In logic and formal semantics , term logic , also known as traditional logic , syllogistic logic or Aristotelian logic , is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics . It was revived after the third century CE by Porphyry 's Isagoge .
58-413: In scholastic logic , differentia (also called differentia specifica ) is one of the predicables ; it is that part of a definition which is predicable in a given genus only of the definiendum ; or the corresponding " metaphysical part " of the object. In the original, logical sense, a differentia is a concept — the notion of "differentia" is a second-order concept, or a "second intention", in
116-663: A denial of Aristotle's notion of diaphora. A theory was only provided by Porphyry's explicit treatment of the predicables presented in his Isagoge . The elaborate scholastic theory of the predicables evolved οn the basis of Boethius' translation of the Isagoge, where the Greek term diaphora was rendered in Latin as "differentia". Although the primary meaning of "differentia" is logical or second-order, it may under certain assumptions have an ontological, first-order application. If it
174-453: A grammatical predicate, as in the sentence "the person coming this way is Callias". But it is still a logical subject. He contrasts universal ( katholou ) secondary substance, genera, with primary substance, particular ( kath' hekaston ) specimens. The formal nature of universals , in so far as they can be generalized "always, or for the most part", is the subject matter of both scientific study and formal logic. The essential feature of
232-488: A part of Catholic theological reasoning. For example, Joyce's Principles of Logic (1908; 3rd edition 1949), written for use in Catholic seminaries, made no mention of Frege or of Bertrand Russell . Some philosophers have complained that predicate logic: Even academic philosophers entirely in the mainstream, such as Gareth Evans , have written as follows: George Boole 's unwavering acceptance of Aristotle's logic
290-590: A professor at the University of Warsaw from 1920 until 1939, when the family house was destroyed by German bombs, and the university was closed by the German occupation. He had been a rector of the university twice during which Łukasiewicz and Stanisław Leśniewski had founded the Lwów–Warsaw school of logic , which was later made famous internationally by Alfred Tarski , who had been a student of Leśniewski. During
348-581: A provisional Polish Scientific Institute. In February 1946, at the invitation of Irish political leader Éamon de Valera (himself a mathematician by profession), Łukasiewicz and his wife relocated to Dublin, where they remained until his death there a decade later. In Ireland, he briefly served as Professor of Mathematical Logic at the Royal Irish Academy (a position created for him). His duties involved giving frequent public lectures. During this period, his book Elements of Mathematical Logic
406-449: A rectangle is a square that is a quadrangle”. Jan Lukasiewicz Jan Łukasiewicz ( Polish: [ˈjan wukaˈɕɛvit͡ʂ] ; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic . His work centred on philosophical logic , mathematical logic and history of logic . He thought innovatively about traditional propositional logic ,
464-528: A significant role in the study of logic. Rather than radically breaking with term logic, modern logics typically expand it. Aristotle 's logical work is collected in the six texts that are collectively known as the Organon . Two of these texts in particular, namely the Prior Analytics and De Interpretatione , contain the heart of Aristotle's treatment of judgements and formal inference , and it
522-440: A word. To assert "all Greeks are men" is not to say that the concept of Greeks is the concept of men, or that word "Greeks" is the word "men". A proposition cannot be built from real things or ideas, but it is not just meaningless words either. In term logic, a "proposition" is simply a form of language : a particular kind of sentence , in which the subject and predicate are combined, so as to assert something true or false. It
580-524: Is S. However, in the Prior Analytics Aristotle rejects the usual form in favour of three of his inventions: Aristotle does not explain why he introduces these innovative expressions but scholars conjecture that the reason may have been that it facilitates the use of letters instead of terms avoiding the ambiguity that results in Greek when letters are used with the linking verb. In his formulation of syllogistic propositions, instead of
638-595: Is an animal." Depending on the position of the middle term, Aristotle divides the syllogism into three kinds: syllogism in the first, second, and third figure. If the Middle Term is subject of one premise and predicate of the other, the premises are in the First Figure. If the Middle Term is predicate of both premises, the premises are in the Second Figure. If the Middle Term is subject of both premises,
SECTION 10
