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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 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 truth or falsity.
A propositions may be universal or particular, and it may be affirmative or negative. Thus there are just four kinds of propositions:
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.
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.
For Aristotle, a term is simply a "thing", a part of a proposition. For early modern logicians like Arnauld (whose Port Royal Logic is the most well-known textbook of the period) it is a psychological entity like an "idea" or "concept". Mill thought it is 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).
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).
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
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
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).
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
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
Let's try to 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
which we rewrite replacing the major term P by "vegetarians", the middle term "M" by "cats", and the minor term S by "domestic felines".
Finally, the conclusion must consist of the minor term followed by the major term. This gives
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 to 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 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")
As explained, Aristotle thought that only in the first or perfect figure was the process of reasoning completely transparent. Their validity of an imperfect syllogism is only evident, when by conversion of its premisses, it can be turned into some mood of the first figure. This was called reduction by the scholastic philosophers.
It is easiest to explain the rules of reduction, using the so-called mnemonic lines first introduced by 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.)
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
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 difficulties. 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 usable form of formal logic, today – in the information age – they are taught nothing.
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