• • • The history of logic deals with the study of the development of the science of valid (). Formal logics developed in ancient times in,, and. Greek methods, particularly (or term logic) as found in the, found wide application and acceptance in Western science and mathematics for millennia. The, especially, began the development of. And philosophers such as (died 524) and (died 1347) further developed Aristotle's logic in the, reaching a high point in the mid-fourteenth century. The period between the fourteenth century and the beginning of the nineteenth century saw largely decline and neglect, and at least one historian of logic regards this time as barren.

Ruled thea day, as evidenced by Sir 's of 1620. Logic revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a rigorous and formal discipline which took as its exemplar the exact method of used in, a hearkening back to the Greek tradition. The development of the modern 'symbolic' or 'mathematical' logic during this period by the likes of,,, and is the most significant in the two-thousand-year history of logic, and is arguably one of the most important and remarkable events in human. Progress in in the first few decades of the twentieth century, particularly arising from the work of and, had a significant impact on and, particularly from the 1950s onwards, in subjects such as,,, and. Main article: Logic began independently in and continued to develop to early modern times without any known influence from Greek logic. 7th century BC) founded the anviksiki school of logic. The (12.173.45), around the 5th century BC, refers to the anviksiki and tarka schools of logic.

5th century BC) developed a form of logic (to which has some similarities) for his formulation of. Logic is described by (c. 350-283 BC) in his as an independent field of inquiry. Two of the six Indian schools of thought deal with logic: and. 2nd century AD) constitute the core texts of the Nyaya school, one of the six orthodox schools of philosophy. This school developed a rigid five-member schema of involving an initial premise, a reason, an example, an application, and a conclusion.

Canadians (French: Canadiens / Canadiennes) are people identified with the country of Canada. This connection may be residential, legal, historical, or cultural.

The became the chief opponent to the Naiyayikas. 150-250 AD), the founder of the ('Middle Way') developed an analysis known as the (Sanskrit), a 'four-cornered' system of argumentation that involves the systematic examination and rejection of each of the 4 possibilities of a proposition, P: • P; that is, being. • not P; that is, not being. • P and not P; that is, being and not being. • not ( P or not P); that is, neither being nor not being. It is interesting to note that under, imply that this is equivalent to the third case ( P and not P), and is therefore superfluous; there are actually only 3 cases to consider. However, (c 480-540 AD) is sometimes said to have developed a formal syllogism, and it was through him and his successor,, that reached its height; it is contested whether their analysis actually constitutes a formal syllogistic system.

In particular, their analysis centered on the definition of an inference-warranting relation, ', also known as invariable concomitance or pervasion. To this end, a doctrine known as 'apoha' or differentiation was developed. This involved what might be called inclusion and exclusion of defining properties. The difficulties involved in this enterprise, in part, stimulated the neo-scholastic school of, which developed a formal analysis of inference in the sixteenth century. This later school began around and, and developed theories resembling modern logic, such as 's 'distinction between sense and reference of proper names' and his 'definition of number,' as well as the Navya-Nyaya theory of 'restrictive conditions for universals' anticipating some of the developments in modern. Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as,, and particularly, as confirmed by his wife, who wrote in 1901 an 'open letter to Dr Bose', which was titled 'Indian Thought and Western Science in the Nineteenth Century' and stated: 'Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan and George Boole on the mathematical atmosphere of 1830-1865'.

Dignāga's famous 'wheel of reason' ( ) is a method of indicating when one thing (such as smoke) can be taken as an invariable sign of another thing (like fire), but the inference is often inductive and based on past observation. Matilal remarks that Dignāga's analysis is much like John Stuart Mill's Joint Method of Agreement and Difference, which is inductive. In addition, the traditional five-member Indian syllogism, though deductively valid, has repetitions that are unnecessary to its logical validity. As a result, some commentators see the traditional Indian syllogism as a rhetorical form that is entirely natural in many cultures of the world, and yet not as a logical form—not in the sense that all logically unnecessary elements have been omitted for the sake of analysis. Logic in China [ ]. Main article: In China, a contemporary of,, 'Master Mo', is credited with founding the, whose canons dealt with issues relating to valid inference and the conditions of correct conclusions. In particular, one of the schools that grew out of Mohism, the, are credited by some scholars for their early investigation of.

