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Inlogic andformal semantics,term logic, also known astraditional logic,syllogistic logic orAristotelian logic, is a loose name for an approach toformal logic that began withAristotle and was developed further inancient history mostly by his followers, thePeripatetics. It was revived after the third century CE byPorphyry'sIsagoge.
Term logic revived inmedieval times, first inIslamic logic byAlpharabius in the tenth century, and later in Christian Europe in the twelfth century with the advent ofnew logic, remaining dominant until the advent ofpredicate logic in the late nineteenth century.
However, even if eclipsed by newer logical systems, term logic still plays 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 theOrganon. Two of these texts in particular, namely thePrior Analytics andOn Interpretation, contain the heart of Aristotle's treatment of judgements and formalinference, and it is principally this part of Aristotle's works that is about termlogic. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment byJan Lukasiewicz of a revolutionary paradigm.[1] Lukasiewicz's approach was reinvigorated in the early 1970s byJohn Corcoran andTimothy Smiley – which informs modern translations ofPrior Analytics by Robin Smith in 1989 andGisela Striker in 2009.[2]
ThePrior Analytics represents 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.[3] In thePrior Analytics, Aristotle identifies valid and invalid forms of arguments calledsyllogisms. A syllogism is an argument that consists of at least three sentences: at least twopremises and a conclusion. Although Aristotle does not call them "categorical sentences", tradition does; he deals with them briefly in theAnalytics and more extensively inOn Interpretation.[4] Each proposition (statement that is a thought of the kind expressible by a declarative sentence)[5] 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 inOn Interpretation is by using a linking verb e.g. P 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.[6] In his formulation of syllogistic propositions, instead of 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..."[7] 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 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.[8]
The fundamental assumption behind the theory is that theformal model ofpropositions are composed of twological symbols called terms – hence the name "two-term theory" or "term logic" – and that thereasoning 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 thefourfold scheme of propositions (seetypes of syllogism for an explanation of the letters A, I, E, and O in the traditional square). Aristotle'soriginalsquare of opposition, however, does not lackexistential import.
The kinds of propositions used for each of the premises and the conclusion combine to create the "mood" of the syllogism. For instance, if the major premise, minor premise, and conclusion are all universal affirmative statements (i.e., A-type), then the syllogism has the mood AAA.
Aterm (Greek ὅροςhoros) is the basic component of the proposition. The original meaning of thehoros (and also of the Latinterminus) is "extreme" or "boundary". The two terms lie on the outside of the proposition, joined by the act of affirmation or denial.
For early modernlogicians like Arnauld (whosePort-Royal Logic was the best-known text of his day), it is a psychological entity like an "idea" or "concept".Mill considers it 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". Aproposition cannot be built from real things or ideas, but it is not just meaningless words either.
The "major term" is thepredicate of the syllogism's conclusion; the "minor term" is thesubject of the syllogism's conclusion.
For Aristotle, the distinction between singular[citation needed] and universal is a fundamentalmetaphysical one, and not merelygrammatical. A singular term for Aristotle isprimary substance, which can only bepredicated of itself: (this) "Callias" or (this) "Socrates" are not predicable of any other thing, thus one does not sayevery Socrates one saysevery human (De Int. 7;Meta. D9, 1018a4). It may feature as a grammatical predicate, as in the sentence "the person coming this way is Callias". But it is still alogical subject.
He contrasts universal (katholou)[9] secondary substance, genera, with primary substance, particular (kath' hekaston)[9][10] specimens. The formal nature ofuniversals, 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.[11]
In term logic, a "proposition" is simply aform of language: a particular kind ofsentence, in which the subject andpredicate are combined, so as to assert something true or false. It is not a thought, nor anabstract entity. The word"propositio" is from the Latin, meaning the first premise of asyllogism. Aristotle uses the word premise (protasis) as a sentence affirming or denying one thing or another (Posterior Analytics 1. 1 24a 16), so apremise is also a form of words.
However, as in modern philosophical logic, it means that which is asserted by the sentence. Writers beforeFrege andRussell, such asBradley, sometimes spoke of 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 anopinion orjudgment, and so is equivalent to "proposition".
Thelogical 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). Thusevery philosopher is mortal is affirmative, since the mortality of philosophers is affirmed universally, whereasno philosopher is mortal is negative by denying such mortality in particular.
Thequantity 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 whereexistential import is assumed,quantification implies the existence of at least one subject, unless disclaimed.
