Epistemic contextualism is one of the most hotly debated topics in contemporary epistemology. Contextualists claim that ‘know’ is a context-sensitive verb associated with different evidential standards in different contexts. Contextualists motivate their view based on a set of behavioural claims. In this paper, I show that several of these behavioural claims are false. I also show that contextualist test cases suffer from a critical confound, which derives from people's tendency to defer to speakers’ statements about their own mental states. My (...) evidence consists in results from several behavioural experiments. I conclude that contextualism is an idle hypothesis and I propose some general methodological lessons. (shrink)

There is a concern that the widespread deployment of autonomous machines will open up a number of ‘responsibility gaps’ throughout society. Various articulations of such techno-responsibility gaps have been proposed over the years, along with several potential solutions. Most of these solutions focus on ‘plugging’ or ‘dissolving’ the gaps. This paper offers an alternative perspective. It argues that techno-responsibility gaps are, sometimes, to be welcomed and that one of the advantages of autonomous machines is that they enable us to embrace (...) certain kinds of responsibility gap. The argument is based on the idea that human morality is often tragic. We frequently confront situations in which competing moral considerations pull in different directions and it is impossible to perfectly balance these considerations. This heightens the burden of responsibility associated with our choices. We cope with the tragedy of moral choice in different ways. Sometimes we delude ourselves into thinking the choices we make were not tragic ; sometimes we delegate the tragic choice to others ; sometimes we make the choice ourselves and bear the psychological consequences. Each of these strategies has its benefits and costs. One potential advantage of autonomous machines is that they enable a reduced cost form of delegation. However, we only gain the advantage of this reduced cost if we accept that some techno-responsibility gaps are virtuous. (shrink)

Researchers recently demonstrated a strong direct relationship between judgments about what a person knows and judgments about how a person should act. But it remains unknown whether actionability judgments cause knowledge judgments, or knowledge judgments cause actionability judgments. This paper uses causal modeling to help answer this question. Across two experiments, we found evidence that actionability judgments cause knowledge judgments.

Synthesizing the domains of investigation highlighted in current research in distributed cognition and related fields, this paper offers an initial taxonomy of the overlapping types of resources which typically contribute to distributed or extended cognitive systems. It then outlines a number of key dimensions on which to analyse both the resulting integrated systems and the components which coalesce into more or less tightly coupled interaction over the course of their formation and renegotiation.

According to factive accounts of the norm of belief and decision-making, you should not believe or base decisions on a falsehood. Even when the evidence misleadingly suggests that a false proposition is true, you should not believe it or base decisions on it. Critics claim that factive accounts are counterintuitive and badly mischaracterize our ordinary practice of evaluating beliefs and decisions. This paper reports four experiments that rigorously test the critic’s accusations and the viability of factive accounts. The results undermine (...) the accusations and provide the best evidence yet of factive norms of belief and decision-making. The results also help discriminate between two leading candidates for a factive norm: truth and knowledge. Knowledge is the superior candidate. (shrink)

This article argues that access to meaningful sexual experience should be included within the set of the goods that are subject to principles of distributive justice. It argues that some people are currently unjustly excluded from meaningful sexual experience and it is not implausible to suggest that they might thereby have certain claim rights to sexual inclusion. This does not entail that anyone has a right to sex with another person, but it does entail that duties may be imposed on (...) society to foster greater sexual inclusion. This is a controversial thesis and this article addresses this controversy by engaging with four major objections to it: the misogyny objection; the impossibility objection; the stigmatisation objection; and the unjust social engineering objection. (shrink)

The idea that technologies can change moral beliefs and practices is an old one. But how, exactly, does this happen? This paper builds on an emerging field of inquiry by developing a synoptic taxonomy of the mechanisms of techno-moral change. It argues that technology affects moral beliefs and practices in three main domains: decisional (how we make morally loaded decisions), relational (how we relate to others) and perceptual (how we perceive situations). It argues that across these three domains there are (...) six primary mechanisms of techno-moral change: (i) adding options; (ii) changing decision-making costs; (iii) enabling new relationships; (iv) changing the burdens and expectations within relationships; (v) changing the balance of power in relationships; and (vi) changing perception (information, mental models and metaphors). The paper also discusses the layered, interactive and second-order effects of these mechanisms. (shrink)

