For over a century now there has been a continuing debate about whether the forms of explanation appropriate to the social sciences are essentially the same as or radically different from those used in the natural sciences. On one side is the empiricist philosophical tradition, ranging at least from John Stuart Mill through the logical positivists. According to this view, the covering law model of explanation appropriate for the natural sciences is equally appropriate for subjects such as history, anthropology, linguistics, economics, and other social sciences.Onthe other side is the interpretivist or hermeneutic tradition which ranges at least from Dilthey in the nineteenth century through the twentieth-century followers of Wittgenstein. According to this tradition, there are special modes of explanation appropriate to human behavior. In the second tradition, for example, Dilthey claims that a special method which he calls Verstehen (literally, understanding) is essential to the social sciences. And more recently, Charles Taylor (1985) claimed that human beings are unique in that events are meaningful to them in a special way and that any mode of explanation adequate to accounting for human behavior must explain this meaning component.

An unstated but underlying feature of this debate is often the assumption that much larger issues are at stake. There is at least the suggestion that the issue is a version of the dispute between materialism, on one hand, and dualism and idealism, on the other.

The Problem

What sorts of systematic explanations should we and can we seek in cognitive science for perception, language comprehension, rational action and other forms of cognition? In broad outline I think the answer is reasonably clear:We are looking for causal explanations, and our subject matter is certain functions of a biological organ, the human and animal brain.

As with any other natural science there are certain assumptions we have to make and certain conditions that our explanations have to meet. Specifically, we have to suppose that there exists a reality totally independent of our representations of it (in a healthier intellectual era it would not be necessary to say that), and we have to suppose that the elements of that reality that we cite in our explanations genuinely function causally.

Not all functions of the brain are relevant to cognition, so we have to be careful to restrict the range of brain functions we are discussing. Cognitive science is about the cognitive functioning of the brain and its relation to the rest of the organism and to the rest of the world in the way that nutrition science is about the digestive functioning of the digestive system and its relation to the rest of the organism and the rest of the world. Like other organs, and indeed like other physical systems, the brain has different levels of description and cognitive science is appropriately concerned with any level of description of the brain that is relevant to the causal explanation of cognition.

The notion of a performative is one that philosophers and linguists are so comfortable with that one gets the impression that somebody must have a satisfactory theory. But I have not seen such a theory and in this essay I want to address the question: how exactly do performatives work? I believe that answering that question is not just a fussy exercise in linguistic analysis but can give us insights into the nature of language and the relation between speech acts and actions generally. Some people who have written about performatives seem to think that it is just a semantic fact about certain verbs that they have performative occurrences, but the puzzle is: how could any verbs have such remarkable properties just as a matter of semantics? I can't fix the roof by saying, “I fix the roof” and I can't fry an egg by saying, “I fry an egg,” but I can promise to come and see you just by saying, “I promise to come and see you” and I can order you to leave the room just by saying, “I order you to leave the room.” Now why the one and not the other? And, to repeat, how exactly does it work? Perhaps the most widely accepted current view is the following: performative utterances are really just statements with truth values like any other statements, and Austin was wrong to contrast performative utterances with some other kind. The only special feature of the performative statement is that the speaker can perform some other speech act indirectly by making the statement.

Recursive Relations

A set of, say, natural numbers is effectively decidable if there is an effective procedure that, applied to a natural number, in a finite amount of time gives the correct answer to the question whether it belongs to the set. Thus, representing the answer ‘yes’ by 1 and the answer ‘no’ by 0, a set is effectively decidable if and only if its characteristic function is effectively computable, where the characteristic function is the function that takes the value 1 for numbers in the set, and the value 0 for numbers not in the set. A set is called recursively decidable, or simply recursive for short, if its characteristic function is recursive, and is called primitive recursive if its characteristic function is primitive recursive. Since recursive functions are effectively computable, recursive sets are effectively decidable. Church's thesis, according to which all effectively computable functions are recursive, implies that all effectively decidable sets are recursive.

The Size and Number of Models

By a model of a sentence or set of sentences we mean an interpretation in which the sentence, or every sentence in the set, comes out true. Thus Γ implies D if every model of Γ is a model of D, D is valid if every interpretation is a model of D, and Γ is unsatisfiable if no interpretation is a model of Γ.

Order in Nonstandard Models

Let ℳ be a model of (true) arithmetic not isomorphic to the standard model N.

The original authors of this work, the late George Boolos and my late colleague Richard Jeffrey, stated in the preface to the first edition that the work was intended for students of philosophy, mathematics, and other fields who desired a more advanced knowledge of logic than is supplied by an introductory course or textbook on the subject, and added the following:

The aim has been to present the principal fundamental theoretical results about logic, and to cover certain other meta-logical results whose proofs are not easily obtainable elsewhere. We have tried to make the exposition as readable as was compatible with the presentation of complete proofs, to use the most elegant proofs we knew of, to employ standard notation, and to reduce hair (as it is technically known).

Such have remained the aims of all subsequent editions, including the present one.

The ‘principal fundamental theoretical results about logic’ are primarily the theorems of Gödel—the completeness theorem and especially the incompleteness theorems—with their attendant lemmas and corollaries. The ‘other meta-logical results’ included have been of two kinds. On the one hand, filling roughly the first third of the book, there is an extended exposition by R.C.J. of the theory of Turing machines, a topic frequently alluded to in the literature of philosophy, computer science, and cognitive studies, but often omitted in textbooks on the level of this one.