In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century. This entry is an introduction to the term logic needed to understand philosophy texts written before it was replaced as a formal logic system by predicate logic. Readers lacking a grasp of the basic terminology and ideas of term logic can have difficulty understanding such texts, because their authors typically assumed an acquaintance with term logic.
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 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. The Jan Lukasiewicz 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 fundamental assumption behind the theory is that propositions are composed of two terms – hence the name "two-term theory" or "term logic" – and that 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.
In the Stanford Encyclopedia of Philosophy article, "The Traditional Square of Opposition", Terence Parsons explains:
One central concern of the Aristotelian tradition in logic is the theory of the categorical syllogism. This is the theory of two-premised arguments in which the premises and conclusion share three terms among them, with each proposition containing two of them. It is distinctive of this enterprise that everybody agrees on which syllogisms are valid. The theory of the syllogism partly constrains the interpretation of the forms. For example, it determines that the A form has existential import, at least if the I form does. For one of the valid patterns (Darapti) is:
- Every C is B
- Every C is A
- So, some A is B
This is invalid if the A form lacks existential import, and valid if it has existential import. It is held to be valid, and so we know how the A form is to be interpreted. One then naturally asks about the O form; what do the syllogisms tell us about it? The answer is that they tell us nothing. This is because Aristotle did not discuss weakened forms of syllogisms, in which one concludes a particular proposition when one could already conclude the corresponding universal. For example, he does not mention the form:
- No C is B
- Every A is C
- So, some A is not B
If people had thoughtfully taken sides for or against the validity of this form, that would clearly be relevant to the understanding of the O form. But the weakened forms were typically ignored...
One other piece of subject-matter bears on the interpretation of the O form. People were interested in Aristotle's discussion of "infinite" negation, which is the use of negation to form a term from a term instead of a proposition from a proposition. In modern English we use "non" for this; we make "non-horse," which is true for exactly those things that are not horses. In medieval Latin "non" and "not" are the same word, and so the distinction required special discussion. It became common to use infinite negation, and logicians pondered its logic. Some writers in the twelfth century and thirteenth centuries adopted a principle called "conversion by contraposition". It states that
- 'Every S is P ' is equivalent to 'Every non-P is non-S '
- 'Some S is not P ' is equivalent to 'Some non-P is not non-S '
Unfortunately, this principle (which is not endorsed by Aristotle) conflicts with the idea that there may be empty or universal terms. For in the universal case it leads directly from the truth:
- Every man is a being
to the falsehood:
- Every non-being is a non-man
(which is false because the universal affirmative has existential import, and there are no non-beings). And in the particular case it leads from the truth (remember that the O form has no existential import):
- A chimera is not a man
To the falsehood:
- A non-man is not a non-chimera
These are [Jean] Buridan's examples, used in the fourteenth century to show the invalidity of contraposition. Unfortunately, by Buridan's time the principle of contraposition had been advocated by a number of authors. The doctrine is already present in several twelfth century tracts, and it is endorsed in the thirteenth century by Peter of Spain, whose work was republished for centuries, by William Sherwood, and by Roger Bacon. By the fourteenth century, problems associated with contraposition seem to be well-known, and authors generally cite the principle and note that it is not valid, but that it becomes valid with an additional assumption of existence of things falling under the subject term. For example, Paul of Venice in his eclectic and widely published Logica Parva from the end of the fourteenth century gives the traditional square with simple conversion but rejects conversion by contraposition, essentially for Buridan's reason.— Terence Parsons, The Stanford Encyclopedia of Philosophy
A term (Greek horos) 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 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 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 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 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 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 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 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.
This 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 is an animal."
The Aristotelian logical system had a formidable influence on the late-philosophy of the French psychoanalyst Jacques Lacan. In the early 1970s, Lacan reworked Aristotle's term logic by way of Frege and Jacques Brunschwig to produce his four formulae of sexuation. While these formulae retain the formal arrangement of the square of opposition, they seek to undermine the universals of both qualities by the 'existence without essence' of Lacan's particular negative proposition.
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 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 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 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 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:
The Latin phrases a priori (lit. "from the earlier") and a posteriori (lit. "from the later") are philosophical terms popularized by Immanuel Kant's Critique of Pure Reason (first published in 1781, second edition in 1787), one of the most influential works in the history of philosophy. However, in their Latin forms they appear in Latin translations of Euclid's Elements, of about 300 BC, a work widely considered during the early European modern period as the model for precise thinking.
