(I noodled around with the first part and so will just post the whole thing. Apologies for the weird gaps and loss of blockquotes that come with just copying and dumping worddocs. Footnote 13 is fun.)
St. Thomas’s Fourth Way is given in a shorthand that omits all the formal accounts of the concepts being used:
In some things are more and less good, true, noble, and other such things. But more and less are said about diverse things so far as they approach in diverse ways something which is most, as what is more hot more approaches what is maximally hot. Therefore something is most true, best, and noblest, and as a consequence, maximally existing (ens).
The first difficulty is that the author gives no indication what formal characteristic he has in mind that ties together the good, true, or noble though he clearly speaks of “other such things”. Just what sort of things are we to include? Richard Dawkins has leapt on this ambiguity and parodies the argument as something that would be just as effective at proving the existence of something maximally stinky. A second difficulty is in identifying in what sense things are more and less to the extent that they approach some maximum. Anyone reading the argument, for example, can easily see that the claim is nonsense when applied to the clearest sense of more and less, sc. in spatial magnitude, number, temporal duration, etc.. Even in the cases where physical science has been able to confirm certain maximum limits of more and less in physical quantities, these limits do not seem to followfrom an axiom about the nature of the more and less but simply from experimental confirmation that finds a maximum when it is there to find. The absolute velocity of light is not a deduction from the nature of velocity, nor is absolute zero from the mere phenomenological experience of things that were more and less warm, still less is either of them a deduction from a general axiom of the sort that St. Thomas appears to be appealing to.
The success of St. Thomas’s proof rests more on the clarification of the second difficulty than the first. Even if we cannot isolate what formal characteristic unifies goodness, truth, and dignity, we can readily admit that they admit of degrees. This leaves us only to articulate a coherent account of the supposed axiom that the more and less are always such with respect to some maximum. Our claim here is that the maximum is that which is per seand primo in the sense that Aristotle articulates in Posterior Analytics I c. 4-6. There are two grades underneath this which constitute the more and less that fall under the maximum. The first is that which is per se but not primo, and last what is neither per se nor primo.
Other Accounts of the More and Less
There are three notionally different accounts of the more and less (1.) that the more and less are combinations of act and potency and the maximum is non-composed act (2.) That the more and less are participated being and the maximum is the essential, and (3.) The more and less are things exemplified by an exterior formal cause, and the maximum is the first exemplar cause. Though we separate them for the sake of clarity most authors will appeal to more than one of these accounts, for example by identifying a non-composed act with something essential, or even weaving together all three accounts.
(1) For the Thomists of the Leonine revival, there was a clear account of the more and less in terms of the relation between potency and act. Thus Grenier rests his account of the Fourth Way on the “principle of causality” that “the cause of any being composed of potency and act is a being which is not composed of potency and act.” O’Brien argues that the more and less are the limited while the maximum is the unlimited, and “limitation” is understood to mean simply act’s composition with potency. Though hardly a Leonine Thomist, Owens appears to concede this an act-potency foundation to the more and less. Edward Feser preserves this tradition by seeing the fundamental account of the principle of causality as being that composite acts are caused by pure or non-composite acts. All of these interpretations are in line with the first of the twenty four Thomistic theses sc. that being is entirely divided into pure act on the one hand and a composition of act and potency on the other, with the first being the cause of the second. Indeed, this interpretation of the Fourth Way can be understood as establishing the truth of the thesis.
While it is entirely possible to demonstratively prove pure actuality from the nature of composite actuality, to restrict ourselves to this interpretation of the fundamental axiom of the Fourth Way would lead to an incoherence in Thomas’s own text since, for him, whatever account we give of this principle must include the more and less hot as one of its concrete instances. All commentators agree that purely material substances are act-potency composites and there is no doubt that heat either is such a substance or is the effect of one. Attempts to wave off St. Thomas’s example are difficult to swallow and even when interpreters of the proof do not seek to minimize the relevance of the example of heat, their interpretations frequently read as though they would have been happier if St. Thomas had not tried to give this concrete example of the principle he was working from.
