Term logic

In philosophy, term logic, also known as traditional logic, is a loose name for the way of doing 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 predicate logic came to be seen as the only formal logic of interest. 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 system

Main article: Organon

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 work that term logic is about.

The basics

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:

• The term is a part of speech representing something, but which is not true or false in its own right, such as "man" or "mortal".
• The proposition consists of two terms, in which one term (the "predicate") is "affirmed" or "denied" of the other (the "subject"), and which is capable of truth or falsity.
• The syllogism is an inference in which one proposition (the "conclusion") follows of necessity from two others (the "premises").

A proposition may be universal or particular, and it may be affirmative or negative. Thus there are just four kinds of propositions:

• A-type: Universal and affirmative or ("All men are mortal")
• I-type: Particular and affirmative ("Some men are philosophers")
• E-type: Universal and negative ("No men are immortal")
• O-type: Particular and negative ("Some men are not philosophers").

This was called the fourfold scheme of propositions. (The origin of the letters A, I, E, and O are explained below in the section on syllogistic maxims.) Aristotle summarised the logical relationship between four types of propositions with his square of oppositions. The syllogistic is a formal theory explaining which combinations of true premises yield true conclusions.

The term

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 Aristotle, a term is simply a "thing", a part of a proposition. 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. None of these interpretations are quite satisfactory. In asserting that something is a unicorn, we are not asserting anything of anything. Nor does "all Greeks are men" say that the ideas of Greeks are ideas 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. This is a problem about the meaning of language that is still not entirely resolved. (See the book by Prior below for an excellent discussion of the problem).

The proposition

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 or anything. 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 of another (Posterior Analytics 1. 1 24a 16), so a premise is also a form of words.

However, in modern philosophical logic, it now means what is asserted as the result of uttering a sentence, and is regarded as something peculiar mental or intentional. 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 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 man is a mortal" is affirmative, since "mortal" is affirmed of "man". "No men are immortals" is negative, since "immortal" is denied of "man".

The quantity of a proposition is whether it is universal (the predicate is affirmed or denied of "the whole" of the subject) or particular (the predicate is affirmed or denied of only "part of" the subject).

Singular terms

For Aristotle, the distinction between singular and universal is a fundamental metaphysical one, and not merely grammatical. A singular term for Aristotle is that which is of such a nature as to be predicated of only one thing, thus "Callias". (De Int. 7). It is not predicable of more than one thing: "Socrates is not predicable of more than one subject, and therefore we do not say every Socrates as we say every man". (Metaphysics D 9, 1018 a4). 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 it with "universal" (katholou - "of a whole"). Universal terms are the basic materials of Aristotle's logic, propositions containing singular terms do not form part of it at all. They are mentioned briefly in the De Interpretatione. Afterwards, in the chapters of the Prior Analytics where Aristotle methodically sets out his theory of the syllogism, they are entirely ignored.

The reason for this omission is clear. The essential feature of term logic is that, of the four terms in the two premises, one must occur twice. Thus

All Greeks are men

All men are mortal.

What is subject in one premise, must be predicate in the other, and so it is necessary to eliminate from the logic any terms which cannot function both as subject and predicate. Singular terms do not function this way, so they are omitted from Aristotle's syllogistic.

In later versions of the syllogistic, singular terms were treated as universals. See for example (where it is clearly stated as received opinion) Part 2, chapter 3, of the Port-Royal Logic. Thus

All men are mortals

All Socrates are men

All Socrates are mortals

This is clearly awkward, and is a weakness exploited by Frege in his devastating attack on the system (from which, ultimately, it never recovered). See concept and object.

The famous syllogism "Socrates is a man ...", is frequently quoted as though from Aristotle. See for example Kapp, Greek Foundations of Traditional Logic, New York 1942, p.17, Copleston A History of Philosophy Vol. I., p. 277, Russell, A History of Western Philosophy London 1946 p. 218. In fact it is nowhere in the Organon. It is first mentioned by Sextus Empiricus in his Hyp. Pyrrh. ii. 164.

