Why does the philosophy concerned with the problem of negative existential statements not make use of mental representations to solve it?

Question

Meinong, Frege, Russel and Kripke all seem to accept the principle, that for a statement to be true, its singular terms must denote an object. This leads to the problem, that a sentence like 'The difference between A and B does not exist' seems to contradict itself, because the term 'difference' must refer to an object, whose existence is denied in the same sentence. Meinong makes wild ontological claims to solve the problem, Frege states that such a sentence is broken and Russel and Kripke somehow try to replace proper names with definite descriptions to solve the problem.

To me, the by far simpler and more intuitive solution seems to be the use of singular terms not to denote objects directly, but rather to refer to thoughts and mental representations we have about objects. Since we have to perceive the object first and make a 'mental image' or representation of it, it seems clear to me that the only thing, we can refer to is the mental representation and not the object itself. This would also conveniently solve the problem of negative existential statements by saying that the difference between A and B is just a thought or a mental image, but doesn't exist in the world. But why doesn't any philosopher take this path to explain the problem?

Answer 1

Some History

Your distinction between "real" objects and merely mental objects in this context has certainly been pursued, as far back as Aquinas (and even further back when you consider Ockham), in his criticism of Anselm's ontological argument. From the IEP article on Anselm's Ontological Argument:

On Aquinas's view, even if we assume that everyone shares the same concept of God as a being than which none greater can be imagined, "it does not therefore follow that he understands what the word signifies exists actually, but only that it exists mentally."

This could be generalized to your position as follows:

Apparently non-denoting singular terms which figure in true sentences refer to objects which exist merely mentally, not actually (or really).

Now, you aren't the first one to adopt such a view. In fact, William of Ockham held precisely this view. He postulated a "mental language" which had concepts to serve as the referents of "fishy" objects like fictional entities or abstract objects. From the SEP article on Ockham:

Ockham was the first philosopher to develop in some detail the notion of “mental language” and to put it to work for him. Aristotle, Boethius and several others had mentioned it before, but Ockham's innovation was to systematically transpose to the fine-grained analysis of human thought both the grammatical categories of his time...[and] the central semantical ideas of signification, connotation and supposition....

The "language of thought" hypothesis has been defended more recently by Jerry Fodor. See the SEP article on the language of thought. He holds, much like Ockham, that nearly all lexical concepts are innate. So, e.g., we're born with a concept of "carburetor" -- this has struck many as implausible. You're likely to have similar problems since you'll have to answer the question of where these concepts with no instances came from. See this paper on radical nativism to see some criticism from a cog sci angle.

Ok, back to Ockham. From the section on his rejection of universals:

For Ockham, the only universal entities it makes sense to talk about are universal concepts, and derivative on them, universal terms in spoken and written language. Metaphysically, these “universal” concepts are singular entities like all others; they are “universal” only in the sense of being “predicable of many.”

Thus, for Ockham, putative abstract objects like universals refer to mental concepts -- and these concepts are innate.

Quine's Solution to the Problem of Negative Existentials

From my answer to this old question about imagining nothing:

Some philosophers, notably the Meinongians, have thought that there are some things that have the property of "not existing". So, they would analyze negative existentials like "There are no unicorns" as expressing the sentence "There is something such that it is a unicorn and it doesn't exist". They could do this because they distinguished between two senses of "there is". One, the one familiar to us from Quine, is to read "there is" as expressing the existential quantifier. Anything that "there is", in this sense, exists. Now, the other sense of "there is" is subsistence. They thought that there are some things (like unicorns, for example) that subsist but do not exist.

Quine thought that this talk of there being things that don't exist was a bunch of nonsense. He held that "there is" only expresses the existential quantifier and that anything there is must exist (as an aside, he famously, but uninformatively, answers the question "What is there?" with "Everything"). But then how did he analyze our earlier sentence about unicorns? He would analyze is thusly: "It is not the case that there exists something such that it is a unicorn" (this would all be regimented in first order logic for Quine). For Quine, this sentence carries no presupposition of anything's existence, much less of a unicorn which subsists but does not exist.

My Own Worries

Ok, now for the reason that I think a view along the lines of what you're suggesting is very problematic. How many concepts for nonexistent things are there? It seems to me that there are infinitely many nonexistent things (if there aren't already I can make up some right now: the round square with a 1 meter area, the round square with a 2 meter area,...).

