I am submitting this essay to a contest, but I thought it would be good to hear some responses from this philosophical community before I continue with the final submission. Feel free to comment on the work. This may prove a valuable resource to revise my essays.

Heterology and Its Impact

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"The correct method in philosophy would really be the following: to say nothing except for what can be said - and then, whenever someone wanted to say something metaphysical, to demonstrate to him that he has failed to give a meaning to certain signs in his propositions."(1) To speak only when speaking can be done, though the prescription far predates Wittgenstein, is the driving force behind logical positivism and this composition. The aim of this work is to refine our lexicon so that our philosophical and scientific dialogues regain some of their sensibility.
When conveying any language, we have two concepts that cover the ways we issue predications to a term - [b]use[/b] and [b]mention[/b]. Use is the act of issuing a predication to the extension(s)(2) to a symbolic term, whereas mention is the act of issuing a predication to the symbol within a symbolic statement. For most, it is just a tool to prevent a use/mention fallacy like that contained in the statement: "Cats have four letters in them." It is a saving grace to acknowledge the use/mention fallacy when it is due. (It saves us from having to dissect our feline friends.) However, there are times when such an evident fallacy appears to contain more than a mere confusion between the use and mention of a term. Let us examine other statements that might meet the criticism of a use/mention fallacy, but may also be ambiguous statements containing factual claims in their use and their mention:
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1.) “Nothing is something.”

2.) “Infinity is finite.”

3.) “Nonexistence exists.”

