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Before Plato While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative science. The systematic study of this seems to have begun with the school of Pythagoras in the late sixth century BC.[4] The three basic principles of geometry are that certain propositions must be accepted as true without demonstration, that all other propositions of the system are derived from these, and that the derivation must be formal, that is, independent of the particular subject matter in question.[4] Fragments of early proofs are preserved in the works of Plato and Aristotle,[7] and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy.[4] Separately from geometry, the idea of a standard argument pattern is found in the Reductio ad absurdum used by Zeno of Elea, a pre-Socratic philosopher of the fifth century BC. This is the technique of drawing an obviously false, absurd or impossible conclusion from an assumption, thus demonstrating that the assumption is false.[8] Plato's Parmenides portrays Zeno as claiming to have written a book defending the monism of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. Other philosophers who practised such dialectic reasoning were the so-called minor Socratics, including Euclid of Megara, who were probably followers of Parmenides and Zeno. The members of this school were called "dialecticians" (from a Greek word meaning "to discuss"). Further evidence that pre-Aristotelian thinkers were concerned with the principles of reasoning is found in the fragment called dissoi logoi, probably written at the beginning of the fourth century BC. This is part of a protracted debate about truth and falsity.[9] In the case of the classical Greek city-states, interest in argumentation was also stimulated by the activities of the Rhetoricians or Orators and the Sophists, who used arguments to defend or attack a thesis, both in legal and political contexts.[10] Plato's logic Mosaic: seven men standing under a tree Plato's academy None of the surviving works of the great fourth-century philosopher Plato (428–347 BC) include any formal logic,[11] but they include important contributions to the field of philosophical logic. Plato raises three questions: What is it that can properly be called true or false? What is the nature of the connection between the assumptions of a valid argument and its conclusion? What is the nature of definition? The first question arises in the dialogue Theaetetus, where Plato identifies thought or opinion with talk or discourse (logos).[12] The second question is a result of Plato's theory of Forms. Forms are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called universals, namely an abstract entity common to each set of things that have the same name. In both The Republic and The Sophist, Plato suggests that the necessary connection between the premisses and the conclusion of an argument corresponds to a necessary connection between "forms".[13] The third question is about definition. Many of Plato's dialogues concern the search for a definition of some important concept (justice, truth, the Good), and it is likely that Plato was impressed by the importance of definition in mathematics.[14] What underlies every definition is a Platonic Form, the common nature present in different particular things. Thus a definition reflects the ultimate object of our understanding, and is the foundation of all valid inference. This had a great influence on Aristotle, in particular Aristotle's notion of the essence of a thing, the "what it is to be" a particular thing of a certain kind.[15] Aristotle's logic Main article: Organon Front cover of book, titled "Aristotelis Logica", with an illustration of eagle on a snake Aristotle's logic was still influential in the Renaissance The logic of Aristotle, and particularly his theory of the syllogism, has had an enormous influence in Western thought.[16] His logical works, called the Organon, are the earliest formal study of logic that have come down to modern times. Though it is difficult to determine the dates, the probable order of writing of Aristotle's logical works is: The Categories, a study of the ten kinds of primitive term. The Topics (with an appendix called On Sophistical Refutations), a discussion of dialectics. On Interpretation, an analysis of simple categorical propositions, into simple terms, negation, and signs of quantity; and a comprehensive treatment of the notions of opposition and conversion. Chapter 7 is at the origin of the square of opposition (or logical square). Chapter 9 contains the beginning of modal logic. The Prior Analytics, a formal analysis of valid argument or "syllogism". The Posterior Analytics, a study of scientific demonstration, containing Aristotle's mature views on logic. These works are of outstanding importance in the history of logic. Aristotle was the first logician to attempt a systematic analysis of logical syntax, into noun (or term), and verb. In the Categories, he attempted to all the possible things that a term can refer to. This idea underpins his philosophical work, the Metaphysics, which also had a profound influence on Western thought. He was the first to deal with the principles of contradiction and excluded middle in a systematic way. He was the first formal logician (i.e. he gave the principles of reasoning using variables to show the underlying logical form of arguments). He was looking for relations of dependence which characterise necessary inference, and distinguished the validity of these relations, from the truth of the premises (the soundness of the argument). The Prior Analytics contains his exposition of the "syllogistic", where three important principles are applied for the first time in history: the use of variables, a purely formal treatment, and the use of an axiomatic system. In the Topics and Sophistical Refutations he also developed a theory of non-formal logic (e.g. the theory of fallacies).[17] Stoic logic Stone bust of a bearded, grave-looking man Chrysippus of Soli The other great school of Greek logic is that of the Stoics.[18] 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. His pupils and successors were called "Megarians", or "Eristics", and later the "Dialecticians". The two most important dialecticians of the Megarian school were Diodorus Cronus and Philo who were active in the late 4th century BC. 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.[19][20] 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 Laertius, Sextus Empiricus, Galen, Aulus Gellius, Alexander of Aphrodisias and Cicero.[21] Three significant contributions of the Stoic school were (i) their account of modality, (ii) their theory of the Material conditional, and (iii) their account of meaning and truth.[22] Modality. According to Aristotle, the Megarians of his day claimed there was no distinction between potentiality and actuality.[23] 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.[24] Diodorus is also famous for his so-called Master argument, that the three propositions "everything that is past is true and necessary", "the impossible does not follow from the possible", and "What neither is nor will be is possible" are inconsistent. Diodorus used the plausibility of the first two to prove that nothing is possible if it neither is nor will be true.[25] Chrysippus, by contrast, denied the second premise and said that the impossible could follow from the possible.[26] Conditional statements. The first logicians to debate conditional statements were Diodorus and his pupil Philo of Megara. Sextus Empiricus refers three times to a debate between Diodorus and Philo. Philo argued that a true conditional is one that does not begin with a truth and end with a falsehood. such as "if it is day, then I am talking". But Diodorus argued that a true conditional is what could not possibly begin with a truth and end with falsehood – thus the conditional quoted above could be false if it were day and I became silent. Philo's criterion of truth is what would now be called a truth-functional definition of "if ... then". In a second reference, Sextus says "According to him there are three ways in which a conditional may be true, and one in which it may be false."[27] Meaning and truth. The most important and striking difference between Megarian-Stoic logic and Aristotelian logic is that it concerns propositions, not terms, and is thus closer to modern propositional logic.[28] 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 is signified, that which signifies, and the object. For example, what signifies is the word Dion, what is signified is what Greeks understand but barbarians do not, and the object is Dion himself.[29] |
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