I think that if we attempted to set out the ‘man on the street's’ view of the nature of time, we would find that the main principle underlying his convictions on this subject might be stated somewhat as follows:

(1) All (and only) things that exist now are real.

Future things (which do not already exist) are not real (on this view); although, of course they will be real when the appropriate time has come to be the present time. Similarly, past things (which have ceased to exist) are not real, although they were real in the past.

Obviously, we shall have to make some assumptions about the concept real if we are to discuss the ‘man on the street's’ view at all. The assumptions I shall make are as follows:

1. I. I-now am real. (Of course, this assumption changes each time I announce that I am making it, since ‘I-now’ refers to a different instantaneous ‘me’.)

2. II. At least one other observer is real, and it is possible for this other observer to be in motion relative to me.

And, the most important assumption, which will be referred to (when properly understood) as the principle that There Are No Privileged Observers:

1. III. If it is the case that all and only the things that stand in a certain relation R to me-now are real, and you-now are also real, then it is also the case that all and only the things that stand in the relation R to you-now are real.

There is no doubt that the examination into the structure of physical geometry and relativity theory initiated by Reichenbach and continued by Grünbaum has extended our understanding and called our attention to matters of philosophical importance. Unfortunately, I find myself unable to agree with Grünbaum on a number of central points. I do agree that the ordinary standard of congruence in physical geometry is the solid rod. I do not agree that one can define ‘congruent’ in terms of solid rods in such a way as to make possible an empirical determination of the metric, especially if the need for correcting for perturbational or differential forces in accordance with some not exactly known system of physical laws is to be taken into account. The prevalent assumption to the contrary is due to an error committed by Reichenbach and frequently found in the literature of the past thirty years (the error of supposing that there must be ‘universal forces’ in any non-standard metric). What appears rather to be the case is that the metric is implicitly specified by the whole system of physical and geometrical laws and ‘correspondence rules’. § No very small subset by itself fully determines the metric; and certainly nothing that one could call a ‘definition’ does this.

Secondly, I believe that it is misleading to say, as Grünbaum does, that physical geometry provides ‘the articulation of the system of relations obtaining between bodies and transported solid rods quite apart from their substance-specific distortions’ (p. 510). What Grünbaum means is not really ‘quite apart from their substance-specific distortions’, but after proper correction has been made for their substance-specific distortions.

Before we say anything about quantum mechanics, let us take a quick look at the Newtonian (or ‘classical’) view of the physical universe. According to that view, nature consists of an enormous number of particles. When Newtonian physics is combined with the theory of the electromagnetic field, it becomes convenient to think of these particles as dimensionless (even if there is a kind of conceptual strain involved in trying to think of something as having a mass but not any size), and as possessing electrical properties – negative charge, or positive charge, or neutrality. This leads to the well-known ‘solar system’ view of the atom – with the electrons whirling around the nucleons (neutrons and protons) just as the planets whirl around the sun. Out of atoms are built molecules; out of molecules, macroscopic objects, scaling from dust motes to whole planets and stars. These latter also fall into larger groupings – solar systems and galaxies – but these larger structures differ from the ones previously mentioned in being held together exclusively by gravitational forces. At every level, however, one has trajectories (ultimately that means the possibility of continuously tracing the movements of the elementary particles) and one has causality (ultimately that means the possibility of extrapolating from the history of the universe up to a given time to its whole sequence of future states).

The philosophy of physics is continuous with physics itself. Just as certain issues in the Foundations of Mathematics have been discussed by both mathematicians and by philosophers of mathematics, so certain issues in the philosophy of physics have been discussed by both physicists and by philosophers of physics. And just as there are issues of a more epistemological kind that tend to concern philosophers of mathematics more than they do working mathematicians, so there are issues that concern philosophers of physics more than they do working physicists. In this brief report I shall try to give an account of the present state of the discussion in America of both kinds of issues, starting with the problems of quantum mechanics, which concern both physicists and philosophers, and ending with general questions about necessary truth and the analytic-synthetic distinction which concern only philosophers.

