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”)

The Sublimity of Logic

PI 89:  A nodal point in PI–a point where numerous intimate connections can be traced.  I am not going to trace them now, not all of them.  But one is that the problem of the sublimity of logic is, at least partially, the result of our subliming of logic, of our relationship to the problem.  We are not wholly confused in subliming logic–logic is sublime.  But its sublimity must square with its not supplying us with new facts, with its investigation of the hardly memorable and easily forgettable.  –Can we so square the sublimity of logic without feeling that Wittgenstein has changed the subject?

These considerations bring us up to the problem: In what sense is logic something sublime?

For there seemed to pertain to logic a peculiar depth–a universal significance. Logic lay, it seemed, at the bottom of all the sciences.–For logical investigation explores the nature of all things. It seeks to see to the bottom of things and is not meant to concern itself whether what actually happens is this or that.—-It takes its rise, not from an interest in the facts of nature, nor from a need to grasp causal connexions: but from an urge to understand the basis, or essence, of everything empirical. Not, however, as if to this end we had to hunt out new facts; it is, rather, of the essence of our investigation that we do not seek to learn anything new by it. We want to understand something that is already in plain view. For this is what we seem in some sense not to understand.

Augustine says in the Confessions “quid est ergo tempus? si nemo ex me quaerat scio; si quaerenti explicare velim, nescio”.–This could not be said about a question of natural science (“What is the specific gravity of hydrogen?” for instance). Something that we know when no one asks us, but no longer know when we are supposed to give an account of it, is something that we need to remind ourselves of. (And it is obviously something of which for some reason it is difficult to remind oneself.)

William Temple on the Frege Channel

Because Psychology studies mental processes, it is very liable to behave as if Logic…were one of its subdivisions.  But in fact Psychology, like every other science, must presuppose the autonomy of Logic; otherwise the writings of psychologists could be no more than their own autobiographies–not nearly so interesting or important as the autobiographies of statesmen, soldiers or artists.  The interest which a psychologist claims for his theory is not that he happens to hold it, but that it is a true account of your experience and mine as well as of his own.  But in this case he must have something to say in support of his theory over and above its psychological history.  For every theory ever held has a psychological history.

Frege and Wittgenstein on the Difficulty of Audience

Please excuse this letter as springing from my unsatisfied need for communication.  I find myself in a vicious circle:  before people pay attention to my Begriffsschrift, they want to see what it can do, and I in turn cannot show this without presupposing familiarity with it.  So it seems I can hardly count on any readers for the book I mentioned…  (Letter to Marty)

This book will perhaps only be understood by those who have themselves already thought the thoughts which are expressed in it–or similar thoughts.   It is therefore not a text-book.  (Preface to TLP)

Quotable and Unquotable Signs (Peter Long)

As Frege was perhaps the first clearly to recognise, the sign for a property or relation…is not a quotable sign:  it is not an isolable piece of language. Of course, it is not incorrect to call ‘(is) white’ a predicative expression or ‘north of’ a relational expression, but what is here being called a predicative or relational expression is logically peripheral to what we should call the sign proper for a property or relation. For the sign that is proper to a property must, of course, have a different form from that which is proper to a relation. We give expression to this difference when we say ‘It is of the essence of a property to be of something’ and ‘It is of the essence of a (dyadic) relation to be between one thing and another’. What these propositions convey could be expressed at the level of language by saying ‘The sign for a property contains the form of a sign for a possible subject of the property’ and ‘The sign for a relation contains the form of signs for possible terms of the relation’. These formulations are not Frege’s, but they express what he meant by calling such signs ‘incomplete’, as opposed to those signs whose form is such that they do not contain the form of other signs, which he calls ‘complete’.  –Peter Long, “Universals:  Logic and Metaphor”, p. 97

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