(2) Interpreting the more an less as degrees of participated being relating to something essential can appeal to a number of very suggestive and powerful Thomistic texts, many of which are laid out by Gerrigou-Legrange in his characteristically thorough devotion to the littera. Consider first De potentia 3.5:
If anything is to be found participated in various degrees by several objects, it must be that, starting with the one in which it is found in the highest degree, it is attributed to all the others in which it is found imperfectly. For those things that are predicated according to more and less, this they have by reason of their greater or less approximation to one of some kind; for if any one of these were to possess this perfection in its own right, then there is no reason why it should be found in a higher degree in the one rather than the other.
Here we have a doctrine of more and less clearly tied to a notion of participation. The very possibility of hierarchy is grounded on the idea that only the maximally such can possess the formal element of the hierarchy essentially. The basis for using the word “essentially” is from ST. q. 3. a. 4: “whatever is found in anything by participation must be caused in it by that to which it belong essentially, as iron is ignited by fire”. Thus this interpretation avoids the interpretive pitfall we saw in (1) by giving an account of how fire is a concrete example of the principle in question.
Use of “participation” language has often been taken as a proof of the Platonic character of the Fourth Way and St. Thomas himself can be quoted in support of this: “all things that are diversified by participation in the more and less perfect are caused by one first being that possesses being most perfectly. Hence Plato said that unity must come before multitude.” There are, however, two difficulties in this. First, using Plato as a supporting authority is not the same thing as making him the basis of one’s opinion, and St. Thomas, both here and elsewhere, seems only to be using Plato as a supporting authority. Secondly, the Fourth Way not only is the only proof that directly quotes Aristotle, but it also seems to go out of its way to establish its bona fides as empirical science. Thomas gives the empirical example of heat not once but twice during the proof, and he directly quotes from a passage in Aristotle that is attempting to prove that the sun is somehow the cause of all being. Such a robust interest in empirical science is hard to square with an account of the proof as “Platonic”, since anything deserving the name of Platonic natural science would not arise until the eighteenth century. Indeed, the impossibility of natural science seems to be the raison d’être of the doctrine of participation, which accounted for any intelligibility of the world not in terms of any immanent feature of the world but in terms of a separate world of forms. Thus, interpreting the proof as Platonic would run into the same problem we saw in (1), sc. it could not account for fire being an example of the principle St. Thomas was appealing to.
A more general problem in the participation account is that it leaves the proof itself as either false or uncertain. Applying the distinction to the proof gives us the following major premises: whatever has X by participation is being caused by what has X essentially. But whatever has X in a merely greater or less degree has it by participation. The difficulty is that multiple beings can have one and the same thing essentially without being the cause of the participated reality. Thus, even if a participated being reduces to something essential, the essential itself is a multitude which turn is explained by a participation in the essential, and so on ad infinitum. St. Thomas’s example is a case in point: fire is essentially hot, but it is not the reason why all other hot things are so. The blood of a living animal (at least under normal circumstances) is essentially hot, along with exothermic reactions and the sun, but not because of any presence of fire within them. And so St. Thomas’s principle is either false, or we need to find some more subtle account of the essential which allows for an order of causality among things that have something essentially.
(3) Appeals to exemplar causality are some of the most ancient accounts of the Fourth Way. Banez gives the briefest explanation by first raising the objection that, if all things that were more an less were caused by some maximum then white would be the cause of all other colors and a man would be the cause of all other animals. Banez responds that white is the cause of other colors as a exemplar cause, because it surpasses others with respect to light, which is the formally the essence of color (formale quid respectu coloris) and because it has more light than the rest of the colors it is the measure of all of them. He also claims that “the same must be said about man with respect to the rest of the animals”, but provides no further account of the matter.
On of the most elegant and forceful accounts of the exemplar causality in the Fourth Way is given by Rebecca Loop in her thesis Exemplary Causality in the First Being. Her argument is as follows:
1.) The more and less exist with respect to a standard from which they really fall short of.
2.) It is only possible to really fall short of a standard if that standard is really possible.
3.) In order for a standard to be really possible, and not just logically possible, it must either exist or be in the power of some agent to bring about.
4.) Thus there is a real standard of goodness, which is not just more or less (i.e. relatively) good, which either exists or is in the power of an agent to make.
5.) Thus, there either is a being that is absolutely good, or who can bring forth absolute goodness from its own power.
6.) Such a being all call God.
