Review of John Kekes' *A Case for Conservatism*.

In this paper, I defend a novel skeptical view about moral disgust. I argue that much recent discussion of moral disgust neglects an important ontological question: is there a distinctive psychological state of moral disgust that is differentiable from generic disgust, and from other psychological states? I investigate the ontological question and propose two conditions that any aspiring account of moral disgust must satisfy: it must be a genuine form of disgust, and it must be genuinely moral. Next, I examine (...) two prominent accounts of moral disgust by John Kekes and Victor Kumar and argue that neither successfully establishes the existence of genuinely moral disgust: Kekes’ account does not satisfy the second condition, and Kumar’s view does not meet the first. I claim that an important general lesson can be drawn from my critiques of Kekes’ and Kumar’s accounts: to establish the existence of moral disgust, one must provide unequivocal evidence that genuinely moral disgust, not generic disgust or anger, is being elicited in response to relevant moral violations. I conclude by considering why we ought to be skeptical about the general prospect of giving a positive answer to the ontological question, given the available evidence. (shrink)

a review of John Kekes' *The Roots of Evil*. I express admiration for the aims and scope of the book, and disagree with some of Kekes' accounts of some historical cases.

This paper provides a principled and elegant solution to the Gettier problem. The key move is to draw a general metaphysical distinction and conscript it for epistemological purposes. Section 1 introduces the Gettier problem. Sections 2–5 discuss instructively wrong or incomplete previous proposals. Section 6 presents my solution and explains its virtues. Section 7 answers the most common objection.

This paper explains what it is to believe something for a reason. My thesis is that you believe something for a reason just in case the reason non-deviantly causes your belief. In the course of arguing for my thesis, I present a new argument that reasons are causes, and offer an informative account of causal non-deviance.

Is it possible to misidentify the object of an episode of bodily awareness? I argue that it is, on the grounds that a person can reasonably be unsure or mistaken as to which part of his or her body he or she is aware of at a given moment. This requires discussing the phenomenon of body ownership, and defending the claim that the proper parts of one’s body are at least no less ‘principal’ among the objects of bodily awareness than (...) is the body as a whole. I conclude with some reasons why this should lead us to think that bodily awareness, unlike introspection, is a form of perception. (shrink)

Supervision of teachers is essential for ensuring effective educational practices, fostering professional development, and achieving student success. The study determined the relationship of program heads' supervisory practices to the teachers’ teaching efficacy. It was conducted in a community college in Tangub City, Misamis Occidental. The descriptive-correlational design was used in the study. There were 146 faculty and 361 students who served as the respondents selected through a stratified random sampling technique. The adapted Program Heads’ Supervisory Practices and researcher-made Teachers’ Teaching (...) Efficacy Questionnaires were used as research instruments. Mean, Standard Deviation, and Pearson Product-Moment Correlation Coefficient were used as statistical tools to analyze the data gathered. Results revealed that the program heads were good in their supervisory practices. The teachers’ teaching efficacy was very high. The program heads’ supervisory practices did not influence the teacher’s teaching efficacy. The strategies employed by program heads in their supervisory roles do not always translate into improved teaching efficiency. The study's findings indicate that program heads need to adopt more responsive and tailored supervisory practices that directly address the specific needs of teachers to enhance teaching efficacy. This may involve integrating targeted professional development focused on practical classroom strategies and fostering peer collaboration to share best practices. Program heads may prioritize open communication and continuous feedback mechanisms to better align their supervisory approaches with the realities of classroom instruction, ensuring that support translates into improved teaching efficiency. (shrink)

Many philosophers have held that it is not possible to experience a spatial object, property, or relation except against the background of an intact awareness of a space that is somehow ‘absolute’. This paper challenges that claim, by analyzing in detail the case of a brain-damaged subject whose visual experiences seem to have violated this condition: spatial objects and properties were present in his visual experience, but space itself was not. I go on to suggest that phenomenological argumentation can give (...) us a kind of evidence about the nature of the mind even if this evidence is not absolutely incorrigible. (shrink)