Decline of term logic

Term logic began to decline in Europe during the Renaissance, when logicians like Agricola and Ramus began to promote place logics. The logical tradition called Port-Royal Logic, or sometimes "traditional logic", claimed that a proposition was a combination of ideas rather than terms, but otherwise followed many of the conventions of term logic and was influential, especially in England, until the 19th century. Spinoza's "way of geometry" was far more influenced by Euclid's Elements than by Aristotelian concepts. Leibniz created a distinctive logical calculus, but nearly all of his work on logic was unpublished and unremarked until Louis Couturat went through the Leibniz Nachlass around 1900, and published many Leibniz manuscripts and a pioneering study of Leibniz's logic.

19th century attempts to algebraize logic, such as the work of Boole and Venn, typically yielded systems highly influenced by the term logic tradition. The first predicate logic was that of Frege's landmark Begriffsschrift, 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 Peirce, who influenced Peano and even more, Ernst Schroder. It reached full fruition in the hands of Bertrand Russell and A. N. Whitehead, whose Principia Mathematica (1910-13) made splendid use of a variant of Peano's predicate logic.

Predicate logic was designed as a form of mathematics, and as such is capable of all sorts of mathematical reasoning beyond the powers of term logic. Predicate logic is also capable of many commonsense inferences that elude term logic. Term logic cannot, for example, explain the inference from "every car is a vehicle", to "every owner of a car is an owner of a vehicle." Syllogistic reasoning cannot explain inferences involving multiple generality. Relations and identity must be treated as subject-predicate relations, which make the identity statements of mathematics difficult to handle. Term logic contains no analog of the singular term and singular proposition, both essential features of predicate logic.

With the ascension of predicate logic, term and syllogistic logic gradually fell into disuse except among students of ancient and medieval philosophy. Since the development of predicate logic, introductory texts on logic have ignored or disparaged term logic, except perhaps as a source of examples for beginning students. A notable exception to this generalization is the four editions of Quine's Methods of Logic (the last edition, dated 1982, is still in print), which discussed term logic (which Quine called "Boolean term schemata") and syllogisms at some length. Quine's writings on logic contain much that is in the spirit of term logic in that they frequently invoke grammatical concepts and examples taken from natural language, even employing bits of scholastic terminology such as "syncategorematic."

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 (1949), written for use in Catholic seminaries, made no mention of Frege or Bertrand Russell. On Aristotle, term logic, and Roman Catholicism, see Copleston's A History of Philosophy.

A revival

Some philosophers have complained that predicate logic:

• Is unnatural in a sense, in that its syntax does not follow the syntax of the sentences that figure in our everyday reasoning. It is, as Quine acknowledged, "Procrustean," employing an artificial language of function and argument, quantifier and bound variable.
• Suffers from embarrassing theoretical problems, probably the most serious being empty names and identity statements.

Even academic philosophers entirely in the mainstream, such as Gareth Evans, have written as follows:

"I come to semantic investigations with a preference for homophonic theories; theories which try to take serious account of the syntactic and semantic devices which actually exist in the language ...I would prefer [such] a theory ... over a theory which is only able to deal with [sentences of the form "all A's are B's"] by "discovering" hidden logical constants ... The objection would not be that such [Fregean] truth conditions are not correct, but that, in a sense which we would all dearly love to have more exactly explained, the syntactic shape of the sentence is treated as so much misleading surface structure" (Evans 1977)

The writings of Fred Sommers (e.g., Sommers 1970) and his students have modified term logic so that it can address these criticisms of predicate logic and overcome the well-known weaknesses of term logic. The result is the "term functor logic" of Sommers (1982), and Sommers and Englebretsen (2000). This logic has a very Boolean appearance, in that '+' and '-' are the sole operational signs and all statements are equations. It has sufficient expressive power to handle relational terms generally, and to capture the validity of arguments that elude syllogistic reasoning. Term functor logic has similarities to Quine's predicate functor logic, an algebraic formalism Quine devised to do first-order logic without quantifiers.