Researchers have debated whether knowledge or certainty is a better candidate for the norm of assertion. Should you make an assertion only if you know it's true? Or should you make an assertion only if you're certain it's true? If either knowledge or certainty is a better candidate, then this will likely have detectable behavioral consequences. I report an experiment that tests for relevant behavioral consequences. The results support the view that assertability is more closely linked to knowledge than to (...) certainty. In multiple scenarios, people were much more willing to allow assertability and certainty to come apart than to allow assertability and knowledge to come apart. (shrink)

Assertion is fundamental to our lives as social and cognitive beings. By asserting we share knowledge, coordinate behavior, and advance collective inquiry. Accordingly, assertion is of considerable interest to cognitive scientists, social scientists, and philosophers. This paper advances our understanding of the norm of assertion. Prior evidence suggests that knowledge is the norm of assertion, a view known as “the knowledge account.” In its strongest form, the knowledge account says that knowledge is both necessary and sufficient for assertability: you should (...) make an assertion if and only if you know that it is true. The knowledge account has been rejected on the grounds that it conflicts with our ordinary practice of evaluating assertions. This paper reports four experiments that address an important objection of this sort, which focuses on a class of examples known as “Gettier cases.” The results undermine the objection and, in the process, provide further evidence for the knowledge account. The findings also teach some important general lessons about intuitional methodology and the curation of genres of thought experiment. (shrink)

Henry Shevlin’s paper—“How could we know when a robot was a moral patient?” – argues that we should recognize robots and artificial intelligence (AI) as psychological moral patients if they are cognitively equivalent to other beings that we already recognize as psychological moral patients (i.e., humans and, at least some, animals). In defending this cognitive equivalence strategy, Shevlin draws inspiration from the “behavioral equivalence” strategy that I have defended in previous work but argues that it is flawed in crucial respects. (...) Unfortunately—and I guess this is hardly surprising—I cannot bring myself to agree that the cognitive equivalence strategy is the superior one. In this article, I try to explain why in three steps. First, I clarify the nature of the question that I take both myself and Shevlin to be answering. Second, I clear up some potential confusions about the behavioral equivalence strategy, addressing some other recent criticisms of it. Third, I will explain why I still favor the behavioral equivalence strategy over the cognitive equivalence one. (shrink)

Compatibilism is the view that determinism is compatible with acting freely and being morally responsible. Incompatibilism is the opposite view. It is often claimed that compatibilism or incompatibilism is a natural part of ordinary social cognition. That is, it is often claimed that patterns in our everyday social judgments reveal an implicit commitment to either compatibilism or incompatibilism. This paper reports five experiments designed to identify such patterns. The results support a nuanced hybrid account: The central tendencies in ordinary social (...) cognition are compatibilism about moral responsibility, compatibilism about positive moral accountability, neither compatibilism nor incompatibilism about negative moral accountability, compatibilism about choice for actions with positive outcomes, and incompatibilism about choice for actions with negative or neutral outcomes. (shrink)

Recent work in philosophy and cognitive science shows that knowledge is the norm of our social practice of assertion, in the sense that an assertion should express knowledge. But why should an assertion express knowledge? I hypothesize that an assertion should express knowledge because the point of assertion is to transmit knowledge. I present evidence supporting this hypothesis.

Through a selective historical, theoretical, and critical survey of the uses of the concept of scaffolding over the past 30 years, this chapter traces the development of the concept across developmental psychology, educational theory, and cognitive anthropology, and its place in the interdisciplinary field of distributed cognition from the 1990s. Offering a big-picture overview of the uses of the notion of scaffolding, it suggests three ways to taxonomise forms of scaffolding, and addresses the possible criticism that the metaphor of scaffolding (...) retains an overly individualist vision of cognition. The chapter is aimed at a broad interdisciplinary audience interested in processes of learning, teaching, and apprenticeship as they apply to the study of memory. (shrink)

Many philosophers claim that interesting forms of epistemic evaluation are insensitive to truth in a very specific way. Suppose that two possible agents believe the same proposition based on the same evidence. Either both are justified or neither is; either both have good evidence for holding the belief or neither does. This does not change if, on this particular occasion, it turns out that only one of the two agents has a true belief. Epitomizing this line of thought are thought (...) experiments about radically deceived “brains in vats.” It is widely and uncritically assumed that such a brain is equally justified as its normally embodied human “twin.” This “parity” intuition is the heart of truth-insensitive theories of core epistemological properties such as justification and rationality. Rejecting the parity intuition is considered radical and revisionist. In this paper, I show that exactly the opposite is true. The parity intuition is idiosyncratic and widely rejected. A brain in a vat is not justified and has worse evidence than its normally embodied counterpart. On nearly every ordinary way of evaluating beliefs, a false belief is significantly inferior to a true belief. Of all the evaluations studied here, only blamelessness is truth-insensitive. (shrink)

I propose and defend the hypothesis that understanding is the norm of explanation. On this proposal, an explanation should express understanding. I call this the understanding account of explanation. The understanding account is supported by social and introspective observations. It is also supported by the relationship between knowledge and understanding, on the one hand, and assertion and explanation, on the other.

