[Edited from Conclusion section:] We have looked at various challenging issues to do with getting connectionism to cope with high-level cognitive activities such a reasoning and natural language understanding. The issues are to do with various facets of generalization that are not commonly noted. We have been concerned in particular with the special forms these issues take in the arena of propositional attitude processing. The main problems we have looked at are: (1) The need to construct explicit representations of generalizations, (...) not just generalize correctly to individual cases; (2) The need to be able to match two or more complex short-term information structures, to enable rapid generalization from recent examples rather than from long-term memories; (3) The need to represent and reason with anomalous combinations of concepts; (4) The need to perform embedded reasoning. This presents special problems for systems using non-concatenative representations (as in mainstream connectionist approaches). We also touched on vague quantification in attitude report complements. Neither this topic nor that of analogies between short-term structures (point 2) has been adequately addressed in the symbolic framework, let alone in connectionism. -/- The opportunities and problems covered are put forward as things worth being optimistic about or pessimistic about, respectively. They are not put forward as decisive arguments for or against connectionism. The hope is that this chapter contributes to a greater understanding of the connectionist/symbolist gap by presenting some unusual issues and by throwing new light on some well known ones. (shrink)

This chapter argues that scientific and philosophical progress in our understanding of the living world requires that we abandon a metaphysics of things in favour of one centred on processes. We identify three main empirical motivations for adopting a process ontology in biology: metabolic turnover, life cycles, and ecological interdependence. We show how taking a processual stance in the philosophy of biology enables us to ground existing critiques of essentialism, reductionism, and mechanicism, all of which have traditionally been associated with (...) substance ontology. We illustrate the consequences of embracing an ontology of processes in biology by considering some of its implications for physiology, genetics, evolution, and medicine. And we attempt to locate the subsequent chapters of the book in relation to the position we defend. (shrink)

I present considerations surrounding pre-reflective self-consciousness, arising in work I am conducting on a new physicalist, process-based account of [phenomenal] consciousness. The account is called the meta-causal account because it identifies consciousness with a certain type of arrangement of meta-causation. Meta-causation is causation where a cause or effect is itself an instance of causation. The proposed type of arrangement involves a sort of time-spanning, internal reflexivity of the overall meta-causation. I argue that, as a result of the account, any conscious (...) process has PRSC. Hence, PRSC does not need to be taken as a stipulation or argued for on purely phenomenological grounds or as a necessary support for reflective consciousness. I also show how it is natural to the account that PRSC is not an additional, peripheral, sort of consciousness, but is intrinsic to all consciousness, thereby fitting claims about self-intimation and co-constitution by various authors, and also being amenable to an adverbial account. As part of this, consciousness of an external object is just the form that current self-consciousness takes, the meta-causal constitution of it being inextricably modulated by the causal relationship with the object. The article also discusses how MCA helps explore issues of for-me-ness, transparency of perceptual consciousness, and possible immediacy and non-relationality of self-consciousness. (shrink)

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)

This paper is a commentary on the target article by Michael Arbib, “Levels of modeling of mechanisms of visually guided behavior”, in the same issue of the journal, pp. 407–465. -/- I focus on the importance of the inclusion of an ability of a system to entertain, at a given time, multiple instantiations of a given schema (situation template, frame, script, action plan, etc.), and complications introduced into neural/connectionist network systems by such inclusion.

How, if at all, consciousness can be part of the physical universe remains a baffling problem. This article outlines a new, developing philosophical theory of how it could do so, and offers a preliminary mathematical formulation of a physical grounding for key aspects of the theory. Because the philosophical side has radical elements, so does the physical-theory side. The philosophical side is radical, first, in proposing that the productivity or dynamism in the universe that many believe to be responsible for (...) its systematic regularities is actually itself a physical constituent of the universe, along with more familiar entities. Indeed, it proposes that instances of dynamism can themselves take part in physical interactions with other entities, this interaction then being “meta-dynamism” (a type of meta-causation). Secondly, the theory is radical, and unique, in arguing that consciousness is necessarily partly constituted of meta-dynamic auto-sensitivity, in other words it must react via meta-dynamism to its own dynamism, and also in conjecturing that some specific form of this sensitivity is sufficient for and indeed constitutive of consciousness. The article proposes a way for physical laws to be modified to accommodate meta-dynamism, via the radical step of including elements that explicitly refer to dynamism itself. Additionally, laws become, explicitly, temporally non-local in referring directly to quantity values holding at times prior to a given instant of application of the law. The approach therefore implicitly brings in considerations about what information determines states. Because of the temporal non-locality, and also because of the deep connections between dynamism and time-flow, the approach also implicitly connects to the topic of entropy insofar as this is related to time. (shrink)

