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this is especially unfortunate given the very restricted range of inferences that are possible within the syl- logistic. But the idea of the symbolismwas epochal in the history of logic.
See: Aristotle and the Stoics; Aristotle and Logic.
4. Stoic Logic
Developed by Chrysippus and the Megarians, Stoic logic constitutes a second major tradition in the sub- ject to be put alongside that engendered by Aristotle. From our perspective, it is difficult to understand fully why the Aristotelian and Stoic traditions should have been thought of as having such separate identities— identities which remained distinct until much nearer the twentieth century. This is because the sorts of subject matter that figured in Stoic logic are now fully integrated with those that concerned Aristotle, and it is not easy to imagine them as distinct. As was men- tioned earlier, perhaps the only way to understand this aspect of the history of logic is by allowing due weight to the fact that the ancient world thought of demonstrative reasoning as a different kind of subject matter from that of philosophical or metaphysical reasoning. It is as if inference was classed in terms of its field of action, rather than, as now, thought of as a unified subject which applied indifferently to different areas of human knowledge. Moreover, it seems clear that there were clashes of personality involved. The Megarian logicians' attacks on Aristotle contributed to the view of their logic as somehow an alternative, when to us it is so clearly a contribution to the same study.
By and large it is fair to describe the main con- tribution of Stoic logic as coming under the headings of philosophy of logic and philosophy of language. There was in their work nothing that compares to the Aristotle's formalization of logical inference. The three main subheadings under which one can classify the Stoic contribution to logic are: the study of logical paradoxes, the study of conditional or hypothetical assertions, and the study of modal notions.
4.1 Paradoxes
The four most famous logical paradoxes discussed in the Megarian and Stoic corpus were:
still has. He has not lost horns, so he still has
them.
Unfortunately, it is not known what the originator (probably Eubulides) or his later followers thought of these paradoxes—what they used them to illustrate or how they saw them as fitting in with logic. However, in their different ways each of them has been influ- ential in the study of logic and language. For, aside from their curiosity value, each of them has important consequences in the study of the semantics of both natural and formal languages. The Liar has been the basis for a great deal of philosophical work on truth and meaning as well as having a crucial role in our construction of modern theories of formal semantics. The Unnoticed Man focuses on the logic of epistemic notions such as knowledge and belief, and treatments of this and cognate puzzles figure centrally in con- temporary philosophy of language. The Heap arises because of the vagueness inherent in predicates of natural language, and it forces us to recognize a cer- tain recalcitrance in such predicates. And the Horned Man can be seen as an early worry about the notion of presupposition.
4.2 Conditionals
The Stoic interest in conditionals (assertions of the form, i f . . . then) probably have their origin in the use of conditional arguments of the form (4):
If P then Q. (4) If P then not Q.
Therefore, it is impossible that P.
This form of argument was used before Aristotle's time in metaphysical contexts by Zeno of Elea, and is of course one form of the reductio adabsurdum argu- ment which entitles us to conclude the negation of a given statement if it can be shown to lead to a contradiction. In any case, it is clear from what we do know about the Stoics, and, in particular, Chrysippus (d. 207 BC) and his follower Philo, that they studied both the conditional form of statement and con- ditional arguments in a strikingly modern way. They seem to have isolated and wrestled with some of the philosophical and linguistic problems that we now think of as problems about the relation between for- mal renderings of the conditional, and natural lan- guage conditionals. Indeed, Philo may well be the first philosopher to have formulated and worried about the truth conditions of the material conditional.
4.3 Modality
What Stoic and Megarian logic is perhaps most closely associated with is its contribution to ideas about modal notions such as possibility and necessity. Most prominent here is Diodorus Cronus whose work dates from some 25 years after the death of Aristotle. Diodorus's name is with the so-called 'Master' argu- ment whose conclusion is that the future is necessary,
(a)
(b)
(c)
The Liar: John says that he is speaking falsely. Is he speaking truly? Clearly, if yes, then no and if no, then yes.
The Unnoticed Man: if John knows Jim, but doesn't recognize him on some occasion then he does not know him.
The Heap: one grain of sand does not make a heap, and if you add another single grain you do not get a heap. So, there is no number of grains you can add to get a heap.
The Horned Man: what John has not lost he
(d)
Logic: Historical Survey
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