In Tulving’s initial characterization, episodic memory was one of multiple memory systems. It was postulated, in pursuit of explanatory depth, as displaying proprietary operations, representations, and substrates such as to explain a range of cognitive, behavioural, and experiential phenomena. Yet the subsequent development of this research program has, paradoxically, introduced surprising doubts about the nature, and indeed existence, of episodic memory. On dominant versions of the ‘common system’ view, on which a single simulation system underlies both remembering and imagining, there (...) are no processes unique to memory to support robust generalizations with inductive potential. Eliminativism about episodic memory seems to follow from the claim that it has no dedicated neurocognitive system of its own. After identifying this under-noticed threat, we push back against modern eliminativists by surveying recent evidence that still indicates specialized mechanisms, computations, and representations that are distinctly mnemic in character. We argue that contemporary realists about episodic memory can retain lessons of the common system approach while resisting the further move to eliminativism. (shrink)

The principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox and Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to higher-order logic. Our assessment (...) of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior’s and Kaplan’s derivations at face value. (shrink)

Robust technological enhancement of core cognitive capacities is now a realistic possibility. From the perspective of neutralism, the view that justifications for public policy should be neutral between reasonable conceptions of the good, only members of a subset of the ethical concerns serve as legitimate justifications for public policy regarding robust technological enhancement. This paper provides a framework for the legitimate use of ethical concerns in justifying public policy decisions regarding these enhancement technologies by evaluating the ethical concerns that arise (...) in the context of testing such technologies on nonhuman animals. Traditional issues in bioethics, as well as novel concerns such as the possibility of moral status enhancement, are evaluated from the perspective of neutralism. (shrink)

This paper applies speech-act theory to craft a new response to Pyrrhonian skepticism and diagnose its appeal. Carefully distinguishing between different levels of language-use and noting their interrelations can help us identify a subtle mistake in a key Pyrrhonian argument.

Dasgupta poses a serious challenge to realism about natural properties. He argues that there is no acceptable explanation of why natural properties deserve the value realists assign to them and are consequently absent of value. In response, this paper defines and defends an alternative non-explanatory account of normativity compatible with realism. Unlike Lewis and Sider, who believe it is sufficient to defend realism solely on realist terms, I engage with the challenge on unfriendly grounds by revealing a tu quoque. Dasgupta (...) and anti-realists face a similar challenge to that directed against realism: one that not only undermines the objection to realism by legitimising non-explanatory normativity but leaves them facing a significant dilemma. (shrink)

The vision of natural kinds that is most common in the modern philosophy of biology, particularly with respect to the question whether species and other taxa are natural kinds, is based on a revision of the notion by Mill in A System of Logic. However, there was another conception that Whewell had previously captured well, which taxonomists have always employed, of kinds as being types that need not have necessary and sufficient characters and properties, or essences. These competing views employ (...) different approaches to scientific methodologies: Mill’s class-kinds are not formed by induction but by deduction, while Whewell’s type-kinds are inductive. More recently, phylogenetic kinds (clades, or monophyletic-kinds) are inductively projectible, and escape Mill’s strictures. Mill’s version represents a shift in the notions of kinds from the biological to the physical sciences. (shrink)

The account of natural kinds as stable property clusters is premised on the possibility of separating the epistemic value of natural kinds from their underlying metaphysics. On that account, the co-instantiation of any sub-cluster of the properties associated with a given natural kind raises the probability of the co-instantiation of the rest, and this clustering of property instantiation is invariant under all relevant counterfactual perturbations. We argue that it is not possible to evaluate the stability of a cluster of properties (...) without taking stock of the metaphysical picture used to account for that stability. Thus, even on the stable property cluster account, the epistemic value of natural kinds remains partly grounded in their metaphysical status. (shrink)

I will develop a new problem for almost all realist theories of colour. The problem involves fluctuations in our colour experiences that are due to visual noise rather than changes in the objects we are looking at.

