The meaning of ‘the’

Manny asked:

I have a very basic question about Frege’s object/ concept distinction. Please don’t make fun of me as I’m new to early analytic philosophy. This question has been bugging me for a while, so I’d appreciate a thorough answer.

In sentences like ‘the cat is grey’ or ‘the cat is in the park,’ do the words ‘the cat’ designate an object? If you were to formalize these sentences, I would think it would go as something like: there is some x such that x is a cat and x is grey/in a park. There wouldn’t be a uniqueness clause, I would think.

If the words that designate an object have to pick out something unique, does that mean the words ‘the cat’ cannot designate an object (since they are not specified enough)? If they don’t designate an object, then what is their logical status?

Answer by Geoffrey Klempner

Thanks, Manny, for this very interesting question. (I know that on the ‘Ask a Question’ page you called yourself ‘Kant student’, but we thought Kant = Immanuel = Manny. OK?)

The real topic is Bertrand Russell’s classic article, ‘On Denoting’ (1905), which is essential reading for anyone looking to explore the roots of analytic philosophy. But more about that in a minute.

Gottlob Frege (1848-1925) is a philosopher worth taking the effort to study for a Kant student. Kant’s notion of a concept as ‘bringing intuitions under a rule’ corresponds to Frege’s idea that a concept expression is a unique kind of function. When you apply the mathematical function ‘+2’ to the natural number 5, you get 7. The number 5 is the argument of the function +2 (I’m not bothering with precise use of quotation marks because it just gets silly) while 7 is the value of the function for that argument.

In a similar way, Frege thought, when you apply the function T/F to the proposition, ‘Paris is the capital of France’ you get the value T. When you apply it to ‘Paris is the capital of Germany’ you get the value F. Propositional functions have just two values T or F. Let’s say it is true that Felix is in the park. The concept ‘…is in the park’ is true of Felix but, say,  false of Fido. For Felix, the value of the corresponding proposition is T, while for Fido the value is F. In Kantian terms, to grasp the concept, ‘…is in the park’, is to be able to apply this ‘rule’ to different objects and decide whether the rule applies or fails to apply in a given case.

Frege would not formalize your example, ‘the cat is in the park’ as ‘there is some x…’ because this states that there is ‘a’ cat in the park, which is a very different claim. Frege’s groundbreaking discovery of first-order predicate calculus (in ‘Begriffsschrifft’ 1879) parses expressions such as ‘a cat’ as not being referring expressions at all, despite appearances, but as functioning as ‘second-order’ concepts, that is to say, concepts that are true or false of other ‘first-order’ concepts. If Felix is in the park, then the second order concept, ‘some x’ is true of the first-order concept, ‘…is a cat and is in the park’.

Syntactically, the definite description ‘the F’ appears to function as a referring expression. The Pastor is in the garden. Pastor James is in the garden. James is in the garden. All these say the same thing (they are all true or all false depending on the facts) although, as Frege argues in ‘On Sense and Reference’ (1892) we can detect a difference in sense, but not reference, in the three referring expressions I have just used. (We don’t need to go into that here.)

But what about referring expressions that lack a reference? In mathematics,  you cannot talk of ‘the x’ unless you have already proved the existence of x. Any referring expression in mathematics must be guaranteed a reference. For example, if you use an expression for a natural number, then you know that the number in question exists, provided you are following the rules for writing down numbers, e.g. in Arabic notation, 1,2,3… .

Frege saw the existence of definite descriptions lacking a referent in natural language as merely showing that natural language is ‘defective’. In many ways, it is not a precise instrument — words can be vague, ambiguous, elliptical — but it works well enough for practical purposes. He wasn’t bothered by that, as his main interest was the language of mathematics. For Russell, by contrast, this was a glaring problem and an obstacle to any serious attempt to understand the nature of thought itself.

I’m not going to attempt to summarize Russell’s article ‘On Denoting’ — I don’t want to spoil the pleasure. The key claim is that, following the example of Frege who saw that ‘a cat’ is not a referring expression even though syntactically it appears to be one, neither is ‘the cat’ a referring expression. When you state, ‘The cat belonging to my sister is in the park’ (better example), you are making two claims, that my sister owns just one cat, and that the cat in question is in the park. So it turns out that ‘the cat…’ is treated similarly to ‘a cat…’ as both involving second-order concepts. (Russell doesn’t give an altogether fair account of Frege in his article, but the language of ‘propositional functions’ in the article is wholly derived from Frege.)

Remarkably, analytic philosophers are still debating Russell’s theory of definite descriptions. You wouldn’t believe the amount of literature that has grown around this one topic. What would Frege have said? I don’t think he would have got the point. Why all the fuss? In that difference in attitude, lies the gap that separates Frege’s limited view of philosophy and its uses, from the rise of analytic ambition.