On this account, God is a “maximal being” in the sense of being a standard by which all other things are judged, and so is the exemplar of all things that are, by definition, more or less good, true, and noble in relation to it. The proof works by translating “more and less” into “the deficient” and then appealing to the axiom that deficiency is a relation to a really possible standard. For all its simplicity, however, and in spite of the argument being to all appearances a sound one, it is clearly Leibnizian and not Thomistic, given its appeal to the principle that the possible is that which either exists or falls in the power of an agent. Without this crucial premise, we are left with a Thomistic account of exemplar causality as a form by which an agent brings forth an effect, and so the Fourth Way can only be seen as collapsing into the Second Way or as presupposing the argument of the Fifth Way. But the argument clearly cannot appeal to something proven afterwards, and it gives no indication that it presupposes an argument from efficient causality. This is all the more remarkable since, for St. Thomas, the causal principle is formally responsible for the truth of his theistic argument, and so to leave off mentioning the reality of a first agent cause would render a proof from exemplar causality unsound. If the proof is nothing but an appeal to exemplar causes, then it is simply a corollary to God’s agent causality of the world. Why would one bother to reestablish the proof ab initio from sensible data if it is a logical consequence of an earlier proof?
The appeal to exemplar causes shares in the same general problem of the previous two accounts. While all can serve as the basis of sound cosmological arguments, they fail to give an adequate account of the empirical application of the principles St. Thomas is working from in the Fourth Way. It is striking how often interpretations of the Fourth Way are belied by the text of the proof itself: it is called the most Platonic of the proofs in spite of being the only one to quote Aristotle directly, and the proof is frequently presented as purely metaphysical in spite of twice claiming it is working from principles that have physical application. Given that St. Thomas will say, later in the proof, that the more and less are things that relate to something maximal in a genus, one might even suspect that St. Thomas is starting with an explicitly empirical premise about things with a generic unity that he takes as able to be generalized to things with an analogical unity. As we will see, the ability to make this generalization is, for St. Thomas, a feature of the very logic of a scientific explanations, and so will apply first to empirical science and by extension to metaphysics.
Perseity is a logical feature of predicates in systematic accounts of a subject, one which is above all necessary in such accounts that seek to explain what some subject is, what properties it has, how it came about, or to give some other genus of cause for it. The need to articulate perseity first arose in response to sophistry, and in both Plato and Aristotle the per se is viewed as the proper opposite to sophistical predication and argumentation. In fact, the first discussion of perseity arises in Plato’s Sophist, and is used as his final and most damning account of sophistry. The argumentation of Sophist is prolix and obscure, but we can get the general sense of it as a response to an argument like this:
If two things are joined together into one class – i.e. given the same description – then they are the same.
But motion and rest, straight and curved, good and evil, etc. can each be joined in one class, sc. “being” or “lines” or “human acts”.
Therefore, motion and rest are the same, as are straight and curved, good and evil, etc.
Plato’s response is that combining things in a class does not mean that any possible combination is allowed; rather, it makes them the same just so far as they are so combined and no further. Black and white can each have color predicated of them, and so far as this goes they are one and the same, but neither can be predicated of the other. In later, more formalized terminology, black is the same as white qua color but not qua black, with qua signaling the presence of what Plato called the ability for combination or communion. Plato sees the failure to discern these differing modes of communion as to live in “non-being”, and he makes this the definitive mark of the sophist.
Aristotle’s develops Plato’s account of perseity as “communion” into an account of the various relationships of universality between the subject and the predicate.
 Del Prado, N. De veritate fundamentali philosophiae christianae Saint Paul’s, Freiburg, 1911. pp 227-232. See also the forceful and thorough weaving together of these notions by Edward Feser in Aquinas One World, Oxford, 2009. pp 99-109
 O’Brien, Thomas C, Metaphysics and the Existence of God. The Thomist Press. Washington D.C. 1960.
 Owens, Joseph St. Thomas on the Existence of God.ed. John R. Catan. St. University of New York Press. Albany 1980 P. 136. originally published in Monist, v. 58, 1974. p. 203-215.
