Judgments which assign relative importance to the causes of particular results can be objective. Historians usually do and can use a factual principle of selection to distinguish between causes and conditions and between more and less important causes. The judgments which distinguish between causes and conditions and the judgments which distinguish between more and less important causes require radically different analyses. In A. M. Jones's work on the decline and fall of Rome, he argued that increased barbarian pressure on the (...) West was the most important cause of Rome's fall. It is possible to understand this statement as an instantiation of the following: A was a more important cause of P relative to 0 than was B if 1) A and B were each a cause of P relative to 0 and 2) either A was necessary for P or B was not necessary for P and 3) had B not occurred, something would have occurred which more closely approximates P than had A not occurred. (shrink)

In this entertaining and challenging collection of logic puzzles, Raymond Smullyan-author of Forever Undecided-continues to delight and astonish us with his gift for making available, in the thoroughly pleasurable form of puzzles, some of the most important mathematical thinking of our time.

More on propositional and first-order logic -- More on propositional logic -- More on first-order logic -- Recursion theory and metamathematics -- Some special topics -- Elementary formal systems and recursive enumerability -- Some recursion theory -- Doubling up -- Metamathematical applications -- Elements of combinatory logic -- Beginning combinatory logic -- Combinatorics galore -- Sages, oracles, and doublets -- Complete and partial systems -- Combinators, recursion, and the undecidable -- Where to go from here.

FRENCH philosophers from Descartes on have been particularly concerned with the philosophical significance of the human body. Perhaps, as Hyppolite declares, Merleau-Ponty most clearly articulates an ontology of the animate body. Such an interest is by no means confined to philosophers in the French tradition. Surprising, as it may seem, the early Hegel also had a strong interest in the theme of embodiment, the significance of which even Merleau-Ponty in his essay ‘Hegel’s Existentialism’ fails to grasp. The French phenomenologist argues (...) that once Hegel embarks on his discussion of Observational Reason the ‘existential’ viewpoint ceases. Thus, the existential problem of death becomes transmuted into a higher form of life through such ‘altitude thinking’. Even Zaner’s The Problem of Embodiment fails to explore the early Hegelian foundations of incarnate-consciousness in the thought of Marcel, Sartre and Merleau-Ponty. It seems appropriate therefore to briefly examine Hegel’s conception of embodiment in The Phenomenology of Mind on the 200th anniversary of his birth. (shrink)

Generally speaking, it appears correct to say that in a formulation of first order logic in which a large number of connectives are taken as primitive which allows us to have our cake and eat it too.

Philosophers since antiquity have argued the merits of mathematics as a normative aid in ethical decision-making and of the mathematization of ethics a theoretical discipline. Recently, Anagnostopoulos, Annas, Broadie and Hutchinson have probed such issues said to be of interest to Aristotle. Despite their studies, the sense in which Aristotle either opposed or proposed a mathematical ethics in subject-matter and method remains unclear. This paper attempts to clarify the matter. It shows Aristotle’s matrix of exactness and inexactness for ethical subject-matter (...) and ethical method in the Nicomachean Ethics. Then it probes a resultant puzzle from the matrix, namely, the HL model of the happy life without consideration of mathematical justice and the HJL model of the happy life with such consideration. Finally, it examines Aristotle’s twofold rationale for differentiating these two models in his overall moral feedback loop system: differences in the intellectual virtue of good deliberation; the priority of friendship over justice for the happy life. This suggests Aristotle saw no objection either to using mathematics as an aid to ethical decision-making for a happy life, or to mathematizing at least some parts of an ethical theory of eudaimonism. (shrink)

Self-referential sentences have played a key role in Tarski's proof [9] of the non-definibility of arithmetic truth within arithmetic and Gödel's proof [2] of the incompleteness of Peano Arithmetic. In this article we consider some new methods of achieving self-reference in a uniform manner.

After examining c.s. peirce's concept, taxonomy, and hierarchy of the theoretical sciences of discovery as well as his notion of the economy of research as an objective of science, a study is made of his occasional but distinct use of the term 'system' in conceptual and procedural contexts. the study shows that peirce's views on systems theory resemble current views held by some proponents of the systems concept and the systems approach to problem-solving.

Written by a creative master of mathematical logic, this introductory text combines stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic. Author Raymond Smullyan offers clear, incremental presentations of difficult logic concepts. He highlights each subject with inventive explanations and unique problems. Smullyan's accessible narrative provides memorable examples of concepts related to proofs, propositional logic and first-order logic, incompleteness theorems, and incompleteness proofs. Additional topics include undecidability, combinatoric logic, and recursion theory. Suitable for undergraduate and (...) graduate courses, this book will also amuse and enlighten mathematically minded readers. (shrink)

The hypothesis that our current psychological forms of description and explanation will one day be replaced by biological ones, while not universally held, is wide-spread and highly influential in both the scientific community and the broader culture. The purpose of this paper is to examine this hypothesis. It will be argued that, while biology has had and will undoubtedly continue to have many extremely valuable and illuminating findings, it cannot and will not replace psychological explanations and concepts in our understanding, (...) scientific and otherwise, of human behavioral phenomena. That is to say, the science of biology will not replace or subsume that of psychology. 2012 APA, all rights reserved). (shrink)

Although generally supportive of our overall position, both Zachar and Gala and Laughon raise questions about our research design. Herein, we respond to these questions by presenting counterarguments that support the soundness of this design.Subsequent to stating some broad agreement with our central thesis that mental disorder is best viewed as a disability concept and not a behavioral one, Dr. Zachar expresses a number of reservations about our work. We focus on the following discussion on what seem to be the (...) most significant of these.Dr. Zachar states the following : “In 8 of their 12 conditions, the dependent variable is evaluation of problematic behavior and the independent variable is disability... (shrink)

A collection of paradoxes, dialogues, problems, and essays discusses aspects of philosophy, including the natures of reality, truth, existence, and death.

This article is written for both the general mathematican and the specialist in mathematical logic. No prior knowledge of metamathematics, recursion theory or combinatory logic is presupposed, although this paper deals with quite general abstractions of standard results in those three areas. Our purpose is to show how some apparently diverse results in these areas can be derived from a common construction. In Section 1 we consider five classical fixed point arguments (or rather, generalizations of them) which we present as (...) problems that the reader might enjoy trying to solve. Solutions are given at the end of the section. In Section 2 we show how all these solutions can be obtained as special cases of a single fixed point theorem. In Section 3 we consider another generalization of the five fixed point results of Section 1 and show that this is of the same strength as that of Section 2. In Section 4 we show some curious strengthenings of results of Section 3 which we believe to be of some interest on their own accounts. (shrink)

Certain things should never be taken for granted, among them... the precise meaning of words that are at the heart of your discipline.For a large class of cases - though not for all - in which we employ the word ‘‘meaning’’ it can be defined thus: the meaning of a word is its use in the language.The prevailing state of affairs in the mental health field is one in which we have been unable to agree on a definition of our (...) central concept, that of ‘mental disorder’. That is to say, we have not achieved consensus in the matter of what criteria constitute the... (shrink)

Collects a variety of mathematics and logic puzzles, some based on the theorems of the mathematician Kurt Godel.

An entertaining series of logic problems and puzzles of increasing difficulty, and all relating important mathematical and logical concepts, includes mind-benders, paradoxes, metapuzzles, number exercises, and a mathematical novel.