Recent work has shown that knowledge attributions affect how people think others should behave, more so than belief attributions do. This paper reports two experiments providing evidence that knowledge attributions also affect behavioral predictions more strongly than belief attributions do, and knowledge attributions facilitate faster behavioral predictions than belief attributions do. Thus, knowledge attributions play multiple critical roles in social cognition, guiding judgments about how people should and will behave.

Sex is an important part of human life. It is a source of pleasure and intimacy, and is integral to many people’s self-identity. This chapter examines the opportunities and challenges posed by the use of AI in how humans express and enact their sexualities. It does so by focusing on three main issues. First, it considers the idea of digisexuality, which according to McArthur and Twist (2017) is the label that should be applied to those ‘whose primary sexual identity comes (...) through the use of technology’, particularly through the use of robotics and AI. While agreeing that this phenomenon is worthy of greater scrutiny, the chapter questions whether it is necessary or socially desirable to see this as a new form of sexual identity. Second, it looks at the role that AI can play in facilitating human-to-human sexual contact, focusing in particular on the use of self-tracking and predictive analytics in optimising sexual and intimate behaviour. There are already a number of apps and services that promise to use AI to do this, but they pose a range of ethical risks that need to be addressed at both an individual and societal level. Finally, it considers the idea that a sophisticated form of AI could be an object of love. Can we be truly intimate with something that has been ‘programmed’ to love us? Contrary to the widely-held view, this chapter argues that this is indeed possible. (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)

A philosopher once wrote the following words:If I examine the PTOLOMAIC and COPERNICAN systems, I endeavour only, by my enquiries, to know the real situation of the planets; that is, in other words, I endeavour to give them, in my conception, the same relations, that they bear towards each other in the heavens. To this operation of the mind, therefore, there seems to be always a real, though often an unknown standard, in the nature of things; nor is truth or (...) falsehood variable by the various apprehensions of mankind. Though all human race should for ever conclude, that the sun moves, and the earth remains at rest, the sun stirs not an inch from his place for all these reasonings; and such conclusions are eternally false and erroneous. (shrink)

We are sometimes unsure of the moral status of our relationships with other entities. Recent case studies in this uncertainty include our relationships with artificial agents (robots, assistant AI, etc.), animals, and patients with “locked-in” syndrome. Do these entities have basic moral standing? Could they count as true friends or lovers? What should we do when we do not know the answer to these questions? An influential line of reasoning suggests that, in such cases of moral uncertainty, we need meta-moral (...) decision rules that allow us to either minimize the risks of moral wrongdoing or improve the choice-worthiness of our actions. One particular argument adopted in this literature is the “risk asymmetry argument,” which claims that the risks associated with accepting or rejecting some moral facts may be sufficiently asymmetrical as to warrant favoring a particular practical resolution of this uncertainty. Focusing on the case study of artificial beings, this article argues that this is best understood as an ethical-epistemic challenge. The article argues that taking potential risk asymmetries seriously can help resolve disputes about the status of human–AI relationships, at least in practical terms (philosophical debates will, no doubt, continue); however, the resolution depends on a proper, empirically grounded assessment of the risks involved. Being skeptical about basic moral status, but more open to the possibility of meaningful relationships with such entities, may be the most sensible approach to take. (shrink)

