I hear a lot of people say things like "The set of all sets does not exist". But this seems to be improper to say. I think one should really say that "There does not exist a set of all sets". Anyway, what I am really asking is, are there any philosophers who think that talk of non-existent objects is merely a language game? And if so, can anyone direct me to such a paper? I am only using the set theory example merely as one example. I could just as well have used "unicorns", "Santa Claus", or "Harry Potter".
Speaking of non-existent objects has been a major source of confusion. Bertrand Russell caught this language bug and developed his famous theory of description. Basically, it says:
A name that does not have a corresponding object is called nonsense. It is not OK to speak nonsense.
If an object is spoken of in the form of "the so-and-so," that object is mentioned by description, where "the so-and-so" can be interpreted as "the x that satisfies ϕ(x)."
"The so-and-so does not exist" means "for all the x's, ϕ(x) is false." It is OK to speak of non-existent object by means of description.
The set theory example is a very good one in my opinion (I am biased), and can serve as a proxy for many other versions. What are we doing in our heads when we 'form a set'?
Naively and intuitively, we are collecting together some objects (possibly an infinite number of them) and trying to consider them all at once. So when I say 'the set of all real numbers', I would like to consider all real numbers together at once as opposed to some individual real number. Similarly, when I consider 'a real number' I am actually considering an infinite subset of rational numbers, since each real number is isomorphic to a Dedekind cut of rational numbers.
It turns out that we can naively form sets like this for a long time without much issue -- this was what mathematicians did prior to the revolution in set theory that occurred in the 20th century. But at a certain point, the collections we're considering become so large that paradoxes are foisted upon us if we do not explicitly create a new notion of 'forming a large set' in our head that is distinct from the 'forming of smaller sets' we were intuitively doing earlier.
This is the basis for a distinction between a 'proper class' and a set. In MK class theory, a prominent set theory capable of handling and predicating upon proper classes, we say that all collections are classes -- sets are those classes which are already members of some other class, and a class is a proper class whenever it is not a set.
This seemingly technical process allows us to legitimately avoid paradoxes like the one you mention -- the class of all sets which are not members of themselves is not a member of itself and not a set. We don't actually have to stop here though; we can then define a notion of 'super-classes' which contain proper classes as their members, and 'super-duper classes' which contain super-classes as their members, etc. This is all legitimate, however if we prove a statement A true for all sets that does not make it true for any proper class a-priori, and proving it true for all proper classes does not make it true for any super-classes, etc.
All of this business is driving at the idea that we can sometimes create solid and rigorous notions of objects which are not well defined when approached naively, if enough care is taken. That being said, I would argue that we cannot coherently discuss objects with properties that are true and false at the same time in the same sense. If we try to form the set of all sets which do not contain themselves, then it is not a member of itself which implies that it is a member of itself which implies that it is not a member of itself... This object is incoherent in my opinion and accordingly ineligible for rational discussion, as is the class of all classes not members of themselves, and the super-class of all super-classes not members of themselves -- the super-class of all classes not members of themselves, however, is probably a well defined object.
Some objects are only 'non-existent' as a result of our poor decision making for the logical context in which they are defined, like the class of all sets not members of themselves or unicorns or Harry Potter (in our universe). There are particular objects, however, which essentially do not exist outside of my ability to ask you to consider them and as such are actually logical red herrings, like the set of all sets not members of themselves.
To answer the first part of the question: The set of all sets does not exist in certain variants of set theory (for instance Zermelo-Fraenkel set theory ZF) but in other variants it is accepted. In most accepting variants it is called a class because certain special properties of the notion set do not apply. But if a set theory is applicable to reality, then all things that exist must exist in this theory, and there cannot be anything excluded. The set of all physical objects and all ideas derived from them does certainly exist. So it is somehow justified to talk about the set of all sets.
But we talk even about the pink elephant although such an animal does not exist. The "the" is here justified as hinting to something that is not a real object but is, as an idea, common knowledge of most people.
In most cases of similar nature, including the (among mathematicians) well-known although not always existing set of all sets, the "the" is used because of this general knowledge, I presume.
The question: "Is it proper to speak of non-existent objects? ... Anyway, what I am really asking is, are there any philosophers who think that talk of non-existent objects is merely a language game?"
Talk of non-existent objects is much more than merely a language game. Please refer to the following where I motivate my answer.
It seems the metaphysics of mathematics with regard to the question, relate to cosmology, The "set of all sets" and Parmenides's "One" as opposed to Descartes's ontological view below.
