By Laurence R. Horn
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Extra info for A Natural History of Negation
OFa - O-Fa (by detachment) (or equivalently OFa - -OFa, given that not necessary = possible not) Thus whatever is necessarily true (Socrates is Socrates, 2 + 2 = 4) would be possibly not true, that is, not necessarily true~a modal log~ cian's nightmare. Aristotle himself was not unaware of thiS result of hiS 14 Negation and Opposition in Classical Logic reasoning: 'It comes about, therefore, that the thing which must necessarily be need not be, which is absurd' (De Int. 22bI6); 'And thus it would follow that a thing which must necessarily be may possibly not be; which is false' (22b34).
26) a. Pleasure is not good. b. Some pleasure is not good. c. It is light and it is not good. For the Stoics, as for Aristotle and the Peripatetics, every proposition is either negative or affirmative, so the propositions in (26) count as affirmative. Apuleius, accepting this as his starting point, distinguishes the ABDlCATIVA (negative) proposition from the DEDlCATlVA (assertive). He may have been the first to recognize what was to become a recurring motif for the developers of the negative theme: the observation that an affirmative proposition may be logically equivalent to a negative counterpart, as in the pair (27) a.
The former claim, however, has often struck critics as a reductio of the proposal that the predicate calculus can adequately model natural language semantics. Do we really want to count All unicorns are equine as true? And as true because of the fact that no unicorns exist? n Wlt~ the logical form of (23A) does not entail the corresponding propoSitIOn With the form of (231): if nothing satisfies F, the former is true and the latter false. 1 Negation and the Legacy of Aristotle 29 (under (18i)) or vacuously true (under (18ii)) merely because no ogres exist.
A Natural History of Negation by Laurence R. Horn