I take as my text propostion 4.0312 of the Tractatus : The possibility of propositions is based on the principle that objects have signs as their representatives. My fundamental idea is that the ‘logical constants’ are not representatives; that there can be no representatives of the logic of facts. Practically the same words occur in Wittgenstein's Notebook for 25 December 1914, where Miss Anscombe translates them: The possibility of the proposition is, of course, founded on the principle of signs as (...) going proxy for objects. Thus in the proposition something has something else as its proxy. But there is also the common cement. My fundamental thought is that the logical constants are not proxies. That the logic of the fact cannot have anything as its proxy. (shrink)

I develop one account of propositions as pictures sharing logical form with what they depict. Two concepts of simplicity in the "tractatus" are then isolated. Since characterization of sachverhalten as configurations of referential simples does not entail their inferential simplicity, By rejecting the tractarian theory of inference, I retain the picture theory without commitment to atomistic ontology. Interpretation of inference as performance then gives rise to a second sense of picturing.

The projection latticesP(ℳ1),P(ℳ2) of two von Neumann subalgebras ℳ1, ℳ2 of the von Neumann algebra ℳ are defined to be logically independent if A ∧ B≠0 for any 0≠AεP(ℳ1), 0≠BP(ℳ2). After motivating this notion in independence, it is shown thatP(ℳ1),P(ℳ2) are logically independent if ℳ1 is a subfactor in a finite factor ℳ andP(ℳ1),P(ℳ2 commute. Also, logical independence is related to the statistical independence conditions called C*-independence W*-independence, and strict locality. Logical independence ofP(ℳ1,P(ℳ2 turns out to be equivalent to the (...) C*-independence of (ℳ1,ℳ2) for mutually commuting ℳ1,ℳ2 and it is shown that if (ℳ1,ℳ2) is a pair of (not necessarily commuting) von Neumann subalgebras, thenP(ℳ1,P(ℳ2 are logically independent in the following cases: ℳ is a finite-dimensional full-matrix algebra and ℳ1,ℳ2 are C*-independent; (ℳ1,ℳ2) is a W*-independent pair; ℳ1,ℳ2 have the property of strict locality. (shrink)