Questions about knowledge in expert sport are not only of applied significance: they also take us to the heart of foundational and heavily-disputed issues in the cognitive sciences. To a first (rough and far from uncontroversial) approximation, we can think of expert ‘knowledge’ as whatever it is that grounds or is applied in (more or less) effective decision-making, especially when in a competitive situation a performer follows one course of action out of a range of possibilities. In these research areas, (...) studies of motor expertise have for many years actively contributed to broader debates in philosophy and psychology (Abernethy, Burgess-Limerick, & Parks, 1994; Williams, Davids, & Williams, 1999). When we navigate the world flexibly and more or less successfully, how much is this due to a capacity to represent it? In considering alternative options, or planning future actions, we seem to transcend our present environment in some way: what is the balance or relation here between highly-tuned bodily dispositions and background knowledge of the world and its patterns? What changes in these regards as we gain experience and adapt to more complex and challenging environments? Is know-how fundamentally different in kind from ordinary factual knowledge of the world, or knowledge-that? And if expertise in a domain does involve or depend on a knowledge base that is somehow more organized or deeper than that of novices, how is this knowledge selectively and appropriately deployed, often under severe time constraints? (shrink)

This article argues that the creation of artificial offspring could make our lives more meaningful. By ‘artificial offspring’ I mean beings that we construct, with a mix of human and non-human-like qualities. Robotic artificial intelligences are paradigmatic examples of the form. There are two reasons for thinking that the creation of such beings could make our lives more meaningful and valuable. The first is that the existence of a collective afterlife—i.e. a set of human-like lives that continue after we die—is (...) likely to be an important source and sustainer of meaning in our present lives. The second is that the creation of artificial offspring provides a plausible and potentially better pathway to a collective afterlife than the traditional biological pathway. Both of these arguments are defended from a variety of objections and misunderstandings. (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)

The paper explores the handling of singular analogy in quantitative inductive logics. It concentrates on two analogical patterns coextensive with the traditional argument from analogy: perfect and imperfect analogy. Each is examined within Carnap’s λ-continuum, Carnap’s and Stegmüller’s λ-η continuum, Carnap’s Basic System, Hintikka’s α-λ continuum, and Hintikka’s and Niiniluoto’s K-dimensional system. Itis argued that these logics handle perfect analogies with ease, and that imperfect analogies, while unmanageable in some logics, are quite manageable in others. The paper concludes with a (...) modification of the K-dimensional system that synthesizes independent proposals by Kuipers and Niiniluoto. (shrink)

The primary goal of this research was to determine the influence of social media marketing (SMM) on the customer purchasing behavior (CPB) of senior high school (SHS) students. Utilizing the non-experimental quantitative method of research and validated questionnaires in data analysis with Mean, Person Product-Moment Correlation Coefficient (Pearson-r), and Multiple Linear Regression Analysis as statistical tools, the outcome displayed that the levels of social media marketing and customer purchasing behavior through the lens of SHS students are high, which means that (...) these two variables are oftentimes manifested. There is a strong positive relationship between social media marketing and customer purchasing behavior. It was also found out that quality content, as a domain of social media marketing, best influences customer purchasing behavior. (shrink)

Klaming and Vedder (2010) have argued that enhancement technologies that improve the epistemic efficiency of the legal system (“epistemic enhancements”) would benefit the common good. But there are two flaws to Klaming and Vedder’s argument. First, they rely on an under-theorised and under-specified conception of the common good. When theory and specification are supplied, their CGJ for enhancing eyewitness memory and recall becomes significantly less persuasive. And second, although aware of such problems, they fail to give due weight and consideration (...) to the tensions between the individual and common good. Taking these criticisms onboard, this article proposes an alternative, and stronger, CGJ for epistemic enhancements. The argument has two prongs. Drawing from the literature on social epistemology and democratic legitimacy, it is first argued that there are strong grounds for thinking that epistemic enhancements are a desirable way to improve the democratic legitimacy of the legal system. This gives prima facie but not decisive weight to the CGJ. It is then argued that due to the ongoing desire to improve the way in which scientific evidence is managed by the legal system, epistemic enhancement is not merely desirable but perhaps morally necessary. Although this may seem to sustain tensions between individual and common interests, I argue that in reality it reveals a deep constitutive harmony between the individual good and the common good, one that is both significant in its own right and one that should be exploited by proponents of enhancement. (shrink)

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.

Some moral disagreements are so persistent that we suspect they are deep : we would disagree even when we have all relevant information and no one makes any mistakes. The possibility of deep disagreement is thought to drive cognitivists toward relativism, but most cognitivists reject relativism. There is an alternative. According to divergentism, cognitivists can reject relativism while allowing for deep disagreement. This view has rarely been defended at length, but many philosophers have implicitly endorsed its elements. I will defend (...) it. (shrink)

Thomas Reid distinguished between natural and artificial language and argued that natural language has a very specific sort of priority over artificial language. This paper critically interprets Reid's discussion, extracts a Reidian explanatory argument for the priority of natural language, and places Reid's thought in the broad tradition of Cartesian linguistics.

