1. Beyond syllogisms of the three figures Aristotle (sometimes): every valid inference can be reduced to the „figures” A different claim : Assumption: If (not a single potentiality for contraries), then (not a single science of the contraries). Proved: Not (always) a single potentiality of contraries. Necesary to agree: Not a single science of the contraries. By the assumption (+ a modus ponens).

2. Inferences that cannot led back (anagein)/resolved (analuein) to the figures: those containing some assumptions (?!) The assumption is a compound proposition (at least in this example). Compound propositions are called in the late antique and mediaeval tradition hypothetical, the inferences connected with them are hypothetical syllogisms. In the peripatetic tradition, there are two types of hypothetical propositions: the conditional (sunekhés) and the disjunctive (diairetikos).

3. Theophrastus Source: Alexander of Aphrodisias Systematisation and some rectifications of Aristotle’s work „Fourth figure” syllogisms Syllogisms in inference scheme form (‚therefore’ [epei] instead of ‚if-then’) Modifications in modal logic suggesting a de dicto reading (e.g. „the conclusion follows the weaker premiss” [deteriorem-principle]) Hypothetical syllogisms: From a conditional, a disjunction, a negated conjunction: similar to the Stoic tropoi. „Totally hypothetical” (di’ holón hupothetikoi) syllogisms, in three figures. Examples: I.If A, then B; if B, then C; therefore if A, then C. II.If A, then B; if not A, then C; therefore if not C, then B. III.If A, then C; if B, then not C; therefore if B, then not A. Obvious analogy with the three figures of categorical syllogisms. Not in the Stoic propositional logic. Monograph: Bocheński 1947 Later peripatetic propositional logic: a supplement to categorical syllogisms, „hypotheticals are reducible to categoricals”.

4. The other school: (Megarians and) Stoics Łukasiewicz 1935: „Zur Geschichte der Aussagenlogik” – rediscovery, against Prantl System of natural deduction, based on five undemonstrated tropoi and four metarules (themata). (We know the 1. and 3. thema only.) Ordinals are used as propositional variables. There are claims of completeness. Reports about huge numbers of inferences reduced to the 5 anapodeiktoi. But we know only 7. Benson Mates 1950: semantical ideas, apparently similar to Frege and Carnap. Bocheński, Martha Kneale: Megarian-Stoic tradition. Michael Frede 1970: monograph, hypothetical reconstruction of the missing themata. Research based on H. von Arnim, Stoicorum Veterum Fragmenta (1903-5) K.-H. Hülsner, 1987: Fragmente zur Dialektik der Stoiker

5. The Socratic family tree: Socrates Euclides of MegaraPlato Aristotle Theophrastos Eubulides paradoxes ? Apollonios Chronus Stilpo Diodoros Chronus conditionals, modalities Philo of Megara conditionals, modalities Zeno of Citium Cleanthes Chrysippus(280-207 B.C.) the five anapodeiktoi, conditionals, modalities, …

6. Paradoxes, atributed to Eubulides Let us call a sentence paradoxical if it leads from a reasonable assumption to contradictory consequences. 1.The Liar Known after Paul, Tit. I. 12 as „the paradox of (Epimenides ) the Cretan” : „All Cretans are liars.” You need to assume some context, e.g. that no other Cretan said a word. „If I am lying and I say that I am lying, do I lie or do I tell the truth?” (Aulus Gellius, Noctes Atticae) In this form it is paradoxical, independently of any context. Dozens of other mentions, first of them: Aristotle, Soph. Elench. 2.Heap or Sorites Would you describe a man with one hair on his head as bald? Yes. Would you describe a man with two hairs on his head as bald? Yes. … You must refrain from describing a man with ten thousand hairs on his head as bald, so where do you draw the line? Other paradoxes: intensional contexts, presuppositions.