Against the topic-transparency of logical operators

There are some problematic cases for defenders of the hypothesis that logical form is topic-transparent, i.e. the claim that two statements that differ only in the logical composition off its atomic components cannot differ in topic: 

On the one hand, many people, myself included, claim that tautologies are not about anything in the content of the atomic components that occur in them, but about the logical operators themselves. Presumably, sentences of the form (P→(Q→ P)) are not about whatever P and Q are about, but about material implication: in particular, they tell us that if the consequent of an implication is true, the whole implication is true as well. The basic argument for this later claim is that whatever P and Q are about makes no difference to the content of the tautology. 

Based on Wittgenstein, Lazerowitz and Ambrose, and myself, we have argued that even though sentences like “Triangles are my favorite geometrical figures” are about triangles, implicit analytic sentences like “Triangles have three sides” are not: they are about trianglehood (of, if you are an anti-realist, the concept of triangle). Under the simple assumption, that these two sentences have the same logical form (universal implications), only the former is about whatever its sentential components are about. 

Then, consider phenomena like meta-linguistic denials (Clapp 2013). Presumably, negations can be used for meta-linguistic denials and normal denials, yet while one may concede that the negation of a sentence is about whatever the negated sentence is about, the meta-linguistic denial of a statement is about the negated statement itself.

Furthermore, since topic containment is usually considered (for example, by Berto himself) to be a necessary (but not sufficient) condition for relevance, there are also cases where the logical form of a sentence is relevant to some dialectical connections like:

"I am very certain that Tesla never uttered such a thing, but it is hard to prove a negative."

Here, it is clear that it is the fact that "Tesla never uttered such a thing" is a negative sentence that makes the second conjunct relevant to the former. The second conjunct is obviously about negative statements, so it seems like the former sentence must be also somehow about its own negativity.

One may also share Plato's intuition that, at least some time, negation makes a statement be about something drastically different –– the opposite! –– of what its affirmative would be about. For example, we do not want to say that a statement like the next is about white people:

"Non-white people were not allowed in Sun City at the time Queen played their concert there"

One might even add other issues that, for example, have been raised in the literature on natural language semantics. For example, it has been argued that a conditional like "If England know what they’re doing then I’m a monkey’s uncle." (Smullyan 1978, McCawley 1981, Noh 1996, Verbrugge & Smessaert 2010, etc.) has nothing to do with its absurd consequent (and only shares topic with its antecedent), i.e., it is not actually about the speaker being a monkey's uncle at all.

In response to issues like this, we can appeal to Berto's use of Wittgenstein’s distinction between tell and show (4.0312). We can interpret his notion of “telling” as corresponding to the relation between a sentence and the thick-proposition it has as content. Thus, the idea here would be something like: these sentences still tell us what they are about (after all, that is their content), and this does not contain anything about its logical form, even if they show their logical form and sometimes this is important. But even in the cases where this is important, what they show is not part of their content. One would have to say something like: no: it has nothing to do with content, but with form, and form is still there, that is why it manifests some times, but that does not make it content.

However, in order for this response to work, as I have just mentioned, we need to accept that what is shown in a statement is not part of its content which is just rejecting the above semantic claims. But as Estrada has just mentioned, these claims are not backed just by intuitions about what these statements are about, but also by how they work under intensional operators: For example, presumably, we can know that triangles have three sides without knowing anything about any particular triangle, but we cannot know that they are our favorite geometric figures if we have not seen or somehow being acquainted with particular triangles. This difference is naturally captured by saying that only the later statement is about triangles, while the former is about trianglehood – since, presumably, we could not know that it is true without understanding what triangles are.

My comments on the second and third sessions of Francesco Berto's Cáedra Gaos on hyperintensionality 

References:

Barceló Aspeitia, A. A. (2002). Universalidad y aplicabilidad de las matemáticas en Wittgenstein y el empirismo logicista. Theoría. Revista del Colegio de Filosofía, (13), 119–136. https://doi.org/10.22201/ffyl.16656415p.2002.13.289

Clapp, L. (2013). Metalinguistic Denials and Negative Existentials. Diánoia, 58(70), 133-157.

Noh, E. J. (1996). A relevance-theoretic account of metarepresentative uses in conditionals. UCL Working Papers in Linguistics, 8, 125-163.