We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive (...) proofs. Our axiom systems will not be strong enough to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the "Tangibility Reflection Principle". We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further. (shrink)

Let us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13], [18]. We will generalize the semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π 1 sentence, valid in the standard model of the Natural Numbers and denoted as V, such that if α is any finite consistent extension of Q + (...) V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property. Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ 0 , as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie [15].). (shrink)

This paper will introduce the notion of a naming convention and use this paradigm to both develop a new version of the Second Incompleteness Theorem and to describe when an axiom system can partially evade the Second Incompleteness Theorem.

Based on an unfinished manuscript by the late philosopher Dallas Willard, this book makes the case that the 20th century saw a massive shift in Western beliefs and attitudes concerning the possibility of moral knowledge, such that knowledge of the moral life and of its conduct is no longer routinely available from the social institutions long thought to be responsible for it. In this sense, moral knowledge--as a publicly available resource for living--has disappeared. Via a detailed survey of main developments (...) in ethical theory from the late 19th through the late 20th centuries, Willard explains philosophy's role in this shift. In pointing out the shortcomings of these developments, he shows that the shift was not the result of rational argument or discovery, but largely of arational social forces--in other words, there was no good reason for moral knowledge to have disappeared. The Disappearance of Moral Knowledge is a unique contribution to the literature on the history of ethics and social morality. Its review of historical work on moral knowledge covers a wide range of thinkers including T.H Green, G.E Moore, Charles L. Stevenson, John Rawls, and Alasdair MacIntyre. But, most importantly, it concludes with a novel proposal for how we might reclaim moral knowledge that is inspired by the phenomenological approach of Knud Logstrup and Emmanuel Levinas. Edited and eventually completed by three of Willard's former graduate students, this book marks the culmination of Willard's project to find a secure basis in knowledge for the moral life. (shrink)

A letter to The Times of London, May 9, 1992 protesting the Cambridge University proposal to award an honorary degree to M. Jacques Derrida.

This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 (...) and Σ1 formulae. Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions. (shrink)

In Part II of Husserl’s “great last work,” as the Crisis is sometimes called, we find two passages in which he comments upon what he took to be an important and utter failure on the part of “die modernen Logistiker”—or, as Carr well translates it, “modern mathematical logicians.” At the end of subsection 36 he says.

In 1981, Paris and Wilkie raised the open question about whether and to what extent the axiom system did satisfy the Second Incompleteness Theorem under Semantic Tableaux deduction. Our prior work showed that the semantic tableaux version of the Second Incompleteness Theorem did generalize for the most common definition of appearing in the standard textbooks.However, there was an alternate interesting definition of this axiom system in the Wilkie–Paris article in the Annals of Pure and Applied Logic 35 , pp. 261–302 (...) which we did not examine in our year-2002 article in the Journal of Symbolic Logic. Our first goal is to show that the incompleteness results of our prior paper can generalize in this alternate context. We will also develop a formal analysis, using a new technique called Passive Induction, that is simpler than the formalism we had used before.A further reason our results are of interest is that we have shown in a companion paper published in Electronic Notes in Theoretical Computer Science 165 , pp. 213–226 that some very unorthodox axiomizations for are anti-thresholds for the Herbrandized version of the Second Incompleteness Theorem. Thus, different axiomizations for have nearly fully opposite incompleteness properties.This paper is self-contained. It will not require a knowledge of our earlier results. (shrink)

For the Society of Christian Philosophers, APA San Francisco, April 4, 2007.

Gödel’s Second Incompleteness Theorem states axiom systems of sufficient strength are unable to verify their own consistency. We will show that axiomatizations for a computer’s floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics. This paper will include both new generalizations of the Second Incompleteness Theorem and techniques for evading it.

We argue for the mind’s independence from the body. We do so by making several moves. First, we analyze two popular kinds of reasons which have swayed many to adopt the independence of the mind from the body. Second, we advance an argument from the ontology of intentionality against the identity thesis, according to which the mind is identical to the brain. We try to show how intentionality is not reducible to or identical to the physical. Lastly, we argue that, (...) contrary to what many materialists contend, the concept of a mind, understood as an immaterial substance, existing independently of the body is both coherent and empirically evidenced. (shrink)

There is an inherent phenomenological tendency in British moral theory, especially from John Locke onward. The purpose of his Essay was, he said, to consider the discerning faculties of a man, as they are employed about the objects which they have to do with. This is language that might serve well in a general description of the work of Husserl and other phenomenologists.