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For some reason, these statements seem to convey messages of philosophic, mathematical, and physical interest. They may even be true, even if the statements initially sound absurd. If there are any truths to these three statements, what do they manage to convey? I will return to each of these statements with more detail later. The only present need is a momentary reflection on the statements as they read.
A wary philosopher will probably edit these statements to account only for the mention of the statements. He would alter them into forms that make the words [i]nothingness[/i], [i]infinity[/i], and [i]nonexistence[/i] truthfully carry their respective predicates - something-ness, finitude, and (barring a skeptic's concerns) existence. The move is safe, but the statements are hardly informative with such an edit. Such things can be said of every symbol or term in the human lexicon. Every letter and word is a finite, existent something. Such ideas are critical to our primitive understanding of how languages work. If it takes infinity to express [i]infinity[/i], for instance, then these words falter to our human faculties, which are likewise finite, existent, and something.
Despite the philosopher's intentions in making such a correction, he has overlooked the interconnectedness of the intended extension of a term and the capacity to mention it. Borrowing from the work of Kurt Grelling and Leon Nelson, we can show just how dire use and mention are to each other for consistent conveyances of ideas. Grelling and Nelson provided a reinterpretation of Russell's paradox in 1908, but in doing so, shifted the terrain of their discussion from mathematical logic to philosophy of language. Their means of accomplishing this was coining the terms [b]autology[/b] and [b]heterology[/b], predicating them adjectivally to certain terms, and then exposing a self-referential paradox with the use of [i]heterology[/i]. The paradox works like this:
A symbol is autological if and only if the use of a symbol applies to that symbol's mention (e.g. [i]pentasyllabic[/i]). A symbol is heterological if and only if the use of a symbol does not apply to that symbol's mention (e.g. [i]monosyllabic[/i]). These two words are mutually exclusive to each other, so no word can be consistently autological and heterological, and if a word is not heterological, then it must be autological and vice versa. From our examples, the following three statements are sound:
[i]Pentasyllabic[/i] cannot be both pentasyllabic and monosyllabic.
[i]Monosyllabic[/i] is pentasyllabic, and therefore not monosyllabic.
[i]Pentasyllabic[/i] is pentasyllabic, and therefore not monosyllabic.
These are clear, sound examples to an autology/heterology dichotomy. The paradox arises when applying these principles self-referentially, since the following two statements contradict themselves:
"[i]Heterological[/i] is heterological."
"[i]Heterological[/i] is autological."
If [i]heterological[/i] is heterological, then that means the use of the symbol, [i]heterological[/i], fits the mention of its opposite, making it autological. Yet if [i]heterological[/i] is autological, then it is heterological by virtue of its being autological.
The problem of this paradox as far as the problems of statements 1-3 are relevant is less one of a remodeling of Russell's paradox,(3) but points to a firm epistemic principle regarding the use of any symbol. The problem of [i]heterological[/i] being heterological is that heterological must contradict the intended use of itself (i.e. heterology) whenever we convey it. If we intend to use [i]heterological[/i] to denote: "Failure of the extension or definition of a symbol to apply to the symbols comprising that term," then heterology definitively fails to ever be adequately conveyed because the definition necessitates it. If we tried to pose the definition of [i]heterological[/i] to any newly devised synonym, then that term's definition, whatever that new symbol may be, destines the failure of its consistent symbolization. The failure results because all words carry the predicate of "being a symbol," making that failure a matter of deduction. [i]Heterological[/i] and other heterological terms refute the possibility of our formation and adequate communication of those terms in a way that suits their definitions. Such terms are impossible to make autological, and so they are always [b]necessarily heterological[/b].
The heterological term to our first example, however, is [b]possibly autological[/b]. A term is possibly autological when there is a possible state of the world where the symbol's mention can remain consistent with the definition by changing the term's symbolization. [i]Monosyllabic[/i] is synonymous with "containing only one syllable," and when inquiring whether [i]monosyllabic[/i] is monosyllabic, we accurately conclude its falsehood. [i]Monosyllabic[/i] is polysyllabic and pentasyllabic, but not monosyllabic. Therefore, [i]monosyllabic[/i] is heterological. But if it was in our desire to change the entire lexicon to avoid all heterological terms, it would be possible to simply change our symbol, [i]monosyllabic[/i], to make it contain only one syllable. In every instance, we can replace [i]monosyllabic[/i] with "[i]monned[/i]" and define it as "containing only one syllable." Henceforth, we could truthfully state that [i]monned[/i] is monned, and therefore autological. We retain the definition, but alter the symbol that conveys the definition. This action has minimal consequence.(4)
The problem of necessary heterology is the worst affront to the terms predicated in statements 1-3, and they lead us to one of two conclusions. Either [b]a.)[/b] there is knowledge we possess, but conveying it to others necessarily contradicts its definition, or [b]b.)[/b] all such knowledge is a farce and the terms used to refer to such knowledge are merely a foolishness we stubbornly persist. In spite of the potential epistemological conclusions, the pragmatics of the matter reduces our expression of necessarily heterological terms to speciousness. Moreover, the problem of necessary heterology prompts us to consider the ability of a term to ever become autological. This is the only objective necessary to know if a heterological term can possibly sidestep the problem of heterology and worth uttering in talks of philosophy and the sciences.
[b]1.)[/b] Consider nothingness. There is an entire philosophical branch of nothingness (nihilism), a group that simply negates any asserted value or judgment, excepting of course, the judgment and evaluation: "All is nothing." We already know that: "All is nothing," is false because, "All is nothing," means, " 'All is nothing,' is nothing." But what of an existential claim rather than a universal one? Could there ever be a statement: "This is nothing?" It would be a stretch of logic, considering how our consideration of this would lead us to seek a Kaplanian character(5) for the word, [i]this[/i].(6) Yet the character to [i]this[/i] presupposes a "something-ness" to whatever we predicate as "nothing." That means we implicitly say, "Something is nothing," when we say, "This is nothing." Something cannot both be something and nothing at the same time, so there is a major contradiction when predicating nothingness to anything, whether the subject is universal or particular.
