Penetrating and illuminating, these essays firmly install Merleau-Ponty among the most innovative and critically debated thinkers of the past half century.

There are theoretical limitations to what can be implemented by a computer program. In this paper we are concerned with a limitation on the strength of computer implemented deduction. We use a version of the Curry paradox to arrive at this limitation.

We develop a functional abstraction principle for the type-free algorithmic logic introduced in our earlier work. Our approach is based on the standard combinators but is supplemented by the novel use of evaluation trees. Then we show that the abstraction principle leads to a Curry fixed point, a statement C that asserts C ⇒ A where A is any given statement. When A is false, such a C yields a paradoxical situation. As discussed in our earlier work, this situation leaves (...) one no choice but to restrict the use of a certain class of implicational rules including modus ponens. (shrink)

There is significant interest in type-free systems that allow flexible self-application. Such systems are of interest in property theory, natural language semantics, the theory of truth, theoretical computer science, the theory of classes, and category theory. While there are a variety of proposed type-free systems, there is a particularly natural type-free system that we believe is prototypical: the logic of recursive algorithms. Algorithmic logic is the study of basic statements concerning algorithms and the algorithmic rules of inference between such statements. (...) As shown in [1], the threat of paradoxes, such as the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. The first part of the paper develops a rich collection of inference rules that do not lead to paradox. The second part identifies traditional rules of logic that are paradoxical in algorithmic logic, and so should be viewed with suspicion in type-free logic generally. (shrink)

α-recursion lifts classical recursion theory from the first transfinite ordinal ω to an arbitrary admissible ordinal α [10]. Idealized computational models for α-recursion analogous to Turing machine models for classical recursion have been proposed and studied [4] and [5] and are applicable in computational approaches to the foundations of logic and mathematics [8]. They also provide a natural setting for modeling extensions of the algorithmic logic described in [1] and [2]. On such models, an α-Turing machine can complete a θ-step (...) computation for any ordinal θ < α. Here we consider constraints on the physical realization of α-Turing machines that arise from the structure of physical spacetime. In particular, we show that an α-Turing machine is realizable in a spacetime constructed from R only if α is countable. While there are spacetimes where uncountable computations are possible and while they may even be empirically distinguishable from a standard spacetime, there is good reason to suppose that such nonstandard spacetimes are nonphysical. We conclude with a suggestion for a revision of Church’s thesis appropriate as an upper bound for physical computation. (shrink)

_Leading scholars engage the later contributions of Maurice Merleau-Ponty._.

In this note, we consider constraints on the physical possibility of transfinite Turing machines that arise from how one models the continuous structure of space and time in one's best physical theories. We conclude by suggesting a version of Church's thesis appropriate as an upper bound for physical computation given how space and time are modeled on our current physical theories.

ABSTRACT:There is considerable overlap between the interests of business ethicists and those of political philosophers. Questions about the moral justifiability of the capitalist system, the basis of property rights, and the problem of inequality in the distribution of income have been of central importance in both fields. However, political philosophers have developed, especially over the past four decades, a set of tools and concepts for addressing these questions that are in many ways quite distinctive. Most business ethicists, on the other (...) hand, consider their field to be primarily a domain of applied ethics, and so adopt methods and conceptual frameworks developed by moral philosophers. In this paper, we discuss some of the salient differences between these two approaches, and suggest some ways in which business ethicists could benefit from taking a more “political philosophy” approach to these questions. Throughout, we underline the importance of seeking greater compatibility among the principles used in normative theorizing about markets, regulations, corporate governance, and business practices. (shrink)