The title is meant to emphasize the immense loss of status I take logic to have undergone in recent decades, and to suggest something about its causes. The loss is most obvious in the context of higher education, where almost no post-secondary institutions now have effectual general requirements in standard formal logic, as that was easily understood thirty or more years ago. Courses in so-called 'critical thinking' are, with rare and noble exceptions, only a further illustration of the point, for (...) many of them, if not most, say nothing at all about logical form and formal logic, and proceed as if thought and discourse could be critically understood and appraised in total ignorance of their formal aspects. (shrink)

When making inferences about similar others, people anchor and adjust away from themselves. However, research on relational self theory suggests the possibility of using knowledge about others as an anchor when they are more similar to a target. We investigated whether social inferences are made on the basis of significant other knowledge through an anchoring and adjustment process, and whether anchoring on a significant other is more effortful than anchoring on the self. Participants answered questions about their likes and habits, (...) as well as the likes and habits of a significant other, a target similar to their significant other, and a yoked control. We found that prediction differences between the significant other and similar target led to longer response times, and we found the opposite effect for self and target differences, suggesting anchoring and adjustment from the significant other rather than the self. These effects were moderated by the source-relative salience of the dimension being evaluated. The evidence was mixed with respect to the question of whether anchoring on a significant other is more effortful than anchoring on the self. (shrink)

I undertake to explain how the well known laws of formal logic – Barbara Syllogism, modus ponens, etc. – relate to experience by developing Edmund Husserl's critique ofFormalism and Psychologism in logical theory and then briefly explaining his positive views of the laws of logic. His view rests upon his understanding of the proposition as a complex, intentional property. The laws of formal logic are, on his view (and mine), statements about the truth values of propositions as determined by their (...) formal character and relationships alone. The laws thus understood explain how algorithms set up to mirror them can accomplish what they do to advance knowledge, even though they operate purely mechanically. Further, they explain the proper sense in which formal laws "govern," and may guide, processes of actual thinking. Husserl's theory is a realist theory in the sense that, on his interpretation, the laws of pure or formal logic hold true regardless of what any individual, culture or species may or may not think, or even if no thinking ever occurs. (shrink)

Jesus The Logician ABSTRACT: In understanding how discipleship to Jesus Christ works, a major issue is how he automatically presents himself to our minds. It is characteristic of most 20th century Christians that he does not automatically come to mind as one of great intellectual power: as Lord of universities and research institutes, of the creative disciplines and scholarship. The Gospel accounts of how he actually worked, however, challenge this intellectually marginal image of him and help us to see him (...) at home in the best of academic and scholarly settings of today, where many of us are called to be his apprentices. (shrink)

Published by InterVarsity Press in Faith, Film and Philosophy: Big Ideas on the Big Screen, edited by R. Douglas Geivett & James S. Spiegel, 2007.

Metaphysical and epistemological commitments seem to determine the course of research in the field of logic as well as its theoretical interpretation. What we take the objects of logical investigation to be determines our views on how they are to be known, and our view of the possible types of knowledge in turn places restrictions on what kinds of things those objects could be. Perhaps it is true that logical studies can be pursued to great lengths without indulging in general (...) declarations about the nature of reality or of human knowledge. But to say this is very far from admitting the possibility of “logic without ontology.” The thoughtful beginning student in logic quickly wants to know what it is he or she is really studying and what, precisely, it has to do with the real worlds. The slightest philosophical stimulation will lead him or her, further, to ask about the evidence supporting the various sorts of claims made as the course progresses, and about the methodology of logic from a scientific point of view. Rarely is any answer given to such questions, much less a reasonable and well elaborated one. Questions of this type are usually discouraged and avoided by text books and instructors, and there may even be some suggestion that to ask them evidences confusion and stupidity. But when they are pressed it turns out that logic as a field of knowledge and inquiry remains in chaos at its philosophical foundations. I mean by this that the leading investigators cannot agree—indeed, they disagree in dimensions categorial—concerning what the subject of logical knowledge is, and concerning the modes of logical knowledge. (shrink)

In response to the common assumption of perpetual moral and spiritual failure in human life, this article offers a view of personal transformation and, in particular, of the role of the human will in such formation, that clears a way forward in progressive conformity to the character of Christ. Drawing from St. Paul's understanding of the “flesh” and the human “spirit,” distinctions are made between the impulsive, the reflective, and the embodied will. It is the embodied/reflective will of the person (...) that is captured by Christ through inner transformation, such that the crucifixion of the flesh and walking by the Spirit brings about routine, easy obedience to Christ, from the inside out. (shrink)