I think abstract concepts could be distinguished with higher-order logic (= simple type theory). For example, the logical predicate “is spherical” (the property of being a sphere) applies to concrete objects. But the predicate “is a shape” applies to properties, like the property of being a sphere. And properties/concepts are abstract objects. So the shape concept is of a higher logical type than the sphere concept. Or take the “color” concept, the property of being a color. In its extension are not concrete objects, but other properties, like being red. Again, concrete objects can be red, but only properties (like redness), which are abstract objects, can be colors. A tomato is not a color, nor can any other concrete (physical or mental) object be a color. There is a type mismatch.
Formally: Let the type of concrete objects be e (for “entity”), and the type of the two truth values (TRUE and FALSE) be t (for “truth value”), and let functional types, which take an object of type x and return an object of type y, be designated with (x,y). Then the type of “is a sphere” is (e,t), and the type of “is a shape” is ((e,t),t). Only objects of type e are concrete, so objects of type (e,t) (properties) are abstract. Even if there weren’t any physical spheres, no spherical things like planets or soccer balls, you could still talk about the abstract sphere: the sphere concept, the property of being spherical.
Now the question is whether all the (intuitively) abstract objects can indeed, in principle, be formalized as being of some complex logical type. I think yes. Because: What else could they be? (I know a way of analyzing natural numbers, the prototypical examples of abstract objects, as complex logical types. Namely as numerical quantifiers. Though the analysis in that case is significantly more involved than in the “color” and “shape” examples.)
I think abstract concepts could be distinguished with higher-order logic (= simple type theory). For example, the logical predicate “is spherical” (the property of being a sphere) applies to concrete objects. But the predicate “is a shape” applies to properties, like the property of being a sphere. And properties/concepts are abstract objects. So the shape concept is of a higher logical type than the sphere concept. Or take the “color” concept, the property of being a color. In its extension are not concrete objects, but other properties, like being red. Again, concrete objects can be red, but only properties (like redness), which are abstract objects, can be colors. A tomato is not a color, nor can any other concrete (physical or mental) object be a color. There is a type mismatch.
Formally: Let the type of concrete objects be e (for “entity”), and the type of the two truth values (TRUE and FALSE) be t (for “truth value”), and let functional types, which take an object of type x and return an object of type y, be designated with (x,y). Then the type of “is a sphere” is (e,t), and the type of “is a shape” is ((e,t),t). Only objects of type e are concrete, so objects of type (e,t) (properties) are abstract. Even if there weren’t any physical spheres, no spherical things like planets or soccer balls, you could still talk about the abstract sphere: the sphere concept, the property of being spherical.
Now the question is whether all the (intuitively) abstract objects can indeed, in principle, be formalized as being of some complex logical type. I think yes. Because: What else could they be? (I know a way of analyzing natural numbers, the prototypical examples of abstract objects, as complex logical types. Namely as numerical quantifiers. Though the analysis in that case is significantly more involved than in the “color” and “shape” examples.)