#1732766242214696-513: Is assumed that the structuring of an essence into "determining" and "determinable" metaphysical parts (which corresponding to a differentia and a genus respectively) exists in reality independently of its being conceived, one can apply the notion "differentia" also to the determining metaphysical part itself, and not just to the concept that expresses it. This is common in Scotism , where the metaphysical parts are said to be formally distinct . If, on
754-514: Is assumed, quantification implies the existence of at least one subject, unless disclaimed. For Aristotle, the distinction between singular and universal is a fundamental metaphysical one, and not merely grammatical . A singular term for Aristotle is primary substance , which can only be predicated of itself: (this) "Callias" or (this) "Socrates" are not predicable of any other thing, thus one does not say every Socrates one says every human ( De Int. 7; Meta. D9, 1018a4). It may feature as
812-421: Is clearly awkward, a weakness exploited by Frege in his devastating attack on the system. The famous syllogism "Socrates is a man ...", is frequently quoted as though from Aristotle, but in fact, it is nowhere in the Organon . Sextus Empiricus in his Hyp. Pyrrh (Outlines of Pyrronism) ii. 164 first mentions the related syllogism "Socrates is a human being, Every human being is an animal, Therefore, Socrates
870-512: Is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought . According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were “to go under, over, and beyond” Aristotle's logic by: More specifically, Boole agreed with what Aristotle said; Boole's ‘disagreements’, if they might be called that, concern what Aristotle did not say. First, in
928-703: Is evidence that Aristotle knew of fourth-figure syllogisms. In the Prior Analytics translated by A. J. Jenkins as it appears in volume 8 of the Great Books of the Western World, Aristotle says of the First Figure: "... If A is predicated of all B, and B of all C, A must be predicated of all C." In the Prior Analytics translated by Robin Smith, Aristotle says of the first figure: "... For if A
986-432: Is mortal is affirmative, since the mortality of philosophers is affirmed universally, whereas no philosopher is mortal is negative by denying such mortality in particular. The quantity of a proposition is whether it is universal (the predicate is affirmed or denied of all subjects or of "the whole") or particular (the predicate is affirmed or denied of some subject or a "part" thereof). In case where existential import
1044-479: Is not a thought, or an abstract entity . The word "propositio" is from the Latin, meaning the first premise of a syllogism . Aristotle uses the word premise ( protasis ) as a sentence affirming or denying one thing or another ( Posterior Analytics 1. 1 24a 16), so a premise is also a form of words. However, as in modern philosophical logic, it means that which is asserted by the sentence. Writers before Frege and Russell , such as Bradley , sometimes spoke of
1102-465: Is predicated of every B and B of every C, it is necessary for A to be predicated of every C." Taking a = is predicated of all = is predicated of every , and using the symbolical method used in the Middle Ages, then the first figure is simplified to: Or what amounts to the same thing: When the four syllogistic propositions, a, e, i, o are placed in the first figure, Aristotle comes up with
1160-466: Is principally this part of Aristotle's works that is about term logic . Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm. Lukasiewicz's approach was reinvigorated in the early 1970s by John Corcoran and Timothy Smiley – which informs modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009. The Prior Analytics represents
1218-648: Is regarded as one of the most important historians of logic. He was born in Lemberg in Austria-Hungary (now Lviv , Ukraine ; Polish : Lwów ) and was the only child of Paweł Łukasiewicz, a captain in the Austrian army, and Leopoldina, née Holtzer, the daughter of a civil servant. His family was Roman Catholic . He finished his gymnasium studies in philology and in 1897 went on to Lemberg University , where he studied philosophy and mathematics. He
SECTION 20
#17327662422141276-427: Is the basic component of the proposition. The original meaning of the horos (and also of the Latin terminus ) is "extreme" or "boundary". The two terms lie on the outside of the proposition, joined by the act of affirmation or denial. For early modern logicians like Arnauld (whose Port-Royal Logic was the best-known text of his day), it is a psychological entity like an "idea" or " concept ". Mill considers it
1334-524: The recursive stack , a last-in, first-out computer memory store proposed by several researchers including Turing , Bauer and Hamblin , and first implemented in 1957. In 1960, Łukasiewicz's notation concepts and stacks were used as the basis of the Burroughs B5000 computer designed by Robert S. Barton and his team at Burroughs Corporation in Pasadena, California . The concepts also led to