Approaching this matter through the problem of heterology, we see that the statement: "[i]Nothingness[/i] is nothingness," is necessarily heterological. If [i]nothingness[/i] really [i]was[/i] nothingness, then it would not be anything, not even the syntactical devices we use to symbolically refer to it. All of the letters and the tones they incite a reader to produce and the definition to which it aims to refer would all have to be nothing in order for there to be an autological [i]nothing.[/i] Basic empirical observation repeatedly demonstrates the facts of letters and words being some things, as well as nothingness containing affirmative qualities (or "something-ness") to our conception of it - vacuums, blackness, and abysses, to name some popular notions. The only way we can hold worthwhile conceptions and perceptions in a way that enables us to predicate nothingness is to hold those conceptions and perceptions, an action that rejects nothingness at outset. Thus, the statement: "Nothingness is something," is sound because any assent to a thing, even nothingness, is an assent to something.
[b]2.)[/b] Infinity, in order to grasp it in its entirety, deserves to be taken as its mathematical command rather than its etymological deconstruction. Our thoughts of the etymology of [i]infinity[/i] may lead us to reject it in its entailment of "nothing to end its progression." This method allows for a syntactical loophole. A clever mathematician could reconstruct the definition of infinity in the hopes of making it autological. Rather than defining [i]infinity[/i] as "not finite," and therein dodging the problem of nothingness, he can reword the inferential command of infinity to mean, "The addition of a number to any original quantity such as that its new absolute value is greater than the absolute value of the original quantity." We can shorten this idea with the notation, [i]n+1[/i], which serves just as well as an affirmative symbol to replace the hopelessly nothing-toting infinity. With these strengthened ideas, is [i]n+1[/i] really n+1? Certainly not. In order for [i]n+1[/i] to be n+1, it would have to continue forever, a result of the denotation of n+1. In other words, n+1, is itself, another n', which would also need to follow the rule of n+1 in order to maintain consistent with n+1's definition. After five repetitions in our effort to convey n+1 in this fashion, we would produce a symbol like [i](((((n+1)+1)+1)+1)+1) - [/i] and would barely scratch the surface toward effectively communicating an n+1 idea fully and completely.
While that approach to autology is a failure, this method may provide inductive defense for our belief in infinity. One might try to show that it is possible for many of our numbers to follow n+1, and then induce that n+1 is probably a consistent view. This consideration, at first glance, is convincing. The command of n+1 applies to any given numerical value, and so it should follow universally that n+1 is universally true. When we observe the consequences of such a defense, though, the inductive proof fails in like manner to the +1' regress covered previously. It merely transfers the problem of conveyance to expressing every original value rather than every possible outcome from a single original value. We would perpetuate a regress of the following fashion: [i](n+1 = n'), (n'+1 = n' '), (n' ' +1 = n' ' ' ) - [/i] Merely assuming to follow the rule for every future value fails to account for the fact that our means of conveying the idea of infinity are contradictory to the idea of infinity. Our ability to make sense of what infinity means is grossly limited, as any method of representation of infinity is necessarily finite. To do otherwise, to actually follow the course toward adequately conveying infinity, would be a wasted a life muttering some strings of ones until blue in the face and dead in the grave.
The loss of infinity, though, is less grave than one might imagine. Universal commands of a presented value to follow a specific equation can account for what infinity failed to do. To insert any mathematical command for an extended string of finite values suffices to explain mathematically represented trends, as we could just as well introduce more numbers to further produce a trend of resultants from those numbers. This also saves other equations from my criticism of infinity. [i]2+1[/i] is 2+1 autologically because [i]2+1[/i] symbolizes what is 2+1, that being 3. If [i]2[/i] is replaced with a variable ([i]x[/i]), and we insert many different numbers in [i]x[/i]'s place, then we arrive at many separate mathematical conclusions, while retaining its autology. For an integer set 1 through 10 to be put in place of x, [i]x+1[/i] symbolizes the expression x+1, all of the values 2 through 11. The only limitation is our willingness to continue to present data that follows our given formula. Infinity is finite, but a large finitude is persuasive enough.
3.) And for nonexistence? We can follow a similar line of heterological criticism with that of nothingness, necessitate existence of our terms in order to convey them, and be done with the matter of physical nonexistence altogether. Everything, it would seem, has to exist at some level in order to have anything said of it, including nonexistence. Thus, we arrive at the paradox of nonexistence being existent and the sense in stating that nonexistence exists.
Further examination exposes a larger issue behind non-existential statements. One of the rules of replacement in predicate logic is that we can rewrite the negation of an existential quantity to a given as a universal negation of that proposition. An example of this comes in the following two logically tautologous statements:
"There are no such things as married bachelors."
"For all things, those things are not married bachelors."
This challenges upon the primitive symbol of negation, itself. All of the terms we have considered contain a negation somewhere in its definition. Strictly speaking, the negation symbol is a tool of transference of truth-values, changing truth to falsehood and falsehood to truth. However, if we are declaring a falsehood of a statement (that of there being married bachelors), we must examine [i]falsehood[/i]'s capability to denote "not truth," and of truth's ability to denote "not falsehood." [i]Truth[/i] is true, thus autological, but is [i]falsehood[/i] false? This is a liar's paradox. If [i]falsehood[/i] is false, then it is a truth by double negation; but if [i]falsehood[/i] is a truth, then it is a falsehood. The problem expressed in necessary heterologies now advances to all definitively negative ideas, whether they are expressed through negative etymologies (e.g. inconceivability, incomprehensibility, independence, and unreality) or positive ones (e.g. sameness ["not different"] and eternity ["no end in time"]).
In order to even retain the truth-functionality to our understanding of logic and language, we must have an adequate definition for the primitive operations of truth and falsehood. I am reserved to assuredly posit any one sturdy definition of truth or falsehood.(7) Though these arguments, I have posed that there are two requirements for us to deem a term's extension adequate.
The definition must avoid the problem of heterology. If the expression of a term contradicts its definition, that term must be possibly autological or we must remove it from the human vocabulary for serious conversations in philosophy or science.
We must affirm the extensions/definitions for those autological terms (in potentiality or actuality), but not negate them. Any negative definition is subject to the problem of heterology because it proposes a falsehood as a definition, therein containing a self-referential paradox.
In addition to saying only what can be said, compliance with these two rules narrows our available objects of argument, enabling us to conclude only what can be most assuredly concluded. Our most pragmatic solution for all terms that fail these two tests is to reserve them for our aesthetic exaggerations. For our more critical inquiries, we know now to keep them finite, existent, and - something. If an argument fails to uphold even these three (and other) basic requirements, it is better that we just pass over them in silence.(8)
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Footnotes:

