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2016年9月24日(土曜日)

Russell と伝統論理学についてもう少し覚書

カテゴリー: - susumuhayashi @ 15時30分02秒

Russell, The principles of mathematicsにおける伝統論理学への言及で、講義で使えそうなところを幾つか記録:

Claas と class-concept (extension v.s. intension)
Chap. II
B. The Calculus of Classes
§21. The insistence on the distinction between and the relation of
whole and part between classes is due to Peano, and is of very great
importance to the whole technical development and the whole of the
applications to mathematics. In the scholastic doctrine of the syllogism,
and in all previous symbolic logic, the two relations are confounded,
except in the work of Frege*. The distinction is the same as that
between the relation of individual to species and that of species to
genus, between the relation of Socrates to the class of Greeks and the
relation of Greeks to men. On the philosophical nature of this distinc-
tion I shall enlarge when I come to deal critically with the nature of
classes; for the present it is enough to observe that the relation of
whole and part is transitive, while e is not so : we have Socrates is a
a man, and men are a class, but not Socrates is a class. It is to be
observed that the class must be distinguished from the class-concept
or predicate by which it is to be defined: thus men are a class, while
man is a class-concept. The relation e must be regarded as holding
between Socrates and men considered collectively, not between Socrates
and man. I shall return to this point in Chapter VI. Peano holds
that all prepositional functions containing only a single variable are
capable of expression in the form “x is an a,” where a is a constant
class; but this view we shall find reason to doubt.

Categorical proposition についての議論:
命題の基本を subject, copula, predicate で考えるのはおかしい。
動詞を無視している。という議論。
 "Socrates is a man" を、
 Socrates | is | a man ではなく、
 Socrates | is a man と divide する。
Chap. III: Implication and formal implication
§43. Assertions
It has always been customary to divide propositions into 
subject and predicate ; but this division has the defect of omitting the
verb. It is true that a graceful concession is sometimes made by loose
talk about the copula, but the verb deserves far more respect than is
thus paid to it. We may say, broadly, that every proposition may be
divided, some in only one way, some in several ways, into a term (the
subject) and something which is said about the subject, which something
I shall call the assertion. Thus “Socrates is a man” may be divided
into Socrates and is a man. The verb, which is the distinguishing mark
of propositions, remains with the assertion ; but the assertion itself,
being robbed of its subject, is neither true nor false. In logical dis
-cussions, the notion of assertion often occurs, but as the word proposition
is used for it, it does not obtain separate consideration. Consider, for
example, the best statement of the identity of indiscernibles: “If x and y
be any two diverse entities, some assertion holds of x which does not
hold of y.” But for the word assertion^ which would ordinarily be
replaced by proposition, this statement is one which would commonly
pass unchallenged. Again, it might be said: “Socrates was a philo=
sopher, and the same is true of Plato.” Such statements require the
analysis of a proposition into an assertion and a subject, in order that
there may be something identical which can be said to be affirmed of
two subjects.

Termについての議論:Chap. IV: Proper names, adjectives and verbs.
§46. Proper names, adjectives and verbs distinguished
§47. Terms
Whatever may be an object of thought, or may occur in any true
or false proposition, or can be counted as one, I call a term. This,
then, is the widest word in the philosophical vocabulary. I shall use
as synonymous with it the words unit, individual, and entity. The
first two emphasize the fact that every term is one, while the third is
derived from the fact that every term has being, i.e. is in some sense.
A man, a moment, a number, a class, a relation, a chimaera, or anything
else that can be mentioned, is sure to be a term ; and to deny that such
and such a thing is a term must always be false.
§48. Things and concepts
 Among terms, it is possible to distinguish two kinds, which
 I shall call respectively things and concepts. The former are the termsindicated
 by proper names, the latter those indicated by all other words.
§49. Concepts as such and as terms


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