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.

The book is thought-provoking and informative, wide in scope while also being technically detailed, and still relevant to modem AI [at least as of 1998, the time of writing this review, but probably also at the time of posting this entry here, 2023] even though it was published in 1989. This relevance lies mainly in the book’s advocacy of distributed computation at multiple levels of description, its combining of neural networks and other techniques, its emphasis on the interplay between action (...) and perception, and its particular approach to natural language processing. The criticisms I have occasionally made above are relatively minor, and one of the major shortcomings of schema theory at least at the time of the book’s publication -- namely the lack of a detailed, preferably neural, mechanism for schema instantiation -- is one of which Arbib is well aware. The book is well produced, for the most part clearly written, and has useful summaries at the beginning of each chapter. Both general readers and AI researchers (mainly but not exclusively those working on robotics, vision and neural networks) could benefit from reading the book. (shrink)

Closure is the epistemological thesis that if S knows that P and knows that P implies Q, then if S infers that Q, S knows that Q. Fred Dretske acknowledges that closure is plausible but contends that it should be rejected because it conflicts with the plausible thesis: Conclusive reasons (CR): S knows that P only if S believes P on the basis of conclusive reasons, i.e., reasons S wouldn‘t have if it weren‘t the case that P. Dretske develops an (...) analysis of knowing that centers on CR, and argues that the requirement undermines skepticism by implying the falsity of closure. We develop a Dretske-style analysis of knowing that incorporates CR, and we argue that this analysis not only accords with closure, but also implies it. In addition, we argue that the analysis accounts for the prima facie plausibility of closure-invoking skeptical arguments, and nonetheless implies that they are fallacious. If our arguments turn out to be sound, the acceptability of Dretske‘s analysis of knowing will be significantly enhanced by the fact that, despite implying closure, it undermines closure-based skepticism. (shrink)

The representation of propositional attitudes (beliefs, desires, etc.) and the analysis of natural-language, propositional-attitude reports presents difficult problems for cognitive science and artificial intelligence. In particular, various representational approaches to attitudes involve the incorrect “imputation,” to cognitive agents, of the use of artificial theory-laden notions. Interesting cases of this problem are shown to occur in several approaches to attitudes. The imputation problem is shown to arise from the way that representational approaches explicate properties and relationships, and in particular from the (...) way they explicate propositional attitudes themselves. Another factor contributing to imputation is the compositional nature of typical semantic approaches to propositional-attitude reports. Some strategies for avoiding undesirable imputation are examined. One of the main conclusions is that the importance of imputations that arise in a representation scheme depends strongly on the use to which the scheme is put -- on whether, for instance, the scheme is used as part of a formal, objective account of natural language, or is used rather as a representational tool within an agent. (shrink)