1392-603: The Analytics and more extensively in On Interpretation . Each proposition (statement that is a thought of the kind expressible by a declarative sentence) of a syllogism is a categorical sentence which has a subject and a predicate connected by a verb. The usual way of connecting the subject and predicate of a categorical sentence as Aristotle does in On Interpretation is by using a linking verb e.g. P
1450-615: The Polish Information Processing Society established the Jan Łukasiewicz Award, to be presented to the most innovative Polish IT companies. From 1999 to 2004, the Department of Computer Science building at UCD was called the Łukasiewicz Building, until all campus buildings were renamed after the disciplines they housed. His model of 3-valued logic allowed for formulating Kleene's ternary logic and
1508-677: The Tsarist government in the 19th century. In 1919, Łukasiewicz left the university to serve as Polish Minister of Religious Denominations and Public Education in Paderewski 's government until 1920. Łukasiewicz led the development of a Polish curriculum replacing the Russian, German and Austrian curricula that had been used in partitioned Poland. The Łukasiewicz curriculum emphasized the early acquisition of logical and mathematical concepts. In 1928, he married Regina Barwińska. He remained
1566-480: The reasoning process is in turn built from propositions: A proposition may be universal or particular, and it may be affirmative or negative. Traditionally, the four kinds of propositions are: This was called the fourfold scheme of propositions (see types of syllogism for an explanation of the letters A, I, E, and O in the traditional square). Aristotle's original square of opposition , however, does not lack existential import . A term (Greek ὅρος horos )
1624-427: The syllogism is that, of the four terms in the two premises, one must occur twice. Thus The subject of one premise, must be the predicate of the other, and so it is necessary to eliminate from the logic any terms which cannot function both as subject and predicate, namely singular terms. However, in a popular 17th-century version of the syllogism, Port-Royal Logic , singular terms were treated as universals: This
1682-422: The "judgment" as something distinct from a sentence, but this is not quite the same. As a further confusion the word "sentence" derives from the Latin, meaning an opinion or judgment , and so is equivalent to " proposition ". The logical quality of a proposition is whether it is affirmative (the predicate is affirmed of the subject) or negative (the predicate is denied of the subject). Thus every philosopher
1740-624: The Middle Ages, for mnemonic reasons, these six forms were called respectively: "Darapti", "Felapton", "Disamis", "Datisi", "Bocardo" and "Ferison". Term logic began to decline in Europe during the Renaissance , when logicians like Rodolphus Agricola Phrisius (1444–1485) and Ramus (1515–1572) began to promote place logics. The logical tradition called Port-Royal Logic , or sometimes "traditional logic", saw propositions as combinations of ideas rather than of terms, but otherwise followed many of
1798-525: The Prior Analytics, "... If one term belongs to all and another to none of the same thing, or if they both belong to all or none of it, I call such figure the third." Referring to universal terms, "... then when both P and R belongs to every S, it results of necessity that P will belong to some R." Simplifying: When the four syllogistic propositions, a, e, i, o are placed in the third figure, Aristotle develops six more valid forms of deduction: In
Differentia - Misplaced Pages Continue
1856-597: The Royal Irish Academy in Dublin hosted an exhibition on his life and work. Łukasiewicz's papers (post-1945) are held by the University of Manchester Library . A number of axiomatizations of classical propositional logic are due to Łukasiewicz. A particularly elegant axiomatization features a mere three axioms and is still invoked to the present day. He was a pioneer investigator of multi-valued logics ; his three-valued propositional calculus , introduced in 1917,
1914-461: The Standpoint of Modern Formal Logic , he mentions that the principle of his notation was to write the functors before the arguments to avoid brackets (i.e., parentheses) and that he had employed his notation in his logical papers since 1929. He then goes on to cite, as an example, a 1930 paper he wrote with Alfred Tarski on the sentential calculus . This notation is the root of the idea of
1972-518: The University of Lemberg. That year, he was appointed a lecturer at the University of Lemberg, where he was eventually appointed Extraordinary Professor by Emperor Franz Joseph I. He taught there until the First World War . In 1915, he was invited to lecture as a full professor at the University of Warsaw , which the German occupation authorities had reopened after it had been closed down by
