition. As is evident from Section VII, “Acknowledgements”, Corcoran’s publications benefited from consultation with other scholars, most notably George Boger, Charles Kahn, John Mulhern, Mary Mulhern, Anthony Preus, Timothy Smiley, Michael Scanlan, Roberto Torretti, and Kevin Tracy. All of Corcoran’s Greek translations were done in collaboration with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. (shrink)

Peter H. Hare (1935-2008) developed informed, original views about the proposition: some published (Hare 1969 and Hare-Madden 1975); some expressed in conversations at scores of meetings of the Buffalo Logic Colloquium and at dinners following. The published views were expository and critical responses to publications by Curt J. Ducasse (1881-1969), a well-known presence in American logic, a founder of the Association for Symbolic Logic and its President for one term.1Hare was already prominent in the University of Buffalo's Philosophy Department in (...) 1969 when I was appointed. Soon after, he became Chair. As his Associate Chair from 1971to 1975, I spent many hours with him in Buffalo and on professional trips .. (shrink)

History witnesses alternative approaches to “the proposition”. The proposition has been referred to as the object of belief, disbelief, and doubt: generally as the object of propositional attitudes, that which can be said to be believed, disbelieved, understood, etc. It has also been taken to be the object of grasping, judging, assuming, affirming, denying, and inquiring: generally as the object of propositional actions, that which can be said to be grasped, judged true or false, assumed for reasoning purposes, etc. The (...) proposition has also been taken to be the subject of truth and falsity: generally as the subject of propositional properties, that which can be said to be true, false, tautological, informative, inconsistent,etc. It has also been taken as the subject and object of logical relations, e.g. that which can be said to imply, be implied, contradict, be contradicted, etc. Prima facie, such properties and relations are non-mental and objective. It has also been taken to be the resultants or products of propositional operations, usually mental or linguistic; e.g. judging, affirming, and denying have been held to produce propositions called judgments, affirmations, and negations, respectively. Propositions have also been taken to be certain declarative sentences. Finally,propositions have been taken to be meanings of certain declarative sentences. This essay is an informal, selective, and incomplete survey of alternative approaches to “the proposition” with special attention to the views of the late American philosopher Peter Hare (1935–2008)and of those who influenced him. DOI: 10.5007/1808-1711.2011v15n1p51. (shrink)

JUNE 2015 UPDATE: A BIBLIOGRAPHY: JOHN CORCORAN’S PUBLICATIONS ON ARISTOTLE 1972–2015 By John Corcoran -/- This presentation includes a complete bibliography of John Corcoran’s publications relevant to his research on Aristotle’s logic. Sections I, II, III, and IV list 21 articles, 44 abstracts, 3 books, and 11 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article from Corcoran’s Philadelphia period that antedates his Aristotle studies and the Journal of Symbolic (...) Logic article from his Buffalo period first reporting his original results; it ends with works published in 2015. A few of the items are annotated as listed or with endnotes connecting them with other work and pointing out passages that in-retrospect are seen to be misleading and in a few places erroneous. In addition, Section V, “Discussions”, is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 8 items published in the 1970s, 23 in the 1980s, 42 in the 1990s, 56 in the 2000s, and 69 in the current decade. The secondary bibliography is also annotated as listed or with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. Section VI, “Alternatives”, lists recent works on Aristotle’s logic oblivious of Corcoran’s research and, more generally, of the Lukasiewicz-initiated tradition. As is evident from Section VII, “Acknowledgements”, Corcoran’s publications benefited from consultation with other scholars, most notably Timothy Smiley, Michael Scanlan, Roberto Torretti, and Kevin Tracy. All of Corcoran’s Greek translations were done in collaboration with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. -/- REQUEST: Please send errors, omissions, and suggestions. I am especially interested in citations made in non-English publications. Also, let me know of passages I should comment on. (shrink)

Making sense of MacKinnon’s claim that pornography silences women requires attention to the discursive and interpretive frameworks that pornography establishes and promotes. Treating pornography as a form of hate speech is promising, but also limited. A close examination of a legal case, in which pornographic images were used to sexually harass, focuses on the hate speech analogy while illustrating the broad and lasting impact of such depictions when targeted at an individual. Applying the distinction between Absolutist and Reclaimer approaches to (...) hate speech and derogatory terms, the article concludes with a discussion of the possibility of creative development of discursive practices within a context that undermines the semantic authority of members of certain groups. (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)

Derogatory terms (racist, sexist, ethnic epithets) have long played various roles and achieved diverse ends in works of art. Focusing on basic aspects of an aesthetic object or work, this article examines the interpretive relation between point of view and content, asking how aesthetic contextualization shapes the impact of such terms. Can context, particularly aesthetic contexts, detach the derogatory force from powerful epithets and racist and sexist images? What would it be about aesthetic contexts that would make this possible? The (...) article puts into aesthetic contexts both Absolutist and Reclaimer views about the power of derogatory terms, applying an inferential role semantics, developed for language, to both words and images. The Reclaimer’s position, that derogation is a matter of pragmatics, of use, emphasizes the context-sensitivity of the term, that is matters who says it, to whom, when, where, and why. Understanding derogatory terms used as hate speech helps to make this position clear. The Absolutist’s position, that no contextual variations can detach the derogation from these deeply derogatory terms, is developed here through attention to subordinating images in pornography. The article argues that art may be able to achieve the deviant interpretations that the Reclaimer needs to undermine the derogations while maintaining sensitivity and awareness of the history of oppression. (shrink)

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 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.

One of the deepest assumptions of Judaism and its offspring, Christianity, is that there is an important difference between human persons and everything else that exists in Creation. We alone are made in God’s image. We alone are the stewards of the earth. It is said in Genesis that we have “dominion over the fish of the sea, and over the birds of the air, and over the cattle, and over all the earth, and over every creeping thing that creeps (...) upon the earth.” It is difficult to see how a traditional theist could deny the significance of the difference between human persons and the rest of Creation. We human persons are morally and ontologically special. (shrink)

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.