Editing Chrysippus (section)

== Logic ==
{{See also|Stoic logic}}
Chrysippus wrote much on the subject of logic and created a system of [[propositional logic]]. [[Aristotle]]'s [[term logic]] had been concerned with the interrelations of [[Syllogism#Terms in syllogism|terms]] such as "Socrates" or "man" ("all men are mortal, Socrates is a man, so Socrates is mortal"). Stoic logic, on the other hand, was concerned with the interrelations of [[proposition]]s such as "it is day" ("if it is day, it is light: but it is day: so it is light").<ref name="sharples24">{{Harvnb|Sharples|2014|p=24}}</ref><ref name="Mueller1978">[[Ian Mueller]] (1978) An Introduction to Stoic Logic {{Harvnb|Rist|1978|pp=2–13}}</ref> Though the earlier [[Megarian school|Megarian dialecticians]] –&nbsp;[[Diodorus Cronus]] and [[Philo the Dialectician|Philo]]&nbsp;– had worked in this field and the pupils of [[Aristotle]] –&nbsp;[[Theophrastus]] and [[Eudemus of Rhodes|Eudemus]]&nbsp;– had investigated [[hypothetical syllogism]]s,<ref name="johansen466">{{harvnb|Johansen|Rosenmeier|1998|p=466}}</ref> it was Chrysippus who developed these principles into a coherent system of propositional logic.<ref name="johansen466"/><ref name="sharples2425">{{harvnb|Sharples|2014|pp=24–25}}</ref>

===Propositions===
Chrysippus defined a proposition as "that which is capable of being denied or affirmed as it is in itself" and gave examples of propositions such as "it is day" and "Dion is walking."<ref>Diogenes Laërtius, vii. 65</ref><ref>{{harvnb|Gould|1970|pp=69–70}}</ref> He distinguished between simple and non-simple propositions, which in modern terminology are known as [[Atomic sentence|atomic]] and [[Molecular sentence|molecular]] propositions.<ref name="johansen466"/> A simple proposition is an elementary statement such as "it is day."<ref name="gould71">{{harvnb|Gould|1970|p=71}}</ref> Simple propositions are linked together to form non-simple propositions by the use of [[logical connective]]s. Chrysippus enumerated five kinds of molecular propositions according to the connective used:<ref name="gould71"/>

{| class="wikitable" style="margin:1.0em auto;border:1px;background:white;"
!colspan=2 style="background:white;text-align:center;"| Logical connectives
|- style="background:white"
! Type                !! Example
|-
| if            || '''if''' it is day, it is light
|-
| and           || it is day '''and''' it is light
|-
| either ... or || '''either''' it is day '''or''' it is night
|-
| because       || '''because''' it is day, it is light
|-
| | more/less likely ... than || '''more likely''' it is day '''than''' it is night
|}

Thus several types of molecular propositions, familiar to modern logic, were listed by Chrysippus, including the [[Logical conjunction|conjunction]], the [[Logical disjunction|disjunction]], and the [[Material conditional|conditional]],<ref name="johansen467">{{harvnb|Johansen|Rosenmeier|1998|p=467}}</ref> and Chrysippus studied their criteria of [[Logical truth|truth]] closely.<ref name="johansen467"/>

===Conditional propositions===
The first logicians to debate conditional statements were [[Diodorus Cronus]] and his pupil [[Philo the Dialectician|Philo]]. Writing five-hundred years later, [[Sextus Empiricus]] refers to a debate between Diodorus and Philo.<ref name="sextus1">[[Sextus Empiricus]], ''Pyr. Hyp.'' ii. 110–112; ''Adv. Math.'' viii. 112–117</ref> Philo regarded all conditionals as true except those which with a correct [[antecedent (logic)|antecedent]] had an incorrect [[consequent]], and this meant a proposition such as "if it is day, then I am talking," is true unless it is day and I fall silent.<ref name="sextus2">Sextus Empiricus, ''Pyr. Hyp.'', ii. 110–112</ref> But Diodorus argued that a true conditional is one in which the antecedent clause could never lead to an untrue conclusion{{snd}}thus, because the proposition "if it is day, then I am talking" can be false, it is invalid.<ref name="sextus1"/> However, [[Paradoxes of material implication|paradoxical propositions]] were still possible such as "if atomic elements of things do not exist, atomic elements exists."<ref name="sextus2"/> Chrysippus adopted a much stricter view regarding conditional propositions, which made such paradoxes impossible:{{efn|When Sextus Empiricus reports the different criteria offered by ancient philosophers for the truth of conditional propositions, he does not mention Chrysippus by name, but modern scholars believe that Chrysippus authored, or, at least, held this view.<ref>See {{Harvnb|Gould|1970|pp=72–82}}</ref>}} to him, a conditional is true if denial of the consequent is logically incompatible with the antecedent.<ref name="johansen468">{{Harvnb|Johansen|Rosenmeier|1998|p=468}}</ref> This corresponds to the modern-day [[strict conditional]].<ref name="johansen468"/>

===Syllogistic===
Chrysippus developed a syllogistic or system of deduction in which he made use of five types of basic arguments or [[argument form]]s called indemonstrable syllogisms,<ref>Diogenes Laërtius, vii. 79</ref> which played the role of axioms, and four [[Rule of inference|inference rules]], called ''themata'' by means of which complex syllogisms could be reduced to these axioms.<ref name="Kneale">{{Harvnb|Kneale|Kneale|1962|pages=158–174}}</ref><ref>[[Susanne Bobzien]], Stoic Syllogistic, ''Oxford Studies in Ancient Philosophy'' 14, 1996, pp. 133–192</ref> The forms of the five indemonstrables were:<ref>Diogenes Laertius, vii. 80-81; Sextus Empiricus, ''Hyp. Pyr.'' ii. 156–159; cf. ''Adv. Math.'' viii. 223ff.</ref><ref name="Mates">{{Harvnb|Mates|1953|pages=67–73 }}</ref>