Amiga Explorer Serial Number. Due to the harsh rule of in the subsequent, this line of investigation disappeared in China until the introduction of Indian philosophy. Logic in the West [ ] Prehistory of logic [ ] Valid reasoning has been employed in all periods of human history. 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, which originally meant the same as 'land measurement'.

The discovered, including the formula for the volume of a. Was also skilled in mathematics. 's medical Diagnostic Handbook in the 11th century BC was based on a logical set of and assumptions, while in the 8th and 7th centuries BC employed an within their predictive planetary systems, an important contribution to the. C Puzzles By Alan R Feuer Pdf Writer. Ancient Greece before Aristotle [ ] While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative.

Both and of the seem aware of geometry's methods. Fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and the.

The proofs of are a paradigm of Greek geometry. The three basic principles of geometry are as follows: • Certain propositions must be accepted as true without demonstration; such a proposition is known as an of geometry. • Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry; such a demonstration is known as a or a 'derivation' of the proposition. • The proof must be formal; that is, the derivation of the proposition must be independent of the particular subject matter in question. Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called, probably written at the beginning of the fourth century BC. This is part of a protracted debate about truth and falsity.

In the case of the classical Greek city-states, interest in argumentation was also stimulated by the activities of the or Orators and the, who used arguments to defend or attack a thesis, both in legal and political contexts. Thales Theorem Thales [ ] It is said Thales, most widely regarded as the first philosopher in the, measured the height of the 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 just as Pythagoras had the.

Thales is the first known individual to use applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has been attributed. And Babylonian mathematicians knew his theorem for special cases before he proved it. It is believed that Thales learned that an angle inscribed in a is a right angle during his travels to. Pythagoras [ ]. Proof of the Pythagorean Theorem in Euclid's Elements Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c. 54 years older Thales. The systematic study of proof seems to have begun with the school of Pythagoras (i. The Pythagoreans) in the late sixth century BC.

Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize form rather than matter. Heraclitus and Parmenides [ ] The writing of (c. 475 BC) was the first place where the word was given special attention in ancient Greek philosophy, Heraclitus held that everything changes and all was fire and conflicting opposites, seemingly unified only by this Logos. 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. Parmenides has been called the discoverer of logic.

In contrast to Heraclitus, held that all is one and nothing changes. He may have been a dissident Pythagorean, disagreeing that One (a number) produced the many. 'X is not' must always be false or meaningless. 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, For this view, that That Which Is Not exists, can never predominate. You must debar your thought from this way of search, nor let ordinary experience in its variety force you along this way, (namely, that of allowing) the eye, sightless as it is, and the ear, full of sound, and the tongue, to rule; but (you must) judge by means of the Reason () the much-contested proof which is expounded by me. (B 7.1–8.2), a pupil of Parmenides, had the idea of a standard argument pattern found in the method of proof known as. This is the technique of drawing an obviously false (that is, 'absurd') conclusion from an assumption, thus demonstrating that the assumption is false. Therefore, Zeno and his teacher are seen as the first to apply the art of logic. Plato's dialogue portrays Zeno as claiming to have written a book defending the of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. Zeno famously used this method to develop his in his arguments against motion.

Such dialectic reasoning later became popular. The members of this school were called 'dialecticians' (from a Greek word meaning 'to discuss'). Plato [ ] Let no one ignorant of geometry enter here. Plato's academy None of the surviving works of the great fourth-century philosopher (428–347 BC) include any formal logic, but they include important contributions to the field of. 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, where Plato identifies thought or opinion with talk or discourse ( logos). The second question is a result of Plato's. Forms are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called, namely an abstract entity common to each set of things that have the same name. In both the and the, Plato suggests that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary connection between 'forms'. The third question is about. 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.

What underlies every definition is a Platonic Form, the common nature present in different particular things. 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, in particular Aristotle's notion of the of a thing. Aristotle [ ].

Aristotle The logic of, and particularly his theory of the, has had an enormous influence in. Aristotle was the first logician to attempt a systematic analysis of, of noun (or ), and of verb. He was the first formal logician, in that he demonstrated the principles of reasoning by employing variables to show the underlying of an argument. He sought relations of dependence which characterize necessary inference, and distinguished the of these relations, from the truth of the premises (the of the argument). He was the first to deal with the principles of and in a systematic way. Aristotle's logic was still influential in the The Organon [ ] His logical works, called the, 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: •, a study of the ten kinds of primitive term.