A syllogism comprises a conclusion derived from two premises. The essential feature of thesyllogism is that, of the three terms in the two premises, one must occur twice: this is called the "middle term", and it does not appear in the conclusion. The premise that contains the middle term and the major term (i.e., the predicate of the conclusion) is called the "major premise". The premise that contains the middle term and the minor term (i.e., the subject of the conclusion) is called the "minor premise". For instance:
Depending on the roles of the middle term in each of the premises, Aristotle divides the syllogism into three kinds: syllogism in the first, second, and third figure.[12] 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, the premises are in the Third Figure.[13] 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 pupilTheophrastus and does not occur in Aristotle's work, although there is evidence that Aristotle knew of fourth-figure syllogisms.[14]
Symbolically, Aristotle's Three Figures may be represented as follows:[15]
| First figure | Second figure | Third figure | |
|---|---|---|---|
| Predicate — Subject | Predicate — Subject | Predicate — Subject | |
| Major premise | A ------------ B | B ------------ A | A ------------ B |
| Minor premise | B ------------ C | B ------------ C | C ------------ B |
| Conclusion | A ********** C | A ********** C | A ********** C |
In thePrior Analytics, Aristotle says of the first figure: "... For if A is predicated of every B and B of every C, it is necessary for A to be predicated of every C."[16]
Takinga =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:[17]
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 the following valid forms of deduction for the first figure:
In the Middle Ages, formnemonic reasons (based on the pattern of vowels used to represent each of the propositions) they were called "Barbara", "Celarent", "Darii" and "Ferio" respectively.[18]
The difference between the first figure and the other two figures is that the syllogism of the first figure is "perfect" (i.e., obviously valid without any further steps or premises) while that of the second and third is not. This is important in Aristotle's theory of the syllogism for 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.[19]
Aristotle says of the second figure: "... 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 the first figure has again come about)."[20]
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".[21]
Aristotle says of the third figure: "... 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."[22]
Simplifying:
When the four syllogistic propositions, a, e, i, o are placed in the third figure, Aristotle develops four more valid forms of deduction:
In the Middle Ages, for mnemonic reasons, these four forms were called respectively: "Disamis", "Datisi", "Bocardo" and "Ferison".[23]
In the following table, a syllogism with "conditional" validity is one whose validity depends on existential import (i.e., it is valid only if at least one instance of the minor term or middle term actually exists). An unconditionally valid syllogism does not depend on existential import. All valid syllogisms must contain at least one universal premise (i.e., an A-type or E-type proposition), but need not contain a particular premise (i.e., an I-type or O-type proposition).
| Figure | Major premise | Minor premise | Conclusion | Mood–Figure | Mnemonic name | Validity |
|---|---|---|---|---|---|---|
| First Figure | AaB | BaC | AaC | AAA-1 | Barbara | Unconditional |
| AaB | BaC | AiC | AAI-1 | Barbari | Conditional (minor) | |
| AaB | BiC | AiC | AII-1 | Darii | Unconditional | |
| AeB | BaC | AeC | EAE-1 | Celarent | Unconditional | |
| AeB | BaC | AoC | EAO-1 | Celaront | Conditional (minor) | |
| AeB | BiC | AoC | EIO-1 | Ferio | Unconditional | |
| Second Figure | MaN | MeX | NeX | AEE-2 | Camestres | Unconditional |
| MaN | MeX | NoX | AEO-2 | Camestros | Conditional (middle) | |
| MaN | MoX | NoX | AOO-2 | Baroco | Unconditional | |
| MeN | MaX | NeX | EAE-2 | Cesare | Unconditional | |
| MeN | MaX | NoX | EAO-2 | Cesaro | Conditional (middle) | |
| MeN | MiX | NoX | EIO-2 | Festino | Unconditional | |
| Third Figure | PaS | RaS | PiR | AAI-3 | Darapti | Conditional (middle) |
| PaS | RiS | PiR | AII-3 | Datisi | Unconditional | |
| PeS | RaS | PoR | EAO-3 | Felapton | Conditional (middle) | |
| PeS | RiS | PoR | EIO-3 | Ferison | Unconditional | |
| PiS | RaS | PiR | IAI-3 | Disamis | Unconditional | |
| PoS | RaS | PoR | OAO-3 | Bocardo | Unconditional |
Term logic began to decline inEurope during theRenaissance, when logicians likeRodolphus Agricola Phrisius (1444–1485) andRamus (1515–1572) began to promote place logics. The logical tradition calledPort-Royal Logic, or sometimes "traditional logic", saw propositions as combinations of ideas rather than of terms, but otherwise followed many of the conventions of term logic. It remained influential, especially in England, until the 19th century.Leibniz created a distinctivelogical calculus, but nearly all of his work onlogic remained unpublished and unremarked untilLouis Couturat went through the LeibnizNachlass around 1900, publishing his pioneering studies in logic.
19th-century attempts to algebraize logic, such as the work ofBoole (1815–1864) andVenn (1834–1923), typically yielded systems highly influenced by the term-logic tradition. The firstpredicate logic was that ofFrege's landmarkBegriffsschrift (1879), little read before 1950, in part because of its eccentric notation. Modernpredicate logic as we know it began in the 1880s with the writings ofCharles Sanders Peirce, who influencedPeano (1858–1932) and even more,Ernst Schröder (1841–1902). It reached fruition in the hands ofBertrand Russell andA. N. Whitehead, whosePrincipia Mathematica (1910–13) made use of a variant of Peano's predicate logic.
Term logic also survived to some extent in traditionalRoman Catholic education, especially inseminaries. Medieval Catholictheology, especially the writings ofThomas Aquinas, had a powerfullyAristotelean cast, and thus term logic became a part of Catholic theological reasoning. For example, Joyce'sPrinciples of Logic (1908; 3rd edition 1949), written for use in Catholic seminaries, made no mention ofFrege or ofBertrand Russell.[24][page needed][need quotation to verify]
Some philosophers have complained that predicate logic:
Even academic philosophers entirely in the mainstream, such asGareth Evans, have written as follows:
George Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logicJohn Corcoran in an accessible introduction toLaws of Thought[25] Corcoran also wrote a point-by-point comparison ofPrior Analytics andLaws of Thought.[26] 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 whatAristotle said; Boole's ‘disagreements’, if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced the four propositional forms ofAristotle'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 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 a rectangle is a square that is a quadrangle”.
At the foundation of Aristotle's syllogistic is a theory of a specific class of arguments: arguments having as premises exactly two categorical sentences with one term in common.
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