The notion of Simulative Reasoning in the study of propositional attitudes within Artificial Intelligence (AI) is strongly related to the Simulation Theory of mental ascription in Philosophy. Roughly speaking, when an AI system engages in Simulative Reasoning about a target agent, it reasons with that agent’s beliefs as temporary hypotheses of its own, thereby coming to conclusions about what the agent might conclude or might have concluded. The contrast is with non-simulative meta-reasoning, where the AI system reasons within a detailed (...) theory about the agent’s (conjectured) reasoning acts. The motive within AI for preferring Simulative Reasoning is that it is more convenient and efficient, because of a simplification of the representations and reasoning processes. The chapter discusses this advantage in detail. It also sketches the use of Simulative Reasoning in an AI natural language processing system, ATT-Meta, that is currently being implemented. This system is directed at the understanding of propositional attitude reports. In ATT-Meta, Simulative Reasoning is yoked to a somewhat independent set of ideas about how attitude reports should be treated. Central here are the claims that (a) speakers often employ commonsense (and largely metaphorical) models of mind in describing agents’ attitudes, (b) the listener accordingly needs often to reason within the terms of such models, rather than on the basis of any objectively justifiable characterization of the mind, and (c) the commonsense models filter the suggestions that Simulative Reasoning comes up with concerning target agents’ reasoning conclusions. There is a yet tighter connection between the commonsense models and the Simulative Reasoning. It turns out that Simulative Reasoning can be rationally reconstructed in terms of a more general type of reasoning about the possibly-counterfactual “world” that the target agent believes in, together with an assumption that that agent has a faithful representation of the world. In the ATT-Meta approach, the reasoner adopts that assumption when it views the target agent through a particular commonsense model (called IDEAS-AS-MODELS). (shrink)

We consider the varieties and directions of influence that the source and target domains involved in a conceptual metaphor can have on each other during the course of understanding metaphorical utterances based on the metaphor. Previous studies have been restricted both as to direction of influence and as to type of influence. They have been largely confined to the “forward” (source to target) direction of influence, and they have concentrated on the transfer of features or propositions and (to some extent) (...) the highlighting of aspects of a domain. By contrast, this article stresses the importance both of other varieties of influence (e.g., transfer of queries and uncertainty effects) and of “reverse” influence (target to source). We seek to curb the natural tendency to think that, because metaphor involves an overall move from source to target, therefore all inter-domain influences in the course of understanding go in that direction. The bulk of the article explores the theoretical issues involved. These issues have arisen out of a reasoning-based approach to metaphor, seeking to make best use of information from metaphorical utterance, context, source and target domains, and known metaphorical mappings, by applying a complex overall reasoning process. The article briefly explains the thorough implementation of reverse influence in a computer program for metaphorical reasoning called ATT-Meta, although the theoretical considerations are relevant to all disciplines concerned with metaphor. We make some suggestions for further computational and psychological research on metaphor. (shrink)

The science of the mind, and of consciousness in particular, needs carefully to consider people's common-sense views of the mind, not just what the mind really is. Such views are themselves an aspect of the nature of (conscious) mind, and therefore part of the object of study for a science of mind. Also, since the common-sense views allow broadly successful social interaction, it is reasonable to look to the common-sense views for some rough guidance as to the real nature of (...) the mind. On the other hand, to the extent that common-sense views are inaccurate, and perhaps even in gross conflict with the true nature of the mind, one interesting scientific question is: why do we hold such views, given our access to our own minds? Why should introspection be limited in a way that allows inaccurate views to hold sway? Now, common-sense views of the mind are revealed in natural language discourse that describes mental states, and such descriptions are largely metaphorical. The metaphors are used within thinking about the mind as well as in language. Therefore the study of metaphor is central to the study of mind. The present article is a preliminary study of the importance of metaphor in the scientific study of consciousness. It concentrates on analysing the nature of a range of important metaphors of mind, briefly discussing the extent to which they can be used to describe or qualify states of consciousness, and pointing to important questions about the nature of consciousness that the study of the metaphors raises. The article further conjectures that the reason people use metaphors in describing themselves is often not (just) that they have intellectually worked out some structural analogy --- e.g., between interactions of physical objects in physical space and interactions of ideas in the mind --- but rather that they {\it feel} their own minds to be as described by the metaphor --- e.g., they feel that the ideas in their minds are interacting physical entities. This matter of how a mind feels to itself is an aspect of the central issue of consciousness, namely its phenomenal quality. -/- (NB: The archived file is a preprint. However, it is substantially the same as the published version.). (shrink)