2030-516: The concept of differentia when he conceived his method of diairesis . Aristotle was the first to use the term diaphora (διαφορά) in a systematic fashion; but he had no explicit theory about it, and his understanding of the term is controversial. Adiaphora - the negation of diaphora - is an important term in Hellenistic philosophy . However, only in Pyrrhonism does it appear to be
2088-521: The conventions of term logic. It remained influential, especially in England, until the 19th century. Leibniz created a distinctive logical calculus , but nearly all of his work on logic remained unpublished and unremarked until Louis Couturat went through the Leibniz Nachlass around 1900, publishing his pioneering studies in logic. 19th-century attempts to algebraize logic, such as
2146-425: The copula ("All/some... are/are not..."), Aristotle uses the expression, "... belongs to/does not belong to all/some..." or "... is said/is not said of all/some..." There are four different types of categorical sentences: universal affirmative (A), universal negative (E), particular affirmative (I) and particular negative (O). A method of symbolization that originated and was used in the Middle Ages greatly simplifies
2204-645: The design of the English Electric multi-programmed KDF9 computer system of 1963, which had two such hardware register stacks. A similar concept underlies the reverse Polish notation ( RPN , a postfix notation) of the Friden EC-130 calculator and its successors, many Hewlett-Packard calculators, the Lisp and Forth programming languages, and the PostScript page description language. In 2008
2262-518: The first figure has again come about)." The above statement can be simplified by using the symbolical method used in the Middle Ages: When the four syllogistic propositions, a, e, i, o are placed in the second figure, Aristotle comes up with the following valid forms of deduction for the second figure: In the Middle Ages, for mnemonic reasons they were called respectively "Camestres", "Cesare", "Festino" and "Baroco". Aristotle says in
2320-553: The first figure is axiomatic while the second and third require proof. The proof of the second and third figure always leads back to the first figure. This is what Robin Smith says in English that Aristotle said in Ancient Greek: "... If M belongs to every N but to no X, then neither will N belong to any X. For if M belongs to no X, neither does X belong to any M; but M belonged to every N; therefore, X will belong to no N (for
2378-506: The first formal study of logic, where logic is understood as the study of arguments. An argument is a series of true or false statements which lead to a true or false conclusion. In the Prior Analytics , Aristotle identifies valid and invalid forms of arguments called syllogisms. A syllogism is an argument that consists of at least three sentences: at least two premises and a conclusion. Although Aristotle does not call them " categorical sentences", tradition does; he deals with them briefly in
Differentia - Misplaced Pages Continue
2436-463: The following valid forms of deduction for the first figure: In the Middle Ages, for mnemonic reasons they were called "Barbara", "Celarent", "Darii" and "Ferio" respectively. The difference between the first figure and the other two figures is that the syllogism of the first figure is complete while that of the second and third is not. This is important in Aristotle's theory of the syllogism for
2494-399: The hands of Bertrand Russell and A. N. Whitehead , whose Principia Mathematica (1910–13) made use of a variant of Peano's predicate logic. Term logic also survived to some extent in traditional Roman Catholic education, especially in seminaries . Medieval Catholic theology , especially the writings of Thomas Aquinas , had a powerfully Aristotelean cast, and thus term logic became
2552-532: The notion of differentia. Scholastic logic Term logic revived in medieval times, first in Islamic logic by Alpharabius in the tenth century, and later in Christian Europe in the twelfth century with the advent of new logic , remaining dominant until the advent of predicate logic in the late nineteenth century. However, even if eclipsed by newer logical systems, term logic still plays
2610-477: The other hand, any mind-independent structuring on the part of the essence is denied (like in Thomism or Suárezianism ), then the partitioning of the essence into a generic and a differentiating part must be considered as merely "conceptua", whereas the actual realities corresponding to the differentia and to the genus evade as really the same. These assumptions therefore do not permit any ontological application of
2668-615: The premises are in the Third Figure. Symbolically, the Three Figures may be represented as follows: In Aristotelian syllogistic ( Prior Analytics , Bk I Caps 4-7), syllogisms are divided into three figures according to the position of the middle term in the two premises. The fourth figure, in which the middle term is the predicate in the major premise and the subject in the minor, was added by Aristotle's pupil Theophrastus and does not occur in Aristotle's work, although there
