Abstract
The distributive concept of class, secured in the theory of sets against the antinomies connected with its pre-theoretical stage, plays a very important role in the language of modern science. It turned out to be an irreplaceable tool for the formulation of theorems of different branches of mathematics and made possible the development of many new theoretical constructions or even whole new disciplines of great scientific importance. This concept has also applications in the philosophy of language, especially in the so-called theory of referential meaning of expressions, which describes the relation between language and what language can refer to.
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Notes
R. Carnap, Meaning and Necessity, enlarged edition, Chicago, 1956.
J.St. Mill, System of Logic …, vol. I, sixth edition, London, 1865, p. 32.
See B. Stanosz, ‘The Problem of Intensionality’, Studia Filozoficzne, foreign language edition, 3, 1966.
It seems that at any rate all simple predicates, which are not definitional abbreviations of compound predicates, have such components in their meanings.
An equivalence relation in the set X is any relation, which is reflexive, symmetric and transitive in X. Each equivalence relation in the set X is connected with some classification of X, and conversely: such connection holds between the equivalence relation R in X and the classification of X if and only if for every x, y belonging to X, R holds between x and y if and only if x and y both belong to the same member of. Then we say, that R makes of X.
The product of two classifications 1 = X 1, …,X n, 2 = Y 1, …, Y m of a given universe is the set of non-empty products X t. Y j, where i = 1, 2, …, n and j = 1, 2, …, m.
If we symbolize as R (Φ) the relation with which the predicate Φ is connected by its meaning, then we can formulate the simple rules of these operations as follows: 1) R (~ Φ) = R (Φ)’; 2) R (Φ1 ∧ Φ2) = R (Φ1)⋂R (Φ2); 3) R (Φ1 ∨ Φ2) = R (Φ1)⋃R (Φ2). The symbols: ~, ⋀, ⋁ are the negation, conjunction and disjunction (in nonexclusive sense) connectives, respectively; the signs:’,., + are the symbols of settheoretical complement, product and union operations, respectively.
Improper attributes can be differentiated further with respect to the differences of the formal properties of corresponding non-equivalence relation. Especially, it seems natural to distinguish the attributes expressed by those predicates which are connected by their meanings with the so-called resemblances, i.e. with the reflexive, symmetric and non-transitive relations.
Let us notice that the operations —, ⊓, and ⊔ are governed by Boolean laws. The set of attributes expressed by all predicates of a given language (connected by their meanings with the relations defined in the same universe) is a Boolean algebra; the contradictory attribute plays the role of 0, and the tautological attribute plays the role of 1.
One may doubt whether the predicate ‘is a friend of’ is really connected by its meaning with that classification (or even whether it is connected with the classification of that universe). Similar doubts arise in the case of other (one-or manyplace) predicates. These doubts show the ambiguity of predicates of the common language.
Of course, the relation R** is not an equivalence relation, so the attribute expressed by the predicate ‘has a friend’ is improper.
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© 1979 PWN — Polish Scientific Publishers — Warszawa
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Stanosz, B. (1979). The Attribute and the Class. In: Pelc, J. (eds) Semiotics in Poland 1984–1969. Synthese Library, vol 119. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9777-6_33
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