This article arises indirectly out of the development of a particular approach, called ATT-Meta, to the understanding of some types of metaphorical utterance. However, the specifics of the approach are not the focus of the present article, which concentrates on some general issues that have informed, or arisen from, the development of the approach. The article connects those issues to the questions of metaphorical meaning and truth. -/- A large part of the exploration of metaphor in fields such as Cognitive (...) Linguistics and natural language Pragmatics takes metaphor to rest on complex mappings between a target subject matter and a source subject matter (see, e.g., Lakoff 1993, 2008). This is one main way in which some aspects of the source, generally including some structural aspects, can have parallels in the target. It is typical for much, and perhaps the whole, of the meaning of a metaphorical expression to be explained by means of these parallels. We will talk about “parallelism” rather than “isomorphism,” or even “analogy,” because “isomorphism” implies a strict one-to-one correspondence between items in the source scenario and items in the target scenario, and analogy theorists often hold that analogy rests on isomorphism. We wish not to prejudge the question of whether looser, messier forms of parallelism are sometimes needed. Also, we will talk about parallels rather than mappings, because our discussion will embrace the question of parallelism that may be discernible, albeit implicitly, in accounts of metaphor that are not presented as being based on mappings. Three main cases of such accounts are Relevance Theory (RT: e.g., Sperber & Wilson 2008, Wilson & Carston 2006), the categorization-based or "class-inclusion“ approach (Glucksberg 2008), and Ritchie’s (2006) CLST account (Context-Limited Simulation Theory). -/- One main issue we will address is that of substantial non-parallelism that can exist between source and target subject-matters in metaphor: more precisely, the issue that there may be source aspects that are exploited by the utterance and have a deep effect on metaphorical interpretation but that do not themselves have a natural parallel in the target subject-matter (although we will point out that artificial parallels can generally be created). This is part of a broader point that the content of metaphorical discourse should often be seen as being derived in a rather holistic way from several metaphorical bits of the discourse, which conspire to describe some source-domain scenario, rather than being derived by putting together metaphorical meanings of each metaphorical bit (even if each bit could in principle be assigned its own separate metaphorical meaning). As a result, we suggest that it is misguided to think of the propositions making up the content (or, if you like, "meaning“) about the target scenario that is being described is a matter of metaphorical meaning of specific grammatical units such as sentences, clauses or other constituents that a traditional semantic theory would assign propositional meaning to. Rather, grammatical units (that are to be taken metaphorically) have meanings in source-domain terms; content in target-domain terms is derived from the source-domain scenario depicted by those units and fleshed out through inference; and the target-domain content is only (in general) fuzzily relatable to particular grammatical units. This view is on a spectrum at whose extreme point we could place Davidson’s (1984) view that metaphors only have literal (i.e., source-domain) meanings, with other effects on the understander not being a matter of propositional content. However, on our view it is proper to take metaphorical discourse as having non-literal meaning, couched as a collection of propositions, among other things possibly; it is just that the propositions are not to be thought of (in general) as meaning of specific sentences, clauses or other grammatical units rather than of a possibly only fuzzily delineated piece of discourse. -/- A second main concern of the article is the question of how much, and what types of, inference are involved in the derivation of metaphorical meaning. While our own ATT-Meta approach accords with, amongst other accounts, Relevance Theory (RT) as to the centrality of inference in metaphor interpretation, we disagree with RT on its claim that metaphor interpretation is a matter of (a relatively high degree of) concept "broadening“ and (often) "narrowing“. We argue that almost all inference can in fact be theoretically re-described as concept broadening and/or narrowing, and what is left over should be included in the metaphor interpretation process anyway. Thus, the RT claim about broadening and narrowing really just says that inference, of unrestricted type, is involved in metaphor understanding, in accord with what ATT-Meta claims. (shrink)

This paper is a commentary on the target article by Lokendra Shastri & Venkat Ajjanagadde [S&A]: “From simple associations to systematic reasoning: A connectionist representation of rules, variables and dynamic bindings using temporal synchrony” in same issue of the journal, pp.417–451. -/- It puts S&A's temporal-synchrony binding method in a broader context, comments on notions of pointing and other ways of associating information - in both computers and connectionist systems - and mentions types of reasoning that are a challenge to (...) S&A's system (amongst many other connectionist systems, recognizing that S&A's is much more capable as regards reasoning than most other contemporary systems). (shrink)

The paper is a commentary on the target article by Christine A. Skarda & Walter J. Freeman, “How brains make chaos in order to make sense of the world”, in the same issue of the journal, pp.161–195. -/- I confine my comments largely to some philosophical claims that Skarda & Freeman make and to the relationship of their model to connectionism. Some of the comments hinge on what symbols are and how they might sit in neural systems.

