Modal Confusion in Rand/Peikoff

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Ayn Rand's Introduction to Objectivist Epistemology includes an essay by Leonard Peikoff entitled "The Analytic-Synthetic Dichotomy."  The section "Necessity and Contingency" concludes with the following paragraph:

Truth is the identification of a fact with reality. Whether the fact in question is metaphysical or man-made, the fact determines the truth: if the fact exists, there is no alternative in regard to what is true. For instance, the fact that the U.S. has 50 states was not metaphysically necessary – but as long as this is men's choice, the proposition that "The U.S. has 50 states" is necessarily true.  A true proposition must describe the facts as they are.  In this sense, a "necessary truth" is a redundancy, and a "contingent truth" a self-contradiction. (Introduction to Objectivist Epistemology, 2nd ed., eds. Binswanger and Peikoff, NAL Books, 1990, p. 111, emphasis in original.) 

I have no objection to part of what is being said in this passage, in fact I heartily agree with it, namely, that facts determine truths.  The non-man-made fact of the moon's having craters makes-true the proposition expressed by 'The moon has craters.'  And similarly for the man-made fact regarding the 50 states cited by Peikoff.  So I cheerfully agree that "if the fact exists, there is no alternative in regard to what is true."  We can put the point as follows given that there is a fact F and a proposition p that records F:

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