Hawking, in his book, A Brief History of Time, concludes with a conditional remark: If we find a complete theory to explain the physical world, then we will come to understand God’s mind. With Goedel in mind, we can raise questions about the completeness of our scientific understanding and the nature of our understanding with regard to God’s mind. We need to ask about the higher order of our understanding when we move to knowing God’s mind. We go onto develop (...) these problems with Nietzsche’s thoughts on the death of God, and the emergence of super human intelligences. Then, we come to Buddha, a logical genius, to see how the Buddhistic enlightenment exemplifies the super human intelligence, as well as the higher order understanding in our knowledge of God’s mind. (shrink)

Gödel’s Ontological argument is distinctive because it is the most sophisticated and formal of ontological arguments and relies heavily on the notion of _positive property_. Gödel uses a third-order modal logic with a property abstraction operator and property quantification into modal contexts. Gödel describes _positive property_ as "independent of the accidental structure of the world"; "pure attribution," as opposed to privation; "positive in the 'moral aesthetic sense.'" _Pure attribution_ seems likely to be related to the Leibnizian concept (...) of perfection. By a careful examination of the formal semantics of third-order modal logic with property abstraction together with a Completeness result for third-order modal logic with property abstraction for faithful models that I previously developed in 2000 in my work, _Gödel’s Ontological Argument_, I argue that it is not possible to develop a sufficient applied third-order modal semantics for Gödel’s ontological argument. As I explore possible approaches for an applied semantics including anti-Realist accounts of the semantics of modal logic compatible with Actualism, I argue that Gödel makes implicit philosophical assumptions which commit him to both possibilism and modal realism. (shrink)

Gödel’s Ontological argument is distinctive because it is the most sophisticated and formal of ontological arguments and relies heavily on the notion of positive property. Gödel uses a third-order modal logic with a property abstraction operator and property quantification into modal contexts. Gödel describes positive property as "independent of the accidental structure of the world"; "pure attribution," as opposed to privation; "positive in the 'moral aesthetic sense.'" Pure attribution seems likely to be related to the Leibnizian concept (...) of perfection.By a careful examination of the formal semantics of third-order modal logic with property abstraction together with a Completeness result for third-order modal logic with property abstraction for faithful models that I previously developed in 2000 in my work, Gödel’s Ontological Argument, I argue that it is not possible to develop a sufficient applied third-order modal semantics for Gödel’s ontological argument. As I explore possible approaches for an applied semantics including anti-Realist accounts of the semantics of modal logic compatible with Actualism, I argue that Gödel makes implicit philosophical assumptions which commit him to both possibilism (the belief in merely possible objects) and modal realism (the belief in possible worlds). (shrink)

This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most (...) striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic. (shrink)

The Jewish religious tradition summons its adherents to save life. For religious Jews preservation of life is the ultimate religious commandment. At the same time Jewish law recognizes that the agony of a moribund person may not be stretched. When the time to die has come this has to be respected. The process of dying should not needlessly be prolonged. We discuss the position of two prominent Orthodox Jewish authorities – the late Rabbi Moshe Feinstein and Rabbi J David Bleich (...) – towards the role of life-sustaining treatment in end-of-life care. From the review, the characteristic halachic and heterogeneous character of Jewish ethical reasoning appears. The specificity of Jewish dealing with ethical dilemmas in health care indicates the importance for contemporary healthcare professionals of providing care which is sensitive to a patient’s culture and worldview. (shrink)

The Diagnostic and Statistical Manual of Mental Disorders (5th ed.; DSM–5; American Psychiatric Association [APA], 2013) defines melancholia as “A mental state characterized by very severe depressi...

The volume contains twenty essays devoted to the philosophy of mathematics and the history of logic. They have been divided into four parts: general philosophical problems of mathematics, Hilbert's program vs. the incompleteness phenomenon, philosophy of mathematics in Poland, mathematical logic in Poland. Among considered problems are: epistemology of mathematics, the meaning of the axiomatic method, existence of mathematical objects, distinction between proof and truth, undefinability of truth, Goedel's theorems and computer science, philosophy of mathematics in Polish mathematical and logical (...) schools, beginnings of mathematical logic in Poland, contribution of Polish logicians to recursion theory. (shrink)

This review concludes that if the authors know what mathematical logic is they have not shared their knowledge with the readers. This highly praised book is replete with errors and incoherency.

Information?rich environments are created to promote data use in schools for the purpose of self?evaluation and quality assurance. However, providing feedback does not guarantee that schools will actually put it to use. One of the main stumbling blocks relates to the interpretation and diagnosis of the information. This study examines the relationship between data literacy competences, support given in interpreting the information, actual use of the feedback and potential school improvement effect. A randomised field experiment with 188 school principals from (...) primary education was set up and a post?test was used to investigate the effects of a support initiative. The results revealed that a minority of schools invested significantly in the interpretation and diagnosis of the school performance feedback (SPF), despite the fact that most of the respondents showed an interest in the SPF report. In addition, data competence support and the subsequent use of feedback were found to be limited. (shrink)

This paper examines the relationships between the many-valued logics G~ and Gn~ of Esteva, Godo, Hajek, and Navara, i.e., Godel logic G enriched with Łukasiewicz negation, and neighbors of intuitionistic logic. The popular fragments of Rauszer's Heyting-Brouwer logic HB admit many-valued extensions similar to G which may likewise be enriched with Łukasiewicz negation; the fuzzy extensions of these logics, including HB, are equivalent to G ~, as are their n-valued extensions equivalent to Gn~ for any n ≥ 2. (...) These enriched systems extend Wansing's logic I4C4, showing that Łukasiewicz negation is a species of Nelson's negation of constructible falsity and yielding a Kripke-style semantics for G~ and Gn~ to complement the many-valued semantics. (shrink)

I read Stefan Collini’s What are Universities For? last week with very mixed feelings. In the past, I’ve much admired his polemical essays on the REF, “impact”, the Browne Report, etc. in the London Review of Books and elsewhere: they speak to my heart. If you don’t know those essays, you can get some of their flavour from his latest article in the Guardian yesterday. But I found the book a disappointment. Perhaps the trouble is that Collini is too decent, (...) too high-minded, has too unrealistically exalted a view about what actually happens in universities which is too coloured by attitudes appropriate to the traditional humanities. And he is optimistic to the point of fantasy if he thinks that people are so susceptible to “the romance of ideas and the power of beauty” that they will want, or can be brought to want, lots and lots of universities in order to promote the ideas (as if they would suppose that the task of “conserving, understanding, extending and handing on to subsequent generations the intellectual, scientific, and artistic heritage of mankind” was clearly ill-served when there were only forty universities in England, as opposed to a hundred and whatever). (shrink)

A common reaction to Hegel’s suggestion that we collapse Kant’s distinction between form and content is that, since such a move would also deprive us of any way of distinguishing the merely logical from the real possibility of our concepts, it is incoherent and ought to be rejected. It is true that these two distinctions are intimately related in Kant, such that if one goes, the other does as well. But it is less obvious that giving them up as Kant (...) conceives them is as incoherent a proposal as many of Hegel’s critics think. It has been the point of a recent account of Hegel’s idealism to demonstrate that his critique of Kant’s dichotomy between form and content or concept and intuition does not commit him to the view that human cognition is materially creative of its content in the manner of a God-like or intuitive intellect—does not, in other words, signify a return to a pre-Critical metaphysics. This is the interpretative stance I adopt here as well, in hope of giving defenders of Kant reasons for taking Hegel more seriously. (shrink)