{| class="wikitable" style="margin:1.0em auto;"
!colspan=2| Name{{efn|These [[Latin]] names, unknown to Chrysippus, date from the Middle Ages.<ref>{{harvnb|Sharples|2014|p=24}}</ref>}}
! Description
! Example
|-
|colspan=2| [[Modus ponens]]
|style="white-space:nowrap;"| If A, then B.&nbsp; A.&nbsp; Therefore, B.
| ''If it is day, it is light. It is day. Therefore, it is light.''
|-
|colspan=2| [[Modus tollens]]
|style="white-space:nowrap;"| If A, then B.&nbsp; Not B.&nbsp; Therefore, not A.
| ''If it is day, it is light. It is not light. Therefore, it is not day.''
|-
|rowspan=2| [[Modus ponendo tollens]]&nbsp;
|style="width:1.5em;text-align:center;"| i
|style="white-space:nowrap;"| Not both A and B.&nbsp; A.&nbsp; Therefore, not B.&nbsp;
| ''It is not both day and night. It is day. Therefore, it is not night.''&nbsp;
|-
|style="text-align:center;"| ii
|style="white-space:nowrap;"| Either A or B.&nbsp; A.&nbsp; Therefore, not B.
| ''It is either day or night. It is day. Therefore, it is not night.''
|-
|colspan=2| [[Modus tollendo ponens]]
|style="white-space:nowrap;"| Either A or B.&nbsp; Not A.&nbsp; Therefore, B.
| ''It is either day or night. It is not day. Therefore, it is night.''
|}

Of the four inference rules (themata, θέματα),<ref name="L&S">{{Harvnb|Long |Sedley|1987|}}, §36 HIJ</ref> only two survived. One, the so-called first ''thema'', was a rule of antilogism. The other, the third ''thema'', was a cut rule by which chain syllogisms could be reduced to simple syllogisms.<ref name="Kneale2">{{Harvnb|Kneale|Kneale|1962|p=169}}</ref><ref>{{cite SEP |url-id=logic-ancient/#StoSyl |title=Ancient Logic: Stoic Syllogistic |last=Bobzien |first=Susanne}}</ref> The purpose of Stoic syllogistic was not merely to create a formal system. It was also understood as the study of the operations of reason, the divine reason (''[[logos]]'') which governs the [[universe]], of which human beings are a part.<ref name="sharples26">{{harvnb|Sharples|2014|p=26}}</ref> The goal was to find valid rules of inference and forms of proof to help people find their way in life.<ref name="johansen466"/>
[[File:New Guinea Singing Dog sniffing the ground.jpg|thumb|130px|Chrysippus argued dogs reason.]]

According to Sextus Empiricus, Chrysippus held that dogs use disjunctive syllogism, such as when using scent to pick which path to run down. This was in contrast to a tradition since Aristotle, who saw reasoning (and reasoning deductively) as man's defining aspect.<ref>Sextus Empiricus, Outlines of Pyrrhonism, I.69</ref>

===Other logical work===
Chrysippus analyzed speech and the handling of names and terms.<ref name="davidson614"/> He also devoted much effort in refuting fallacies and paradoxes.<ref name="davidson614"/> According to Diogenes Laërtius, Chrysippus wrote twelve works in 23 books on the [[Liar paradox]]; seven works in 17 books on [[amphiboly]]; and another nine works in 26 books on other conundrums.<ref name="barnes71">{{Harvnb|Barnes|1999|p=71}}</ref> In all, 28 works or 66 books were given over to puzzles or paradoxes.<ref name="barnes71"/>
Chrysippus is the first Stoic for whom the third of the four [[Stoic categories]], i.e. the category ''somehow disposed'' is attested.<ref>Stephen Menn, "The Stoic Theory of Categories", in ''Oxford Studies in Ancient Philosophy'': Volume XVII: 1999, 215–247.</ref> In the surviving evidence, Chrysippus frequently makes use of the categories of ''substance'' and ''quality'', but makes little use of the other two Stoic categories (''somehow disposed'' and ''somehow disposed in relation to something'').<ref name="gould107">{{Harvnb|Gould|1970|p=107}}</ref> It is not clear whether the categories had any special significance for Chrysippus, and a clear doctrine of categories may be the work of later Stoics.<ref name="gould107"/>

===Later reception===
Chrysippus came to be renowned as one of the foremost logicians of ancient Greece. When [[Clement of Alexandria]] wanted to mention one who was master among logicians, as [[Homer]] was master among [[poet]]s, it was Chrysippus, not Aristotle, he chose.<ref>[[Clement of Alexandria]], ''Stromata'', vii. 16</ref> [[Diogenes Laërtius]] wrote: "If the gods use [[dialectic]], they would use none other than that of Chrysippus."<ref>Diogenes Laërtius, vii. 180.</ref> The logical work by Chrysippus came to be neglected and forgotten. Aristotle's logic prevailed, partly because it was seen as more practical, and partly because it was taken up by the [[Neoplatonists]].<ref name="sharples26"/> As recently as the 19th century, Stoic logic was treated with contempt, a barren formulaic system, which was merely clothing the logic of Aristotle with new terminology.<ref name="otoole403">{{Harvnb|O'Toole|Jennings|2004|p=403}}</ref> It was not until the 20th century, with the advances in logic, and the modern [[propositional calculus]], that it became clear that Stoic logic constituted a significant achievement.<ref name="johansen466"/>