Ancient Logic

Logic as a discipline starts with the transition from the more or less unreflective use of logical methods and argument patterns to the reflection on and inquiry into these methods and patterns and their elements, including the syntax and semantics of sentences. In Greek and Roman antiquity, discussions of some elements of logic and a focus on methods of inference can be traced back to the late 5th century BCE. The Sophists, and later Plato (early 4th c.) displayed an interest in sentence analysis, truth, and fallacies, and Eubulides of Miletus (mid-4th c.) is on record as the inventor of both the Liar and the Sorites paradox. But logic as a fully systematic discipline begins with Aristotle, who systematized much of the logical inquiry of his predecessors. His main achievements were his theory of the logical interrelation of affirmative and negative existential and universal statements and, based on this theory, his syllogistic, which can be interpreted as a system of deductive inference. Aristotle's logic is known as term-logic, since it is concerned with the logical relations between terms, such as ‘human being’, ‘animal’, ‘white’. It shares elements with both set theory and predicate logic. Aristotle's successors in his school, the Peripatos, notably Theophrastus and Eudemus, widened the scope of deductive inference and improved some aspects of Aristotle's logic.
In the Hellenistic period, and apparently independent of Aristotle's achievements, the logician Diodorus Cronus and his pupil Philo (see the entry Dialectical school) worked out the beginnings of a logic that took propositions, rather than terms, as its basic elements. They influenced the second major theorist of logic in antiquity, the Stoic Chrysippus (mid-3rd c.), whose main achievement is the development of a propositional logic, crowned by a deductive system. Regarded by many in antiquity as the greatest logician, he was innovative in a large number of topics that are central to contemporary formal and philosophical logic. The many close similarities between Chrysippus' philosophical logic and that of Gottlob Frege are especially striking. Chrysippus' Stoic successors systematized his logic, and made some additions.
The development of logic from c. 100 BCE to c. 250 CE remains mostly in the dark, but there can be no doubt that logic was one of the topics regularly studied and researched. At some point Peripatetics and Stoics began taking notice of each other's logical systems, and we witness some conflation of both terminologies and theories. Aristotelian syllogistic became known as ‘categorical syllogistic’ and the Peripatetic adaptation of Stoic syllogistic as ‘hypothetical syllogistic’. In the 2nd century CE, Galen attempted to synthesize the two traditions; he also professed to have introduced a third kind of syllogism, the ‘relational syllogism’, which apparently was meant to help formalize mathematical reasoning. The attempt of some Middle Platonists (1st c. BCE–2nd c. CE) to claim a specifically Platonic logic failed, and in its stead, the Neo-Platonists (3rd–6th c. CE) adopted a scholasticized version of Aristotelian logic as their own. In the monumental—if rarely creative—volumes of the Greek commentators on Aristotle's logical works we find elements of Stoic and later Peripatetic logic as well as Platonism, and ancient mathematics and rhetoric. Much the same holds for the Latin logical writings by Apuleius (2nd c. CE) and Boethius (6th c. CE), which pave the way for Aristotelian logic, thus supplemented, to enter the Medieval era.

[size=30]1. Pre-Aristotelian Logic

1.1 Syntax and Semantics

Some of the Sophists classified types of sentences (logoi) according to their force. So Protagoras (485–415 BCE), who included wish, question, answer and command (Diels Kranz (DK) 80.A1, Diogenes Laertius (D. L.) 9.53–4), and Alcidamas (pupil of Gorgias, fl. 4th BCE), who distinguished assertion (phasis), denial (apophasis), question and address (prosagoreusis) (D. L. 9.54). Antisthenes (mid-5th–mid-4th cent.) defined a sentence as ‘that which indicates what a thing was or is’ (D. L. 6.3, DK 45) and stated that someone who says what is speaks truly (DK49). Perhaps the earliest surviving passage on logic is found in the Dissoi Logoi or Double Arguments (DK 90.4, c. 400 BCE). It is evidence for a debate over truth and falsehood. Opposed were the views (i) that truth is a—temporal—property of sentences, and that a sentence is true (when it is said), if and only if things are as the sentence says they are when it is said, and false if they aren't; and (ii) that truth is an atemporal property of what is said, and that what is said is true if and only if the things are the case, false if they aren't the case. These are rudimentary formulations of two alternative correspondence theories of truth. The same passage displays awareness of the fact that self-referential use of the truth-predicate can be problematic—an insight also documented by the discovery of the Liar paradox by Eubulides of Miletus (mid-4th c. BCE) shortly thereafter.
Some Platonic dialogues contain passages whose topic is indubitably logic. In the Sophist, Plato analyzes simple statements as containing a verb (rhêma), which indicates action, and a noun(onoma), which indicates the agent (Soph. 261e–262a). Anticipating the modern distinction of logical types, he argues that neither a series of nouns nor a series of verbs can combine into a statement (Soph. 262a–d). Plato also divorces syntax (‘what is a statement?’) from semantics (‘when is it true?’). Something (e.g. ‘Theaetetus is sitting’) is a statement if it both succeeds in specifying a subject and says something about this subject. Plato thus determines subject and predicate as relational elements in a statement and excludes as statements subject-predicate combinations containing empty subject expressions. Something is a true statement if with reference to its subject (Theaetetus) it says of what is (e.g. sitting) that it is. Something is a false statement if with reference to its subject it says of something other than what is (e.g. flying), that it is. Here Plato produces a sketch of a deflationist theory of truth (Soph. 262e–263d; cf. Crat.385b). He also distinguished negations from affirmations and took the negation particle to have narrow scope: it negates the predicate, not the whole sentence (Soph. 257b–c). There are many passages in Plato where he struggles to explain certain logical relations: for example his theory that things participate in Forms corresponds to a rudimentary theory of predication; in theSophist and elsewhere he grapples with the class relations of exclusion, union and co-extension; also with the difference between the ‘is’ of predication (being) and the ‘is’ of identity (sameness); and in Republic 4, 436bff., he anticipates the law of non-contradiction. But his explications of these logical questions are cast in metaphysical terms, and so can at most be regarded as proto-logical.[/size]

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Ancient Logic

[size=30]1. Pre-Aristotelian Logic

1.1 Syntax and Semantics

Ancient Logic :: تعاليق

Ancient Logic

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