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We shall be concerned with a natural one-sorted language L for talking about probability spaces, where quantifiers are intended to range over events in a given space. In this language one can express statements like `for every event E, if P(E) > 0, then there exists an event F such that 0 < P(F) < P(E)' — here P denotes the measure in question. It is known that the validity problem for arbitrary L-sentences is highly undecidable. So one may wish to know more about prefix fragments of L. It turns out that the validity problem for AE-sentences in L is decidable, while that for EA-sentences is not. This solves the decision problem for prefix fragments of L. Moreover, similar results can be obtained for some natural one-sorted fragments of Halpern's first-order logics of probability.
