Algebra and A General Theory of Logic

Michael Friedman’s Reconsidering Logical Positivism, which I have read before but recently got in the mail, is a very good and important book which argues, persuasively, that the dominant interpretation of logical positivism within the Anglo sphere is wrong. It’s not just the empiricists who have got logical positivism wrong, it is also the social constructivists.

Briefly, Friedman argues that the dominant interpretation of logical positivism, what he refers to as radical empiricism, is wrong because it has the verification theory of meaning at the core of logical positivist thought. Rather, Friedman shows, the logical positivists were concerned with the “relativised a priori,” and its role in empirical knowledge. Kant had geometry as the example par excellence of synthetic a priori knowledge. The problem was that mathematicians soon developed non Euclidean geometries, and moreover, non Euclidean geometries went on to form the mathematical basis of General Relativity.

This was a puzzle because you get a type of relativism; different geometries based on different axioms can be developed, and what’s more they each play an important role in decoding nature. That was the central issue animating the logical positivists, not verificationism, and Friedman goes on to discuss this in the context of 20th century philosophy. That’s an interesting topic but not one for today.

One aspect of the “relative a priori” is different systems of logic, propositional, predicate and non classical that have proliferated especially from Frege onward. This has interested me from time to time, especially at the epistemological, not necessarily logical, level. I’ve kind of thought of traditional logic as “natural” logic as it seems to be an innate, intuitive, form of knowledge that logicians have developed into formal systems. Non classical logics are another matter, however. They are not nearly as “natural,” cognitively speaking, for us. So I’ve thought of logic in terms of “natural” logic and “formal” logic like the way Chomsky contrasts “natural” from “formal” languages.

But an interesting question becomes; what if really, in so far as logic is concerned, the concept of the relative a priori is not appropriate? That is, that underlying the proliferation of logical systems there exists a hidden unity? A general theory of logic, as it were.

There’s an interesting link between logic and algebra. Boolean algebra and propositional logic are two sides of the same coin. There exist algebraic constructions of other logical systems. Developing algebraic formulations of all known logical systems opens the prospect of logical unification through deeper abstraction.

If logic, of whatever variety, is algebra then it is possible to imagine there being algebraic structures or categories that underpin the diversity of logical systems, meaning that they are all essentially the same. One of the important developments of modern mathematics, for example with the anarchist Alexander Grothendieck, is abstract algebra which looks at algebraic structures with reference to category theory. Algebra and category theory might open up the door to unification. I’d like to think that there exists such unity between logical systems.

There could very well be a plan of attack here that unites logic once more. At this point it’s not hard to be reminded of David Hilbert.

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