{ !?!? } / { } !?!?

In my previous post, I included an animated gif which used three panels: an initial one which occupies the full space of the image, and two others which appear, one after the other, inside the larger one. (The larger one does undergo some change of its own, too, when the smaller ones are embedded.) The smaller, embedded panels function as ‘footnotes.’ When the second of the two smaller panels is embedded, Batman’s text in the larger panel reads ” { !?!? }**” and his ‘footnoted’ text in the embedded panel reads “** = { } !?!?”.

I am (affectionately) parodying here logical formulae from systems like modal logic that show whether and how operators interact, formulae like ◊∃x(Fx)↔∃x◊(Fx), (possibly, there is an x such that Fx if and only there is an x, such that possibly Fx), the conjunction of the Barcan formula and its converse, which claims one can reverse the ordering of the modal operator ◊ (possibly) and the existential quantifier ∃x (there is an x).

After I had worked out the general idea of the meme in my head, and came actually to create it, I encountered a problem which greatly delighted me. I became unsure whether the footnoted text should read “= { } !?!?” or “if and only if { } !?!?”. Why the confusion?

The presence of the curly brackets (parentheses) suggests that the text in each of the panels is a singular terms referring to a set. That makes the form of the equivalence asserted into an identity, hence “=”. But the presence of the punctuation, particularly in the footnoted text, suggests a sentence, since it is, canonically, sentences that take exclamation and question marks. If the texts were sentences, the form of their equivalence should be a sentential operator such as “if and only if”. Of course, in the main panel, the punctuation appears inside the curly brackets. This is part of the joke in Batman’s text since they occur inside, but not as indicating members of the set, which Robin’s main text implies should be the empty set. But if the equivalence is correct, then the appearance of the punctuation inside the curly brackets is interchangeable with their appearance after the brackets, in the usual place for the sentential punctuation marks “!” and “?”. This is the other part of the joke in Batman’s text. But how to tell the joke? With “=” or with “if and only if”? As you can see, I opted for the former.