• (with an appendix called ), a discussion of dialectics. •, an analysis of simple into simple terms, negation, and signs of quantity. •, a formal analysis of what makes a (a valid argument, according to Aristotle). •, a study of scientific demonstration, containing Aristotle's mature views on logic. This diagram shows the contradictory relationships between in the of. 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, which itself had a profound influence on Western thought.

He also developed a theory of non-formal logic ( i.e., the theory of ), which is presented in Topics and Sophistical Refutations. On Interpretation contains a comprehensive treatment of the notions of and conversion; chapter 7 is at the origin of the (or logical square); chapter 9 contains the beginning of. 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. Stoics [ ] The other great school of Greek logic is that of the.

Stoic logic traces its roots back to the late 5th century BC philosopher, a pupil of and slightly older contemporary of Plato, probably following in the tradition of Parmenides and Zeno. His pupils and successors were called ', or 'Eristics', and later the 'Dialecticians'. The two most important dialecticians of the Megarian school were and, who were active in the late 4th century BC. Of Soli The Stoics adopted the Megarian logic and systemized it. The most important member of the school was (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.

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,,,,, and. Three significant contributions of the Stoic school were (i) their account of, (ii) their theory of the, and (iii) their account of and. According to Aristotle, the Megarians of his day claimed there was no distinction between.

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. Diodorus is also famous for what is known as his, which states that each pair of the following 3 propositions contradicts the third proposition: • Everything that is past is true and necessary. • The impossible does not follow from the possible.

• What neither is nor will be is possible. Diodorus used the plausibility of the first two to prove that nothing is possible if it neither is nor will be true. Chrysippus, by contrast, denied the second premise and said that the impossible could follow from the possible. • Conditional statements. The first logicians to debate were Diodorus and his pupil Philo of Megara.

Sextus Empiricus refers three times to a debate between Diodorus and Philo. Philo regarded a conditional as true unless it has both a true and a false. Questions on the Old Logic 'Medieval logic' (also known as 'Scholastic logic') generally means the form of Aristotelian logic developed in throughout roughly the period 1200–1600. For centuries after Stoic logic had been formulated, it was the dominant system of logic in the classical world. When the study of logic resumed after the, the main source was the work of the Christian philosopher, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics. Until the twelfth century, the only works of Aristotle available in the West were the Categories, On Interpretation, and Boethius's translation of the of (a commentary on the Categories).

These works were known as the 'Old Logic' ( Logica Vetus or Ars Vetus). An important work in this tradition was the Logica Ingredientibus of (1079–1142). His direct influence was small, but his influence through pupils such as was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed. By the early thirteenth century, the remaining works of Aristotle's Organon (including the,, and the ) had been recovered in the West. Logical work until then was mostly paraphrasis or commentary on the work of Aristotle.

The period from the middle of the thirteenth to the middle of the fourteenth century was one of significant developments in logic, particularly in three areas which were original, with little foundation in the Aristotelian tradition that came before. These were: • The theory of.

Supposition theory deals with the way that predicates ( e.g., 'man') range over a domain of individuals ( e.g., all men). In the proposition 'every man is an animal', does the term 'man' range over or 'supposit for' men existing just in the present, or does the range include past and future men? Can a term supposit for a non-existing individual?

Some medievalists have argued that this idea is a precursor of modern. 'The theory of supposition with the associated theories of copulatio (sign-capacity of adjectival terms), ampliatio (widening of referential domain), and distributio constitute one of the most original achievements of Western medieval logic'. • The theory of. Syncategoremata are terms which are necessary for logic, but which, unlike categorematic terms, do not signify on their own behalf, but 'co-signify' with other words. Examples of syncategoremata are 'and', 'not', 'every', 'if', and so on. • The theory of. A consequence is a hypothetical, conditional proposition: two propositions joined by the terms 'if.

For example, 'if a man runs, then God exists' ( Si homo currit, Deus est). A fully developed theory of consequences is given in Book III of 's work. There, Ockham distinguishes between 'material' and 'formal' consequences, which are roughly equivalent to the modern and respectively.

