The theme of the present work is the way in which modal logic, a branch of logic first studied by Aristotle, has been found to shed light on the mathematical study of mathematical reasoning, a study begun by David Hilbert and brought to fruition by Kurt Gödel.

Modal logic

The basic concepts of modal logic are those of necessity and possibility: A statement is called “possible” if it might be true (or might have been true) and “necessary” if it must be true (or could not have been untrue). E.g., since there might be a war in the year 2000, the statement that there will be a war then is possible; but the statement is not necessary, for there might not be one. On the other hand, the statement that there will or won't be a war in 2000 is necessary.

Necessity and possibility are interdefinable: a statement is necessary iff (if and only if) its negation is not possible, and, therefore, a statement is possible iff its negation is not necessary.

The customary sign for necessity in modal logic is the box, ‘□’, read ‘necessarily’, or ‘it is necessary that…’; the sign for possibility is the diamond ‘◊’, read ‘possibly,’ or ‘it is possible that…’. Thus like ∧ and ∨ and ∀ and ∃, either one of □ and ◊ can be regarded as defined from the other, □ as ¬◊¬ and ◊ as ¬□¬. We shall usually take □ as primitive and ◊ as defined: ‘◊A’ will abbreviate: ‘¬□¬A’.

Introduction

We recall from Chapter 3 the definition of the ω-inconsistency of a theory T (whose language contains 0 and s): T is ω-inconsistent iff for some formula A(x), T⊢∃xA(x), and for every natural number n, T⊢¬A(n). T is ω-consistent iff it is not ω-inconsistent. If T is ω-consistent, then T⊬∃xx ≠ x, and therefore T is consistent.

It is easy to show, however, that the converse does not hold: Let T be the theory that results when Bew(┌⊥┐) is added to PA. Since PA does not prove ¬Bew(┌⊥┐), T is consistent and for every n, the Δ sentence ¬Pf(n, ┌⊥┐) is true. Thus for every n, PA⊢¬Pf(n, ┌⊥┐), and so for every n, T⊢¬Pf(n, ┌⊥┐) (T extends PA). But T⊢Bew(┌⊥┐), that is, T⊢∃yPf(y, ┌⊥┐). So, despite its consistency, T is ω-inconsistent.

As a sentence S is said to be inconsistent with T if the theory whose axioms are those of T together with S itself is inconsistent, so S is ω-inconsistent (with T) if the theory whose axioms are those of T together with S is ω-inconsistent. S is ω-consistent iff not ω-inconsistent.

We call a sentence S ω-provable in T iff ¬S is ω-inconsistent with T. So if S is provable in T, S is ω-provable in T.

We are going to investigate a system of propositional modal logic, which we call ‘GL’, for Gödel and Löb. GL is also sometimes called provability logic, but the term is also used to mean modal logic, as applied to the study of provability. By studying GL, we can learn new and interesting facts about provability and consistency, concepts studied by Gödel in “On formally undecidable propositions of Principia Mathematica and related systems I”, and about the phenomenon of self-reference.

Like the systems T (sometimes called ‘M’), S4, B, and S5, which are four of the best-known systems of modal logic, GL is a normal system of propositional modal logic. That is to say, the theorems of GL contain all tautologies of the propositional calculus (including, of course, those that contain the special symbols of modal logic); contain all distribution axioms, i.e., all sentences of the form □(A→B)→(□A→□B); and are closed under the rules of modus ponens, substitution, and necessitation, according to which □A is a theorem provided that A is. Nor does GL differ from those other systems in the syntax of its sentences: exactly the same sequences of symbols count as well-formed sentences in all five systems.

GL differs greatly from T, S4, B, and S5, however, with respect to basic questions of theoremhood. All sentences □(□A→A)→□A are axioms of GL.

In a series of papers and a recent book, Fred Dretske has been working out an innovative account of how reasons explain behavior. His starting point is what we may call “the causal thesis”, often associated with Davidson, that reasons rationalize behavior by being its cause. With Davidson, therefore, Dretske takes rationalizing explanations to be a species of causal explanation, explanations that specify the causal antecedents of their explananda. Reasons are beliefs, desires, and other assorted “contentbearing” states, and these are among the paradigmatic instances of intentional mental states. Thus, the problem of explaining how reasons rationalize (that is, explain by providing reasons) is, for Dretske, the problem of giving an account of how intentional states can be causes, that is, the problem of intentional or rational causation. If we further assume, with Dretske, that the behavior to be rationalized is, or often involves, bodily events and processes, our problem is seen as a special case of the problem of psychophysical causation, that of understanding how mental events or states can enter into causal relations with physical events, as their causes or their effects. There is of course an even broader problem of mental causation, the problem of explaining how mental events can enter into any sort of causal relation, either as causes or as effects, whether with physical events or with other mental events.

