In this book Saul Kripke brings his powerful philosophical intelligence to bear on Wittgenstein's analysis of the notion of following a rule.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Everyset. for example, is decidable by such machines, and the semi-decidable sets form a portion of thesets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators.

Accelerating Turing machines have attracted much attention in the last decade or so. They have been described as “the work-horse of hypercomputation” (Potgieter and Rosinger 2010: 853). But do they really compute beyond the “Turing limit”—e.g., compute the halting function? We argue that the answer depends on what you mean by an accelerating Turing machine, on what you mean by computation, and even on what you mean by a Turing machine. We show first that in the current literature the term (...) “accelerating Turing machine” is used to refer to two very different species of accelerating machine, which we call end-stage-in and end-stage-out machines, respectively. We argue that end-stage-in accelerating machines are not Turing machines at all. We then present two differing conceptions of computation, the internal and the external, and introduce the notion of an epistemic embedding of a computation. We argue that no accelerating Turing machine computes the halting function in the internal sense. Finally, we distinguish between two very different conceptions of the Turing machine, the purist conception and the realist conception; and we argue that Turing himself was no subscriber to the purist conception. We conclude that under the realist conception, but not under the purist conception, an accelerating Turing machine is able to compute the halting function in the external sense. We adopt a relatively informal approach throughout, since we take the key issues to be philosophical rather than mathematical. (shrink)

Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of contains n consecutive 7s, for any n; solve the Turing-machine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary (...) to a recent paper by Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle's Chinese room argument. (shrink)

In the history of modern philosophy systematic connections were assumed to hold between the modal concepts of logical possibility and necessity and the concept of conceivability. However, in the eyes of many contemporary philosophers, insuperable objections face any attempt to analyze the modal concepts in terms of conceivability. It is important to keep in mind that a philosophical explanation of modality does not have to take the form of a reductive analysis. In this paper I attempt to provide a response-dependent (...) account of the modal concepts in terms of conceivability along the lines of a nonreductive model of explanation. (shrink)

There is a long tradition in philosophy of using a priori methods to draw conclusions about what is possible and what is necessary, and often in turn to draw conclusions about matters of substantive metaphysics. Arguments like this typically have three steps: first an epistemic claim , from there to a modal claim , and from there to a metaphysical claim.

A number of popular arguments for dualism start from a premise about an epistemic gap between physical truths about truths about consciousness, and infer an ontological gap between physical processes and consciousness. Arguments of this sort include the conceivability argument, the knowledge argument, the explanatory-gap argument, and the property dualism argument. Such arguments are often resisted on the grounds that epistemic premises do not entail ontological conclusion. My view is that one can legitimately infer ontological conclusions from epistemic premises, if (...) one is very careful about how one reasons. To do so, the best way is to reason first from epistemic premises to modal conclusions , and from there to ontological conclusions. Here, the crucial issue is the link between the epistemic and modal domains. How can one reason from theses about what is knowable or conceivable to theses about what is necessary or possible? To bridge the epistemic and modal domains, the framework of two-dimensional semantics can play a central role. I have used this framework in earlier work to mount an argument against materialism. Here, I want to revisit the argument, laying it out in a more explicit and careful form, and responding to a number of objections. In what follows I will concentrate mostly on the conceivability argument. I think that very similar considerations apply to the other arguments mentioned above, however. In the final section of the paper, I show how this analysis might yield a unified treatment of a number of anti-materialist arguments. (shrink)