Wholly Hypothetical Syllogisms

Theophrastus is further credited with the invention of a system of the later so-called ‘wholly hypothetical syllogisms’ (Theophrastus fr. 113 Fortenbaugh). These syllogisms were originally abbreviated term-logical arguments of the kind

اقتباس :: If [something is] A, [it is] B.
If [something is] B, [it is] C.
Therefore, if [something is] A, [it is] C.

and at least some of them were regarded as reducible to Aristotle's categorical syllogisms, presumably by way of the equivalences to ‘Every A is B’, etc. In parallel to Aristotle's syllogistic, Theophrastus distinguished three figures; each had sixteen modes. The first eight modes of the first figure are obtained by going through all permutations with ‘not X’ instead of ‘X’ (with X forA, B, C); the second eight modes are obtained by using a rule of contraposition on the conclusion:

اقتباس :: (CR) From ‘if X, Y’ infer ‘if the contradictory of Y then the contradictory of X’

The sixteen modes of the second figure were obtained by using (CR) on the schema of the first premise of the first figure arguments, e.g.

اقتباس :: If [something is] not B, [it is] not A.
If [something is] B, [it is] C.
Therefore, if [something is] A, [it is] C.

The sixteen modes of the third figure were obtained by using (CR) on the schema of the second premise of the first figure arguments, e.g.

اقتباس :: If [something is] A, [it is] B.
If [something is] not C, [it is] not B.
Therefore, if [something is] A, [it is] C.

Theophrastus claimed that all second and third figure syllogisms could be reduced to first figure syllogisms. If Alexander of Aphrodisias (2nd c. CE Peripatetic) reports faithfully, any use of (CR) which transforms a syllogism into a first figure syllogism was such a reduction. The large number of modes and reductions can be explained by the fact that Theophrastus did not have the logical means for substituting negative for positive components in an argument. In later antiquity, after some intermediate stages, and possibly under Stoic influence, the wholly hypothetical syllogisms were interpreted as propositional-logical arguments of the kind

اقتباس :: If p, then q.
If q, then r.
Therefore, if p, then r.

4. Diodorus Cronus and Philo the Logician

In the later 4th to mid 3rd centuries BCE, contemporary with Theophrastus and Eudemus, a loosely connected group of philosophers, sometimes referred to as dialecticians (see entry ‘Dialectical School’) and possibly influenced by Eubulides, conceived of logic as a logic of propositions. Their best known exponents were Diodorus Cronus and his pupil Philo (sometimes called ‘Philo of Megara’). Although no writings of theirs are preserved, there are a number of later reports of their doctrines. They each made groundbreaking contributions to the development of propositional logic, in particular to the theories of conditionals and modalities.
A conditional (sunêmmenon) was considered a non-simple proposition composed of two propositions and the connecting particle ‘if’. Philo, who may be credited with introducing truth-functionality into logic, provided the following criterion for their truth: A conditional is falsewhen and only when its antecedent is true and its consequent is false, and it is true in the three remaining truth-value combinations. The Philonian conditional resembles material implication, except that—since propositions were conceived of as functions of time that can have different truth-values at different times—it may change its truth-value over time. For Diodorus, a conditional proposition is true if it neither was nor is possible that its antecedent is true and its consequent false. The temporal elements in this account suggest that the possibility of a truth-value change in Philo's conditionals was meant to be improved on. With his own modal notions (see below) applied, a conditional is Diodorean-true now if and only if it is Philonian-true at all times. Diodorus' conditional is thus reminiscent of strict implication. Philo's and Diodorus' conceptions of conditionals lead to variants of the ‘paradoxes’ of material and strict implication—a fact the ancients were aware of (Sextus Empiricus [S. E.] M. 8.109–117).
Philo and Diodorus each considered the four modalities possibility, impossibility, necessity and non-necessity. These were conceived of as modal properties or modal values of propositions, not as modal operators. Philo defined them as follows: ‘Possible is that which is capable of being true by the proposition's own nature … necessary is that which is true, and which, as far as it is in itself, is not capable of being false. Non-necessary is that which as far as it is in itself, is capable of being false, and impossible is that which by its own nature is not capable of being true.’ Diodorus' definitions were these: ‘Possible is that which either is or will be [true]; impossible that which is false and will not be true; necessary that which is true and will not be false; non-necessary that which either is false already or will be false.’ Both sets of definitions satisfy the following standard requirements of modal logic: (i) necessity entails truth and truth entails possibility; (ii) possibility and impossibility are contradictories, and so are necessity and non-necessity; (iii) necessity and possibility are interdefinable; (iv) every proposition is either necessary or impossible or both possible and non-necessary. Philo's definitions appear to introduce mere conceptual modalities, whereas with Diodorus' definitions, some propositions may change their modal value (Boeth. In Arist. De Int., sec. ed., 234–235 Meiser).
Diodorus' definition of possibility rules out future contingents and implies the counterintuitive thesis that only the actual is possible. Diodorus tried to prove this claim with his famous Master Argument, which sets out to show the incompatibility of (i) ‘every past truth is necessary’, (ii) ‘the impossible does not follow from the possible’, and (iii) ‘something is possible which neither is nor will be true’ (Epict. Diss. II.19). The argument has not survived, but various reconstructions have been suggested. Some affinity with the arguments for logical determinism in Aristotle's De Interpretatione 9 is likely.
On the topic of ambiguity, Diodorus held that no linguistic expression is ambiguous. He supported this dictum by a theory of meaning based on speaker intention. Speakers generally intend to say only one thing when they speak. What is said when they speak is what they intend to say. Any discrepancy between speaker intention and listener decoding has its cause in the obscurity of what was said, not its ambiguity (Aulus Gellius 11.12.2–3)

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Wholly Hypothetical Syllogisms

4. Diodorus Cronus and Philo the Logician

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Wholly Hypothetical Syllogisms

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