This article covers four types of hedonism: ancient hedonism; ethical hedonism; axiological hedonism; and psychological hedonism. It concentrates on the latter two types, both by clarifying them and by discussing arguments in their behalf. It closes with a few words about the relevance of those positions to applied ethics.

This paper calls for a re-appraisal of McGee's analysis of the semantics, logic and probabilities of indicative conditionals presented in his 1989 paper Conditional probabilities and compounds of conditionals. The probabilistic measures introduced by McGee are given a new axiomatisation built on the principle that the antecedent of a conditional is probabilistically independent of the conditional and a more transparent method of constructing such measures is provided. McGee's Dutch book argument is restructured to more clearly reveal that it introduces a (...) novel contribution to the epistemology of semantic indeterminacy, and shows that its more controversial implications are unavoidable if we want to maintain the Ramsey Test along with the standard laws of probability. Importantly, it is shown that the counterexamples that have been levelled at McGee's analysis|generating a rather wide consensus that it yields `unintuitive' or `wrong' probabilities for compounds |fail to strike at their intended target; for to honour the intuitions of the counterexamples one must either give up the Ramsey Test or the standard laws of probability. It will be argued that we need to give up neither if we take the counterexamples as further evidence that the indicative conditional sometimes allows for a non-epistemic `causal' interpretation alongside its usual epistemic interpretation. (shrink)

I objected that the defeasibility theory of knowledge prohibits you from knowing that you know that p if your knowledge that p is a posteriori. Rodrigo Borges claims that Peter Klein has already satisfactorily answered a version of my objection. He attempts to defend Klein’s reply and argues that my objection fails because a principle on which it is based is false.I will show that my objection is not a version of the old one that Klein attempts (unsuccessfully) to address, (...) that Borges’ defence of Klein’s reply fails and that his argument against my new objection leaves it untouched. (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)

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)

Rachael Briggs and Daniel Nolan have recently proposed an improved version of Nozick’s tracking account of knowledge. I show that, despite its virtues, the new proposal suffers from three serious problems.

Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the last: the (...) parallel postulate. -/- Euclid’s axioms are general principles of magnitude: they concern geometric magnitudes and magnitudes of other kinds as well even numbers. The first is often translated “Things that equal the same thing equal one another”. -/- There are other differences that are or might become important. -/- Aristotle [fl. 350 BCE] meticulously separated his basic principles [archai, singular archê] according to subject matter: geometrical, arithmetic, astronomical, etc. However, he made no distinction that can be assimilated to Euclid’s postulate/axiom distinction. -/- Today we divide basic principles into non-logical [topic-specific] and logical [topic-neutral] but this too is not the same as Euclid’s. In this regard it is important to be cognizant of the difference between equality and identity—a distinction often crudely ignored by modern logicians. Tarski is a rare exception. The four angles of a rectangle are equal to—not identical to—one another; the size of one angle of a rectangle is identical to the size of any other of its angles. No two angles are identical to each other. -/- The sentence ‘Things that equal the same thing equal one another’ contains no occurrence of the word ‘magnitude’. This paper considers the problem of formalizing the proposition Euclid intended as a principle of magnitudes while being faithful to the logical form and to its information content. (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)

In its strongest, unqualified form the principle of wholistic reference is that each and every proposition refers to the whole universe of discourse as such, regardless how limited the referents of its non-logical or content terms. Even though Boole changed from a monistic fixed-universe framework in his earlier works of 1847 and 1848 to a pluralistic multiple-universe framework in his mature treatise of 1854, he never wavered in his frank avowal of the principle of wholistic reference, possibly in a slightly (...) weaker form. Indeed, he took it as an essential accompaniment to his theory of concept formation and proposition formation. Similar views are found in later logicians, and some of the most recent formulations of standard, one-sorted first-order logic seem to be in accord with a form of it, if they do not actually imply the principle itself. (shrink)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...) At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

