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?
Frege did think that definite descriptions, such as "the President of the United States" or "the cat," are (what he called) proper names, or what are more usually now called singular terms. And he thought that singular terms do denote objects. So I think he did believe that "the cat" refers to an object. The logical structure of the sentence "the cat is in the park" would be something like: Pc. The formalization you offer, "there is some x such that x is a cat and x is in the park," would schematize rather the sentence: some cat is in the park. Perhaps Frege would suggest that "the cat," used properly, is really elliptical for something like "the cat my mother owns." If the definite description really doesn't designate anything, then (in Frege's terminology), it might have a sense but lack a reference. Any sentence containing it might express a thought but would lack a truth value. (The latter claim needs to be qualified to deal with non-extensional contexts.) Hope this helps a bit!