(1) Wittgenstein, L. (1921). Tractatus Logico-Philosophicus (D.F. Pears & B.F. McGuinness, Trans.). London: Routledge. p.89 (6.53)

(2) To account for the remaining essay, it is necessary to clarify the extensions of extensions, which can be a physical entity to which a word refers (its object) or the reference of a word to an idea (its definition). For the entirety of this essay, all definitions are extensions, the extension of an idea.

(3) It still deserves some glossing over. The relation between the heterology and “the class such as that all of its subclasses are not members to themselves” is striking. Heterology, defined, is itself a predication (or class) to which all of its subclasses (including the expression of the word, heterology) is not a member to it.

(4) If a social community is dreadfully concerned over it, they could eliminate all heterological terms, even those that are possibly autological. I have presented only one method to doing this (“rewording”) because it is easier. Another method is “reabsorbing,” where we replace possibly autological terms with one or more of its autological synonyms, enough to carry the whole definition through the conjunction of already present terms. For an example of the second, we could take the word nonchalant and replace it with a conjunction of the words cool, relaxed, and detached to fully convey the idea to which nonchalant refers.

(5) Kaplan, D. (1989). Demonstratives. In J. Almog, J. Perry, & H. Wettstein, (Ed.), Themes from David Kaplan (pp.481-563). Oxford: Oxford University Press.

(6) A decent character for this is “the proximal subject or object of a sentence.”

(7) Though it is beyond the scope of this essay, an affirmative falsehood will probably have to reintroduce psychologism in one way or another.

(8) Wittgenstein, L. (1921). Tractatus Logico-Philosophicus (D.F. Pears & B.F. McGuinness, Trans.). London: Routledge. p.89 (7.0)

Bibliography:

1.) Wittgenstein, L. (1921). Tractatus Logico-Philosophicus (D.F. Pears & B.F. McGuinness, Trans.). London: Routledge. p.89 (7.0).

2.) Kaplan, D. (1989). Demonstratives. In J. Almog, J. Perry, & H. Wettstein, (Ed.), Themes from David Kaplan (pp.481-563). Oxford: Oxford University Press.

3.) Martin, R.L. (1968). On Grelling’s Paradox. The Philosophical Review, 77(3), pp.321-331.

4.) Grelling, K. (1936). The Logical Paradoxes. Mind, 45(180), pp.481-486.