Notions of Logical Independence

In Prague this past week, David Miller gave a talk in which (among many other interesting things) he distinguished two notions of logical independence. One he credits to Moore (the mathematician, not the philosopher) and Wittgenstein, and that’s the notion of independence at work when we say, e.g., that an axiom system is independent. A set Γ is independent if for each A ∈ Γ, Γ\A is consistent with ¬A. Moore’s notion of complete independence is a generalization of that, where we require that for each Δ ⊆ Γ, Γ\Δ is consistent with ¬Δ.

The other notion he credits to H. M. Sheffer, and that’s the notion of maximal independence: Γ is maximally independent if any two A, B ∈ Γ have no consequences in common, other than tautologies.

{p, q}, for instance, is (completely) independent in the first sense, but not maximally independent (p and q have the non-tautological consequence pq in common).

I think these are interesting concepts, and I should find out more about them. David makes use of them in comparing (false) theories in a 1974 paper. I hadn’t heard of Sheffer’s notion before; maybe that’s because the paper he defines it in is unpublished. But from David’s paper I see that Tarski uses it as well.

David Miller, 1974. On the comparison of false theories by their bases. The British Journal for the Philosophy of Science 25(2) 178–188.

Eliakim Hastings Moore, 1910. Introduction to a form of general analysis. The New Haven Mathematical Colloquium 1–150.

Henry Maurice Sheffer, 1921. The general theory of notational relativity. (Mimeograph)

3 thoughts on “Notions of Logical Independence

  1. Woo Hoo! Logic blogging, and some I even fairly well understood. Posted by Matt

  2. Interesting…isn’t it the case that any two sentences that are neither tautologies nor subcotrariae (i.e. the negation of one of them doesn’t entail the other) are not maximally independent? Posted by Rafal

  3. Interesting… So is the following true: a maximally independent set either (i) contains two elements A, B where B is equivalent with -A (not A) (being subcontrary formulae is not enough) or (ii) if it contains more than two elements, say n where n is at least 2, then it contains at least n-2 tautologies, and at most one contingent formula A, and a formula B equivalent with -A. Posted by Kristof

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