The Ways of Paradox and Other Essays
Selected Logic Papers
Modern analytic philosophy is descended from a fertile if temporary union between the revived formal logic of our age and empiricist philosophy. In the last decades of the nineteenth century logic underwent, at the hands of Frege, its most important developments since Aristotle started it off as a systematic discipline. Frege’s work seemed to realize the prophetic dreams of Leibniz. With it formal logic came to cover a vastly wider field than the syllogistic logic of Aristotle had ever done. Where the logic of Aristotle was largely confined to the study of inferences owing their validity to the way the words “all,” “some,” and “not” occurred in them (as in the old favorite: all men are mortal, all Greeks are men, so all Greeks are mortal), the logic of Frege also covered inferences hinging on “and,” “or,” and “if.” Aristotle’s theory of the syllogism turned out to be a rather small, elementary segment of the second main part of Frege’s system. Frege’s logic was expressed with unprecedented rigor, and as a crowning achievement, seemed to afford a basis of indubitable certainties from which the whole of mathematics could be derived, effecting a unification of the two disciplines.
At much the same time Mach in Austria, and W. K. Clifford and Karl Pearson in England, were adapting features of the traditional empiricism of Mill to the interpretation of physics, a science that Mill knew only as an intelligent general reader. For this new philosophy of science the basis of all knowledge of fact was the reports of immediate, subjective sense-experience. All the statements of common observation and scientific theory owed their significance and truth to these basic assertions, of which they were held to be convenient abbreviations. Concepts of ordinary material things and of the theoretical entities of science were understood as shorthand for the concepts of direct perception: color patches in the visual field, felt pressures and so forth. Science in this view was an application of Mach’s principle of the economy of thought, a concise and convenient notation for the common patterns and regularities to be found in the streams of individual sense-experience.
The union of logic and empiricism was solemnized in the first really independent philosophical writings of the first man to combine the requisite logical and philosophical expertise, in Our Knowledge of the External World (1914) and Philosophy of Logical Atomism (1918) of Bertrand Russell. In these works the world is seen as an array of individual events of sense-experience, some related so as to constitute minds, some (including many of the previous group) as material things. For Russell the task of philosophy was to use the new resources of logic to analyze the stock of received knowledge, to reveal its ultimate constitution in basic empirical terms where this was possible, and to discard it as illusion where no such reduction could be effected.
WITTGENSTEIN’S Tractatus Logico-Philosophicus (1922) was the first brilliant wayward child of the marriage, but the parental lineaments were more obvious in the logical positivism of the Vienna Circle. Wittgenstein’s chief service to the tradition at this early stage of his career was to provide a congruous account of logic itself, the instrument of analysis. Formal research in the most elementary and fundamental part of the discipline led him to the view that the laws of logic are tautologies, statements that owe their necessary truth not to some unalterable structural features of the world which they might be thought to describe but to the conventions of language by which they are endowed with meaning. The central idea here is that all necessary truths are at bottom alike in nature. The most sophisticated theorems of mathematics are true for the same reason that “all bachelors are unmarried” is. That simple truism records no substantial truth about the world. It reflects a fact about language, that the same rules are in force for the distinct expressions “bachelor” and “unmarried man.” We do not discover that, as a matter of objective fact, all bachelors are unmarried men. The possibility of an exception is ruled out in advance by the meanings that are conventionally assigned to the words in question.
In its standard form, as expounded in the 1930s by Schlick and Carnap, logical positivism consisted of three main doctrines, multiply related to each other: phenomenalism—a theory of factual knowledge, conventionalism—a theory of logic or formal knowledge, and verificationism—a theory of meaning. Phenomenalism holds that all knowledge of fact is or can be reduced to knowledge of immediate experience, conventionalism that the necessary truth of logic and mathematics is due to conventions of language, verificationism that any utterance that is neither phenomenal nor conventional is beyond the pale, metaphysical, without meaning as a statement, and at best an effusion of feeling. Thus for the positivist “there is an apple here” is really a compact way of referring to a pattern of visual, tactual, and olfactory experiences that could be obtained here. “4 × 7 = 28” is made true by the meanings conventionally given to the numerals, and “God loves us all,” being neither reducible to immediate sense experience nor true in virtue of its meaning, is not really a statement at all but perhaps the expression of a subconscious wish that everything will turn out all right for us in the end.