The problem of ‘measurement’ in quantum mechanics

Quantum mechanics asserts that if A and B are any two possible ‘states’ of a physical system, then there exists at least one state (and in fact a continuous infinity of states) which can be described as ‘superpositions of A and B’. If A and B are the sorts of states talked about in classical physics – definite states of position, or momentum, or kinetic energy, etc. – then their superpositions may not correspond to classically thinkable states.

The Liar and related paradoxes

The importance of the Liar paradox to the theory of truth has already become apparent; for Tarski's formal adequacy conditions on definitions of truth are motivated, in large part, by the need to avoid it. It is time, now, to give the Liar and related paradoxes some direct attention on their own account.

Why the ‘Liar paradox’? Well, the Liar sentence, together with apparently obvious principles about truth, leads, by apparently valid reasoning, to contradiction; that is why it is called a paradox (from the Greek, ‘para’ and ‘doxa’ ‘beyond belief’).

The Liar comes in several variants; the classic version concerns the sentence:

Suppose S is true; then what it says is the case; so it is false. Suppose, on the other hand, that S is false; then what it says is not the case, so it is true. So S is true iff S is false. Variants include indirectly self-referential sentences, such as:

Another variant, the ‘Epimenides’ paradox, concerns a Cretan called Epimenides, who is supposed to have said that all Cretans are always liars.

Assessing arguments

Arguments are assessed in a great many ways; some, for instance, are judged to be more persuasive or convincing than others, some to be more interesting or fruitful than others, and so forth. The kinds of assessment that can be made can be classified, in a rough and ready way, like this:

1. (i) logical: is there a connection of the appropriate sort between the premises and the conclusion?

2. (ii) material: are the premises and conclusion true?

3. (iii) rhetorical: is the argument persuasive, appealing, interesting to the audience?

I have given only the vaguest indication of the kinds of question characteristic of each dimension of assessment, but a rough indication should be adequate for present purposes. The separate category given to rhetorical considerations is not intended to suggest that the validity of an argument, or the truth of its premises, is quite irrelevant to its persuasiveness; it is intended, rather, to allow for the fact that, though if people were completely rational they would be persuaded only by valid arguments with true premises, in fact often enough they are persuaded by invalid arguments or arguments with false premises, and not persuaded by sound (cf. p. 14 below) arguments (see e.g. Thouless 1930, Stebbing 1939, Flew 1975, Geach 1976 for discussion of, and advice on how to avoid, such failures of rationality).

In what follows I shall be almost exclusively concerned with the first, logical, dimension of assessment.

Logic, philosophy of logic, metalogic

The business of philosophy of logic, as I understand it, is to investigate the philosophical problems raised by logic – as the business of the philosophy of science is to investigate the philosophical problems raised by science, and of the philosophy of mathematics to investigate the philosophical problems raised by mathematics.

A central concern of logic is to discriminate valid from invalid arguments; and formal logical systems, such as the familiar sentence and predicate calculi, are intended to supply precise canons, purely formal standards, of validity. So among the characteristically philosophical questions raised by the enterprise of logic are these: What does it mean to say that an argument is valid? that one statement follows from another? that a statement is logically true? Is validity to be explained as relative to some formal system? Or is there an extra-systematic idea that formal systems aim to represent? What has being valid got to do with being a good argument? How do formal logical systems help one to assess informal arguments? How like ‘and’ is ‘&’, for instance, and what should one think of ‘p’ and ‘q’ as standing for? Is there one correct formal logic? and what might ‘correct’ mean here? How does one recognise a valid argument or a logical truth? Which formal systems count as logics, and why?

A summary sketch

The object of this section is to sketch the main kinds of theories of truth which have been proposed, and to indicate how they relate to each other. (Subsequent sections will discuss some theories in detail.)

