B. Jack Copeland

To compute is to execute an algorithm. More precisely, to say that a device or organ computes is to say that there exists a modelling relationship of a certain kind between it and a formal specification of an algorithm and supporting architecture. The key issue is to delimit the phrase of a certain kind. I call this the problem of distinguishing between standard and nonstandard models of computation. The successful drawing of this distinction guards Turing's 1936 analysis of computation against a difficulty that has persistently been raised against it, and undercuts various objections that have been made to the computational theory of mind.

Ignoring the temporal dimension, an object such as a railway tunnel or a human body is a three-dimensional whole composed of three-dimensional parts. The four-dimensionalist holds that a physical object exhibiting identity across time—Descartes, for example—is a four-dimensional whole composed of 'briefer' four-dimensional objects, its temporal parts. Peter van Inwagen (1990) has argued that four-dimensionalism cannot be sustained, or at best can be sustained only by a counterpart theorist. We argue that different schemes of individuation of temporal parts are available, which undermines van Inwagen's argument.

Fuzzy logic extends deductive methods to situations in which the information available may be only partly or approximately true. Fuzzy logic has often been championed as a logic of vague terms, and it does indeed provide an intuitive analysis of what goes wrong in Sorites reasoning. Here a fuzzy semantics is given for a language containing the quasi-modal operators “Determinately” (Delta) and “Indeterminately” (Nabla) and the identity predicate (=). The semantics is sensitive to higher-order vagueness. For example, the semantics distinguishes between Herbert’s being a clear borderline case of a bald man and his being a borderline borderline case of a bald man. I show that a famous reductio ad absurdum of the statement “Indeterminately (a=b)”, due to Gareth Evans, is not valid when the background logic is fuzzy logic. Moreover, an improved form of Evans’s reductio due to Harold Noonan is also not valid. The proposed fuzzy semantics appears to provide a promising route to resolving Sorites-type paradoxes.

This article provides a detailed analysis of the transfer of a key cluster of ideas from mathematical logic to computing. We demonstrate the impact of certain of Turing’s logico-philosophical concepts from the mid-1930s on the emergence of the modern electronic computer—and so, in consequence, Turing’s impact on the direction of modern philosophy, via the computational turn. We explain why both Turing and von Neumann saw the problem of developing the electronic computer as a problem in logic, and we describe their joint journey from logic to electronic computation. While much has been written about Turing’s and von Neumann’s individual contributions to the development of the computer, this article investigates less well-known terrain: their interactions and mutual influences. Along the way we argue against ‘logic skeptics’ and ‘Turing skeptics’, who claim that neither logic nor Turing played any significant role in the creation of the modern computer.

Famous examples of conceivability arguments include (i) Descartes’ argument for mind-body dualism, (ii) Kripke's ‘modal argument’ against psychophysical identity theory, (iii) Chalmers’ ‘zombie argument’ against materialism, and (iv) modal versions of the ontological argument for theism. In this paper, we show that for any such conceivability argument, C, there is a corresponding ‘mirror argument’, M. M is deductively valid and has a conclusion that contradicts C's conclusion. Hence, a proponent of C—henceforth, a ‘conceivabilist’—can be warranted in holding that C's premises are conjointly true only if she can find fault with one of M's premises. But M's premises are modelled on a pair of C's premises. The same reasoning that supports the latter supports the former. For this reason, a conceivabilist can repudiate M's premises only on pain of severely undermining C's premises. We conclude on this basis that all conceivability arguments, including each of (i)–(iv), are fallacious.

What are the limits of physical computation? In his ‘Church’s Thesis and Principles for Mechanisms’, Turing’s student Robin Gandy proved that any machine satisfying four idealised physical ‘principles’ is equivalent to some Turing machine. Gandy’s four principles in effect define a class of computing machines (‘Gandy machines’). Our question is: What is the relationship of this class to the class of all (ideal) physical computing machines? Gandy himself suggests that the relationship is identity. We do not share this view. We will point to interesting examples of (ideal) physical machines that fall outside the class of Gandy machines and compute functions that are not Turing-machine computable.

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.

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.