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 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.

## Contents

## 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 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.

## 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. 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

*form does. For one of the valid patterns (Darapti) is:*

**I***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

*form is to be interpreted. One then naturally asks about the*

**A***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:*

**O***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

*S*is

*P*' is equivalent to 'Every non-

*P*is non-

*S*'

*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 beingto 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):

To the falsehood:

A non-man is not a non-chimera*Logica Parva*

## 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 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.

## 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. 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.

## 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 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 syllogistic is that, of the four terms in the two premises, one must occur twice. Thus

All Greeks are**men**All

**men**are mortal.

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 syllogistic, Port-Royal Logic, singular terms were treated as universals:

All men are mortalsAll Socrates are menAll Socrates are mortalsThis is clearly awkward, 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, but 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."

## Influence on philosophy

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.

## Decline of term logic

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.

## Revival

Some philosophers have complained that predicate logic:

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)