Looking at the first proposition, “Is there a fallacy in someone making the claim that nothing has inherent meaning?” One must define the terms. In this case, it seems the operative word is “nothing.”
What is the meaning of “nothing.” You are accustomed to using the concept of “nothing” to indicate the absence of specific things. For instance, you may say, “I have nothing in my pocket,” meaning you have no physical objects in your pocket—or, “There is nothing my glass,” meaning your glass is empty.
However, the metaphysical meaning of “nothing” is absolute non-existence, the literal void. Non-existence does not exist. If non-existence existed, then it would not be non-existence because it existed. If it existed, it would be part of existence. “Nothing” is a concept pertaining exclusively to a relationship. It has meaning only in relation to something. No-thing means the absence of all some-things. Nothing by itself is nothing.
Michael Shermer put it: "Imagine nothing. Go ahead. What do you see? I picture dark empty space devoid of galaxies, stars and planets. But not only would there be no matter, there would be no space or time either. Not even darkness. And no sentient life to observe the nothingness. Just … nothing. Picture that. You can’t."
Indeed, the entire concept of nothing is challenging to comprehend. But, if you grasp the meaning and the difference between these two concepts, “something” and “nothing,” then you have learned the two broadest fundamentals of philosophy: existence and non-existence.
Therefore, the word “nothing” does have a meaning. This is not a logical question, but one of defining one’s terms.
Nothing means: not to exist.
Nothing is not a different kind of something. It is nothing.
However, as philosopher Quentin Smith pointed out, it is a logical fallacy to talk about “nothing” as if it were “something.” He says that doing so suggests that “there might have been nothing” and implies “it is possible that there is nothing.”
To carry his explanation a step further, in the standard usage of English, to say “there is” means “something is.’’ Therefore, to restate the proposition, to say “there is nothing” means “something is nothing.” That is a logical contradiction. A fallacy.
The famous Robert Kuhn illustrates the contradiction by exchanging the word “nothing” and replacing it with “not-something.” The proposition then reads, “not-something is something,” and makes the contradiction more apparent.
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Now for the 2nd question: “Is there a fallacy in claiming that words specifically have no inherent meaning?”
Depending upon the context, this statement may be correct. Words have the meaning assigned to them. Therefore, if the propounder of the statement is saying only this, then the statement is accurate. Of note, though, is that one primary function of definitions is to tell us what is and is not included in a concept by giving us a test or rule for membership.
A second function of a definition is to clarify the relationships among concepts. Concepts are not isolated, self-contained units; they form networks of interrelated ideas. They fit together into genus-species hierarchies.
But there is more to it than that. A concept groups things together into classes based on similarities. In some cases, like TABLE, the objects and their similarities are perceptible. You can literally see the similarity among tables. In such cases, we can employ an ostensive definition. That's a fancy name for pointing and saying, "Things like that." More often, however, the referents of a concept and the attributes they have in common are not directly observable, and we have to learn about them using other concepts that we already understand.
It is only possible to think beyond the perceptual level with words, and to think clearly, those words must have clear definitions.