Similar accounts are given by and. The last great works in this tradition are the Logic of John Poinsot (1589–1644, known as ), the Metaphysical Disputations of (1548–1617), and the Logica Demonstrativa of (1667–1733). Traditional logic [ ] The textbook tradition [ ]. 's Art of Logic (1584) Traditional logic generally means the textbook tradition that begins with 's and 's Logic, or the Art of Thinking, better known as the. Published in 1662, it was the most influential work on logic after Aristotle until the nineteenth century. The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather than terms, for example) within a framework that is broadly derived from Aristotelian and medieval. Between 1664 and 1700, there were eight editions, and the book had considerable influence after that.

The Port-Royal introduces the concepts of and. The account of that gives in the Essay is essentially that of the Port-Royal: 'Verbal propositions, which are words, [are] the signs of our ideas, put together or separated in affirmative or negative sentences. So that proposition consists in the putting together or separating these signs, according as the things which they stand for agree or disagree.' Helped popularize logic, a reaction against Aristotle. Another influential work was the by, published in 1620. The title translates as 'new instrument'. This is a reference to 's work known as the.

In this work, Bacon rejects the syllogistic method of Aristotle in favor of an alternative procedure 'which by slow and faithful toil gathers information from things and brings it into understanding'. This method is known as, a method which starts from empirical observation and proceeds to lower axioms or propositions; from these lower axioms, more general ones can be induced. For example, in finding the cause of a phenomenal nature such as heat, 3 lists should be constructed: • The presence list: a list of every situation where heat is found. • The absence list: a list of every situation that is similar to at least one of those of the presence list, except for the lack of heat. • The variability list: a list of every situation where heat can vary. Then, the form nature (or cause) of heat may be defined as that which is common to every situation of the presence list, and which is lacking from every situation of the absence list, and which varies by degree in every situation of the variability list. Other works in the textbook tradition include 's Logick: Or, the Right Use of Reason (1725), 's Logic (1826), and 's A System of Logic (1843).

Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lie in introspection influenced the view that logic is best understood as a branch of psychology, a view which dominated the next fifty years of its development, especially in Germany. Logic in Hegel's philosophy [ ]. George Boole Modern logic begins with what is known as the 'algebraic school', originating with Boole and including,,, and. Their objective was to develop a calculus to formalise reasoning in the area of classes, propositions, and probabilities. The school begins with Boole's seminal work Mathematical Analysis of Logic which appeared in 1847, although (1847) is its immediate precursor. The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an usher in a private school in.

For example, let x and y stand for classes let the symbol = signify that the classes have the same members, xy stand for the class containing all and only the members of x and y and so on. Boole calls these elective symbols, i.e. Symbols which select certain objects for consideration. An expression in which elective symbols are used is called an elective function, and an equation of which the members are elective functions, is an elective equation.

The theory of elective functions and their 'development' is essentially the modern idea of and their expression in. Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between 'primary propositions' which are the subject of syllogistic theory, and 'secondary propositions', which are the subject of propositional logic, and showed how under different 'interpretations' the same algebraic system could represent both. An example of a primary proposition is 'All inhabitants are either Europeans or Asiatics.' An example of a secondary proposition is 'Either all inhabitants are Europeans or they are all Asiatics.'

These are easily distinguished in modern propositional calculus, where it is also possible to show that the first follows from the second, but it is a significant disadvantage that there is no way of representing this in the Boolean system. In his Symbolic Logic (1881), used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's methods could be mechanised, and constructed a 'logical machine' which he showed to the the following year. In 1885 proposed an electrical version of the machine that is still extant (). Alfred Tarski, a pupil of, is best known for his definition of truth and, and the semantic concept of.

In 1933, he published (in Polish) The concept of truth in formalized languages, in which he proposed his: a sentence such as 'snow is white' is true if and only if snow is white. Tarski's theory separated the, which makes the statement about truth, from the, which contains the sentence whose truth is being asserted, and gave a correspondence (the ) between phrases in the object language and elements of an.

Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of. Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as,, and. According to Anita Feferman, Tarski 'changed the face of logic in the twentieth century'.

And proposed formal models of computability, giving independent negative solutions to Hilbert's in 1936 and 1937, respectively. The Entscheidungsproblem asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the as a key example of a mathematical problem without an algorithmic solution.

Church's system for computation developed into the modern, while the became a standard model for a general-purpose computing device. It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the that any deterministic that can be carried out by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both and are. Later work by and in the 1940s extended the scope of computability theory and introduced the concept of.

The results of the first few decades of the twentieth century also had an impact upon and, particularly from the 1950s onwards, in subjects such as,,, and. Logic after WWII [ ].