In a less formal vein, term logic has acquired a following among those advocating a return to educational methods grounded in the medieval Trivium: grammar, logic, and rhetoric. Advocates of the Trivium include the Paideia Proposal by philosopher Mortimer J. Adler, and some homeschoolers. The Trivium views logic not as a form of mathematics, but as part of a classical education in language. Those advocating this line see predicate logic as excessively nominalistic, as primarily concerned with the manipulation of symbols (syntax) and not with the whys and essences of things (ontology and metaphysics).

In a similar vein the extremely complex symbolic logic that developed in the 20th century and its formal strictness can sometimes mislead students by giving them the impression that there is something "truer" about it because its machinery is complex, however, it in fact lacks truth and processes lies and falsehoods equally. It may turn out in fact to be another form of systematised modernism.

A variant of term logic, probabilistic term logic, which assigns a probability value and a confidence value to the truth of both terms and propositions, is gaining popularity in artificial intelligence systems. Variants include both Pei Wang's "Non-Axiomatic Reasoning System" (NARS) and Ben Goertzel's "OpenCog" system.

See also

• Aristotle
• Contraposition
• Conversion (logic)
• De Interpretatione
• Obversion
• Organon

• Port-Royal Logic
• Prior Analytics
• Propositional calculus
• Syllogism
• Transposition (logic)

References

• Bocheński, I. M., 1951. Ancient Formal Logic. North-Holland.
• Louis Couturat, 1961 (1901). La Logique de Leibniz. Hildesheim: Georg Olms Verlagsbuchhandlung.
• Gareth Evans, 1977, "Pronouns, Quantifiers and Relative Clauses," Canadian Journal of Philosophy.
• Peter Geach, 1976. Reason and Argument. University of California Press.
• Hammond and Scullard, 1992. The Oxford Classical Dictionary. Oxford University Press, ISBN 0-19-869117-3.
• Joyce, George Hayward, 1949 (1908). Principles of Logic, 3rd ed. Longmans. A manual written for use in Catholic seminaries. Authoritative on traditional logic, with many references to medieval and ancient sources. Contains no hint of modern formal logic. The author lived 1864-1943.
• Jan Łukasiewicz, 1951. Aristotle's Syllogistic, from the Standpoint of Modern Formal Logic. Oxford Univ. Press.
• John Stuart Mill, 1904. A System of Logic, 8th ed. London.
• Parry and Hacker, 1991. Aristotelian Logic. State University of New York Press.
• Arthur Prior, 1962. Formal Logic, 2nd ed. Oxford Univ. Press. While primarily devoted to modern formal logic, contains much on term and medieval logic.
• --------, 1976. The Doctrine of Propositions and Terms. Peter Geach and A. J. P. Kenny, eds. London: Duckworth.
• Willard Quine, 1986. Philosophy of Logic 2nd ed. Harvard Univ. Press.
• Rose, Lynn E., 1968. Aristotle's Syllogistic. Springfield: Clarence C. Thomas.
• Sommers, Fred, 1970, "The Calculus of Terms," Mind 79: 1-39. Reprinted in Englebretsen, G., ed., 1987. The new syllogistic New York: Peter Lang. ISBN 0-8204-0448-9
• --------, 1982. The logic of natural language. Oxford University Press.
• --------, 1990, "Predication in the Logic of Terms," Notre Dame Journal of Formal Logic 31: 106-26.
• -------- and Englebretsen, George, 2000. An invitation to formal reasoning. The logic of terms. Aldershot UK: Ashgate. ISBN 0-7546-1366-6

External links

• Stanford Encyclopedia of Philosophy:
  • Aristotle's Logic -- by Robin Smith.
  • Traditional Square of Opposition -- by Terence Parsons.
• Internet Encyclopedia of Philosophy: "Aristotle." Discusses how Aristotle's logic was viewed by his many successors.
• Terence Parsons "History of Pre-Fregean Logic." Lecture notes on traditional logic.
• Annotated bibliographies of writings by:
  • Fred Sommers.
  • George Englebretsen.
• PlanetMath: Aristotelian Logic.