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# Term logic

Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived broadly unchanged 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 period without an elementary grasp of the terminology and ideas that were assumed by all philosophers until then. This article provides a basic introduction to the traditional system, with suggestions for further reading.

 Table of contents 1 The Basics 2 The Term 3 The Proposition 4 Singular terms 5 Predicables 6 Influence on philosophy 7 The Syllogism 8 Mood and Figure 9 Examples 10 Conversion and reduction 11 Syllogistic maxims 12 Decline of term logic 13 Revisionist Logic 14 External links 15 Sources for this article

## The Basics

The fundamental assumption behind the theory is that propositions are composed of two terms - whence the name "two-term theory" or "term logic" – and that the reasoning process is in turn built from propositions:

A propositions may be universal or particular, and it may be affirmative or negative. This 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 mnemonics.) The syllogistic is a formal theory explaining which combinations of true premisses yield true conclusions.

## The Term

A term (Greek horos) is the basic component of the proposition. The original meaning of the horos and also 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.

## The Proposition

In term logic, a "proposition" is simply a form of language: a particular kind of sentence, in 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 premiss of a syllogism. Aristotle uses the word premiss (protasis) as a sentence affirming or denying one thing of another (AP 1. 1 24a 16), so a premiss 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-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

The distinction between singular and universal is fundamental to Aristotle's metaphysics, and not merely grammatical. A singular term for Aristotle 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 premisses, one must occur twice. Thus

All greeks are men
All men are mortal.

What is subject in one premiss, 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 is nowhere in the Organon. It is first mentioned by Sextus Empiricus (Hyp. Pyrrh. ii. 164).

## The Syllogism

There can only be three terms in the syllogism, since the two terms in the conclusion are already in the premisses, and one term is common to both premisses. This leads to the following definitions:

The syllogism is always written major premiss, minor premiss,conclusion. Thus the syllogism of the form AII is written as

A M-P All cats are carnivorous
I S-M Some mammals are cats
I S-P Some mammals are carnivorous

## Mood and Figure

The mood of a syllogism is distinguished by the quality and quantity of the two premisses. There are eight valid moods: AA, AI, AE, AO, IA, EA, EI, OA.

The figure of a syllogism is determined by the position of the middle term. In figure 1, which Aristotle thought the most important, since it reflects our reasoning process most closely, the middle term is subject in the major, predicate in the minor. In figure 2, it is predicate in both premisses. In figure 3, it is subject in both premisses. In figure 4 (which Aristotle did not discuss, however) , it is predicate in the major, subject in the minor. Thus

```Figure 1  Figure 2  Figure 3  Figure 4
M-P         P-M       M-P       P-M
S-M         S-M       M-S       M-S
S-P         S-P       S-P       S-P
```

## Examples

Let's try and write a syllogism of the first figure, of the mood EAE (known as "Celarent") . This must have the first premiss beginning "no", the second beginning "all" and the conclusion beginning "no". And it must have a major term (P), let's say "vegetarians", a middle term, say "cats", and a minor term, say "domestic felines". The order of terms for the premisses in first figure (see table above) is

M-P
S-M

which we rewrite replacing the major term P by "vegetarians", the middle term "M" by "cats", and the minor term S by "domestic felines".

No cats are vegetarians
All domestic felines are cats

Finally, the conclusion must consist of the minor term followed by the major term. This gives

No domestic felines are vegetarians

But note the logic we were following was strictly based on the rules above. You didn’t have to think what the premisses were saying at all. It could have been a syllogism in a foreign language, you still could have reached the conclusion by following the rules. Now, as an exercise, read the premisses, think what they mean, and try and think what that implies. No cats are vegetarians. None of these things we are thinking about, eat vegetables. But all domestic felines are wholly included in these things, cats. So none of those domestic felines can eat vegetables either. But wait, that's the conclusion we reached by the other method. And it seemed so natural. Perhaps that's why Aristotle thought the syllogism (and particularly the first figure), was so natural. It's a mechnical rule-based process, yet something deeply embedded, in a way that seems quite un-rule-like, in our heads.

As an amusement: write a program to make syllogisms from random terms, moods and figures. Hint: only use common nouns in the plural, as adjectives are harder to convert into the traditional term structure.

## Conversion and reduction

Conversion is the process of changing one proposition into another simply by re-arranging the terms. Simple conversion is a change which preserves the meaning of the proposition. Thus

Some S is a P converts to Some P is an S No S are P converts to no P are S

Conversion per accidens involves changing the proposition into another which is implied it,but not the same. Thus

All S are P converts to Some S are P

(Notice that for conversion per accidens to be valid, there is an existential assumption involved in "all S are P")

It is easiest to explain the rules of reduction, using the so-called mnemonic lines first introduced by William of Shyreswood in a manual written in the first half of the thirteenth century.

Barbara, Celarent, Darii, Ferioque, prioris
Cesare, Camestres, Festino, Baroco, secundae
Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet
Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison.