These terms are used with respect to reasoning (epistemology) to distinguish "necessary conclusions from first premises" (i.e., what must come before sense observation) from "conclusions based on sense observation" which must follow it. Thus, the two kinds of knowledge, justification, or argument, may be glossed:
A priori knowledge or justification is independent of experience, as with mathematics (3 + 2 = 5), tautologies ("All bachelors are unmarried"), and deduction from pure reason (e.g., ontological proofs).
A posteriori knowledge or justification depends on experience or empirical evidence, as with most aspects of science and personal knowledge.There are many points of view on these two types of knowledge, and their relationship gives rise to one of the oldest problems in modern philosophy.
The terms a priori and a posteriori are primarily used as adjectives to modify the noun "knowledge" (for example, "a priori knowledge"). However, "a priori" is sometimes used to modify other nouns, such as "truth". Philosophers also may use "apriority" and "aprioricity" as nouns to refer (approximately) to the quality of being "a priori".Although definitions and use of the terms have varied in the history of philosophy, they have consistently labeled two separate epistemological notions. See also the related distinctions: deductive/inductive, analytic/synthetic, necessary/contingent.Apodicticity
"Apodictic" or "apodeictic" (Ancient Greek: ἀποδεικτικός, "capable of demonstration") is an adjectival expression from Aristotelean logic that refers to propositions that are demonstrably, necessarily or self-evidently the case. Apodicticity or apodixis is the corresponding abstract noun, referring to logical certainty.
Apodictic propositions contrast with assertoric propositions, which merely assert that something is (or is not) true, and with problematic propositions, which assert only the possibility of something being true. Apodictic judgments are clearly provable or logically certain. For instance, "Two plus two equals four" is apodictic. "Chicago is larger than Omaha" is assertoric. "A corporation could be wealthier than a country" is problematic. In Aristotelian logic, "apodictic" is opposed to "dialectic," as scientific proof is opposed to philosophical reasoning. Kant contrasted "apodictic" with "problematic" and "assertoric" in the Critique of Pure Reason, on page A70/B95.Aristotelianism
Aristotelianism ( ARR-i-stə-TEE-lee-ə-niz-əm) is a tradition of philosophy that takes its defining inspiration from the work of Aristotle. This school of thought, in the modern sense of philosophy, covers existence, ethics, mind and related subjects. In Aristotle's time, philosophy included natural philosophy, which preceded the advent of modern science during the Scientific Revolution. The works of Aristotle were initially defended by the members of the Peripatetic school and later on by the Neoplatonists, who produced many commentaries on Aristotle's writings. In the Islamic Golden Age, Avicenna and Averroes translated the works of Aristotle into Arabic and under them, along with philosophers such as Al-Kindi and Al-Farabi, Aristotelianism became a major part of early Islamic philosophy.
Moses Maimonides adopted Aristotelianism from the Islamic scholars and based his famous Guide for the Perplexed on it and that became the basis of Jewish scholastic philosophy. Although some of Aristotle's logical works were known to western Europe, it was not until the Latin translations of the 12th century that the works of Aristotle and his Arabic commentators became widely available. Scholars such as Albertus Magnus and Thomas Aquinas interpreted and systematized Aristotle's works in accordance with Christian theology.
After retreating under criticism from modern natural philosophers, the distinctively Aristotelian idea of teleology was transmitted through Wolff and Kant to Hegel, who applied it to history as a totality. Although this project was criticized by Trendelenburg and Brentano as non-Aristotelian, Hegel's influence is now often said to be responsible for an important Aristotelian influence upon Marx.
Recent Aristotelian ethical and "practical" philosophy, such as that of Gadamer and McDowell, is often premissed upon a rejection of Aristotelianism's traditional metaphysical or theoretical philosophy. From this viewpoint, the early modern tradition of political republicanism, which views the res publica, public sphere or state as constituted by its citizens' virtuous activity, can appear thoroughly Aristotelian.