1971. Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value, Journal of Structural Learning 3, #2, 1–16. REPRINTED 1976. Structural Learning II Issues and Approaches, ed. J. Scandura, Gordon & Breach Science Publishers, New York, MR56#15263. -/- This is the second of a series of three articles dealing with application of linguistics and logic to the study of mathematical reasoning, especially in the setting of a concern for improvement of (...) mathematical education. The present article presupposes the previous one. Herein we develop our ideas of the purposes of a theory of proof and the criterion of success to be applied to such theories. In addition we speculate at length concerning the specific kinds of uses to which a successful theory of proof may be put vis-a-vis improvement of various aspects of mathematical education. The final article will deal with the construction of such a theory. The 1st is the 1971. Discourse Grammars and the Structure of Mathematical Reasoning I: Mathematical Reasoning and Stratification of Language, Journal of Structural Learning 3, #1, 55–74. https://www.academia.edu/s/fb081b1886?source=link . (shrink)

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)

Contrary to dictionaries, a non sequitur isn’t “any statement that doesn’t follow logically from previous statements”. Otherwise, every opening statement would be a non sequitur: a non sequitur is a statement claimed to follow from previous statements but that doesn’t follow. If the sentence making a given statement doesn’t contain ‘thus’, ‘so’, ‘hence’, ‘therefore’, or something else indicating an implication claim, the statement isn’t a non sequitur in this sense. But this is only one of several senses of that expression, (...) which, contrary to appearances, is a unitary common noun more properly written as one word: non-sequitur or nonsequitur. Above a nonsequitur was, roughly speaking, a conclusion of an invalid premise- conclusion argument, but not every such, only those that were stated to follow from premises that had been stated. Every informative proposition is the conclusion of an invalid argument. Anyway, nonsequiturs were not arguments. However, the expression ‘nonsequitur’ also means “invalid premise- conclusion argument”. It also means something like “fallacious reasoning” as in ‘Abe’s proof of the big-bang hypothesis is a nonsequitur’. The English noun is also used for an irrelevancy, an abrupt change of topic, or for any of many other expressions that are gratuitous, inappropriate, incongruous, or in some other way out of place. The English noun is derived from the Latin expression ‘non sequitur’, which is not a common noun but a sentence variously meaning “it isn’t in sequence ”, “it doesn’t follow”, “it is out of order”, etc. A statement that begins ‘in the second place’ is a nonsequitur if the previous sentence lacked implication of something like ‘in the first place’ or if it contained something similar to ‘in the third place’. In fact, a true implication claim can be a nonsequitur if there is no reason for making it. If Ben says ‘Hence, Abe swims and Abe swims’ in reply to Carl’s statement ‘Abe swims‘, then Ben may well be committing a nonsequitur. Many conversation stoppers are nonsequiturs. This lecture surveys and analyses a wide selection of uses of ‘nonsequitur’ in logic and logic-related literature. -/- END OF DRAFT 4 -/- NOTE 091215: When Rick Perry was discussing possible withdrawal from the race—unfortunately not the human race but the Republican race—a reporter reminded him that just last week Donald Trump predicted the Perry withdrawal. Perry interrupted with “A broken clock is right once a day”. -/- If you did not know Perry you might think his reply was a brilliant diversionary nonsequitur. If you know Perry you might think his reply was a stupid nonsequitur. -/- For the record, he did say ‘once a day’, not ‘twice a day’. BTW, my interjection ‘unfortunately not the human race but the Republican race’ was also a nonsequitur. (shrink)

This 4-page review-essay—which is entirely reportorial and philosophically neutral as are my other contributions to MATHEMATICAL REVIEWS—starts with a short introduction to the philosophy known as mathematical structuralism. The history of structuralism traces back to George Boole (1815–1864). By reference to a recent article various feature of structuralism are discussed with special attention to ambiguity and other terminological issues. The review-essay includes a description of the recent article. The article’s 4-sentence summary is quoted in full and then analyzed. The point (...) of the quotation is to make clear how murky, incompetent, and badly written the paper is. There is no way to determine from the article whether the editor or referees suggests improvements. (shrink)