This paper is a commentary on the target article by Dana H. Ballard, “Cortical connections and parallel processing: Structure and function”, in the same issue of the journal, pp. 67–120. -/- I raise some issues about the connectionist or neural-network implementation of information and information processing. Issues include the sharing of information by different parts of a connectionist/neural network, the copying of complex information from one place to another in a network, the possibility of connection weights not being synaptic weights, (...) and the possibility of fast communication within single neurons as complex systems in their own right. (shrink)

Although Paul Churchland and Jerry Fodor both subscribe to the so-called theory-theory– the theory that folk psychology (FP) is an empirical theory of behavior – they disagree strongly about FP’s fate. Churchland contends that FP is a fundamentally flawed view analogous to folk biology, and he argues that recent advances in computational neuroscience and connectionist AI point toward development of a scientifically respectable replacement theory that will give rise to a new common-sense psychology. Fodor, however, wagers that FP will be (...) largely preserved and vindicated by scientific investigations of behavior. Recent findings by developmental psychologists, I argue, will push both Churchlandians and Fodorians toward the pessimistic view that FP is a misguided theory that will never be displaced, because it is, so to speak, built into our cognitive systems. I explore the possibility of preserving optimism by rejecting the theory-theory and adopting the simulation theory, a competing view developed by Robert Gordon, Alvin Goldman, and Jane Heal. According to simulationists, common-sense interpretation of behavior is accomplished by means of pretense-like operations that deploy the cognitive system’s own reasoning capabilities in a disengaged manner. Since on this view no theory-like set of principles would be needed, the simulation theory seems to enjoy a simplicity advantage over the theory-theory. Steven Stich and Shawn Nichols, however, contend that as the cognitive system would require special mechanisms for disengaged operation, the simplicity question cannot be resolved until suitable computational models are developed. I describe a set of models I have constructed to meet this need, and I discuss the contribution such models can make to determining FP’s fate. (shrink)

Reasoning about mental states and processes is important in various subareas of the legal domain. A trial lawyer might need to reason and the beliefs, reasoning and other mental states and processes of members of a jury; a police officer might need to reason about the conjectured beliefs and reasoning of perpetrators; a judge may need to consider a defendant's mental states and processes for the purposes of sentencing and so on. Further, the mental states in question may themselves be (...) about the mental states and processes of other people. Therefore, if AI systems are to assist with reasoning tasks in law, they may need to be able to reason about mental states and processes. Such reasoning is riddled with uncertainty, and this is true in particular in the legal domain. The article discusses how various different types of uncertainty arise, and shows how they greatly complicate the task of reasoning about mental states and processes. The article concentrates on the special case of states of belief and processes of reasoning, and sketches an implemented, prototype computer program (ATT-Meta) that copes with the various types of uncertainty in reasoning about beliefs and reasoning. In particular, the article outlines the system's facilities for handling conflict between different lines of argument, especially when these lie within the reasoning of different people. The system's approach is illustrated by application to a real-life mugging example. [NB: The date on the archived preprint avail via this Phil* page has an incorrect date, 2011, on it. The correct date is 2001.]. (shrink)

In metaphor research there is usually some notion of transfer of aspects of the source domain to the target domain. More rarely, transfers in the opposite direction are countenanced, affecting one’s perception of source as well as target. This paper argues that, even without this aim, transfers from target to source should happen. One radical claim here is that it is often better to translate information from literal sentences into prevailing metaphorical terms than to translate the information from metaphorical sentences (...) into literal terms. The issues have been obscured by confusion between intuitive directions of static source/target mappings, directions of individual transfer actions, and direction of main intended information flow. Relevance to an implemented AI system for metaphorical reasoning, ATT-Meta, and to Blending Theory are briefly mentioned. Asymmetry of metaphor is also addressed. (shrink)

