Logical Paradoxes

The Stoics recognized the importance of both the Liar and the Sorites paradoxes (Cicero Acad. 2.95–8, Plut. Comm.Not. 1059D–E, Chrys. Log. Zet. col.IX). Chrysippus may have tried to solve the Liar as follows: there is an ineliminable ambiguity in the Liar sentence (‘I am speaking falsely’, uttered in isolation) between the assertibles (i) ‘I falsely say I speak falsely’ and (ii) ‘Iam speaking falsely’ (i.e. I am doing what I'm saying, viz. speaking falsely), of which, at any time the Liar sentence is uttered, precisely one is true, but it is arbitrary which one. (i) entails (iii) ‘Iam speaking truly’ and is incompatible with (ii) and with (iv) ‘I truly say I speak falsely’. (ii) entails (iv) and is incompatible with (i) and (iii). Thus bivalence is preserved (cf. Cavini 1993). Chrysippus' stand on the Sorites seems to have been that vague borderline sentences uttered in the context of a Sorites series have no assertibles corresponding to them, and that it is obscure to us where the borderline cases start, so that it is rational for us to stop answering while still on safe ground (i.e. before we might begin to make utterances with no assertible corresponding to them). The latter remark suggests Chrysippus was aware of the problem of higher order vagueness. Again, bivalence of assertibles is preserved (cf. Bobzien 2002). The Stoics also discussed various other well-known paradoxes. In particular, for the paradoxes of presupposition, known in antiquity as the Horned One, they produced a Russellian-type solution based on a hidden scope ambiguity of negation (cf. Bobzien 2012)

6. Epicurus and the Epicureans

Epicurus (late 4th–early 3rd c. BCE) and the Epicureans are said to have rejected logic as an unnecessary discipline (D. L. 10.31, Usener 257). This notwithstanding, several aspects of their philosophy forced or prompted them to take a stand on some issues in philosophical logic. (1)Language meaning and definition: The Epicureans held that natural languages came into existence not by stipulation of word meanings but as the result of the innate capacities of humans for using signs and articulating sounds and of human social interaction (D. L. 10.75–6); that language is learnt in context (Lucretius 5.1028ff); and that linguistic expressions of natural languages are clearer and more conspicuous than their definitions; even that definitions would destroy their conspicuousness (Usener 258, 243); and that philosophers hence should use ordinary language rather than introduce technical expressions (Epicurus On Nature 28). (2)Truth-bearers: the Epicureans deny the existence of incorporeal meanings, such as Stoic sayables. Their truth-bearers are linguistic items, more precisely, utterances (phônai) (S. E. M8.13, 258; Usener 259, 265). Truth consists in the correspondence of things and utterances, falsehood in a lack of such correspondence (S. E. M 8.9, Usener 244), although the details are obscure here. (3) Excluded middle: with utterances as truth-bearers, the Epicureans face the question what the truth-values of future contingents are. Two views are recorded. One is the denial of the Principle of Excluded Middle (‘p or not p’) for future contingents (Usener 376, Cicero Acad. 2.97, Cicero Fat. 37). The other, more interesting, one leaves the Excluded Middle intact for all utterances, but holds that, in the case of future contingents, the component utterances ‘p’ and ‘not p’ are neither true nor false (Cicero Fat. 37), but, it seems, indefinite. This could be regarded as an anticipation of supervaluationism. (4) Induction: Inductive logic was comparatively little developed in antiquity. Aristotle discusses arguments from the particular to the universal (epagôgê) in the Topics and Posterior Analytics but does not provide a theory of them. Some later Epicureans developed a theory of inductive inference which bases the inference on empirical observation that certain properties concur without exception (Philodemus De Signis).

7. Later Antiquity

Very little is known about the development of logic from c. 100 BCE to c. 250 CE. It is unclear when Peripatetics and Stoics began taking notice of each others' logical achievements. At some point during that period, the terminological distinction between ‘categorical syllogisms’, used for Aristotelian syllogisms, and ‘hypothetical syllogisms’, used not only for those introduced by Theophrastus and Eudemus, but also for the Stoic propositional-logical syllogisms, gained a foothold. In the first century BCE, the Peripatetics Ariston of Alexandria and Boethus of Sidon wrote about syllogistic. Ariston is said to have introduced the so-called ‘subaltern’ syllogisms (Barbari, Celaront, Cesaro, Camestrop and Camenop) into Aristotelian syllogistic (Apuleius Int. 213.5–10), i.e. the syllogisms one gains by applying the subalternation rules (that were acknowledged by Aristotle in his Topics)

اقتباس :: From ‘A holds of every B’ infer ‘A holds of some B’