The reality of the mental is closely tied to the possibility of mental causation, and anyone who takes a realist attitude toward the mental must be prepared with an account of how mental causation is possible.

EPIPHENOMENAL CAUSATION

Jonathan Edwards held the doctrine that ordinary material things do not persist through time but are at each moment created, and recreated, by God ex nihilo. He writes:

Thus, the present “time slice” of this table, although it is very much like the one preceding it, has no causal connection with it; for each slice is a wholly distinct creation by God. The temporal parts of this table are successive effects of an underlying persisting cause, God's creative activity. In arguing for this doctrine, Edwards offers the following striking analogy:

Suppose we could create an exact physical replica of a living human being - exactly like him cell for cell, molecule for molecule, atom for atom. Such a replica would be indistinguishable, at least physically, from the original. For we are supposing that the replica is a perfect physical copy in every detail. The idea of such a replica, whether artificially created or naturally found, is a perfectly coherent one; in fact, it is consistent with all known laws of nature. The idea of course is a commonplace in science fiction.

Given that your replica and you are exactly alike physically, will you also share your psychological life with him? Will your replica have your psychological traits and dispositions, intellectual powers and artistic gifts, anxieties and depressions, likes and dislikes, and virtues and vices? Will it feel pain, remorse, joy and elation exactly in the way you do? That is, if two organisms have identical physical features, will they be identical in psychological characteristics as well?

According to many moral theorists, any two things sharing the same ‘naturalistic’ or ‘descriptive’ features cannot differ in respect of moral or evaluative properties. Thus, it has been said that if St. Francis is a good man, anyone who is just like him in all naturalistic respects - in this case, broadly psychological properties, such as traits of character and personality - must of necessity be a good man.

I want to reopen the question whether the same bit of behavior, say an action we perform such as climbing a ladder, can be given both a “mechanistic” explanation, in terms of physiological processes and laws, and a “purposive” explanation, in terms of “reasons” (e.g., goals and beliefs). In a paper published in 1968, Norman Malcolm defended a negative answer. He argued that once an action has been explained by setting forth its physiological causal antecedents it is no longer open to us to explain it by citing the agent's reasons, that is, his beliefs, desires, intentions, and the like. Alvin Goldman immediately replied to Malcolm, arguing that mechanistic and purposive explanations are indeed compatible, that we can in fact characterize a type of situation in which one and the same behavior can be seen to be explainable both physiologically and rationally.

I want to reopen this debate not only because there is more to be said on this issue but also, and more importantly, because the issue has significant implications for some problems of much current interest in the philosophy of mind. A proper appreciation of the broader methodological issues and options involved will, I believe, help us to get clearer about some matters of current controversy. As we shall see, the question of explanatory compatibility leads us to more general questions about the pos-sibility of multiple explanations of a single explanandum, and the relationship between two distinct explanatory theories covering overlapping domains of phenomena.

RELATIONAL SUPERVENIENCE

Accounts of supervenience to date have almost exclusively focused on properties (that is, monadic attributes), although relations are informally mentioned sometimes in connection with supervenience. What happens if relations are explicitly taken into consideration in characterizing supervenience?

Let A be the supervening set of attributes, and B the base set. Consider first the case in which A includes an n-adic relation R, but B includes only monadic properties. It is evident that for R to supervene on B, the following condition is necessary and sufficient:

Depending on whether the n-tuples compared are restricted to a single world or may berecruited from different worlds, this will yield either “weak” or “strong” supervenience (Essay 5). But what is it for two n-tuples,Xn and Yn, to be indiscernible from eachother with respect to B? Since B is assumed to include only properties and norelations, the answer is simple: Xn is indiscernible fromYn in B just in case for each i (1' i' n) xiis indiscernible from yiin respect of B-properties.

The essays selected for this volume have been written over a period of approximately twenty years since the early 1970s, and are reprinted here without changes except for typographical and minor stylistic corrections and the updating of footnotes. Part I consists of papers on the metaphysical issues of events, causation, and supervenience; Part II includes papers on issues in the metaphysics of mind - in particular, mind-body supervenience and mental causation. Each part ends with a set of postscripts indicating my current thoughts on some of the central problems discussed therein.

I wish I could say that I stand by everything I said in these papers; on some issues I do of course, but on others my views have changed, rather significantly in a few instances, and I expect them to continue to change and evolve. On some of the issues I am not even clear just what I am now prepared to defend. This is the case, for example, with the theory of events. In Essays 1 and 3, I formulated and argued for what is now standardly called the “property exemplification” account of events, and I still think that it is a viable approach. However, I am now inclined to think that ontological schemes are by and large optional, and that the main considerations that should govern the choice of an ontology are those of utility, simplicity, elegance, and the like.