The social welfare functional approach to social choice theory fails to distinguish a genuine change in individual well-beings from a merely representational change due to the use of different measurement scales. A generalization of the concept of a social welfare functional is introduced that explicitly takes account of the scales that are used to measure well-beings so as to distinguish between these two kinds of changes. This generalization of the standard theoretical framework results in a more satisfactory formulation of welfarism, (...) the doctrine that social alternatives are evaluated and socially ranked solely in terms of the well-beings of the relevant individuals. This scale-dependent form of welfarism is axiomatized using this framework. The implications of this approach for characterizing classes of social welfare orderings are also considered. (shrink)

Certain passages in Kaplan’s ‘Demonstratives’ are often taken to show that non-vacuous sentential operators associated with a certain parameter of sentential truth require a corresponding relativism concerning assertoric contents: namely, their truth values also must vary with that parameter. Thus, for example, the non-vacuity of a temporal sentential operator ‘always’ would require some of its operands to have contents that have different truth values at different times. While making no claims about Kaplan’s intentions, we provide several reconstructions of how such (...) an argument might go, focusing on the case of time and temporal operators as an illustration. What we regard as the most plausible reconstruction of the argument establishes a conclusion similar enough to that attributed to Kaplan. However, the argument overgenerates, leading to absurd consequences. We conclude that we must distinguish assertoric contents from compositional semantic values, and argue that once they are distinguished, the argument fails to establish any substantial conclusions. We also briefly discuss a related argument commonly attributed to Lewis, and a recent variant due to Weber. (shrink)

Joint attention is a social-cognitive phenomenon in which two or more agents direct their attention together towards the same object. Definitions range from this rather broad conception to more specific definitions which require that, in addition, attention be directed to the same aspect of that object and that agents need to be mutually aware of their jointly attending. Joint attention is an important coordination mechanism in joint action. The capacity for engaging in joint attention, in particular in the sense of (...) this narrower definition, is frequently taken to indicate the presence of theory of mind in the participating agents, and is also implicated in the development of theory of mind. However, the prominence of sensorimotor components in establishing and sustaining episodes of joint attention — e.g., eye and head movements, pointing and vocalizations — suggests that sensorimotor approaches to social interaction may be well-placed to account for important parts of joint attention without invoking theory-of-mind abilities, and may therefore provide a valuable alternative to approaches emphasizing theory of mind. In particular, sensorimotor accounts may explain aspects of joint attention through the dynamics and regularities in the sensorimotor processes that coordinate the activities of the agents. (shrink)

In “Beat the (Backward) Clock,” we argued that John Williams and Neil Sinhababu’s Backward Clock Case fails to be a counterexample to Robert Nozick’s or Fred Dretske’s Theories of Knowledge. Williams’ reply to our paper, “There’s Nothing to Beat a Backward Clock: A Rejoinder to Adams, Barker and Clarke,” is a further attempt to defend their counterexample against a range of objections. In this paper, we argue that, despite the number and length of footnotes, Williams is still wrong.

The traditional use of conventional media by businesses for audience targeting has shifted with the rise of influencer marketing, notably on platforms like TikTok, posing challenges in content adaptation and technological adaptation. Albert Bandura's Social Cognitive Theory examines factors shaping purchasing behavior, particularly relevant for young professionals. A quantitative correlational study focused on young professionals engaging with TikTok and influenced by Filipino TikTok creators, revealing education level as a key determinant of purchasing behavior. Extended TikTok engagement positively correlates with increased (...) purchasing likelihood. Filipino TikTok influencers significantly impact young professionals' purchasing decisions, emphasizing the need for tailored marketing strategies targeting this demographic. Recommendations were provided for effective influencer and digital marketing strategies targeting young professionals. (shrink)