Iffication, Preiffication, Qualiffication, Reiffication, and Deiffication. -/- Roughly, iffication is the speech-act in which—by appending a suitable if-clause—the speaker qualifies a previous statement. The clause following if is called the qualiffication. In many cases, the intention is to retract part of the previous statement—called the preiffication. I can retract part of “I will buy three” by appending “if I have money”. This initial study focuses on logical relations among propositional contents of speech-acts—not their full conversational implicatures, which will be treated (...) elsewhere. The modified statement—called the iffication—is never stronger than the preiffication. A degenerate iffication is one logically equivalent to its preiffication. There are limiting cases of degenerate iffications. In one, the qualiffication is tautological, as “I will buy three if three is three”. In another, the negation of the qualiffication implies the preiffication, as “I will buy three if I will not buy three”. Reiffication is iffication of an iffication. “I will buy three if I have money” is reifficated by appending “if there are three left”. Deiffication is the speech-act in which—by appending a suitable and-clause—the effect of an iffication is cancelled so that the result implies the preiffication. “I will buy three if I have money” is deifficated by appending “and I have money”. All further examples come from standard (one-sorted, tenseless, non-modal) first-order arithmetic. All theorems are about first-order arithmetic propositions. An easy theorem, hinted above, is that an iffication is degenerate if and only if the negation of the qualiffication implies the preiffication. The iffication of a conjunction using one of the conjuncts as qualiffication need not imply the other conjunct: “Two is an even square if two is square” does not imply “Two is even”. END OF PRINTED ABSTRACT. -/- Some find that the last statement provides a surprise in logic. https://www.academia.edu/s/a5a4386b75?source=link Acknowledgements: Robert Barnes, William Frank, Amanda Hicks, David Hitchcock, Leonard Jacuzzo, Edward Keenan, Mary Mulhern, Frango Nabrasa, and Roberto Torretti. (shrink)

This paper will annoy modern logicians who follow Bertrand Russell in taking pleasure in denigrating Aristotle for [allegedly] being ignorant of relational propositions. To be sure this paper does not clear Aristotle of the charge. On the contrary, it shows that such ignorance, which seems unforgivable in the current century, still dominated the thinking of one of the greatest modern logicians as late as 1831. Today it is difficult to accept the proposition that Aristotle was blind to the fact that, (...) for example, incommensurability is a relation and not a property: that the proposition “In every square, the diagonals are incommensurable with the sides” is relational and not categorical. This paper asks the reader to do something more difficult: to accept the proposition that as late as 1831 De Morgan was blind to the same fact. This paper shows conclusively that in 1831, De Morgan was still in the grips of the allegedly Aristotelian paradigm. (shrink)

CORCORAN RECOMMENDS COCCHIARELLA ON TYPE THEORY. The 1983 review in Mathematical Reviews 83e:03005 of: Cocchiarella, Nino “The development of the theory of logical types and the notion of a logical subject in Russell's early philosophy: Bertrand Russell's early philosophy, Part I”. Synthese 45 (1980), no. 1, 71-115 .

This is a brief reference article on William Wollaston's moral theory, including some influential objections to it.

The premise-fact confusion in Aristotle’s PRIOR ANALYTICS. -/- The premise-fact fallacy is talking about premises when the facts are what matters or talking about facts when the premises are what matters. It is not useful to put too fine a point on this pencil. -/- In one form it is thinking that the truth-values of premises are relevant to what their consequences in fact are, or relevant to determining what their consequences are. Thus, e.g., someone commits the premise-fact fallacy if (...) they think that a proposition has different consequences were it true than it would have if false. C. I. Lewis said that confusing logical consequence with material consequence leads to this fallacy. See Corcoran’s 1973 “Meanings of implication” [available on Academia. edu]. -/- The premise-fact confusion occurs in a written passage that implies the premise-fact fallacy or that suggests that the writer isn’t clear about the issues involved in the premise-fact fallacy. Here are some examples. -/- E1: If Abe is Ben and Ben swims, then it would follow that Abe swims. -/- Comment: The truth is that from “Abe is Ben and Ben swims”, the proposition “Abe swims” follows. Whether in fact Abe is Ben and Ben swims is irrelevant to whether “Abe swims” follows from “Abe is Ben and Ben swims”. -/- E1 suggests that maybe “Abe swims” wouldn’t follow from “Abe is Ben and Ben swims” if the latter were false. -/- E2: The truth of “Abe is Ben and Ben swims” implies that Abe swims. -/- E3: Indirect deduction requires assuming something false. -/- Comment: If the premises of an indirect deduction are true the conclusion is true and thus the “reductio” assumption is false. But deduction, whether direct or indirect, does not require true premises. In fact, indirect deduction is often used to determine that the premises are not all true. -/- Anyway, the one-page paper accompanying this abstract reports one of dozens of premise-fact errors in PRIOR ANALYTICS. In the session, people can add their own examples and comment on them. For example, is the one at 25b32 the first? What is the next premise-fact error after 25b32? Which translators or commentators discuss this? -/- . (shrink)