ABSTRACT Drawing inspiration from Fred Dretske, L. S. Carrier, John A. Barker, and Robert Nozick, we develop a tracking analysis of knowing according to which a true belief constitutes knowledge if and only if it is based on reasons that are sensitive to the fact that makes it true, that is, reasons that wouldn’t obtain if the belief weren’t true. We show that our sensitivity analysis handles numerous Gettier-type cases and lottery problems, blocks pathways leading to skepticism, and validates (...) the epistemic closure thesis that correct inferences from known premises yield knowledge of the conclusions. We discuss the plausible views of Ted Warfield and Branden Fitelson regarding cases of knowledge acquired via inference from false premises, and we show how our sensitivity analysis can account for such cases. We present arguments designed to discredit putative counterexamples to sensitivity analyses recently proffered by Tristan Haze, John Williams and Neil Sinhababu, which involve true statements made by untrustworthy informants and strange clocks that sometimes display the correct time while running backwards. Finally, we show that in virtue of employing the paradox-free subjunctive conditionals codified by Relevance Logic theorists instead of the paradox-laden subjunctive conditionals codified by Robert Stalnaker and David Lewis. (shrink)

(for Adam Morton's half) I argue that if we take the values of persons to be ordered in a way that allows incomparability, then the problems Broome raises have easy solutions. In particular we can maintain that creating people is morally neutral while killing them has a negative value.

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)

This study investigates the significant relationship between procrastination and academic burnout among first-year college students. Employing correlational design and standardized tests, the statistical analysis reveals that the r coefficient of 0.33 indicates a low positive correlation between the variables. The p-value of 0.00, which is less than 0.05, leads to the decision to reject the null hypothesis. Hence, a significant relationship exists between procrastination and academic burnout among college students.

In a recent very interesting and important challenge to tracking theories of knowledge, Williams & Sinhababu claim to have devised a counter-example to tracking theories of knowledge of a sort that escapes the defense of those theories by Adams & Clarke. In this paper we will explain why this is not true. Tracking theories are not undermined by the example of the backward clock, as interesting as the case is.

Chalmers and Hájek argue that on an epistemic reading of Ramsey’s test for the rational acceptability of conditionals, it is faulty. They claim that applying the test to each of a certain pair of conditionals requires one to think that one is omniscient or infallible, unless one forms irrational Moore-paradoxical beliefs. I show that this claim is false. The epistemic Ramsey test is indeed faulty. Applying it requires that one think of anyone as all-believing and if one is rational, to (...) think of anyone as infallible-if-rational. But this is not because of Moore-paradoxical beliefs. Rather it is because applying the test requires a certain supposition about conscious belief. It is important to understand the nature of this supposition. (shrink)

I offer a novel account of the absurdity of Moore-paradoxical assertion in terms of an interlocutor’s fully conscious beliefs. This account starts with an original argument for the principle that fully conscious belief collects over conjunction. The argument is premised on the synchronic unity of consciousness and the transparency of belief.

In this paper I will take it for granted that Zagzebski's position articulates a broadly Catholic perspective, and that Plantinga's position accurately represents a broadly Calvinist one. But I will argue that so construed, the Catholic and the Calvinist are much closer than Zagzebski implies: both views are person-based in an important sense of that term; both are internalist on Zagzebski's usage and externalist on the standard usage; and Plantinga's position is consistent with the social elements that Zagzebski stresses in (...) her view. In the second part of the paper I will identify what I think is the real issue between Zagzebski and Plantinga. Namely, Zagzebski thinks that knowledge requires epistemic responsibility, in the sense that instances of knowledge must be appropriately praiseworthy. Plantinga thinks that no such condition is required for knowledge or warranted belief. I will argue that on this issue Zagzebski is right, and that her virtue approach gives us resources for seeing why. Finally, I will look at the consequences for religious knowledge. Here I will argue that the consequences are minimal. Even if knowledge requires responsibility, Plantinga can still make a good case that religious belief is properly basic. And even if religious belief is properly basic, natural theology can still have an important role in the justification of religious belief. (shrink)