The pressure of history brought most leading European positivists to the United States at the end of the 1930s, notably Carnap and Reichenbach. They got an intellectual welcome over and above the political asylum open to them as refugees from fascism. To start with, there is a certain broad affinity between positivism and pragmatism, in Dewey’s version at any rate, since both are pro-scientific and anti-theological. Peirce, who was preeminently a logician and a philosopher of science, shared the leading interests of the positivists and his version of pragmatism closely anticipated the verification principle. For Peirce the meaning of a statement lay in the observable difference that would result from its being true rather than false. Such a view is remote from the pragmatism of folklore, the idea that the true is what works, which is not all that much of a travesty of the position of William James. Then there is a strongly pragmatic flavor to the idea that all concepts but those of immediate sensory qualities are abbreviative constructions to be chosen on grounds of convenience and economy. The leading American theorist of knowledge of the interwar period, C. I. Lewis, had come, from a pragmatic starting point, very close to positivism by 1929 when the European movement was only just under way. Always a phenomenalist, he was at that time a conventionalist and he was an anti-metaphysician in practice if not by profession.
BUT JUST AS TRANSPLANTATION seemed to be giving new life to the most developed form of the union between logic and empiricism, the union itself was coming apart. Logic was withdrawing itself from philosophy and assimilating itself to mathematics. A good theoretical reason for this was Gödel’s proof, in his epoch-making paper of 1931, that the Frege-Russell ideal of constructing a complete system of logic and mathematics was incapable of completion. So the interest of logicians turned from the construction of one ideal system to the disciplined study of deductive systems of all kinds from outside. Another reason was the increasing bulk and sophistication of logic itself which converted its skilled practitioners into an autonomous profession, one that had a direct technical application in the theory of electrical circuits and computer engineering.
On the other side, philosophy of a broadly empiricist, non-edifying kind was undergoing a revolt against formalism, that is against the conviction that discourse achieves its ideal form in the propositions of mathematics and natural science and against the connected principle of method which takes translation into the notation of formal logic to be the proper way to distill the substantial content from any kind of thought or speech. The uncharitable explain this revolt by the fact that philosophers in Britain, where it began, ordinarily know a good deal of classics and rather little mathematics or science. But this was more the fertilizer than the seed of the new development.
At any rate since 1945 analytic or empirical philosophy has become less and less formal, more and more grammatical, at first in Britain, more recently in the United States. It has not tried to solve the traditional problems of philosophy in the manner of positivism by a formalistic regimentation of language but has aimed rather to undermine the problems, to expose them as the outcome of hidden, seductive deviations from the ordinary use of words by close study of the way in which language is actually employed. Mathematics and natural science, the ideal forms of thought for positivism, are viewed by linguistic philosophers, usually pretty much out of the corners of their eyes, as just two out of the many going concerns within the whole field of discourse. Wittgenstein, in the later part of his career, and J. L. Austin, the most influential linguistic philosophers, laid great stress on the many other purposes that language can serve over and above the communication of knowledge. For them statement is not the prime function of language but is on a level with advice, incitement, the expression of feeling, and a host of others. Secondly, they dismissed the claims of formal logic to be a canon of rationality. For them it is not a display of the essential structure of rational discourse, but is rather a diagram or even caricature of properly conducted intellectual processes, a highly selective, not a literal, representation of them. Their philosophical method does not pursue economy and system but seeks to make explicit in all their complex variety the rules which govern established uses of words. This informal logic is a conservative undertaking, for it does not judge the forms of discourse it examines by a formal ideal of the communication of truth. It seeks simply to understand the different kinds of generally prevailing discourse and to codify them only to the extent that is necessary for the relief of philosophical perplexity and is possible in view of the complexity of language. Positivism, by contrast, is radical and critical. It dismisses metaphysics and theology altogether and looks down on common speech and the everyday beliefs it is used to express as first crude approximations to the language and the theories of science.
From a distance the two schools may seem as hard to distinguish as are Stalinists and revisionists to the eye of a stockbroker but there is no love lost between them. To their opponents the positivists look rigid and unenlightened followers of Procrustes, the linguistic philosophers look desultory, amateurish, and constitutionally inconclusive. An example may help to convey the difference between their procedures. Both are interested in the concept of cause. But where the positivist seeks for a single defining formula in logical notation for the sense of the word central to science, the linguistic philosopher compiles an open-ended album of its possible uses, registering every nuance and inflection with taxonomical dedication.