Name relation or meaning relation
Name relation or meaning relation
The relation between a symbol (formula, word, phrase) and that which it denotes or of which it is the name.
Where a particular (Interpreted) system does not contain symbols for formulas, it may be desirable to employ Gdel’s device for associating (positive integral) numbers with formulas, and to consider the relation between a number and that which the associated formula denotes. This we shall call the numerical name relation and distinguish it from the relation between a formula and that which it denotes by calling the latter the semantical name relation.
In many (interpreted) logistic systems — including such as contain, with their usual interpretations, the Zermelo set theory, or the simple theory of types with axiom of infinity, or the functional calculus of second order with addition of Peano’s postulates for arithmetic — it is impossible without contradiction to introduce the numerical name relation with its natural properties, because Grelling’s paradox or similar paradoxes would result (see paradoxes, logical). The same can be said of the semantical name relation in cases where symbols for formulas are present.
Such systems may, however, contain partial name relations which function as name relations in the case of some but not all of the formulas of the system (or of their associated Gdel numbers).
In particular, it is normally possible — at least it does not obviously lead to contradiction in the case of such systems as the Zermelo set theory or the simple theory of types (functional calculus of order omega) with axiom of infinity — to extend a system L1 into a system L2 (the semantics of L1 in the sense of Tarski), so that L2 shall contain symbols for the formulas of L1, and for the essential syntactical relations between formulas of L1, and for a relation which functions as a name relation as regards all the formulas of L1 (or, in the case of the theory of types, one such relation for each type), together with appropriate new primitive formulas. Then L2 may be similarly extended into L3, and so on through a hierarchy of systems each including the preceding one as a part.
Or, if L1 contains symbols for positive integers, we may extend L1 into L2 by merely adding a symbol for a relation which functions as a numerical name relation as regards all numbers of formulas of L1 (or one such relation for each type) together with appropriate new primitive formulas; and so on through a hierarchy of systems L1, L2, L3, . . . .
See further Semantics; Semtotic 2; Truth, Semantical. — A.C.