The most famous contemporary Aristotelian philosopher is Alasdair MacIntyre. Especially famous for helping to revive virtue ethics in his book After Virtue, MacIntyre revises Aristotelianism with the argument that the highest temporal goods, which are internal to human beings, are actualized through participation in social practices. He juxtaposes Aristotelianism with the managerial institutions of capitalism and its state, and with rival traditions — including the philosophies of Hume and Nietzsche — that reject Aristotle's idea of essentially human goods and virtues and instead legitimate capitalism. Therefore, on MacIntyre's account, Aristotelianism is not identical with Western philosophy as a whole; rather, it is "the best theory so far, [including] the best theory so far about what makes a particular theory the best one." Politically and socially, it has been characterized as a newly "revolutionary Aristotelianism". This may be contrasted with the more conventional, apolitical and effectively conservative uses of Aristotle by, for example, Gadamer and McDowell. Other important contemporary Aristotelian theorists include Fred D. Miller, Jr. in politics and Rosalind Hursthouse in ethics.Assertoric
An assertoric proposition in Aristotelian logic merely asserts that something is (or is not) the case, in contrast to problematic propositions which assert the possibility of something being true, or apodeictic propositions which assert things which are necessarily or self-evidently true or false. For instance, "Chicago is larger than Omaha" is assertoric. "A corporation could be wealthier than a country" is problematic. "Two plus two equals four" is apodeictic.Dictum de omni et nullo
In Aristotelian logic, dictum de omni et nullo (Latin: "the maxim of all and none") is the principle that whatever is affirmed or denied of a whole kind K may be affirmed or denied (respectively) of any subkind of K. This principle is fundamental to syllogistic logic in the sense that all valid syllogistic argument forms are reducible to applications of the two constituent principles dictum de omni and dictum de nullo.Enthymeme
An enthymeme (Greek: ἐνθύμημα, enthumēma) is a rhetorical syllogism (a three-part deductive argument) used in oratorical practice. Originally theorized by Aristotle, there are four types of enthymeme, at least two of which are described in Aristotle's work.Aristotle referred to the enthymeme as "the body of proof", "the strongest of rhetorical proofs...a kind of syllogism" (Rhetoric I.I.3,11). He considered it to be one of two kinds of proof, the other of which was the paradeigma. Maxims, Aristotle thought, were a derivative of enthymemes. (Rhetoric II.XX.1)Middle term
In logic, a middle term is a term that appears (as a subject or predicate of a categorical proposition) in both premises but not in the conclusion of a categorical syllogism. Example:
Major premise: All men are mortal.
Minor premise: Socrates is a man.
Conclusion: Socrates is mortal.The middle term is bolded above.Monadic predicate calculus
In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symbols. All atomic formulas are thus of the form , where is a relation symbol and is a variable.
Monadic predicate calculus can be contrasted with polyadic predicate calculus, which allows relation symbols that take two or more arguments.Polysyllogism
A polysyllogism (also called multi-premise syllogism, sorites, climax, or gradatio) is a string of any number of propositions forming together a sequence of syllogisms such that the conclusion of each syllogism, together with the next proposition, is a premise for the next, and so on. Each constituent syllogism is called a prosyllogism except the very last, because the conclusion of the last syllogism is not a premise for another syllogism.Port-Royal Logic
Port-Royal Logic, or Logique de Port-Royal, is the common name of La logique, ou l'art de penser, an important textbook on logic first published anonymously in 1662 by Antoine Arnauld and Pierre Nicole, two prominent members of the Jansenist movement, centered on Port-Royal. Blaise Pascal likely contributed considerable portions of the text. Its linguistic companion piece is the Port Royal Grammar (1660) by Arnauld and Lancelot.
Written in French, it became quite popular and was in use up to the twentieth century, introducing the reader to logic, and exhibiting strong Cartesian elements in its metaphysics and epistemology (Arnauld having been one of the main philosophers whose objections were published, with replies, in Descartes' Meditations on First Philosophy). The Port-Royal Logic is sometimes cited as a paradigmatic example of traditional term logic.
The philosopher Louis Marin particularly studied it in the 20th century (La Critique du discours, Éditions de Minuit, 1975), while Michel Foucault considered it, in The Order of Things, one of the bases of the classical épistémè.
Among the contributions of the Port-Royal Logic is the introduction of a distinction between comprehension and extension, which would later become a more refined distinction between intension and extension. Roughly speaking: a definition with more qualifications or features (the intension) denotes a class with fewer members (the extension), and vice versa.
