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Manfred comments on Walkthrough of "Definability of Truth in Probabilistic Logic" - Less Wrong
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The empty set is tautological because P({}) = P({} and something) + P({} and not-something) = P(something) + P(not-something). Hm, but that's using axioms 1 and 2. Can we get it using just axiom 3 as the paper claims?
When you say that P({} and something) = P(something), you suppose the hypothesis (in addition to using several nontrivial consequences of coherence that So8res mentioned, like P mapping equivalent statements to the same thing).
More importantly, "{} and something" isn't a syntactically correct sentence. I don't think most authors consider the empty sentence syntactically correct either. (Marker, the textbook I used, doesn't.)
Whoops, you're right.