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Two assumptions

  1. At least "hypothetically" or "for the sake of argument," I would like to use the slingshot argument to compare and contrast various sentence-types. So for now, assume that reality adds up to One True Fact {}.

  2. Let parathetic be the adjective for sentences like, "This sentence is true," "It is unknown that this sentence is true," "This sentence ought to be false," "Using method X, this sentence is unprovable," and so on and on. These have the peculiar feature that at least in some sort of relevant logic, they are incapable of being inferred from any premises whatsoever; they also do not merit being introduced axiomatically (just consider, "This sentence is not an axiom"). So by whatever means we might persuade ourselves to believe in them, they are still "parathetic" compared to the thesis behind any of our other beliefs.

This is the puzzle: suppose that a parathetic sentence possibly corresponds to a fact. Then suppose this fact is absorbed into the , since all facts are supposed to really just be subfacts (at best) of the . Now, the sentential map of the is an immense conjunction, so the presence of a parathetic sentence, in the logical manifold of the , transfers the characteristic function of the local parathesis into the thetic attire of the entire . So, for example, if, "This sentence is false," is absorbed into the , then the itself can be interpreted as saying of itself, "I, the , am false," or, "~." Or if the absorbs, "This sentence ought to be true," it makes everything, even the interior moral deficiency of parts of the 's world, so that it ought to be as it is, "which is anathema."

However, it seems as if the Gödel sentence need not be detained outside of the rest of reality. "I can't be proved by method X," when sucked into the , just has the then announce, "I, the , can't be proved by method X," which is true, after all (indeed, the axiomatic components of the are unprovable by any method whatsoever, in the limit).

So what does this mean? Can we use the concept of the as a "filter" on parathetic sentences, indicating the allowed and disallowed ones? My initial idea was a rejoinder to Priest's argument that the apparent dialetheia of the liar sentence then becomes a broader counterexample to the LNC: we counterargue that even if the liar sentence inescapably generates a paradox, it does so in a parathetic form that cannot be absorbed into the rest of factual reality, wherefore it is only an LNC-counterexample outside of the rest of factual reality, which isn't saying much, then.

Reasoning: merging the indexical aspect of the parathetic sentences into the rest of reality reinterprets the internal indexical at play, so that "This" now refers to the factive total, not just one part. It is harder to see that examples like, "L: L is π2% true," would transfer their self-reference to the whole, and maybe they wouldn't transfer this, but I'm assuming (for now) that they would. I don't remember anything to the contrary from my notes comparing the L-liar and the liar index, anyway.

Kristian Berry
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  • You aren't proposing conjoining the parathetic sentence to the OTF; you are conjoining a different sentence of the same form. This is not justified by your premise. For example, if the parathetic sentence is A: "this sentence is true", then you are suggesting on the basis of the truth of A, adding not A to the OTF but rather the sentences B "OTF is true". You can't justify B on the ground that A is true. – David Gudeman Apr 28 '22 at 00:39
  • I was thinking if we used a "logical microscope" on the OTF, we'd find a segment of its sentential map that read something like "... this-or-that-is-such-and-such, so-and-so did what or not, I am a false sentence, they went that way, someone did something..." So suppose that further segment, "I am a false sentence," corresponds to a (sub)fact. If all subfacts are absorbed into the OTF, or rather if every true sentence refers to the same OTF, then the truth of, "I am lying," is in reference to the same OTF. – Kristian Berry Apr 28 '22 at 00:45
  • But how could, "I am lying," refer to the same ultimate fact as every other true sentence without amounting to a negation over every other true sentence? And hence over the OTF "from within" (though "from the outside" the actual OTF would be the same as the internal ~OTF, i.e. the real immense conjunction would be the negation over the illusory/hallucinatory conjunction). – Kristian Berry Apr 28 '22 at 00:48
  • Phrases like "I" and "this sentence" are a category in logic called indexicals. The semantics can get pretty complicated but generally you have to resolve the indexical before you can assign a truth value to a sentence. There are exceptions like "I wrote this" which are always true regardless of how you resolve the indexical, but this isn't one of those cases. Basically, you are resolving the indexical one way to get a truth value and then adding it to the other sentence and resolving it in a different way. It's like interpreting a bound variable different ways in the same binding context. – David Gudeman Apr 28 '22 at 03:27
  • I would be hard pressed to think that indexicals work "just like that" in the joint context of Davidsonian truth-functional semantics and deviant beings like the liar index. At any rate, the question is not whether the OTF absorption-transfer mechanism is "real," since I myself don't even believe in the reality of the OTF itself in the first place. The question is just whether, supposing such transfer takes place, then "counterfactually" some absorbed sentences "go through," others don't, and is there anything useful in the metalogic to be gleaned from this? – Kristian Berry Apr 28 '22 at 09:44
  • Well, I'm at a loss. Maybe I'm just not understanding you, because to me it looks like you are proposing a logical operation that is obviously not valid. – David Gudeman Apr 28 '22 at 13:28

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