Each word represents the formula of a valid mood and is interpreted according to the following rules:

• The first three vowels indicate the quantity and quality of the three propositions, thus Barbara: AAA, Celarent, EAE and so on.
• The initial consonant of each formula after the first four indicates that the mood is to be reduced to that mood among the first four which has the same initial
• "s" immediately after a vowel signifies that the corresponding proposition is to be converted simply during reduction,
• "p" in the same position indicates that the proposition is to be converted partially or per accidens,
• "m" between the first two vowels of a formula signifies that the premisses are to be transposed,
• "c" appearing after one of the first two vowels signifies that the premiss is to be replaced by the negative of the conclusion for reduction per impossibile.

## Syllogistic maxims

There are a number of maxims and verses associated with the syllogistic. Their origin is mostly unknown. For example

The letters A, I, E and O are taken from the vowels of the Latin Affirmo and Nego.

Asserit A, negat E, sed universaliter ambae
Asserit I, negat O, sed particulariter ambo

Shyreswood's version of the "Barbara" verses is as follows:

Barbara celarent darii ferio baralipton
Celantes dabitis fapesmo frisesomorum;
Cesare campestres festino baroco; darapti
Felapton disamis datisi bocardo ferison.

Another runs:

Barbara, Celarent, Darii, Ferioque prioris
Cesare, Camestres, Festino, Baroco secundae
Tertia grande sonans recitat Darapti, Felapton
Disamis, Datisi, Bocardo, Ferison.
Quartae Sunt Bamalip, Calames, Dimatis, Fesapo, Fresison.

## Decline of term logic

Term logic dominated logic throughout most of its history until the advent of modern or predicate logic a century ago, in the late nineteenth and early twentieth century, which led to its eclipse.

The decline was ultimately due to the superiority of the new logic in the mathematical reasoning for which it was designed. Term logic cannot, for example, explain the inference from "every car is a vehicle", to "every owner of a car is an owner of an vehicle ", which is elementary in predicate logic. It is confined to syllogistic arguments, and cannot explain inferences involving multiple generality. Relations and identity must be treated as subject-predicate relations, which makes the identity statements of mathematics difficult to handle, and of course the singular term and singular proposition, which is essential to modern predicate logic, does not properly feature at all.

Note, however, that the decline was a protracted affair. It is simply not true that there was a brief "Frege Russell" period 1890-1910 in which the old logic vanished overnight. The process took more like 70 years. Even Quine's Methods of Logic devotes considerable space to the syllogistic, and Joyce's manual, whose final edition was in 1949, does not mention Frege or Russell at all.)

## Revisionist Logic

The innovation of predicate logic led to an almost complete abandonment of the traditional system. It is customary to revile or disparage it in standard textbook introductions. However, it is not entirely in disuse. Term logic was still part of the curriculum in many Catholic schools until the late part of the twentieth century, and taught in places even today. More recently, some philosophers have begun work on a revisionist programme to reinstate some of the fundamental ideas of term logic. Their main complaint about modern logic is

• that Predicate Logic is in a sense unnatural, in that its syntax does not follow the syntax of the sentences that figure in our everyday reasoning. It is, as Quine acknowledges, "Procrustean" employing an artificial language of function and argument, quantifier and bound variable.
• that there are still embarrassing theoretical problems faced by Predicate Logic. Possibly the most serious are of empty names, and of 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)

Fred Sommers has designed a formal logic which he claims is consistent with our innate logical abilities, and which resolves the philosophical dificulties. See, for example, his seminal work The Logic of Natural Language. The problem, as Sommers says, is that "the older logic of terms is no longer taught and modern predicate logic is too difficult to be taught". School children a hundred years ago were taught a useable form of formal logic, today – in the information age – they are taught nothing.

Joyce, G.H. Principles of Logic, 3rd edition, London 1949. This was amanual written for (Catholic) schools, probably in the early 1910's. It is spendidly out of date, there being no hint even of the existence of modern logic, yet it is completely authoritative within its own subject area. There are also many useful references to medieval and ancient sources.

Lukasiewicz, J., Aristotle's Syllogistic, Oxford 1951. An excellent, meticulously researched book by one of the founding fathers of modern logic, though his propaganda for the modern system comes across, these days, as a little strident.

Prior, A.N The Doctrine of Propositions & Terms London 1976. An excellent book that covers the philosophy around the syllogistic.

Mill, J.S A System of Logic, (8th edition) London 1904. The eighth edition is the best, containing all the original plus later notes written by Mill. Much of it is propaganda for Mill's philosophy, but it contains many useful thoughts on the syllogistic, and it is a historical document, as it was so widely read in Europe and America. It may have been an influence on Frege.

Aristotle, Analytica Posteriora Books I & II, transl. G.R.G.Mure, in The Works of Aristotle ed. Ross Oxford 1924. Ross's edition is still (in this writers view) the best English translation of Aristotle. There are still many copies available on the second hand market, handsomely bound and beautiful.

Evans, G. "Pronouns, Quantifiers and Relative Clauses" Canadian Journal of Philosophy 1977

Sommers, F. The Logic of Natural Language, Oxford 1982. An excellent journey around the myths that have built up around term logic, and an impassioned attack upon the logic of Frege.

To do: Predicables, Influence on philosophy: to include "existence as a predicate", the Kantian categories, and the petitio principii problem, Explanation of reduction per impossibile ''

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