Xenophanes of Colophon (fl. 530 BC) is widely regarded as the first skeptic in the history of Western philosophy, but the character of his skepticism as expressed in his fragment B 34 has long been a matter of debate. After reviewing the interpretations of B 34 defended by Hermann Fränkel, Bruno Snell, and Sir Karl Popper, I argue that B 34 is best understood in connection with a traditional view of the sources and limits of human understanding. If we hold (...) that mortal beings can achieve certain knowledge only with the assistance of divine powers, while rejecting any possibility of bridging the divine-human gap, a broad skepticism concerning the prospects for human knowledge becomes inevitable. It is therefore of some relevance that Xenophanes repudiated the practice of divination through signs (mantikê) and offered entirely naturalistic explanations of phenomena popularly regarded as indications of the divine will (thunder, lightning, eclipses, rainbows, etc.). (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)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows from the premises: (...) there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

William Oliver Martin published "The Order and Integration of Knowledge" in 1957 to address the problem of the nature and the order of various kinds of knowledge; in particular, the theoretical problem of how one kind of knowledge is related to another kind. Martin characterizes kinds of knowledge as being either autonomous or synthetic. The latter are reducible to two or more of the autonomous (or irreducible) kinds of knowledge, viz., history (H), metaphysics (Meta), theology (T), formal logic (FL), mathematics (...) (Math), and generalizations of experimental science (G). Metaphysics and theology constitute the two domains of the ontological context while history and experimental science are the two domains of the phenomenological context. The relation of one kind of knowledge to another may be instrumental, constitutive, and/or regulative. For instance, historical propositions are constitutive of G, metaphysical propositions are regulative of G, and propositions in formal logic and mathematics are instrumental to G. Theological propositions are not related to G and so there is no conflict between science and theology. Martin's work sheds light on the possible areas of incompatibility between science and religion. (shrink)

This article focuses on the criticisms of current approaches in educational research methodology. Itsummarizes rationales for mixed methods and argues that the mixing quantitative paradigm andqualitative paradigm is problematic due to practical and philosophical arguments. It is alsoindicated that the current rise of mixed methods work has increased problems with quantitativeand qualitative methods. In this article we offer a different symbolic system, with differentlogical form for describing educational phenomena based on the philosophical assumptions andnew mathematical reasoning: para-quantitativism. Para-quantitative theory is (...) an approachwhich has been developed in respect to close relationship between paradigm and method, using apostpositivist transcendental realism as a philosophical beginning of the research methodology,taking Operational Logic System or Fuzzy Logic System (OLS/FLS) as logic of scientificresearch in education. (shrink)

Conceptual engineering is now a central topic in contemporary philosophy. Just 4-5 years ago it wasn’t. People were then engaged in the engineering of various philosophical concepts (in various sub-disciplines), but typically not self-consciously so. Qua philosophical method, conceptual engineering was under-explored, often ignored, and poorly understood. In my lifetime, I have never seen interest in a philosophical topic grow with such explosive intensity. The sociology behind this is fascinating and no doubt immensely complex (and an excellent case study for (...) those interested in the dynamics of academic disciplines). That topic, however, will have to wait for another occasion. Suffice it to say that if Fixing Language (FL) contributed even a little bit to this change of focus in philosophical methodology, it would have achieved one of its central goals. In that connection, it is encouraging that the papers in this symposium are in fundamental agreement about the significance and centrality of conceptual engineering to philosophy. That said, the goal of FL was not only to advocate for a topic, but also to defend a particular approach to it: The Austerity Framework. These replies have helped me see clearer the limitations of that view and points where my presentation was suboptimal. The responses below are in part a reconstruction of what I had in mind while writing the book and in part an effort to ameliorate. I’m grateful to the symposiasts for helping me get a better grip on these very hard issues. (shrink)

This article is a defense and explication of Aristotelian substance-accident terminology used in the Catholic dogma of transubstantiation following upon Fr. Thomas Reese's denigration of orthodox terminology and theology. It was reworked from a paper entitled “They Must Fall into Being: The Son’s Power as Quasi-Subject of the Accidents of Bread and Wine in the Sacrament of the Eucharist” which I delivered on Feb. 4, 2023, at The Holiness of God and the Mystery of the Eucharist conference at Ave Maria (...) University co-sponsored by the Aquinas Center for Theological Renewal (Ave Maria, FL) and the St. Paul Center for Biblical Theology (Steubenville, OH). (shrink)

This paper investigates how later medieval intellectuals dealt with perspectiva – the medieval discipline of optics, which had seen considerable popularity in Latin Europe since the 13th century and was epitomized in several “books of knowledge” of differing scopes, levels of difficulty and intended audience. This paper is focused narrowly on one of these intellectuals – Reimbotus de Castro (fl. 1350s–1380s), who was not only personal physician to the Roman Emperor Charles IV but was also a diligent copyist and abbreviator (...) of many quadrivial and medical texts, and the owner of several codices now kept in the Bibliotheca Palatina. One of these codices, Pal. lat. 1380, includes two optical treatises copied by Reimbot himself. A closer reading of these texts, hitherto unnoticed by historians of medieval science, provides fresh insight into the reception of optical knowledge in the intellectual milieu of Paris in the 1360s, when Reimbot resided here. The first text is Reimbot’s reportatio of lectures on the famous optical textbook Perspectiva communis by John Peckham; the second is Reimbot’s redaction of the unknown optical compendium Perspectiva cum sit una. Finally, this paper addresses the issue of why optics was interesting at all for people at the intersection of the scholarly and courtly communities like Reimbot (and late medieval scholars generally). It is suggested that medieval optics, being useful for astronomical observations, could be considered a sort of auxiliary discipline for astrology and astrological medicine. (shrink)