 Thus Gilson: “Thomas’s example of the more and less hot should cause no illusions. It is simply a comparison, a manuducio, to help us understand the principal thesis. Certainly the “supremely hot” is only a relative supreme degree”. Gilson Etienne, Thomism: The Philosophy of Thomas Aquinas. A translation of Le thomisme sixth edition. by Shook, L.K. and Maurer, A. Pontifical Institute of Medieval Studies, Toronto, 2002. p 136. cf. also Elders who claims that the example “is only a comparison and not an instance” or “only an example” Philosophical Theology of Thomas Aquinas, p. 123 and 113. The claims are obscure and even baffling: certainly the point in making a comparison is to identify commonality between things compared. A thing might be only an example, but it still must exemplify the principle. Pace Gilson, to say that the supremely hot is only ‘relative’ is to undercut precisely what would allow it to be similar to the summum bonum. This is supposing that “relative supreme degree” has any coherent sense, to say nothing of having a sense that could be included in an axiom at the basis of Thomas’s proof. How, for example, would something “relatively supreme” be any different from something that was simply “more and less”? In fairness to Elders, he does attempt to integrate the nature of fire into his account of the fourth way by pointing out its active character, which “stresses the active character of efficient causality” (p. 116 ibid). This might, however, create more problems than it solves: for what seems to be a simple proof based on the more and less in a formal characteristic is now seeing as being exemplified by an aspect of efficient causality.
 See The One God. Trans. Dom Bede Rose. B Herder Book Company, St. Louis. 1946. pp. 145-148 .
 The text is from De spiritualibus creaturis. A. 10. It should be noted that St. Thomas here appears to give an alternate proof to the Fourth Way based on the deficient character of the human intellect. Owens appeals not only to this text but also to the proof for a subsistent existence in De ente et essentia. See Owens, Joseph, An elementary Christian metaphysics. Bruce, USA, 1963. One purely ontological presentation of the argument in the Fourth Way is given in II Sententiae D. 1 q. 1 a. 1 which concludes to something whose “natura sit ipsum sum esse.” on the basis of things whose nature can be understood without knowing whether they exist. Thomists in search of a purely metaphysical statement of the Fourth Way would do well to appeal to this text, assuming that St. Thomas did not intentionally set out to make this sort of proof more empirically based by working from principles that had application in physical science – though we claim this is exactly what he does in the Fourth Way.
 Cf. Copleston “[T]his argument puts one in mind at once of Plato’s Symposium and Republic…Aquinas was not immediately acquainted with either work, but the Platonic line of thought was familiar to him from other writers Thomas Aquinas Harper and Rowe, USA, 1976. pp125-6 also Gilson, ibid p 72 “No doubt this inquiry [into the Fourth Way] would be fruitless if we do not introduce the Platonic and Augustinian idea of participation” and Smith, Gerard Natural Theology Macmillan, New York, 1951. p 133
 We use “more and less” as divided from “the maximum”, and so modify the former by “merely”. Certainly, there is another sense in which the maximum is “more”.
 While Cajetan adds some precisions and to some extent disagrees with Banez, he nevertheless concludes his commentary on the Fourth Way saying “All colors, in the measure that they approach white, have something more of light, and consequently have the nature (ratio) of color more perfectly.” Opera Omnia Iussu Impensaque Leonis XIII P. M. Edita. Volume IV, p. 51.
 Banez, Scholastica Commentaria in primam partem summa theologicae. P. I Q. 2 a. 3 reprinted by Brown, Debuque. 1934. Translation is my own.
 In The Aquinas Review, Volume V, no. 1 1998.
 One of the Port-Royal syllogisms plays with the same ambiguity, though it presents it as a Barbara syllogism:
He who calls you an animal speaks the truth
He who calls you a jackass calls you an animal
Therefore, he who calls you a jackass speaks the truth.
The argument generalizes to a sophistical template that can make anything into its opposite. First, find a general class for the two opposites. Now make the major term “speaks the truth”, the middle term the predication of the class of one opposite, and the minor premise the predication of the opposites of each other, e.g.:
He who says black is a color speaks the truth
He who says black is white says black is a color.
Therefore, he who says black is white speaks the truth.
 See Plato Sophist XXXXX. Arisotle concurs with Plato’s judgment in Sophistical Refutations, XXXXsaying that the Sophist deals with non-being so far as he shuns perseity.