Now, the problem here is that the assumption that we could hold a countable infinity of concepts in our head seems problematic -- it isn't obvious there's enough storage space ("memory"). Once you realize that there are different orders of infinity (e.g., the real numbers are uncountably infinite, making them strictly larger than the natural numbers, integers, or rationals; and things get MUCH bigger from there), the assumption that they could all be some sort of mental concept seems incredibly implausible. Even if we can hold countably many concepts, the higher up you go in the orders of infinity, the more it seems that storing that many concepts in a mind is impossible. So, you can't produce enough mental concepts to serve as referents for all of the possible terms appearing to refer to nonexistent objects.

So, maybe at this point you'd say "ok, the concepts don't have to actually be in anyone's mind it's enough that they're merely possible mental concepts". But mental concepts detached from any minds start to look a whole lot like abstract objects, and your view collapses into one of very many different possible views (all of Meinong, Frege, and Kripke could be read as holding such a view; Frege most obviously since he often used mental language to describe propositions ("thoughts"), even though he vehemently rejected this psychologistic picture).

Ok, that's all I can think of off the top of my head, but should give you some idea of where to go for further research.

Answer 2

It is simply not true that people have not investigated this approach. They just tend to classify themselves as something other than philosophers.

Perhaps logic should be a branch of psychology or cognitive science. But, traditionally, it isn't. We would like to imagine that the built-in human language processing system can be approached entirely from within itself: that it is not tied to our bodies, or even our temporal existence. So we have made three very different things out of the study of words: logic, linguistics, and philology, in increasing order of the degree to which they admit psychology matters.

Displacing this issue onto what is effectively a computer model, wherein the mind is viewed as a fact-rewriting machine, works for linguistics. In a Chomskian model, https://en.wikipedia.org/wiki/Deep_structure_and_surface_structure referents are generated as nameless tokens and resolved by transformations until they can be expressed. This kind of approach resolves the whole issue, but at the cost of stepping away from what 'meaning' is, and into a world that presumes the mind works a given way, albeit one that gets a lot of support when we look at how other cultures form and use their languages.

A view more embedded in philology comes from Lacan and Desassure http://nosubject.com/index.php?title=Signifier. It admits that ultimate referents never actually exist, they are players in stories, and the stories do not have meanings, they have effects. In that context, again, something like your approach makes sense. The potential existence of the difference between A and B is a negotiating position, not a fact, so its ultimate resolution is not about meaning, it is about navigating the world. The story of how it might exist matters, temporarily, just as much as the bigger story where we ultimately decide it doesn't.

But traditional ontology tends to want to support both realistic and idealist positions, to some degree. So it avoids explicitly letting logic be reduced to linguistics or philology. Traditionally, the people looking at these issues have always preferred to handle language in terms of other intuitions, and to reify imaginary ideals, rather than relying on actual acts of interaction or processing. We want a model that feels like mathematics, because then we can correctly layer our sciences, and have a single most basic kind of belief, that springs entirely from intuition. Mathematics requires that any solution should be expressible in terms of imaginary objects alone.

I think we know now, that this kind of endeavor is bound to fail. We already know the weakness of models like this from mathematics' study of itself: our natural view of negation, and our equally strong intuition of 'plenitude' (that anything logically entailed by other mental images can always be imagined) don't fit together in absolute terms. Evading that issue in logic is as important as evading it in set-theory. But we keep right on doing mathematics, and we will keep doing this.

Answer 3

The view that we refer to mental entities was actually quite common among past western philosophers, especially in the idealist period in the wake of Berkeley and Kant. Frege,  Husserl, and others rebelled against this tendency, toward to end of the 19th century, for various reasons.

For one thing, the view that we refer to mental concepts commits us to subjectivism. Suppose e.g. that "Pinocchio does not exist" means "Pinocchio exists only as a mental concept". Now since my mind is distinct from yours, my mental concepts are distinct from your mental concepts. Therefore, my Pinocchio will be distinct from your Pinocchio. Therefore, there will be not one Pinocchio, but one Pinocchio per every Pinocchio-reader mind.

Now subjectivism may be correct to some extent. But as on overarching doctrine it seems to be self-defeating. For one thing, if everything is subjective, why do we disagree, we do we argue (rationally)? Without an inter-subjective common ground, if everyone were locked-in with one's own mental concepts, there would be no point in rationally arguing among ourselves.

The later philosophers (Frege et al) came up with alternatives which are less burdened with subjectivism, and also - ontologically speaking - simpler, less committal. There is no need to posit subjective, mental structures, if inter-subjective linguistic structures can do the job just as well, and even better.