According to Ockham's first theory, in what way do 'universals' exist?
In each case, consider a universal to be abstract. This gets complicated because some nominalists (or conceptualists) might consider numbers to be real in the abstract but something like redness does not. For the sake of argument, let's say that universals are abstract and therefore, do not exist by themselves; they only exist as exemplified by particular, real objects. Thus, strength does not exist abstractly by itself as universal 'strength.' But particular, real objects like lions can exhibit strength. Since each exhibition of strength depends on a particular object/thing, it is individual; not universal. So, for Ockham, universals are abstract but does this mean they don't exist at all?
It depends. Ockham is either considered a nominalist, a conceptualist or a terminist in the beginning of his philosophy. All three are similar but slightly different. As a nominalist, Ockham would believe that universals only exist as names (mentally, spoken or written.) 'Humanity' exists only as a word. Humanity in the real world is different for each human. There is no real universal humanity. It is only an abstract name(word) that, like strength, can be exhibited individually each human.
As a conceptualist, Ockham would believe that a universal existed as a mental concept. In a sense, this means that a universal "stood for" or "represented" the Real object - only in thought. So, 'redness' or 'humanity' are mentally produced concepts which reflect a MENTAL common/universal ontology shared by all things red or human. They are concepts produced when supposing their existence. So, they are a bit more than just words (names). This is just a tricky way of saying these concepts are boiled down to the terms we come up with when trying to think of universals. Thus, he is also described as a terminist. The red things and human things in the real world do not share a universal or essential redness or humanity. As a terminist or conceptualist, universals like redness, humanity and strength are just metaphysical concepts or terms produced by the act of thinking of all the things that INDIVIDUALLY EXHIBIT these universals.
So, kind of bridging the three (nominal, conceptual and terminist), Ockham's first theory of universals could be considered a phenomenological theory of thought-intentionality; MEANING THAT UNIVERSALS ARE MENTAL CONSTRUCTS - they are metaphysical/abstract but not real. In other words, Ockham believed universals to be products of the actual process of mentally considering universals. Redness is a term that you mentally consider when you think about things that exhibit redness. Redness is a property of a red thing, but redness, by itself, exists only as a mental consideration. There is no essential 'redness" somewhere out there.