اقتباس :: From ‘A holds of no B’ infer ‘A does not hold of some B’

to the conclusions of the relevant syllogisms. Boethus suggested substantial modifications to Aristotle's theories: he claimed that all categorical syllogisms are complete, and that hypothetical syllogistic is prior to categorical (Galen Inst. Log. 7.2), although we are not told what this priority was thought to consist in. The Stoic Posidonius (c. 135–c. 51 BCE) defended the possibility of logical or mathematical deduction against the Epicureans and discussed some syllogisms he called ‘conclusive by the force of an axiom’, which apparently included arguments of the type ‘As the 1st is to the 2nd, so the 3rd is to the 4th; the ratio of the 1st to the 2nd is double; therefore the ratio of the 3rd to the 4th is double’, which was considered conclusive by the force of the axiom ‘things which are in general of the same ratio, are also of the same particular ratio’ (Galen Inst. Log. 18.8). At least two Stoics in this period wrote a work on Aristotle's Categories. From his writings we know that Cicero (1st c. BCE) was knowledgeable about both Peripatetic and Stoic logic; and Epictetus' discourses (late 1st–early 2nd c. CE) prove that he was acquainted with some of the more taxing parts of Chrysippus' logic. In all likelihood, there existed at least a few creative logicians in this period, but we do not know who they were or what they created.
The next logician of rank, if of lower rank, of whom we have sufficient evidence to speak is Galen (129–199 or 216 CE), who achieved greater fame as a physician. He studied logic with both Peripatetic and Stoic teachers, and recommended availing oneself of parts of either doctrine, as long as it could be used for scientific demonstration. He composed commentaries on logical works by Aristotle, Theophrastus, Eudemus and Chrysippus, as well as treatises on various logical problems and a major work entitled On Demonstration. All these are lost, except for some information in later texts, but his Introduction to Logic has come down to us almost in full. In On Demonstration, Galen developed, among other things, a theory of compound categorical syllogisms with four terms, which fall into four figures, but we do not know the details. He also introduced the so-called relational syllogisms, examples of which are ‘A is equal to B, B is equal to C; therefore A is equal to C’ and ‘Dio owns half as much as Theo; Theo owns half as much as Philo. Therefore Dio owns a quarter of what Philo owns’ (Galen Inst. Log, 17–18). All the relational syllogisms Galen mentions have in common that they are not reducible in either Aristotle's or the Stoic syllogistic, but it is difficult to find further formal characteristics that unite them. In general, in his Introduction to Logic Galen merges Aristotelian Syllogistic with a strongly Peripatetic reinterpretation of Stoic propositional logic. This becomes apparent in particular in Galen's emphatic denial that truth-preservation is sufficient for the validity or syllogismhood of an argument, and his insistence that, instead, knowledge-introduction or knowledge-extension is a necessary condition for something to count as a syllogism.[5]
The second ancient introduction to logic that has survived is Apuleius' (2nd cent. CE) De Interpretatione. This Latin text, too, displays knowledge of Stoic and Peripatetic logic; it contains the first full presentation of the square of opposition, which illustrates the logical relations between categorical sentences by diagram. The Platonist Alcinous (2nd cent. CE), in hisHandbook of Platonism chapter 5, is witness to the emergence of a specifically Platonist logic, constructed on the Platonic notions and procedures of division, definition, analysis and hypothesis, but there is little that would make a logician's heart beat faster. At some time between the 3rd and 6th century CE Stoic logic faded into oblivion, to be resurrected only in the 20thcentury, in the wake of the (re)-discovery of propositional logic.
The surviving, often voluminous, Greek commentaries on Aristotle's logical works by Alexander of Aphrodisias (fl. c. 200 CE), Porphyry (234–c. 305), Ammonius Hermeiou (5th century), Philoponus (c. 500) and Simplicius (6th century) and the Latin ones by Boethius (c. 480–524) are mainly important for preserving alternative interpretations of Aristotle's logic and as sources for lost Peripatetic and Stoic works. They also allow us to trace the gradual development from a Peripatetic exegesis of Aristotle’s Organon to a more eclectic logic that resulted from the absorption and inclusion of elements not just from Stoic and Platonist theories but also from mathematics and rhetoric. Two of the commentators in particular deserve special mention in their own right: Porphyry, for writing the Isagoge or Introduction (i.e. to Aristotle's Categories), in which he discusses the five notions of genus, species, differentia, property and accident as basic notions one needs to know to understand the Categories. For centuries, the Isagoge was the first logic text a student would tackle, and Porphyry's five predicables (which differ from Aristotle's four) formed the basis for the medieval doctrine of the quinque voces. The second is Boethius. In addition to commentaries, he wrote a number of logical treatises, mostly simple explications of Aristotelian logic, but also two very interesting ones: (i) His On Topical Differentiae bears witness to the elaborated system of topical arguments that logicians of later antiquity had developed from Aristotle's Topics under the influence of the needs of Roman lawyers. (ii) HisOn Hypothetical Syllogisms systematically presents wholly hypothetical and mixed hypothetical syllogisms as they are known from the early Peripatetics; it may be derived from Porphyry. Boethius' insistence that the negation of ‘If it is A, it is B’ is ‘If it is A, it is not B’ suggests a suppositional understanding of the conditional, a view for which there is also some evidence in Ammonius, but that is not attested for earlier logicians. Historically, Boethius is most important because he translated all of Aristotle's Organon into Latin, making these texts (except thePosterior Analytics) available to philosophers of the medieval period.

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Logical Paradoxes

6. Epicurus and the Epicureans

7. Later Antiquity

Logical Paradoxes :: تعاليق

Logical Paradoxes

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