2726-568: The principle of non-contradiction and the law of excluded middle , offering one of the earliest systems of many-valued logic . Contemporary research on Aristotelian logic also builds on innovative works by Łukasiewicz, which applied methods from modern logic to the formalization of Aristotle 's syllogistic . The Łukasiewicz approach was reinvigorated in the early 1970s in a series of papers by John Corcoran and Timothy Smiley that inform modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009. Łukasiewicz
2784-399: The realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce “No quadrangle that is a square is a rectangle that is a rhombus” from “No square that is a quadrangle is a rhombus that is a rectangle” or from “No rhombus that is
2842-429: The realm of foundations, Boole reduced the four propositional forms of Aristotle's logic to formulas in the form of equations– by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic– another revolutionary idea –involved Boole's doctrine that Aristotle's rules of inference (the “perfect syllogisms”) must be supplemented by rules for equation solving. Third, in
2900-428: The scholastic nomenclature. In the scholastic theory it is a kind of essential predicate — a predicate that belongs to its subjects de re necessarily. It is distinguished against the species by expressing the (specific) essence of the object only partially and against the genus by expressing the determining rather than the determined part of the essence. In Ancient Greek philosophy , Plato implicitly employed
2958-606: The start of the Second World War , he worked at the Warsaw Underground University . After the German occupation authorities had closed the university, he earned a meager living in the Warsaw city archive. His friendship with Heinrich Scholz (German professor of mathematical logic) helped him, too, and it was Scholz who arranged for the Łukasiewicz family's passage to Germany in 1944 (Łukasiewicz
SECTION 50
#17327662422143016-435: The study of the Prior Analytics. Following this tradition then, let: Categorical sentences may then be abbreviated as follows: From the viewpoint of modern logic, only a few types of sentences can be represented in this way. The fundamental assumption behind the theory is that the formal model of propositions are composed of two logical symbols called terms – hence the name "two-term theory" or "term logic" – and that
3074-491: The work of Boole (1815–1864) and Venn (1834–1923), typically yielded systems highly influenced by the term-logic tradition. The first predicate logic was that of Frege 's landmark Begriffsschrift (1879), little read before 1950, in part because of its eccentric notation. Modern predicate logic as we know it began in the 1880s with the writings of Charles Sanders Peirce , who influenced Peano (1858–1932) and even more, Ernst Schröder (1841–1902). It reached fruition in
3132-576: Was a pupil of the philosopher Kazimierz Twardowski . In 1902, he received a Doctor of Philosophy degree under the patronage of Emperor Franz Joseph I of Austria , who gave him a special doctoral ring with diamonds. He spent three years as a private teacher, and in 1905, he received a scholarship to complete his philosophy studies at the University of Berlin and the University of Louvain in Belgium. Łukasiewicz continued studying for his habilitation qualification and in 1906 submitted his thesis to
3190-538: Was fearful of the Red Army advance). As it became increasingly clear that Germany would lose the war, Łukasiewicz and his wife tried to move to Switzerland , but were unable to get permission from the German authorities. They thus spent the last months of the war in Münster , Germany. After the end of the war, unwilling to return to a Soviet-controlled Poland, they moved first to Belgium, where Łukasiewicz taught logic at
3248-561: Was invented: I came upon the idea of a parenthesis-free notation in 1924. I used that notation for the first time in my article Łukasiewicz (1), p. 610, footnote. The reference cited by Łukasiewicz, i.e., Łukasiewicz (1), is apparently a lithographed report in Polish . The referring paper by Łukasiewicz was reviewed by Henry A. Pogorzelski in the Journal of Symbolic Logic in 1965. In Łukasiewicz's 1951 book, Aristotle's Syllogistic from
3306-584: Was published in English by Macmillan (1963, translated from Polish by Olgierd Wojtasiewicz). Jan Łukasiewicz died on 13 February 1956. He was buried in Mount Jerome Cemetery , in Dublin. At the urging of the Armenian community in Poland, his remains were repatriated to Poland 66 years later. He was reburied on 22 November 2022 in Warsaw's Old Powązki Cemetery . From October to December 2022,
3364-411: Was the first explicitly axiomatized non-classical logical calculus . He wrote on the philosophy of science , and his approach to the making of scientific theories was similar to the thinking of Karl Popper . Łukasiewicz invented the Polish notation (named after his nationality) for the logical connectives around 1920. A quotation from a paper by Jan Łukasiewicz in 1931 states how the notation
#213786