Semiotics in Poland 1984–1969 pp 284-317 | Cite as

# The Semantics of Open Concepts

## Abstract

The main problem to be discussed in this paper has a long philosophical tradition. It emerged for the first time in reflections on vague terms in common language. The term (a) ‘youth’ (in the sense of a young man) may serve as a classical example. There is practically no doubt that a man under 18 is still a youth, and that no one over 30 is any longer a youth. But what about a person who is, for instance, 25 years old? The meaning assigned to that term in common language is such that we are not in a position to answer the question. What then is the nature of a statement that a person aged 25 is a youth? Is it a true or a false, but essentially undecidable statement, as some claim ? Or is it deprived of any truth value, as others would have it? Or is it, perhaps, the case that both the statement and its negation are false, as still others maintain, thereby rejecting the principle of the excluded middle — one of the fundamental laws of logic? These questions lead to further questions. What, in fact, is such a statement about? Does the term ‘youth’ denote any definite set of objects ? And if it does, then what is that set?

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### Notes

- 1.The partial definition of the predicate
*Q*in its general form can be formu lated so as to be a non-creative definition: ∧*x*((P_{1}*x*∧ ~*P*_{2}x →*Qx*) ∧ (*P*_{2}X ∧ ~ P_{1}*x*→ ~*Qx*)). The above definition is logically equivalent to the type (1) statement: ∧*x*((P_{1}*x*≡ ~ P_{2}*x*) → (*Qx*≡ P_{1}*x*)).Google Scholar - 2.For a more detailed discussion see M. Przełęcki, ‘Pojecia teoretyczne a do świadczenie’ (Theoretical Concepts and Experience),
*Studia Logica*, XI, 1961, ‘O pojęciu genotypu’ (On the Concept of Genotype),*Studia Filozoficzne*, 26, 1961.Google Scholar - 3.R. Suszko, ‘Logika formalna a niektóre zagadnienia teorii poznania’ (Formal Logic and Some Problems in Epistemology)
*Myśl Filozoficzna*, 28, 29, 1957.Google Scholar - 4.To simplify matters we are disregarding here those extra-logical expressions which are function symbols. Likewise, the predicates which occur in the examples will always be treated as monadic predicates. But our analysis can easily be generalized so as to cover predicates of
*n*arguments.Google Scholar - 5.A unary relation is identified with a subset of
*U*.Google Scholar - 6.In p1esenting these semantic concepts I have followed R. Suszko’s paper, cited above.Google Scholar
- 7.The relativization to language will be omitted in the definitions that follow; we do this to simplify our formulations as far as possible. and will be treated as models of
*L*and*L*’ respectively. The clause in (EL):*D*(*Q*) → (*Z*(*Q*) ≡*Z*) is true in every model is, of course, equivalent to: for any model if*D*(*Q*) is true in, then*Z*(*Q*) is true in if and only if*Z*is true in. We shall refer to the last formulation from time to time.Google Scholar - 8.
- 9.This can be demonstrated more clearly for a case in which Ψ and
*Q*are simple predicates in*L*, e.g.,*P*_{1}and P_{2}, so that*D*(*Q*) becomes A*x*(*P*_{1}*x*→ (*Q x*≡*P*_{2}*x*)). If = <*U*,*x*_{1}, …,*x*_{n},*X*_{1}, …,*X*_{m},*Y*>, then we take to be <*U*,*x*_{1}, …,*x*_{n},*X*_{1}, …,*X*_{m},*X*_{2}>. It that case the assumption that*D*(*Q*) is true in is equivalent to the tautology ∧*x*∈*U*(*x*∈*X*_{1}→ (*x*∈*X*_{2}≡*x*∈*X*_{2})) and as such may be disregarded. On the other hand, the statement that*Z*(*Q*) is true in is equivalent to the statement that*Z*(*P*_{2}) is true in, and may be replaced by the latter. By way of example let*Z*(*Q*) be*P*_{1}*a*_{1}∧*Qa*_{1}. The latter is true in if and only if*x*_{1}∈*X*_{1}∧*x*∈*X*_{2}, and hence, if and only if P_{1}*a*_{1}∧*P*_{2}*a*_{1}in true inGoogle Scholar - 10.See
*The Reach of Science*, Toronto 1958. A precise analogue of the concept of a “determinate” statement will be introduced later in this paper.Google Scholar - 11.This formulation is equivalent to the condition (EL) as formulated in the text. An analogous observation applies to (ET) and (EM).Google Scholar
- 12.To simplify matters, we omit the necessary relativization to language: a statement
*Z*is true in a language*L*, or in a model of*L*.Google Scholar - 13.This interpretation of the concepts of proper model and true statement is to be found in R. Suszko’s paper cited above and in J. Kemeny, ‘A New Approach to Semantics’,
*Journal of Symbolic Logic*, 21, 1956.Google Scholar - 14.I discuss the language of empirical theories in my paper
*Theoretical Concepts and Experience*(see footnote^{2}above), and the ways of interpreting the specific terms of empirical theories, in ‘Interpretacja systemów aksjomatycznych’ (Interpretation of Axiomatic Systems),*Studia Filozoficzne*, 21, 1960. I have analysed the interpretation of observation terms in ‘O definiowaniu terminów spostrze-żeniowych’ (On Defining Observation Terms), in:*Rozprawy logiczne*(Papers on Logic). Warsaw 1964.Google Scholar - 15.
- 16.Standpoint (I.I) roughly corresponds to the standpoint I adopt in my paper ‘W sprawie terminów nieostrych’ (Concerning Vague Terms),
*Studia Logica*, VIII, 1958.Google Scholar - 17.Standpoint (I.2) coincides with that adopted by H. Mehlberg in
*The Reach of Science*.Google Scholar - 18.The standpoint (II) approximately corresponds to that adopted by W. Rozeboom in ‘The Factual Content of Theoretical Concepts’.
*Minnesota Studies*… Vol. 3, 1962.Google Scholar - 19.I know of no representative of standpoint (III).Google Scholar
- 20.The standpoint adopted by T. Kubiéski in ‘Nazwy nieostre’ (Vague Terms),
*Studia Logica*, VII, 1958, falls within the general outline of standpoint (IV).Google Scholar - 21.
- 22.This is explicitly stated by H. Mehlberg in his book cited above. Standpoint (V) is an expanded and more precise version of the standpoint which I adopt in
*Theoretical Concepts and Experience*and which is tacitly assumed by many authors.Google Scholar - 23.Both the relative and the absolute concepts of designation and denotation require relativization to a given language, which for simplicity is omitted here.Google Scholar
- 24.
- 25.
- 26.Such a non-classical calculus of terms is constructed by T. Kubiéski in his paper cited above.Google Scholar
- 27.The following assumption could be a reason for the adoption of definition (B):
*P*designates*x*if and only if every onewho correctly uses*P*can, under specified circumstances, predicate*P*about*x*.Google Scholar - 28.The proof given in this paper does not apply to the general case.Google Scholar
- 29.Cf. E. Nagel,
*The Structure of Science*, New York 1961.Google Scholar