"For I have certainly no cause to complain that God has not given me an intelligence which is more powerful, or a natural light which is stronger than that which I have received from Him, since it is proper to the finite understanding not to comprehend a multitude of things, and it is proper to a created understanding to be finite; on the contrary, I have every reason to render thanks to God who owes me nothing and who has given me all the perfections I possess, and I should be far from charging Him with injustice, and with having deprived me of, or wrongfully withheld from me, these perfections which He has not bestowed upon me." (Descartes 1641: 22)
The question asks for a generalized answer about referencing non-existing objects. To say generalizing is wrong, is a generalization, and therefore is "self referentially incoherent". A generalization could exist about the problem, but knowing whether such a generalization is correct, will take much deliberation, research and open discussion, if it is at all possible to reach consensus about the problem.
Referring to non-existing objects is a methodology to influence others. It is big business and many people use the methodology to survive. My argument against it is, the methodology puts false puzzle pieces in people's minds, which hinder abilities to think creatively. Creative thinking happens when true puzzle pieces in minds are put together to form new truths. The more false puzzle pieces exist in minds the more difficult it is to think creatively. Creativity is important for sustainability. Functionalism is relevant, especially in the media industry, medical industry and the interrelation between religions and utilitarian economies. Referring to Santa Claus influence children, especially with regard to consumption of utilities, which many people depend on to survive.
Your question is a topical question, which is not openly discussed and I would also like to see papers about the question. I think the reason why papers are not seen is because basically the topic is not open for discussion, due to vested financial interests. "Discussions" about the problem sometimes end in despotic accusations like "you think you are God", because people who use functionalist methodology, oppose the idea Truth (correspondence theory).
The problem relates to religion because Aquinas (1273) i.e. referred cosmologically to "God Himself Who cannot lie". Descartes (1641: 19) wrote: "From this it is manifest that He cannot be a deceiver, since the light of nature teaches us that fraud and deception necessarily proceed from some defect."
Honesty in religion and philosophy relates to "gods" and "goddesses", which is a sensitive issue many do not like to discuss.
Nietzsche (1886: 156) wrote: “Honesty – granted that this is our virtue, from which we cannot get free, we free spirits – well, let us labour at it with all love and malice and not weary of 'perfecting' ourselves in our virtue, the only one we have: may its brightness one day overspread this ageing culture and its dull, gloomy seriousness like a gilded azure mocking evening glow! And if our honesty should one day none the less grow weary, and sigh, and stretch its limbs, and find us too hard, and like to have things better, easier, gentler, like an agreeable vice: let us remain hard, we last of the Stoics! And let us send to the aid of our honesty whatever we have of devilry in us – our disgust at the clumsy and casual, our 'nimitur in vetitum', our adventurer's courage, our sharp and fastidious curiosity, our subtlest, most disguised, most spiritual will to power and world-overcoming which wanders avidly through all the realms of the future – let us go to the aid of our 'god' with all our 'devils'! It is probable that we shall be misunderstood and taken for what we are not: but what of that! People will say: 'Their “honesty” - is their devilry and nothing more!' But what of that! And even if they were right! Have all gods hitherto not been such devils grown holy and been rebaptized?"
The Greek word "chrestotes" appears eight times in the New Testament and was translated as "goodness", "kindness", "integrity" etc. of God (BST). In Plato's works "chrestotes" was translated with "honesty" (Plato 350BC: 412e). Honesty is probably also partly the cause of the functional argument "you think you are God", which Caiaphas used in his utilitarian argument to sacrifice Jesus. The ontological realization; each person has different definitions for words of definitions for words of definitions ad infinitum, motivates to be more honest; to consider others' meanings of words, when communicating.
Caiaphas syndrome is a term, which was first published, according to my research, around 1994. The term relates to the problem, because it is influenced by the idea that "God is" honest and singular.
AQUINAS; T. 1273. Summa theologica: treatise on the theological virtues: of the act of faith, article 4: whether it is necessary to believe those things which can be proved by natural reason? From: http://www.sacred-texts.com/chr/aquinas/summa/sum257.htm on 19 June 2013.
BST. From: http://www.biblestudytools.com/lexicons/greek/kjv/chrestotes.html on 23 Sep 2017.
DESCARTES; R. 1641. Meditations on First Philosophy. In: Internet Encyclopedia of Philosophy, 1996; The Philosophical Works of Descartes, 1911. Cambridge University Press. From: http://selfpace.uconn.edu/class/percep/DescartesMeditations.pdf on 20 Sep 2017.
NIETZSCHE; F. 1886. Beyond Good and Evil, Prelude to a Philosophy of the Future. London: Penguin, 3rd edition, 2003.
PLATO. 350BC. Definitions. In: Plato Complete Works; Cooper & Hutchinson, eds. Cambridge: Hackett, Kindle edition, Location 46914, 1997.