Coherence theories take truth to consist in relations of coherence among a set of beliefs. Coherence theories were proposed e.g. by Bradley 1914, and also by some positivist opponents of idealism, such as Neurath 1932; more recently, Rescher 1973 and Dauer 1974 have defended this kind of approach. Correspondence theories take the truth of a proposition to consist, not in its relations to other propositions, but in its relation to the world, its correspondence to the facts. Theories of this kind were held by both Russell 1918 and Wittgenstein 1922, during the period of their adherence to logical atomism; Austin defended a version of the correspondence theory in 1950. The pragmatist theory, developed in the works of Peirce (see e.g. 1877), Dewey (see e.g. 1901) and James (see e.g. 1909) has affinities with both coherence and correspondence theories, allowing that the truth of a belief derives from its correspondence with reality, but stressing also that it is manifested by the beliefs' survival of test by experience, its coherence with other beliefs; the account of truth proposed in Dummett 1959 has, in turn, quite strong affinities with the pragmatist view.

Many-valued systems

Restrictions of classical logic: deviant logics

One system is a deviation of another if it shares the vocabulary of the first, but has a different set of theorems/valid inferences; a ‘deviant logic’ is a system which is a deviation of classical logic. (A system may involve both an extension and a deviation of classical logic, if it adds new vocabulary and hence new theorems/valid inferences, but at the same time differs from classical logic with respect to theorems/valid inferences involving only shared vocabulary essentially. The system E, considered in ch. 10 §6, would be an example.) Many-valued logics are deviant; sharing its vocabulary, they characteristically lack certain theorems of classical logic, such as the ‘law of excluded middle’, ‘p ∨ −p’. (Some also add new vocabulary and so come in the category of extensions as well.)

The many-valued logics I shall consider in this chapter were devised from two main kinds of motivation: the purely mathematical interest of alternatives to the 2-valued semantics of classical sentence logic; and – of more philosophical interest – dissatisfaction with the classical imposition of an exhaustive dichotomy into the true and the false, and, relatedly, dissatisfaction with certain classical theorems or inferences. The second kind of motivation is characteristic – as I observed in ch. 9 §2 – of proposals for restrictions of classical logic.

Three approaches

A recurrent issue in the philosophy of logic concerns the question, with what kind of item logic deals, or perhaps primarily deals. The alternatives offered are usually sentences, statements and propositions, or, more rarely these days, judgments or beliefs. I have put the question in a deliberately vague way, since more than one issue seems to be involved. Once again, as with the issue about the meanings of connectives, quantifiers, etc., the problem concerns the relation between formal and informal arguments: what in informal arguments corresponds to the well-formed formulae of formal languages? It may be useful to distinguish three approaches to the question:

1. (i) syntactic: what, in natural languages, is the analogue of the ‘p’, ‘q’ of formal logic?

In speaking thus far of ‘sentence calculus’, I did not mean to beg this question. Some prefer to speak of ‘propositional calculus’, ‘propositional variables’, ‘propositional connectives’; and so far I have said nothing to justify my preference for the former usage.

1. (ii) semantic: what kind of item is capable of truth and falsity?

Since formal languages aim to represent those informal arguments which are valid extra-systematically, that is, which are truthpreserving, this will relate closely to the first issue.

1. (iii) pragmatic: what kinds of item should one suppose to be the ‘objects’ of belief, knowledge, supposition, etc.?

(‘Know’, ‘believe’, ‘suppose’, etc. are sometimes called the verbs of ‘propositional attitude’.) Since one can know, believe or suppose either something true or something false, the third will relate quite closely to the second issue.

Singular terms and their interpretation

Some formulations of the predicate calculus employ singular terms (‘a’, ‘b’ … etc.) as well as variables. If the quantifiers are to be interpreted substitutionally, of course, the presence of singular terms in the language to supply the appropriate substitution instances is essential. What, in informal argument, corresponds to singular terms in formal logic? Singular terms are usually thought of as the formal analogues of proper names in natural languages. (Where the variables range over numbers, the numerals would correspond to the singular terms.) The formal interpretation of singular terms assigns to each a specific individual in the domain over which the variables range; and, in natural languages, proper names are thought to work in a similar way, each standing for a particular person (or place or whatever).

