The Development of Logic, Mk III?

It’s now just over fifty years since William and Martha Kneale’s The Development of Logic was first published. The book is still in print, and there remains nothing quite like it. There’s a very great deal of scholarship revealed in the first four-hundred or so pages leading up to Frege, and those pages still read rather well, and remain a very helpful introduction. The pivotal role then given to Frege was not so usual at the time, and William Kneale’s discussion over some eighty pages was a boon to generations of students. The concluding chapters seem much more dated now, but overall the book remains a quite splendid achievement.

It would surely be near impossible now for a pair of authors to emulate the range of the Kneales from-Aristotle-to-the-present. At one end of the task, just think of the explosion of work on later Greek, Roman and medieval philosophy which would have to be engaged with. At the other end of the task, think how far logic has moved forward on how many fronts in the last half-century or so. It’s not surprising, then, a student looking for real breadth is now faced with multi-author blockbusters. The Handbook of the History of Logic has reached eleven volumes and something like 8000 pages(!). As is the way of such things, the quality and level of its entries varies wildly, with authors rather too keen to show off the depth of their expertise, so much will be beyond the ready grasp of (the contemporary equivalent of) the student who once gratefully reached for The Development of Logic.

Even a recent multi-author volume of more restricted scope, Leila Haaparanta’s The Development of Modern Logic, runs at something like three times the length of the Kneale’s coverage of the same period. And this book seems to suffer particularly badly from a lack of editorial vision (how come the estimable Wilfrid Hodges writing on ‘Set Theory, Model Theory, and Computability Theory’ (yes, all three!) is confined to 28 pages, while Andrew Aberdein and Stephen Read get 111 pages on ‘The Philosophy of Alternative Logics’?). There’s some useful material here, but it’s no update of the Kneale’s book.

I rather think, then, that there is a gap here waiting to be filled by, shall we call it, The Development of Modern Logic: from Frege to Tarski — a single-author book aimed at advanced undergraduates or beginning graduate students in philosophy (and mathmos who care about such things). Pre-Frege, the coverage could be fairly minimal. And the book could still stop (in effect) roughly when the original The Development of Logic stops.

Why so? The Kneales write in their preface “our primary purpose has been to record the first appearance of those ideas which seem to us most important in the logic of our own day”. Now I’m not so sure how illuminating it is to dig down to first appearances of ideas, especially given that the founding fathers of modern logic were so ignorant/dismissive of their predecessors. (To be sure, it is quite fun to know something of what there is to know about Chrysippus, for example: but it won’t help you to understand any better Frege or Russell or Hilbert or the rest. Or to jump to another example, Bolzano deserves credit for his story about logical consequence: but it was buried in obscurity and had little direct influence.) On the other hand, however, I do think that the Kneale’s officially Whiggish approach (which they don’t themselves stick to at all closely) can be a helpful way to go. To be sure, it is not the only sort of history worth doing: but it is one illuminating route to take. After all, around forty years ago, more or less, a certain conception of what constitutes the core of modern logic becomes entrenched: modern textbooks of mathematical logic, in their early chapters, look much of a muchness. The question arises: how and why did we get from Frege’s founding document, the Begriffsschrift, to the entrenched modern conception? Exploring this is a narrow enough project to be carried out in a reasonable-sized, accessible, book, but surely interesting enough to be a very worth-while exercise.

After all, it isn’t that easy for students e.g. to find out what happens logically in Principia, or e.g. to discover. just what Hilbert’s contribution to the development of logic was, or e.g. to understand how Gödel could prove the semantic completeness of first-order logic years before Tarski had nailed down the semantics.

So, as I say, I rather think that the project of writing in a student-friendly way about the development of logic from Frege to Tarski would be very worth doing. Maybe I should make a start. What do you think?

7 thoughts on “The Development of Logic, Mk III?”

  1. An emphatic yes.
    As the the period includes Gödel, Hilbert, Heyting, Bernays, Herbrand, Kleene however, any selection of excluded topics in the broadest sense from Frege to today will of necessity be a personal choice, but of no less interest for that reason.

  2. I wonder whether it is especially easy, and especially tempting, to be Whiggish when writing the history of logic – or indeed of mathematics or of the natural sciences. My thought is not merely that knowledge is cumulative. Perhaps more importantly, and especially in logic and in mathematics, knowledge needs to accumulate in a particular order, at least over relatively short spans. There might be a radically different possible path to our current knowledge that we could find, if we went back far enough and created a fork from what turned out to be the actual path of development, but over the short to medium term, that is not really an option. Instead, we see that result X was established, and that this (or the method by which X was established) was necessary in order for result Y to be established. This must tempt the historian to see X as leading up to Y.

  3. André Gargoura

    Dear Peter,

    You are asking for preference, hence my choice would be :

    1. A continuation of your “Introduction to formal logic” via an “Introduction to first and second-order logic” which, added to your “Introduction to Gödel’s Theorems”, would form a modern, comprehensible SUM !
    A sort of continuation of Kleene’s books… I can already hear the comments about Enderton, Mendelson & al. but I have my doubts…

    2. Development of logic from Frege to Tarski.

    3. Ordinals.

    Kind regards


  4. I will vote unconditionally for a future bestselling : Peter Smith’s Development of logic from Frege to Tarski, especially due to your statement : “to understand how Gödel could prove the semantic completeness of first-order logic years before Tarski had nailed down the semantics.” I really need it (as you can see from some of my questions in math.stackexchange).


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