Notes by RBJ on

Foundations without Foundationalism

by Stewart Shapiro

This is a good source of information about second order logic, and some other things, like the history of first order logic.

However, I'm going to criticise it here, in relation to its main title.

Shapiro's "Foundationalism" seems to me a paper tiger. The main characteristic of foundationalism as defined by Shapiro is the desire for absolute security (or "as secure as humanly possible"), in an epistemological sense. He defines two variants, strong and moderate, strength in this case being the additional view that there is exactly one correct foundation system (though this is all relativised to some unspecified field of study).

For all I know this is fully consistent with the rest of the literature on this topic, but I still think its a bad use of language, in that it excludes me, an ardent enthusiast for foundation systems. I feel that I ought to be encompassed by any reasonable use of the term "foundationalist".

Curiously, Shapiro's notion of "foundation" also seems to me odd. I have found no mention of conservative extension or proof theoretic strength. Definitions are of course mentioned, but not their foundational role in the foundational equation:


.. where "foundation-system" here replaces the logicist "logic" for readers who are unable to accept set theory as logic.

This notion of a foundation system (which I consider very important) does not seem to appear anywhere in this book (not even under a different name, so far as I can see). Yet there are many uses of the word "foundation" which seem to me quite inappropriate. As in "the foundational conception of logic".

Here's another thing I fail to comprehend. For Shapiro the repudiation of "foundationalism" and the advocacy of second order logic are connected, for he view's foundationalism as responsible for the modern prejudice in favour of first order logic. I don't see it!

Shapiro undoubtedly knows a great deal about second order logic, and this is certainly a book to read if you are philosophically inclined and interested in second order logic. However, the main impression that his book leaves on me is one of muddle. (which I should admit is not unusual, I think most recent analytic philosophy is completely muddled)

Perhaps the muddle is mine.

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