Super-Humeans argue that the most parsimonious ontology of the natural world compatible with our best physical theories consists exclusively of particles and the distance relations between them. This paper argues by contrast that Super-Humean reduction goes insufficiently far, by showing there to be a more parsimonious ontology compatible with physics: Ultimate-Humeanism. This novel view posits an ontology consisting solely of the particles and distance relations required for the existence of a single brain. Super-Humeans impose conditions on what counts as an (...) ontology of the natural world to avoid their view slipping into this kind of ontology, but these conditions are arbitrarily imposed and once this is exposed, Super-Humeans face a dilemma. Either they can embrace Ultimate-Humeanism as the minimal ontology of the natural world compatible with physics, or (more likely) they can rethink the methodology that got them there. Overall, this paper argues that Super-Humeanism currently lacks principled motivation, outlines a framework for naturalistic ontological reductions, and exposes the consequences of unchecked adherence to a simplicity-driven methodology. (shrink)

I suggest two improvements to Joshua Rasmussen’s intriguing recent argument that a causally powerful being necessarily exists.

I discuss two arguments against the view that reasons are propositions. I consider responses to each argument, including recent responses due to Mark Schroeder, and suggest further responses of my own. In each case, the discussion proceeds by comparing reasons to answers and goals.

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)

A considerable body of recent work in developmental psychology and animal behavior has addressed the cognitive processes required to recognize oneself in a mirror. Most models of such "mirrored self-recognition" (MSR) treat it as the result of inferential processes drawing on the subject’s possession of some sort of mature "self-awareness". The present chapter argues that such an approach to MSR is not obligatory, and suggests some empirical grounds for rejecting it. We also sketch the outlines of an alternative, "embodied" theory (...) of MSR, and propose a way to evaluate it using the tools of adaptive robotics. (shrink)

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

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)

Llull and Leibniz both subscribed to conceptual atomism: the belief that the majority of concepts are compounds constructed from a relatively small number of primitive concepts. Llull worked out techniques for finding the logically possible combinations of his primitives, but Leibniz criticized Llull’s execution of these techniques. This article argues that Leibniz was right about things being more complicated than Llull thought but that he was wrong about the details. The paper attempts to correct these details.

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

This chapter has two objectives. The first is to clarify Aristotle’s view of the first principles of the sciences. The second is to stake out a critical position with respect to this view. The paper sketches an alternative to Aristotle’s intuitionism based in part on the use of quantitative inductive logics.

In a well-known footnote in Book 3 of his Treatise of Human Nature, Hume calls William Wollaston's moral theory a "whimsical system" and purports to destroy it with a few brief objections. The first of those objections, although fatally flawed, has hitherto gone unrefuted. To my knowledge, its chief error has escaped attention. In this paper I expose that error; I also show that it has relevance beyond the present subject. It can occur with regard to any moral theory which, (...) like Wollaston's, locates the wrongness of an act in a property that can reside in non-actions no less than in actions. (shrink)

Some attention has also been devoted to a particular kind of judgment or a particular form of the intellect’s second operation, sometimes named separatio by Thomas. Important editions of questions 5 and 6 of Thomas’s commentary on the De Trinitate of Boethius in 1948 and 1955 and the groundbreaking study by L. B. Geiger in 1947, all have set the stage for further emphasis on this distinctive type of intellectual operation when it comes to one’s discovery of being, or better, (...) of that notion of being that can serve as subject of a science of being as being rather than a science of being as material or as quantified. While this new development has remained largely unnoticed in certain regions of Thomistic scholarship for a number of years, it has been pursued in depth by other writers. At the same time, investigation of the same nicely dovetails with the renewed emphasis on existence and on judgment as the process required to discover being as existing to which we have referred above. For as will be seen below, at least one passage in Thomas’s commentary reinforces the contention that one must pass beyond simple apprehension to the mind’s second operation or to judgment if one is to grasp being explicitly as existing. This particular point, however, is not our primary concern here. (shrink)

