# Term logic

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Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived until the advent of modern predicate logic in the late nineteenth century.

It can sometimes be difficult to understand philosophy before the period of Frege and Russell without an elementary grasp of the terminology and ideas that were assumed by all philosophers until then. This article provides a basic introduction to traditional logic.

## Aristotle's system

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 philosophers are rich")
• 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 (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 creates a logic he called the "way of geometry", influenced far more by Euclidean logical styles than Aristotelian ones. Leibniz created a logical calculus quite distinct from most previous logical styles. But, by the 19th century, attempts to algebrize logic had often yielded systems that were highly influenced by the term logic tradition, such as the work of Boole and Venn. Starting around 1900, the term logic tradition was beginning to be eclipsed by the advent of predicate logic. Predicate logic was the creation of Frege (whose landmark Begriffsschrift was little read before 1950), Charles Peirce, Ernst Schroder, and Peano. It reached full fruition in the hands of Bertrand Russell and A. N. Whitehead, whose Principia Mathematica, published 1910-13, made splendid use of predicate logic. Since the development of predicate logic, term logics have not been popular in the West, except perhaps as simplifications for beginning students.

Predicate logic was designed as a form of mathematics, and as such is capable of all sorts of reasoning about mathematics that is completely beyond the powers of term logic. Moreover, predicate logic is 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." Term logic, confined to syllogistic arguments, 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.

It is not true that, after a brief "Frege-Russell" period (1880-1910), the old term logic simply vanished. Rather, the decline was a protracted affair, taking approximately 70 years. Even Quine's Methods of Logic (1st ed. 1950 and still in print) devotes considerable space to term logic and syllogisms, and Quine was fond of scholastic terminology such as "syncategorematic." Joyce's 1949 manual does not mention Frege or Russell at all.

At the end of the 20th century, Fred Sommers (1982) and George Englebretsen advocated an enhanced form of term logic they called term functor logic. This logic has sufficient expressive power to capture the validity of the above argument, and can handle relational terms generally. It has a very Boolean appearance, employing '+' and '-' as its sole operational signs. All statements take the form of equations. Term functor logic has similarities to Quine's predicate functor logic but has less of a following.

## Revisionist logic

Predicate logic led to the almost complete abandonment of term and syllogistic logic, except among students of ancient and medieval philosophy, and in traditional Roman Catholic education. It is customary to revile or disparage term logic in standard introductory texts. More recently, some philosophers have begun work on a revisionist programme to reinstate some of the fundamental ideas of term logic. Their main complaints about predicate logic are that it:

• 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 orthodox and entirely mainstream philosophers such as Gareth Evans have voiced discontent:

"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)

Heeding the Paideia proposal from philosopher Mortimer J. Adler, homeschooling advocates in recent years have revived the Trivium, a medieval curriculum consisting of the – grammar, logic, and rhetoric – arguing that logic is properly part of a classical education in language, and not of mathematics. The problem, as they see it, is that predicate logic is excessively nominalistic, in that it is primarily concerned with the manipulation of symbols, and not with the whys and essences of things.

## 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 Catholic seminaries. Although containing no hint of modern formal logic, it is authoritative on its subject, classical logic. Many references to medieval and ancient sources. The author lived 1864-1943.
• Jan Łukasiewicz, 1951. Aristotle's Syllogistic, from the Standpoint of Modern Formal Logic. Clarendon Press, Oxford.
• John Stuart Mill, 1904. A System of Logic, 8th ed. London.
• Parry and Hacker, 1991. Aristotelian Logic. State University of New York Press.
• Parsons, Terence. "Traditional Square of Opposition." Stanford Encyclopedia of Philosophy.
• Arthur Prior, 1962. Formal Logic, 2nd ed. Oxford Univ. Press. While primarily devoted to modern formal logic, contains much on term and other classical logics.
• ------, 1976. The Doctrine of Propositions and Terms. Peter Geach and A. J. P. Kenny, eds. London: Duckworth.
• Rose, Lynn E., 1968. Aristotle's Syllogistic. Springfield: Clarence C. Thomas.
• Smith, Robin. "Aristotle's Logic." Stanford Encyclopedia of Philosophy.
• Sommers, Fred, 1970. The Calculus of Terms, in: Mind, vol. 79, 1; pp. 1-39 (1970); Reprinted in: George Englebretsen (ed.) The new syllogistic New York: Peter Lang 1987, ISBN 0-8204-0448-9
• Sommers, Fred: The logic of natural language. Oxford University Press, 1982
• Sommers, Fred: Predication in the Logic of Terms, in: Notre Dame Journal of Formal Logic, vol. 31, 1, Winter 1990, pp. 106–126 [1]
• Sommers, Fred & Englebretsen, George: An invitation to formal reasoning. The logic of terms, Aldershot, Burlington, Singapore, and Sydney: Ashgate 2000 ISBN 0-7546-1366-6