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Which logicians (outside 19th century mathematical logic) adopted, explicitly or implicitly, Philo's idea that the truth conditions of the conditional "If P, then Q" were best expressed as "Either Q or not P (or both)", i.e., ¬P ∨ Q?
Philo the Megarian was reported by Sextus Empiricus in his text against the logicians as having proposed this model. Sextus Empiricus also reported, in the same text, that Diodorus Chronus, Philo's logic teacher, argued against Philo's position.
Anybody else?
To speak of "the best model" is somewhat odd. Logicians have long known that there are different kinds of conditionals. As to whether the truth functional conditional that we now call the material conditional was known to medieval logicians, there is some scholarly debate on the subject.
William of Ockham, Pseudo-Scotus, John Buridan and Walter Burleigh do not appear to have explicitly described what we now call the material conditional, but all gave examples of conditionals and their role within valid arguments that seem to commit them to accepting it as a kind of conditional. Also, some of these logicians can be understood as distinguishing between material conditionals and strict conditionals and accepting both.
Some references:
Philotheus Boehner, "Does Ockham Know of Material Implication?" Franciscan Studies, 1951, Vol 3/4, pp. 203-230.
Marilyn McCord Adams, "Did Ockham Know of Material and Strict Implication? A Reconsideration" Franciscan Studies, 1973, Vol 33, pp. 5-37.
E. A. Moody, Truth and Consequence in Mediaeval Logic, 1953, pp. 64-80.
David Sanford, If P then Q, 2nd edition, 2003, chapter 2, pp. 30-45.
There are stark conceptual discrepancies between the syllogistic theories of the past and the contemporary predicate and propositional calculi. Hence, it would be a faulty analogy to compare them by immediate correspondences. However, we can assess their convergencies and divergencies over the validities they admit in the concrete argument instances.
An exemplary divergency is the conception of disjunction. A or B used to be conceived as exclusive by form, hence the possibility that both A and B hold was not included in the analyses as a matter of logical form. But this should not suggest that the philosophers that mediated on logical matters were not aware of inclusive construal of disjunction. As Benson Mates notes in his Stoic Logic (2nd printing, p. 53)
The Stoics undoubtedly knew of inclusive disjunction, although we possess no clear truth-functional definition of this connective. Galen says: "Also in some propositions it is possible not only for one part to hold, but several, or even all; but it is necessary for one to hold. Some call such propositions 'almost disjunctions,' since disjunctions, whether composed of two atomic propositions or of more, have just one true member."
They were also aware of the connections between the conditional and disjunctive statements. An explicit statement of these connections is given by Boethius in his compendium De Hypotheticis Syllogismis (see Book III 10.4 ff.), which can be listed as
For the syllogistic equivalence of conditional and disjunctive statements to overlap with the inclusive disjunctive form of material implication in contemporary systems, both the antecedent and the consequent should hold, while the negation of the antecedent should be admissible as well.
Boethius's approach is a syncretism of Peripatetic and Stoic doctrines; it is a separate and not easy task to lay out a clear and coherent account of his discussion, so we pass by even a brief explanation of it. But it is conspicuous that the inclusive case obtains when the antecedent and the consequent stand in "incomplete incompatibility" (Galen's term; hence, either it is the case that both do not hold simultaneously or it is the case that both fail to hold simultaneously). This is the case of If it is not A, it is B, which we have sought.
