Universals, for our purposes, are sets, predicates, properties, relations and attributes –the same universal can instantiate or subsume numerically distinct things. Among universals we distinguish between those that are extensional, like sets and predicates and those that are intensional, like properties, relations and attributes.
Roughly, the distinction between these two types of universals is the following. Those that are extensional obey the principle that equivalence implies identity, while those that are intensional are in violation of this principle. In cases of non-deformity, the property of being a creature with a kidney is extensionally the same as the property of being a creature with a heart insofar as the same things have each, but the properties are different. Sets and predicates, on the other hand, do satisfy the principle of extensional equivalence: the set of creatures with a kidney and the set of creatures with a heart have the same members, and are thus identical.
Now, universals in general are taken to fall within the range of functions from the set of predicates of a language to the universal they give expression to , where is the set of universals of . Hellman explains that among such functions there are some that assign universals with much discrimination: two predicates are mapped to the same universal only if the predicates are identical. And others do so with less discrimination: coextensionality is enough. Hellman imagines a partial ordering of universals based on discrimination criteria with predicates being among the most discriminating and extensions among the least.
Since Hellman is concerned with properties and relations used in science, so the first thing we need to do is set up identity criteria for these attributes. For example, we want to say that temperature is the very same magnitude as mean molecular kinetic energy. So two predicates and are mapped to the same universal just in case it is a scientific necessity that these two predicates apply to the same things. In other words, and express the same attribute only in case in every model in , and have the same interpretation. Remember that is the set of structures modeling scientific possibility.
So, how do we express attributes given this way of thinking about them? Hellman suggests using the technique from modal logic, such that the function from members of to the extensions assigned to that predicate by each model in is identified with the attribute expressed by that predicate. In this way, two predicates express the same attribute in case they do so in every structure representing a scientific possibility; thereby getting the desired necessity.
If we need to represent universals that discriminate differently than scientific attributes, we can use sets or collections of structures different from insofar as the universalizing function is no more discriminating than those functions that assign predicates to the same universal in case the predicates are logically equivalent.
For these universals, the identity conditions are the following: functions (i.e., attributes) are identical just in case they assign the same arguments (i.e., members of ) to the same values (i.e., extensions of predicates). Or simply say that two predicates and express the same attribute if, and only if, is a law of science.
This last bit will be what links the discussion of attributes to that of reducibility –since we need to make use of explicit definability.
In the next set of notes I’ll go into the difference that Hellman notes between the ideological and ontological status of universals.