How should we think about the role of visual spatial awareness in perception and perceptual knowledge? A common view, which finds a characteristic expression in Kant but has an intellectual heritage reaching back farther than that, is that an account of spatial awareness is fundamental to a theory of experience because spatiality is the defining characteristic of “outer sense”, of our perceptual awareness of how things are in the parts of the world that surround us. A natural counterpart to this (...) idea is to treat self-consciousness as residing in a kind of sense that is fundamentally “inner”, such as introspection or whatever else gives one privileged access to his own mental states as well as the proprioceptive and kinesthetic awareness of bodily position. This division is compatible, of course, with the idea that inner sense provides an awareness of a distinctive kind of “body space”, but it treats that as importantly different from the awareness of the worldly space around one. -/- In contrast to such a picture, this dissertation proposes an account of visual spatial awareness according to which it is no less a source of self-consciousness than of the awareness of the objects around us, and an account of self-awareness in which visual experience is essentially implicated. I begin by arguing that we should think of visual spatial awareness not as necessary for the individuation of visual sensations but rather as an essential element in the awareness of an experientially objective world. In the subsequent chapters, I argue that in being visually aware of the egocentric positions of the worldly objects around us we are often aware also of our own spatial locations with respect to them, and that the visual experience of the world around one and one’s own situation in it is often an essential component in the knowledge that a human agent will have of his own intentional actions. (shrink)

The paper considers the criticisms that Eleonore Stump has made of Richard Swinburne's account of Christian's revelation, as set out in his book "Revelation: From Metaphor to Analogy." It argues that Stump's criticisms of Swinburne's theory of biblical interpretation are misguided, but that her criticism of his deistic picture of revelation contains a crucial insight. Direct theories of revelation, which see God as communicating propositions directly to believers, are superior to deistic ones, which see God as communicating propositions only to (...) an original group who then hand on the propositions to everyone else. Stump's suggested alternative to a deistic picture is flawed. A better theory would result from incorporating Swinburne's account of the Church into a direct theory, and holding that God communicates propositions directly to believers in the teaching of the Church. This position combines the insights to be found in Stump and Swinburne, while avoiding their mistakes. (shrink)

Ted Sider has shown that my indeterminism argument for comparativist theories of quantity also applies to Mundy's absolutist theory. This is because Mundy's theory posits only "pure" relations, i.e. relations between values of the same quantity (between masses and other masses, or distances and other distances). It is straightforward to solve the problem by positing additional mixed relations.

This review aimed at exploring the psychosocial disorders of Nigeria as a nation, its effect on the citizens and the remedy. Psychosocial disorder is a mental illness induced by life experiences, stress, as well as maladaptive cognitive and behavioural processes. The prevalence of these disorders include depression, schizophrenia, anxiety disorders, substance use disorder, personality disorder and autism spectrum disorders have rapidly increase over the past years in Nigeria with its negative impact on the socioeconomic status, psychological well-being; peaceful coexistence and (...) mental health of the entire society. These have precipitated some social anomalies such as bribery and corruption, antisocial behaviour, religious fanaticism, terrorism and so on. The study therefore recommends the need for government to implement effective mental health interventions program that would facilitate easy access to mental healthcare services in Nigeria. There is also an urgent need to improve the mental health awareness of Nigerians through education and public mental health campaign. (shrink)

This article argues that the word ψυχαγωγία and its cognates which are found in the writings of Philodemus retain a semantic connection to the domain of magic and are best translated in terms of ‘enchantment’ rather than the more generic sense of ‘entertainment’.

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

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)