Predication is an integral part of making a judgement, which is expressed in propositions (such as 'the sun is round'). Predication itself is possible because in some sense something can be said of something else. But this is only possible because something above or distinct in thought from any particular subject can come to be instantiated in a particular subject; that is to say, a universal, not being limited to any particular concrete instance, can be predicated of a subject(s). Thus in our example 'roundness' is predicated of the particular subject 'the sun'. Roundness is considered to be universal at least in some respect in that where the sun can only be said of thing (namely itself), roundness can be said of many.
If we reject such an argument, does this not alter our logic also? It might be best to clarify the question using our example of the sun. The traditional view of the proposition "the sun is round" is that the universal 'roundness' is being predicated of the particular subject 'the sun'. But rejecting the notion that 'roundness' is universal would thus seem to either result in rejecting that 'roundness' is said of many things (although granting that it is said of one thing) or to reject that there is a distinction between 'roundness' and 'the sun'. The first option seems very hard to uphold. For if we allow that some concept of roundness is said of one thing, why should we not hold that it is said of many things. It is by the same logic and inference that we arrive at the conclusion that roundness is said of one thing that we arrive at the conclusion that is said of any other thing it might be thought to be said of. This leaves the second option, which seems to be the far more ambitious but also the far more clear requirement for rejecting universals; that is to reject the distinction between 'roundness' and 'the sun'.
Accompanying this effort seems to be the evidence of perception; in perceiving the sun we do not perceive of its roundness independent of its own being. Instead, in our mental image of the sun, 'roundness' and 'the sun' are inseparable. Thus, our original statement might be converted from "the sun is round" to "the round sun". But in itself, this is not constituent of a judgement. It is only rather a subject, that does not inform us of its validity or truthfulness. One could attempt to formulate a new proposition by adding another predicate as follows: "the round sun is a fact". But this itself seems to face the same problems, namely that the predicate 'fact' would seem to be a universal, since 'the round sun' is not the only thing that can be factual. Furthermore, if the only predicate were 'fact' this seems to have the adverse effect of making propositions ambiguous. For example, in saying "the round sun is a fact" it is unclear what is being affirmed. This is perhaps more clear in negative propositions: "the round sun is not a fact". What is not a fact about such a proposition? That there is no sun or that there is no round sun?
In any case, it seems that universals are both unavoidable in terms of their necessity in making a judgement and in clarifying what is meant by a judgement. But is this correct? Does the absence of universals in some sense result inevitably in the impossibility of judgments?