Frege Betraying Frege

“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 which 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 have in common the expression ‘father of’, the function-argument relation finds expression in complex designations such as ‘the capital of Holland’ and ‘Rembrant’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…

“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 an 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 we allows himself to be persuaded that because ‘falls under’ is a transitive verb, it stands for a relation…

“Since ‘falls under’ is not a relational expression, it follows that phrases of the form ‘the concept F‘ are not singular terms.  Unlike ‘the city of Leeds’, which designates a certain city, ‘the concept malleable‘ does not designate a certain concept.  Hence we cannot regard the verb and accusative of ‘Gold falls under the concept malleable‘ as signs in their own right.  In combination they form an expression for a concept, but in themselves are not expressions for anything.  Frege of course recognized that phrases of the form ‘the concept F’ are not concept-words, but if you take ‘falls under’ to be a genuine Beziehungswort, as Frege did in “On Concept and Object”, you have in consistency to construe such phrases as singular terms.  Frege was thus forced to equivocate:  as a singular term a phrase of this form must stand for an object, so by parity with ‘the city of Leeds’ it should stand for an object that (somehow) represents a concept.  –And yet how easy it is to go astray here!  For in our sentence there is expressed a relation between gold and the concept malleable.  So what is more natural than to assume that ‘falls under’ is an expression for that relation?  And yet the right conception is so close at hand!  For if ‘the concept malleable‘ is not an expression for a concept, it cannot stand for the second term of the relation of an object’s falling under a concept.  And so ‘falls under’ cannot itself be an expression for that relation.  We thus reach the conclusion that the relation expressed in our sentence is not expressed by it.”   (Peter Long, “Formal Relations”)

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