INTRODUCTION

It is part of today's conventional wisdom in philosophy of mind that psychological states are “multiply realizable”, and are in fact so realized, in a variety of structures and organisms. We are constantly reminded that any mental state, say pain, is capable of “realization”, “instantiation”, or “implementation” in widely diverse neural-biological structures in humans, felines, reptiles, mollusks, and perhaps other organisms further removed from us. Sometimes we are asked to contemplate the possibility that extraterrestrial creatures with a biochemistry radically different from the earthlings', or even electro-mechanical devices, can “realize the same psychology” that characterizes humans. This claim, to be called hereafter ‘the Multiple Realization Thesis’ (“MR”, for short), is widely accepted by philosophers, especially those who are inclined to favor the functionalist line on mentality. I will not here dispute the truth of MR, although what I will say may prompt a reassessment of the considerations that have led to its nearly universal acceptance.

And there is an influential and virtually uncontested view about the philosophical significance of MR. This is the belief that MR refutes psychophysical reductionism once and for all. In particular, the classic psychoneural identity theory of Feigl and Smart, the so-called “type physicalism”, is standardly thought to have been definitively dispatched by MR to the heap of obsolete philosophical theories of mind. At any rate, it is this claim, that MR proves the physical irreducibility of the mental, that will be the starting point of my discussion.

The term ‘event’ ordinarily implies change, and most changes are changes in a substance. Whether coming into being and passing away can be construed as changes in substances is a question we shall not consider here. A change in a substance occurs when that substance acquires a property it did not previously have, or loses a property it previously had. Whether fissions and fusions of substances can be considered as cases of losing or acquiring properties is, again, a question we shall not discuss in this paper. By ‘substance’ I mean things like tables, chairs, atoms, living creatures, bits of stuff like water and bronze, and the like; there is no need here to associate this notion with a particular philosophical doctrine about substance.

Besides events, we also speak of “states”. If “events” signal changes,“states” seem to be static things, “unchanges”, to use a term of C. J. Ducasse's; some examples of states would be my body's weighing 140 pounds, the earth's being nearly spherical in shape, and the presence of oxygen in this room. There are, however, good reasons for not taking this dichotomy of changes and unchanges, or of events and states, too seriously at the initial stage of developing a theory of events. For one thing, there are cases that are hard to classify e.g., the whirring of my typewriter, having a throbbing pain in the right elbow.

In an earlier paper, “Concepts of Supervenience,” I characterized two distinct concepts of supervenience, “strong” and “weak,” and compared them with each other and with a third concept, “global supervenience.” In this paper I wish to correct an error in the earlier paper and present further material on supervenience, including a new characterization of strong supervenience, which I believe is particularly perspicuous, and a discussion of the adequacy of global supervenience as a determination relation. I shall also present a strengthened relation of global supervenience based on similarity rather than indiscernibility between worlds, which may well be a more useful concept than the currently popular conception of global supervenience.

A NEW CHARACTERIZATION OF “STRONG SUPERVENIENCE”

Let A and B be two sets of properties (closed under complementation, conjunction, disjunction, and perhaps other property-forming operations). A is said to weakly supervene on B just in case:

1. (I) Necessarily, for any x and y, if x and y share all properties in B, then x and y share all properties in A - that is, indiscernibility in B entails indiscernibility in A.

This corresponds in a straightforward way to the informal characterization of supervenience commonly found in the literature. As was shown in the earlier paper, weak supervenience can be equivalently explained as follows:

1. (II) Necessarily, for any object x and any property F in A, if x has F, then there exists a property G in B such that x has G, and if any y has G, it has F.

In his celebrated discussion of causation Hume identified four prima facie constituents in the relation of causation. As everyone knows, they are constant conjunction, contiguity in space and time, temporal priority, and necessary connection. As ordinarily understood, the causal relation is a binary relation relating causes to their effects, and so presumably are the four relations Hume discerns in it. But what do these four relations tell us about the nature of the entities they relate?

Constant conjunction is a relation between generic events, that is, kinds or types of events; constant conjunction makes no clear or nontrivial sense when directly applied to spatiotemporally bounded individual events. On the other hand, it is clear that the relation of temporal priority calls for individual, rather than generic, events as its relata; there appears to be no useful way of construing ‘earlier than’ as a relation between kinds of classes of events in a causal context.

What of the condition of contiguity? This condition has two parts, temporal and spatial. Temporal contiguity makes sense when applied to events; two events are contiguous in time if they temporally overlap. But spatial contiguity makes best sense when applied not to events but to objects, especially material bodies; intuitively at least, we surely understand what it is for two bodies to be in contact or to overlap. For events, however, the very notion of spatial location often becomes fuzzy and indeterminate.