|Basic Concept: theories differ in what terms they choose as the basic concepts, which are not further defined. A definition of these concepts within the theory would be circular and may cause > paradoxes. E.g. The theory of mind by G. Ryle is based on the concept of disposition, other theories presuppose mental objects. See also paradoxes, theories, terms, definitions, definability, systems, explanations.|
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Basic concept/B.C./Field/(s): it is impossible to say of a basic concept if it is, e.g., semantic or proof theoretical. E.g. implication as a basic concept.
This is the case with natural deduction (ND, Gentzen) (Implication: cannot be considered proof-theoretical in ND, in terms of the derivation procedure, because it occurs in it itself (circular). - Nevertheless, ND is more proof theoretical than semantic. - It is often quite reasonable to consider implication a basic concept.
Basic Concept/Field: (E.g. implication as basic concept) may be two things: a) primitive predicate - b) primitive operator.
Theory/Basic Concept/Predicate/Infinity/Davidson/Field: (Davidson, 1965): no theory can be developed from an infinite number of primitive predicates.
Solution/Field: we can characterize an infinite number of predicates recursively instead by using a final number of axiom schemes.
Quinean Platonism/Field: as the basic concept a certain concept of quantity from which all other mathematical objects are constructed. - So natural numbers and real numbers would actually be sets.
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980