Preface

Dov M. Gabbay, King’s College London

John Woods, University of British Columbia and King’s College London and University of Lethbridge

In designing the Handbook of the History of Logic, we have been guided by two quite general principles of organization. In some cases a volume would show to advantage if organized around schools of thought or theoretical types. Logic and the Modalities in the Twentieth Century and The Many Valued and Nonmonotonic Turn in Logic are examples of this principle at work. In other cases we have found it more natural to assemble volumes around the names of practitioners. The subtitle of The Rise of Mathematical Logic: From Leibniz to Frege, reflects this principle, as does the present volume, Logic From Russell to Church. To a considerable extent, the two organizing principles are a loose convenience, between which the lines of demarcation are somewhat arbitrary. Even so, the developments tracked in this present volume are, by and large, at least as well known by the names of their authors as by their own names.

We have been asked for our rationale in ending volume 3 with Frege and opening volume 5 with Russell. Were not Frege and Russell virtual collaborators in giving to logic the momentum that would drive its mathematical turn? Why, then, would this pair be treated separately in the Handbook? Part of the answer to this question is unapologetically utilitarian. Volumes can only be so big, and chapters can be published only when they are ready. But the answer also has a further part, concerning which a word or two of explanation would be in order. It is well known that Frege and Russell shared a philosophical motivation for producing a new logic. The motivation is logicism, which asserts the reducibility of arithmetic to pure logic. Notwithstanding this commonality, Frege and Russell had quite different positions about the epistemic status of logic. Frege thought that the axioms of logic were known with certainty. Russell’s view was that in some cases an axiom — reducibility, for example - could be justified only inductively (or, as we would say, abductively) or pragmatically. It would not be too much to say that this difference about mathematical epistemology presages the instrumentalism that seeks to mediate the rivalrous pluralisms that would be exhibited by logic in the years ahead. A further difference between the two logicists pertains to the background context of their work. Frege did most of his work on mathematical logic before he knew of the paradox communicated to him by Russell in that fateful letter of 1902. Frege’s adjustment to it was an afterthought published at the page- proof stage as an appendix to volume two of the Grundgesetze. For Russell on the other hand, virtually all of his contributions to logic were fashioned in the aftermath of his paradox. His work was deeply marked by efforts to evade it, and was so in ways that gave further impetus to the instrumental turn in logic, albeit one that Russell tried to resist in the name of “logical common sense”. But it would not be too long before Carnap would proclaim his Principle of Tolerance:

We may say, then, that the short period between the publication of the second volume of the Grundgesetze (1903) and of Principles of Mathematics (1903) marks the beginning of the end for a naïve apriority about the principles of logic, an epistemological shift that would reconfigure the explosive intellectual successes of the first fifty years of the twentieth century.

Logic’s two dominant themes in the period from Russell to Church are its foundational creativity and its procedural limitations and liabilities. One might say that this was a period accentuated by both triumph and crisis. This is the time when logic would acquire its modern face, what with the emergence of model theory, proof theory, recursion theory, set theory and metamathematics. Hence the great names: Russell, Brouwer, Heyting, Skolem, Löwenheim, Herbrand, Wittgenstein, Ramsey, Lesniewski, Hilbert, Gödel, Tarski, Post, Gentzen, Curry and Church. But it was also the time in which these foundational breakthroughs would have to be engineered in the face of, in addition to Russell’s paradox, the semantic paradoxes of Tarski and others and the incompleteness results of Gödel, all of which which tested Hilbert’s insistence that mathematical truth has no home beyond the safe-harbours of formal proof. The problems that attended the emergence and development of mathematical logic were not only difficult; they were also treacherous. The first great technical accomplishment in logic was Aristotle’s all-but-correct proof of the perfectibility of syllogisms. There are those who think that the mediaeval doctrine of suppositio ranks high, as it surely does in the form in which categorical propositions are construed as multiply-quantified statements of identity. The 20th century originators of modern logic faced crises that threatened the collapse of two of its foundations — set theory and model theory — and a third which significantly attenuated the scope of a further foundation — proof theory. The great ambitions of logic would have to be navigated beyond the reach of these perils. Meeting these difficulties demanded, and received, system-building of extraordinary virtuosity, unparalled by anything that had gone before. From this there emerged a stable idea of what it took to be a logic: a precise, and typically a recursively generated, grammar; a well-developed formal semantics; a syntactic theory of formal proof; and the capacity to answer its own metatheoretical questions - soundness, completeness, decidability and the like. In the fifty-year period covered by the present volume, mathematization became a necessary part of what it took to be a logic; in the following fifty years many would see mathematical virtuosity as sufficient. For the most part the founders tried to honour Russell’s principle of logical common sense. While they sought to produce mathematically robust systems, they also wanted these systems to be conceptually sound. The paradoxes were a wrench to such ambitions. It is to the everlasting credit of the founders that the paradoxes were no bar to mathematical virtuosity (indeed they were in large part the provocateurs of it). But the same could not be said of conceptual fidelity. The founders knew this, and said so; the age of intuitive foundations for logic was coming to an end. What for the founders was a philosophical discouragement has been for at least some of their descendents a naïve and dispensable presumption. Concerning the concepts that logicians have an interest in analyzing, what has come to count above all is the mathematical adequacy of the systems that describes them. Conceptual adequacy has become a separate issue at best.

The founders of twentieth-century logic were mainly philosophers, and those who weren’t were respectful of philosophical motivations. Their investigations of logic were driven by philosophical considerations and judged by the standards of philosophical soundness, such as they may be. But as the fortunes of mathematical virtuosity rose and those of philosophical cogency diminished, a corresponding transformation was underway in the administrative arrangements under which intellectual work was institutionalized and professionalized. What began as a philosophical discipline became, in the space of five decades, a mathematical one. The irony is unmistakable. In the hands of Frege and Russell, the role of logic was a take-over of mathematics. Take-overs are tricky, whether the Norman occupation of Britain of Daimler’s absorption of Chrysler. There is always the lurking possibility of “going native”, which is precisely what happened to these two invaders. How could it also not happen to any logic that was purpose-built to capture arithmetic? In its take-over of mathematics, logic went native.

Some will think talk of “take-over” and “going native” too melodramatic for serious scholarship. But this is a stylistic objection, not a substantive one. The transformation of logic into mathematics is one of the momentous events of intellectual history. It is a transformation initiated by the very people who would not have wanted to see it happen. In remarking upon it here, it is not necessary to lament philosophy’s diminution; one might just as well celebrate logic’s liberation. Was it not a philosopher who remarked that logic is an old discipline, and that since 1879 is a great one? Even so, the question remains whether philosophy has a future in the developments in logic that lie ahead.

We should also say a word or two about the inclusion here of Meinongian logic. Meinong was not a logician and never thought of himself as one. So Meinong did not make our list of pantheonic names. Even so, Meinong had an elaborate theory of objects and a concomitant, if largely implicit, theory of reference for them. One of the most distinctive features of Russell’s logic — and of most of those that followed in its wake — is the theory of reference reflected in the doctrine of logically proper names and definite descriptions. In some ways, Russell’s theory of reference was conceived of as an alternative to Meinong’s theory of objects. If we here allow...