Aristotles Modal Logic: Essence and Entailment in the Organon

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Have Institutional Access? Forgot your password? PDF Preview. Table of Contents. Related Content. Alienation The Concept and its Reception. Valid reasoning has been employed in all periods of human history.

The Organon: III. Prior Analytics

However, logic studies the principles of valid reasoning, inference and demonstration. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry , which originally meant the same as "land measurement". Esagil-kin-apli 's medical Diagnostic Handbook in the 11th century BC was based on a logical set of axioms and assumptions, [15] while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems, an important contribution to the philosophy of science.

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While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Both Thales and Pythagoras of the Pre-Socratic philosophers seem aware of geometry's methods. Fragments of early proofs are preserved in the works of Plato and Aristotle, [17] and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy.

The three basic principles of geometry are as follows:. Further evidence that early Greek 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.

Notre Dame Journal of Formal Logic

This is part of a protracted debate about truth and falsity. It is said Thales, most widely regarded as the first philosopher in the Greek tradition , [20] [21] measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. Thales was said to have had a sacrifice in celebration of discovering Thales' theorem just as Pythagoras had the Pythagorean theorem.

Thales is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has been attributed.

Aristotle's modal logic : essence and entailment in the Organon /

The writing of Heraclitus c. He is known for his obscure sayings. This logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. For though all things come to be in accordance with this logos , humans are like the inexperienced when they experience such words and deeds as I set out, distinguishing each in accordance with its nature and saying how it is. But other people fail to notice what they do when awake, just as they forget what they do while asleep.

In contrast to Heraclitus, Parmenides held that all is one and nothing changes. He may have been a dissident Pythagorean, disagreeing that One a number produced the many. What exists can in no way not exist. Our sense perceptions with its noticing of generation and destruction are in grievous error.

Instead of sense perception, Parmenides advocated logos as the means to Truth. He has been called the discoverer of logic, [30] [31]. Zeno of Elea , a pupil of Parmenides, had the idea of a standard argument pattern found in the method of proof known as reductio ad absurdum. This is the technique of drawing an obviously false that is, "absurd" conclusion from an assumption, thus demonstrating that the assumption is false. Zeno famously used this method to develop his paradoxes in his arguments against motion. Such dialectic reasoning later became popular.

Aristotle's Modal Logic: Essence and Entailment in the Organon

The members of this school were called "dialecticians" from a Greek word meaning "to discuss". None of the surviving works of the great fourth-century philosopher Plato — BC include any formal logic, [34] but they include important contributions to the field of philosophical logic. Plato raises three questions:. The first question arises in the dialogue Theaetetus , where Plato identifies thought or opinion with talk or discourse logos.

Ancient Logic |

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 assumptions of a valid argument and its conclusion corresponds to a necessary connection between "forms". 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.

Thus, a definition reflects the ultimate object of understanding, and is the foundation of all valid inference. This had a great influence on Plato's student Aristotle , in particular Aristotle's notion of the essence of a thing. The logic of Aristotle , and particularly his theory of the syllogism , has had an enormous influence in Western thought.

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He was the first formal logician , in that he demonstrated the principles of reasoning by employing variables to show the underlying logical form of an argument. He sought relations of dependence which characterize necessary inference, and distinguished the validity of these relations, from the truth of the premises. He was the first to deal with the principles of contradiction and excluded middle in a systematic way. 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:. These works are of outstanding importance in the history of logic. In the Categories , he attempts to discern all the possible things to which a term can refer; this idea underpins his philosophical work Metaphysics , which itself had a profound influence on Western thought. He also developed a theory of non-formal logic i. On Interpretation contains 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 contains his exposition of the "syllogism", 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. The other great school of Greek logic is that of the Stoics. 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. He is supposed to have written over works, including at least on logic, almost none of which survive. 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.

The works of Al-Kindi , Al-Farabi , Avicenna , Al-Ghazali , Averroes and other Muslim logicians were based on Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West. Ibn Sina Avicenna — was the founder of Avicennian logic , which replaced Aristotelian logic as the dominant system of logic in the Islamic world, [58] and also had an important influence on Western medieval writers such as Albertus Magnus. Fakhr al-Din al-Razi b. In response to this tradition, Nasir al-Din al-Tusi — began a tradition of Neo-Avicennian logic which remained faithful to Avicenna's work and existed as an alternative to the more dominant Post-Avicennian school over the following centuries.

The Illuminationist school was founded by Shahab al-Din Suhrawardi — , who developed the idea of "decisive necessity", which refers to the reduction of all modalities necessity, possibility , contingency and impossibility to the single mode of necessity.

He further claimed that induction itself is founded on a process of analogy. His model of analogical reasoning was based on that of juridical arguments. The Sharh al-takmil fi'l-mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century is the last major Arabic work on logic that has been studied.