So while in the case of the quantifiers the main controversy surrounds the question of the most suitable formal interpretation, in the case of singular terms the problems centre, rather, on the understanding of their natural language ‘analogues’. The formal interpretation of singular terms in straightforward extensional languages is uncontroversial; however, rival views about how to understand proper names in natural languages have been used in support of alternative proposals about the formal interpretation of singular terms in less straightforward, e.g. modal, calculi. Among the disputed questions about how, exactly, proper names work are, for instance: just which expressions are bona fide proper names?

Necessary truth

Modal logic is intended to represent arguments involving essentially the concepts of necessity and possibility. Some preliminary comments about the idea of necessity, therefore, won't go amiss. There is a long philosophical tradition of distinguishing between necessary and contingent truths. The distinction is often explained along the following lines: a necessary truth is one which could not be otherwise, a contingent truth one which could; or, the negation of a necessary truth is impossible or contradictory, the negation of a contingent truth possible or consistent; or, a necessary truth is true in all possible worlds (pp. 188ff. below), a contingent truth is true in the actual but not in all possible worlds. Evidently, such accounts aren't fully explanatory, in view of their ‘could (not) be otherwise’, ‘(im)-possible’, ‘possible world’. So the distinction is sometimes introduced, rather, by means of examples: in a recent book (Plantinga 1974 p. 1) ‘7 + 5 = 12’, ‘If all men are mortal and Socrates is a man, then Socrates is mortal’ and ‘If a thing is red, it is coloured’ are offered as examples of necessary truths, and ‘The average rainfall in Los Angeles is about 12 inches’ as an example of a contingent truth.

The distinction between necessary and contingent truths is a metaphysical one; it should be distinguished from the epistemological distinction between a priori and a posteriori truths.

‘Classical’ and ‘non-classical’ logics

There are a great many formal logical systems. In fact, ever since the ‘classical’ logical apparatus was formulated, there have been those who urged that it should be improved, modified or replaced. An instructive example can be taken from the history of the material conditional; anticipated by the Stoics, ‘material implication’ was formalised by Frege 1879 and Russell and Whitehead 1910 and supplied with a suitable semantics by Post 1921 and Wittgenstein 1922. As early as 1880, however, MacColl had urged the claims of a stricter conditional; ‘strict implication’ was formalised by Lewis 1918; and after that dissatisfaction with its claims to represent entailment led to the introduction of ‘relevant implication’ (see ch. 10 §7).

My present object is to get some perspective on the great variety of logical systems, to approach such questions as how they relate to each other, whether one must choose between them, and, if so, how. My strategy will be to consider the various ways in which the standard logical apparatus has been modified, and the various pressures in response to which such modifications have been made.

The century since the publication of Frege's Begriffsschrift has seen a tremendous growth in the development and study of logical systems. The variety of this growth is as impressive as its scale. One can distinguish four major areas of development, two in formal, two in philosophical studies: (i) the development of the standard logical apparatus, beginning with Frege's and Russell and Whitehead's presentation of the syntax of sentence and predicate calculi, subsequently supplied with a semantics by the work of e.g. Post, Wittgenstein, Löwenheim and Henkin, and studied metalogically in the work of e.g. Church and Gödel; (ii) the development of non-standard calculi, such as the modal logics initiated by C. I. Lewis, the many-valued logics initiated by Łukasiewicz and Post, the Intuitionist logics initiated by Brouwer. Alongside these one has (iii) philosophical study of the application of these systems to informal argument, of the interpretation of the sentence connectives and quantifiers, of such concepts as truth and logical truth; and (iv) study of the aims and capacities of formalisation, by those, such as Carnap and Quine, who are optimistic about the philosophical significance of formal languages, by those, such as F. C. S. Schiller and Strawson, who are sceptical of the pretensions of symbolic logic to philosophical relevance, and by those, such as Dewey, who urge a more psychological and dynamic conception of logic over the prevailing one.