Ever since Darwin people have worried about the sceptical implications of evolution. If our minds are products of evolution like those of other animals, why suppose that the beliefs they produce are true, rather than merely useful? We consider this problem for beliefs in three different domains: religion, morality, and commonsense and scientific claims about matters of empirical fact. We identify replies to evolutionary scepticism that work in some domains but not in others. One reply is that evolution can be (...) expected to design systems that produce true beliefs in some domain. This reply works for commonsense beliefs and can be extended to scientific beliefs. But it does not work for moral or religious beliefs. An alternative reply which has been used defend moral beliefs is that their truth does not consist in their tracking some external state of affairs. Whether or not it is successful in the case of moral beliefs, this reply is less plausible for religious beliefs. So religious beliefs emerge as particularly vulnerable to evolutionary debunking. (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)

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

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

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

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)

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)

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)

It is proposed that our memory is made up of individual, unconnected Records, to each of which is attached a Heading. Retrieval of a Record can only be accomplished by addressing the attached Heading, the contents of which cannot itself be retrieved. Each Heading is made up of a mixture of content in more or less literal form and context, the latter including specification of environment and of internal states (e.g. drug states and mood). This view of memory allows an (...) easy account of a number of natural memory phenomena as well as a variety of laboratory findings such as the differences between recall and recognition. The theory further proposes that Headed Records can neither be deleted nor modified. Data apparently against such a hypothesis can be accounted for in terms of the retrieval process. (shrink)

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

The supervision of teachers is essential for ensuring effective teaching methods, ongoing professional growth, and student success. This study explored the experiences of program heads in supervising teachers. It was conducted in a local college in Tangub City, Misamis Occidental. The phenomenological design was used in the study. Eleven program heads served as the participants selected through the purposive sampling technique. The Interview Guide was used as the research instrument. Moustakas' transcendental phenomenology of data analysis was utilized to analyze the (...) data gathered. Results revealed the following main themes: the critical role of supervision in advancing teacher development and educational excellence, mutual learning and growth, the role of program heads in teacher supervision, overcoming challenges in teacher supervision, and effective communication strategies. Supervision improves teaching and learning quality, which supports professional development. The complex nature of supervisory roles requires adaptive strategies and strong support networks. It is recommended for program heads and teachers to establish open communication to help address the challenges encountered during supervision. (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)

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)

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

Everyday our vision travels across time and space. We see images in the media about atrocities, disruptions, crises, famine, and wars. And in each case our sense of injustice is awakened. We feel outrage and indignation based upon our ideals and value systems which were formed through our traditions and religions. But in this age where the power of media and information is so powerful, what we see is often manufactured to appeal to our values. While these values circulate among (...) the images we see in cyberspace, these manipulations are rooted in certain realities: geography, natural resources, and power relationships. Our values are managed to serve the control of resources and territory. They serve the deeper reality of geography and geopolitics. How then are these ideals and values created, manipulated, and opposed across various pivots or boundaries, between East and West, between the individual and the collective? Through English geographer Sir Halford Mackinder’s concepts of “pivot” and “heartland”; German philosopher Carl Schmitt’s importance of “nomos”; and French writer Victor Segalen’s reflections on the loss of cultural diversity, we outline this priority of geography. By examining these writers, we can begin to ask if our ideals and values have any real moral or theological significance, or if they are merely effects of the competition between powers. Can ideals and values lead to real change and development, or are they merely leashes to guide us based on the aims of power? (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)

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

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

2006. George Boole. Encyclopedia of Philosophy. 2nd edition. Detroit: Macmillan Reference USA. -/- George Boole (1815-1864), whose name lives among modern computer-related sciences in Boolean Algebra, Boolean Logic, Boolean Operations, and the like, is one of the most celebrated logicians of all time. Ironically, his actual writings often go unread and his actual contributions to logic are virtually unknown—despite the fact that he was one of the clearest writers in the field. Working with various students including Susan Wood and Sriram (...) Nambiar, I have written several publications trying to set the record straight—but so far to little avail. This encyclopedia entry is one more attempt to set the record straight in a way that can be appreciated by non-experts.Also see https://www.academia.edu/10161999/Booles_criteria_of_validity_and_invalidity . (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)

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)