The main idea traces back through the scholastic philosophers to Aristotle's ideas about genus and species, and is fundamental in the philosophy of Leibniz. More recently, it has been related to mathematical lattice theory in Formal Concept Analysis, and independently formalized similarly by Yu. Schreider's group in Moscow, Jon Barwise & Jerry Seligman in Information Flow, and others.Premise
A premise or premiss is a statement that an argument claims will induce or justify a conclusion. In other words, a premise is an assumption that something is true.
In logic, an argument requires a set of (at least) two declarative sentences (or "propositions") known as the premises or premisses along with another declarative sentence (or "proposition") known as the conclusion. This structure of two premises and one conclusion forms the basic argumentative structure. More complex arguments can use a series of rules to connect several premises to one conclusion, or to derive a number of conclusions from the original premises which then act as premises for additional conclusions. An example of this is the use of the rules of inference found within symbolic logic.
Aristotle held that any logical argument could be reduced to two premises and a conclusion. Premises are sometimes left unstated in which case they are called missing premises, for example:
Socrates is mortal because all men are mortal.It is evident that a tacitly understood claim is that Socrates is a man. The fully expressed reasoning is thus:
Because all men are mortal and Socrates is a man, Socrates is mortal.In this example, the independent clauses preceding the comma (namely, "all men are mortal" and "Socrates is a man") are the premises, while "Socrates is mortal" is the conclusion.
The proof of a conclusion depends on both the truth of the premises and the validity of the argument. Also, additional information is required over and above the meaning of the premise to determine if the full meaning of the conclusion coincides with what is.For Euclid, premises constitute two of the three propositions in a syllogism, with the other being the conclusion. These propositions contain three terms: subject and predicate of the conclusion, and the middle term. The subject of the conclusion is called the minor term while the predicate is the major term. The premise that contains the middle term and major term is called the major premise while the premise that contains the middle term and minor term is called the minor premise.A premise can also be an indicator word if statements have been combined into a logical argument and such word functions to mark the role of one or more of the statements. It indicates that the statement it is attached to is a premise.Relative term
A relative term is a term that makes two or more distinct references to objects (which may be the same object, for example in "The Morning Star is the Evening Star"). A relative term is typically expressed in ordinary language by means of a phrase with explicit or implicit blanks. Examples:
__ loves __
__ is the same object as __
__ is giver of __ to __.The word is is a relative term when it expresses identity.
The colloquial meaning for a relative term is that it is different for different people or situations. An example: someone who is 5 feet tall might think someone who is 5 feet six inches tall is tall, but someone who is 6 feet would think that that person is short. An atom is big compared to a quark, but it is very small when compared to a body cell. Fast food may be healthier than preserved food, but unhealthy compared to organic produce.Square of opposition
The square of opposition is a diagram representing the relations between the four basic categorical propositions.
The origin of the square can be traced back to Aristotle making the distinction between two oppositions: contradiction and contrariety.
But Aristotle did not draw any diagram. This was done several centuries later by Apuleius and Boethius.Statistical syllogism
A statistical syllogism (or proportional syllogism or direct inference) is a non-deductive syllogism. It argues, using inductive reasoning, from a generalization true for the most part to a particular case.Sum of Logic
The Summa Logicae ("Sum of Logic") is a textbook on logic by William of Ockham. It was written around 1323.
Systematically, it resembles other works of medieval logic, organised under the basic headings of the Aristotelian Predicables, Categories, terms, propositions, and syllogisms. These headings, though often given in a different order, represent the basic arrangement of scholastic works on logic.
This work is important in that it contains the main account of Ockham's nominalism, a position related to the problem of universals.Syncategorematic term
In scholastic logic, a syncategorematic term (or syncategorema) is a word that cannot serve as the subject or the predicate of a proposition, and thus cannot stand for any of Aristotle's categories, but can be used with other terms to form a proposition. Words such as 'all', 'and', 'if' are examples of such terms.Term
Term may refer to:
Terminology, or term, a noun or compound word used in a specific context, in particular:
Technical term, part of the specialized vocabulary of a particular field, specifically:
Scientific terminology, terms used by scientistsTerm (logic)
In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact. In particular, terms appear as components of a formula.
A first-order term is recursively constructed from constant symbols, variables and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, is a term built from the constant 1, the variable x, and the binary function symbols and ; it is part of the atomic formula which evaluates to true for each real-numbered value of x.
Besides in logic, terms play important roles in universal algebra, and rewriting systems.
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