This is something I have to work harder to see in Plato, partly because Strauss is working with a more refined definition of conventionalism than the one I typically use:
Contrary to our first impression, conventionalism does not assert that the meaning of right or justice is altogether arbitrary or that there is no universal agreement of any kind in regard to right or justice (Strauss 108).
I understand where Strauss is ultimately going with this. However, those many times you encounter someone in a dialogue concerned with differences between law codes (cf. Minos, where the companion wonders about why people have different burial customs) or asserting the lawful is not exactly the just (cf. Memorabilia IV.5) do rely on something about conventionalism at least seeming arbitrary. Strauss continues:
On the contrary, conventionalism presupposes that all men understand by justice fundamentally the same thing: to be just means not to hurt others, or it means to help others or to be concerned with the common good.
And yet we can say that the ultimate expression of conventionalism is Thrasymachus’ position in the Republic: justice is the interest of the stronger. (I realize Strauss employs a vulgar vs. philosophic conventionalism distinction later. I am aiming to see more prior.) There certainly seems to be something arbitrary about that! Focus on “interest” and “stronger” helps bring forth the relevant considerations. “Justice” is whatever it takes for self-preservation or community (whether that community is united under a dictator or dictates itself is another question).
Conventionalism rejects natural right on these grounds: (1) justice stands in an inescapable tension with everyone’s natural desire, which is directed solely toward his own good; (2) as far as justice has a foundation in nature – as far as it is, generally speaking, advantageous to the individual – its demands are limited to the members of the city, i.e., of a conventional unit; what is called “natural right” consists of certain rough rules of social expediency which are valid only for the members of the particular group and which, in addition, lack universal validity even in intra-group relations; (3) what is universally meant by “right” or “justice” leaves wholly undetermined the precise meaning of “helping” or “hurting” or “the common good”; it is only through specification that these terms become truly meaningful, and every specification is conventional. The variety of notions of justice confirms rather than proves the conventional character of justice.
To go further: it is plain “conventionalism” in this stronger form, precisely because it is useful and accounts for enough of a variety of/within codes, completely blinds one to “natural right.” You’d have to be unjust in a certain way to see beyond “conventionalism.” You’d need to dissimulate, be a liar, the lie being that you were a true citizen. The want to be more just alone does not bring one to natural right, as we see from people who obsess over the most minute matters of legality and morality. One needs something to pause obedience and cause a “turn.”
How does Plato reduce “conventionalism” to “the good is the pleasant?” It seems the former is a serious theory about law and justice. Strauss talks about how the pre-Socratics, in bringing forth “nature,” make what is natural good and what is conventional false. The pre-Socratics are actually conventionalists. They just don’t confuse the common good with an individual’s good. For an individual’s good, you need to go to nature, and that causes the whole conventionalist thesis to unravel. It literally gets set aside in the face of what could be (and is, given that “science” emerges through the pre-Socratics) a more powerful understanding. This is in addition to any problems, of course, conventionalism may have with its own coherence.
I think one issue worth considering is conventionalism and the “common good.” Is the common good law? Justice? Not really. In our everyday political life, we can spot innumerable conflicts between the common good and law. I’d go so far as to identify a tension between both generally. If you want what is good for the community, it needs to be procured. That can happen through legal or illegal means, but at some point, it has to happen. The law points another direction entirely. For those of us reading Plato, Xenophon, Aristotle, etc. full time, we need to take seriously that the philosopher’s considerations of the good and the just may diverge just as wildly as these phenomena do in everyday life.
Strauss, Leo. Natural Right and History. Chicago: University of Chicago, 1965. p. 108