Reber and Alcock have recently made a sharp attack on the entire psi literature, and in particular a recent overview by Cardeña of the meta-analyses across various categories of psi. They claim the data are inherently fl awed because of their disconnect with our current understanding of the world. As a result, they ignore the data and identify key scientific principles that they argue clash with psi. In this Commentary, I argue that these key principles are diffi cult to apply (...) in areas where our understanding remains poor, especially quantum mechanics and consciousness. I also explore how the psi data may fi t within these two domains. (shrink)

In this paper, we introduce for the first time the neutrosophic quadruple numbers (of the form a + bT + cI + dF) and the refined neutrosophic quadruple numbers. Then we define an absorbance law, based on a prevalence order, both of them in order to multiply the neutrosophic components T, I, F, or their sub-components tJ, Ik, Fl and thus to construct the multiplication of neutrosophic quadruple numbers.

Different reasons give rise to the question, what philosophy really is, and by tradition we know many answers. Plato’s answer can be found by examining his explicit statements about philosophy in his dialogues, or by analyzing his representation of Socrates – philosophy become fl esh. But an other way to fi nd an answer to the question lies in examining the things which – according to Plato – we cannot do without. There are three of them, namely the idea, logos (...) and aporia. These three taken together – the insight that we orient ourselves according to some unity in our different fi elds of life; that we cannot dismiss dialogue and debate; and questions outlive answers – paint a picture of philosophy as «unbehauptendes Denken»1 [undogmatic thought]. Such «unbehauptendes Denken» does not aim to insist on new claims against old ones, but instead seeks to analyze and to refl ect upon old views. (shrink)

This study aimed to determine the experiences of male and female students on Flexible Learning (FL) during the COVID-19. A qualitative method using a phenomenological approach was used to ascertain the students' experiences, while written questionnaires were utilized to gather the information. The 24 students taking Development Communication and Public Administration were chosen based on set criteria. Both male and female flexible learning participants appreciated its benefits, including convenience and reduced academic stress. They both faced technology-related challenges and suggested improvements. (...) However, differences emerged in their experiences. Males found FL fun, emphasizing family time, while females highlighted skill development and concerns about teaching methods. Females also mentioned practical skill development not emphasized by males. These variations may stem from differing learning preferences and experiences. Variations in male and female experiences in Flexible Learning highlight the need for teachers to tailor instruction, offer mental health support, and incorporate skill development. Improved teacher training and clear guidelines are crucial, as are gender-inclusive program design and regular feedback mechanisms to create equitable and effective educational experiences. (shrink)

Glоbalizatiоn is an inеvitablе and оbjеctivе trеnd; it has bоth pоsitivе and nеgativе sidеs and pоsеs fоr all cоuntriеs and pеоplеs bоth оppоrtunitiеs and challеngеs. Rеcоgnizing glоbal issuеs rеquirеs glоbal cоnsciоusnеss. Glоbal cоnsciоusnеss, accоrding tо thе authоr, has thе fоllоwing charactеristics: (i) Rеflеcting thе suprеmacy оf univеrsal valuеs; (ii) It must bе fundamеntally adjustеd in thе sеlf-cоnsciоusnеss оf еach natiоn as wеll as оf еach individual; (iii) Highly sciеntific. Tо build glоbal cоnsciоusnеss, philоsоphy plays a particularly impоrtant rоlе. This rоlе (...) manifеsts itsеlf in its twо basic functiоns: thе wоrldviеw functiоn and thе mеthоdоlоgical functiоn. (shrink)

The revised and polished version of Inwood’s 2011 Carl Newell Jackson at Harvard University, Ethics after Aristotle surveys the ethical teachings of the original “neo-Aristotelians,” namely those self-identified (although not always named) members of the Peripatetic school from the time of Theophrastus (fl. 300 BCE) until that of Alexander of Aphrodisias (fl. 200 CE). An initial chapter surveys the sorts of problems in Aristotle’s ethical corpus which would generate subsequent debate amongst members of the Peripatetic school. Chapter Two examines the (...) views of “Magnus,” the name which Inwood gives to the anonymous 3rd century author of the Magna Moralia (which Inwood takes to be pseudo-Aristotle), and those of Strato of Lampsacus, Lycon, and Hieronymus, 3rd century heads of the Peripatetic school, all of whom show the influences of Epicureanism in their re-articulations of Aristotelian positions. Chapter Three, entitled “The Turning Point,” finds in the work of Critolaus—head of the Peripatetic school in the middle of the 2nd century BCE—a move away from the centrality of activity within Aristotelian ethical thought, which Critolaus instead replaces with the notion of possessing specific goods, namely those of the body, the soul, and what is external. The same chapter argues that at approximately the same historical point Cicero, in the character of Piso in De finibus, articulated an account of Peripatetic ethics that was far more faithful to 4th century Aristotelianism. The final two chapters focus on neo-Aristotelian ethical philosophizing within a new and explicitly Roman cultural setting. (shrink)

Answer Set Programming is a new paradigm based on logic programming. The main component of answer set programming is a system that finds the answer sets of logic programs. During the computation of an answer set, systems are faced with choice points where they have to select a literal and assign it a truth value. Generally, systems utilize some heuristics to choose new literals at the choice points. The heuristic used is one of the key factors for the performance of (...) the system. A new heuristic for answer set programming has been developed. This heuristic is inspired by hierarchical planning. The notion of criticality, which was introduced for generating abstraction hierarchies in hierarchical planning, is used in this heuristic. The resulting system (CSMODELS) uses this new heuristic in a static way. CSMODELS is based on the system SMODELS. The experimental results show that this new heuristic is promising for answer set programming. A comparison of search times with SMODELS demonstrate CSMODELS’ usefulness. (shrink)