Editing History of logic (section)

===Stoics===
{{Main|Stoic logic}}
The other great school of Greek logic is that of the [[Stoicism|Stoics]].<ref>"Throughout later antiquity two great schools of logic were distinguished, the Peripatetic which was derived from Aristotle, and the Stoic which was developed by Chrysippus from the teachings of the Megarians" – Kneale p. 113</ref> Stoic logic traces its roots back to the late 5th century BC philosopher [[Euclid of Megara]], a pupil of [[Socrates]] and slightly older contemporary of Plato, probably following in the tradition of Parmenides and Zeno. His pupils and successors were called "[[Megarian school|Megarians]]", or "Eristics", and later the "Dialecticians". The two most important dialecticians of the Megarian school were [[Diodorus Cronus]] and [[Philo the Dialectician|Philo]], who were active in the late 4th century BC. 
[[File:Chrysippos BM 1846.jpg|alt=Stone bust of a bearded, grave-looking man|thumb|160px|[[Chrysippus]] of Soli]]
The Stoics adopted the Megarian logic and systemized it. The most important member of the school was [[Chrysippus]] (c. 278 – c. 206 BC), who was its third head, and who formalized much of Stoic doctrine. He is supposed to have written over 700 works, including at least 300 on logic, almost none of which survive.<ref>''Oxford Companion'', article "Chrysippus", p. 134</ref><ref>[http://plato.stanford.edu/entries/logic-ancient/] Stanford Encyclopedia of Philosophy: [[Susanne Bobzien]], ''Ancient Logic''</ref> Unlike with Aristotle, we have no complete works by the Megarians or the early Stoics, and have to rely mostly on accounts (sometimes hostile) by later sources, including prominently [[Diogenes Laërtius]], [[Sextus Empiricus]], [[Galen]], [[Aulus Gellius]], [[Alexander of Aphrodisias]], and [[Cicero]].<ref>K. Hülser, Die Fragmente zur Dialektik der Stoiker, 4 vols, Stuttgart 1986–1987</ref>

Three significant contributions of the Stoic school were (i) their account of [[Modal logic|modality]], (ii) their theory of the [[Material conditional]], and (iii) their account of [[Meaning (philosophy of language)|meaning]] and [[truth]].<ref>Kneale 117–158</ref>

* ''Modality''. According to Aristotle, the Megarians of his day claimed there was no distinction between [[Potentiality and actuality (Aristotle)|potentiality and actuality]].<ref>''Metaphysics'' Eta 3, 1046b 29</ref> Diodorus Cronus defined the possible as that which either is or will be, the impossible as what will not be true, and the contingent as that which either is already, or will be false.<ref>[[Boethius]], ''Commentary on the Perihermenias'', Meiser p. 234</ref> Diodorus is also famous for what is known as his [[Master argument (Diodorus Cronus)|Master argument]], which states that each pair of the following 3 propositions contradicts the third proposition:
:* Everything that is past is true and necessary.
:* The impossible does not follow from the possible.
:* What neither is nor will be is possible.
: Diodorus used the plausibility of the first two to prove that nothing is possible if it neither is nor will be true.<ref>[[Epictetus]], ''Dissertationes'' ed. Schenkel ii. 19. I.</ref> Chrysippus, by contrast, denied the second premise and said that the impossible could follow from the possible.<ref>Alexander p. 177</ref>

* ''Conditional statements''. The first logicians to debate [[Material conditional|conditional statements]] were Diodorus and his pupil Philo of Megara. Sextus Empiricus refers three times to a debate between Diodorus and Philo. Philo regarded a conditional as true unless it has both a true [[antecedent (logic)|antecedent]] and a false [[consequent]]. Precisely, let ''T<sub>0</sub>'' and ''T<sub>1</sub>'' be true statements, and let ''F<sub>0</sub>'' and ''F<sub>1</sub>'' be false statements; then, according to Philo, each of the following conditionals is a true statement, because it is not the case that the consequent is false while the antecedent is true (it is not the case that a false statement is asserted to follow from a true statement):
:* If ''T<sub>0</sub>'', then ''T<sub>1</sub>''
:* If ''F<sub>0</sub>'', then ''T<sub>0</sub>''
:* If ''F<sub>0</sub>'', then ''F<sub>1</sub>''
: The following conditional does not meet this requirement, and is therefore a false statement according to Philo:
:* If ''T<sub>0</sub>'', then ''F<sub>0</sub>''
: Indeed, Sextus says "According to &#91;Philo&#93;, there are three ways in which a conditional may be true, and one in which it may be false."<ref name="sextus-adv-math">Sextus Empiricus, ''Adv. Math.'' viii, Section 113</ref> Philo's criterion of truth is what would now be called a [[truth-functional]] definition of "if ... then"; it is the definition used in [[predicate logic|modern logic]].

:In contrast, Diodorus allowed the validity of conditionals only when the antecedent clause could never lead to an untrue conclusion.<ref name="sextus-adv-math"/><ref>Sextus Empiricus, ''Hypotyp.'' ii. 110, comp.</ref><ref>Cicero, ''Academica'', ii. 47, ''de Fato'', 6.</ref>
* ''Meaning and truth''. The most important and striking difference between Megarian-Stoic logic and Aristotelian logic is that Megarian-Stoic logic concerns propositions, not terms, and is thus closer to modern [[propositional logic]].<ref>See e.g. Lukasiewicz p. 21</ref> The Stoics distinguished between utterance (''phone''), which may be noise, speech (''lexis''), which is articulate but which may be meaningless, and discourse (''logos''), which is meaningful utterance. The most original part of their theory is the idea that what is expressed by a sentence, called a ''lekton'', is something real; this corresponds to what is now called a ''proposition''. Sextus says that according to the Stoics, three things are linked together: that which signifies, that which is signified, and the object; for example, that which signifies is the word ''Dion'', and that which is signified is what Greeks understand but barbarians do not, and the object is Dion himself.<ref>Sextus Bk viii., Sections 11, 12</ref>