Formal Relations

Abstract

We speak not only of the relation of a city to a country of which it is the capital or of a man to a child of whom he is the father, but of the relation of an object to a function of which it is the argument. But whereas the first relation finds expression in sentences that have in common the expression 'capital of and the second in sentences that have in common the expression 'father of, the function-argument relation finds expression in complex designations such as 'the capital of Holland' and 'Rembrandt's father', which have no expression in common. For this relation is not one that can be put into words at all. We might say, echoing Tractatus 4.121, that it is not something we can express by means of language, but something which expresses itself in language.Because there is no expression for the function-argument relation, such a relation is quite unlike a relation proper. By a relation proper I mean a relation as it is understood in predicate logic. Not that a relation proper is something for which there always exists an actual sign in the language, as there does for the relation capital of or father of. For the purpose of the logician a (first-level) relation is expressed in any sentence (of ordinary discourse) that contains two or more singular terms functioning as logical subjects. Still, it is something we express by means of language, as is shown by the fact that if the language contains no sign for it one can always be introduced. Thus we could introduce by definition a sign for the relation between Leeds and London that is expressed in 'Leeds is farther from Paris than it is from London'.Frege's thesis that a concept is a particular case of a function embodies the fundamental insight that the sense in which we speak of the relation of an object to a concept it falls under is the same as that in which we speak of the relation of an object to a function of which it is the argument. As there is no expression for the latter, so there is none for the former. If therefore we use the locution 'a falls under the concept F' and write 'Gold falls under the concept malleable' in place of 'Gold is malleable', we do not express in words a relation that is expressed in the shorter sentence without words. Frege thus betrays his own insight when he allows himself to be persuaded that because 'falls under' is a transitive verb, it stands for a relation. 1 In place of 'the capital of Holland' we could likewise write 'the value of the function capital of for the argument Holland'. Suppose someone now thought

On concept and object

“…(For further discussion of the distinction, see [9].) So in so far as part of the point of insisting that properties and relations must be expressed by properties of, and relations between, signs, is to provide a complete account of when something constitutes a sentence-and given that it can only succeed in this by employing the distinction between material and formal relations-then the doctrine of the incompleteness of predicative expressions, on the present account, will embody an insight that holds good also in the realms of sense and reference.…”