al-fārābī and existential import of categorical propositions

Al-fārābī is arguably the first logician inheriting aristotle’s works in logic who gave a systematic discussion of existential import of categorical propositions: universal affirmatives (a), universal negatives (e), particular affirmatives (i), and particular negatives (o). this article examines his views on the existential import of categorical positions in relation to the development of doctrines of existential import by medieval latin logicians, such as abelard and ockham.traditional logic holds that universal affirmatives imply matching universal negatives (the subalternation thesis). modern logic denies this thesis by taking both particular affirmatives and negatives to have existential import, for the subalternation thesis conflicts with all particular propositions having existential import. medieval logicians give two different responses to the conflict. ockham denies that particular negatives (e.g. ‘some man is not white’) have existential import. abelard accepts existential import of all particular propositions by denying that particular negatives are contradictories of universal affirmatives. al-fārābī’s discussion of existential import has a common core of the two responses. in al maqulat (1986), he holds:t1. both universal and particular affirmatives are false, if “their subjects do not exist”.t2. universal affirmatives and their negations are contradictories (“split truth and falsity”) “whether the subjects exist or not”.on these theses, ‘every man is white’ and ‘some man is white’ imply the existence of men but ‘not every man is white’ does not (it is vacuously true if there is no man).these theses are neutral about whether particular negatives have existential import. but his earlier work on syllogisms (al-fārābī 2020) has discussions of particular negatives that relate to their existential import. in giving examples of particular negatives, he gives both the particular negative ‘some humans are not white’ and the negated universal affirmative ‘not all humans are white’ (2010, 119). this indicates the view that particular negatives (in the strict sense) are logically equivalent to negated universal affirmatives. combined with t1 and t2, it yields the thesis that particular negatives lack existential import. but the work also has treatments of particular negatives that conflicts with this thesis, as hodges (2020, 51f) points out. while it is not clear how al-fārābī would have resolved the conflict, his work suggests two different routes: (1) to deny the equivalence between particular negatives and negated universal affirmatives, and (2) to accept the equivalence by denying that particular negatives have existential import. and we can see that these routes match the doctrines of existential import developed by abelard and ockham.