Quine's paper begins in a dogmatic and ends in a conciliatory tone. The reason for this is probably that he holds the existence of Pegasus and the existence of universals to be insupportable, and therefore false, and yet he cannot justify the existence of the various entities in mathematics, though he is not disposed to condemn them also.
Pegasus comes first. Some philosophers apparently hold that we cannot significantly use singular terms without presupposing the existence of the things they purport to name. Quine first inspects the accounts of these philosophers and illustrates problems which arise in these accounts. He then goes on to say that the theory of descriptions together with a little device for reducing names to descriptions, enables us to say these things without using any singular terms.
Next universals. Quine denies the existence of universals and also of meanings. His position is that the use of adjectives, and the use of the word "meaning" can be adequately accounted for without supposing that properties exist or that meanings exist.
Now Quine gives his criteria of ontological commitment. This seems to be that we are only committed to the existence of an entity or class of entities when we explicitly assert that existence by use of some utterance such as "there is something which ...". The range of entities which our utterances assert or presuppose is the range over which the bound variables in our assertions must extend in order to make them true.
This leaves open the question of what exactly Quine's position is in relation to, for example, mathematical entities. And in the last few pages of his paper Quine suddenly acquires "tolerance and an experimental spirit". He becomes ontologically relativistic and concedes not only that we can adopt whatever ontology suits us but even that we can switch from the one to the other at the drop of a hat.
Quine in this paper exhibits an inclination to nominalism. He is ambivalent with respect to the question whether ontological questions are absolute or relative. Predominantly his remarks suggest that he believes that ontological questions have objective answers, a certain class of entities either do or do not exist. This may be regarded as implicit in the notion of "ontological commitment". The idea is that if one uses a certain form of language which appears to quantify over a kind of entity, and there is no apparent way of restating the proposition which avoids the quantification, then we are committed to entities of that kind. Thus one might suppose that by studying someones claims we might reconstruct a more or less definite account of the kinds of things to the existence of which he is "committed".
The possibility seems to be excluded that someone might use ontology rather flexibly, at one moment talk as if certain entities exist and at another talk as if they did not. This is rather like Carnap did, in adopting at one moment a phenomenalistic, another a physicalistic and yet again a theoretical language, and rather like what he endorsed with his "principal of tolerance".
From this point of view, Quine is an ontological absolutist, and Carnap a relativist.
However, there are multiple contrary indications. When, as the paper comes to a close, he moves from talking around the topic of commitment, which is not about what is, but about what particular assertions should be taken as asserting that there is, to really talking about what exists, he proposes that these questions should be addressed in a manner similar to the assessment of scientific theories. This he takes to involve adopting the simplest theory which is consistent with our "raw experience". Specifically he asserts that ontological questions are no more "a matter of language" than is any "system of scientific theory". Here he shows willing to accept alongside each other as equally legitimate the conceptual schemes and ontologies which appear in Carnap's phenomenalistic and physicalistic languages. In this he accepts that a more lavish ontology (e.g. admitting the physical as well as the phenomenal, or admitting reals as well as rationals) may be a valuable simplification. However, he describes the supplementary ontologies as "convenient myth", thus appearing to retain an ontological absolutism while endorsing ontologically relativistic manners of speaking. We are close here to losing the idea that quantification involves commitment, for Quine.
[this bit needs a rewrite now] In criticism of the paper one might apply the ontological commitment criteria which Quine puts forward to normal utterances relating to the entities which Quine dismisses. It would seem to be uncontentious in ordinary language to assert, for example, that there are seven colours in the spectrum, or that a certain statement has a meaning, or that a possibility of something or other exists. The most obvious way to interpret these statements is such that they constitute an example of ontological commitment by Quine's criterion, and though it may be possible to re-interpret such utterances in such a way as to avoid the ontological commitment, Quine gives us no good reason why we should. The nearest he comes is when he puts up various odd questions about "possible men" to show that their existence raises problems. It is however, not necessarily a great problem that not all questions can significantly be asked of a given entity, and there seems little reason to suppose that a question which can reasonably be asked of a material entity must also make sense when asked of an abstract entity.