Oftentimes in arguments, we introduce sentences (or propositions?) via prescriptions like, "Let it be that..." or, "Assume that..." or so on. How would we introduce the liar sentences as premises as such, though? The direct route is ill-formed because, "Assume that this sentence is false," being a prescription, is not truth-apt as such (same goes for, "Assume that this sentence is true"). (Or consider, "L: Assume L is true," etc.)

Would we say, "Assume that, 'This sentence is false,' is true"? I guess we could do that; but the inability to be directly introduced on its own terms is a peculiar property of a given liar (or honest) sentence, one I can't help but think has something to do with resolving the liar paradox.

  • How is "Assume that <any other sentence>" being a prescription with no truth value different for any other sentence?
    – Conifold
    Mar 12 '21 at 16:53
  • These are known as self-referential statements plato.stanford.edu/entries/self-reference Various solutions have been proposed like excluding them from formalization (good enough for math), building hierarchies of languages etc.
    – Fizz
    Mar 12 '21 at 17:23
  • Idk what difference it makes, just that it has to make one. That's what I'm trying to understand. This reminds me of how, "Is this sentence true?" also seems "off" since "this sentence" refers to a question. Mar 12 '21 at 18:12
  • I am not sure we want to introduce Liar sentences as premises at all, it is doubtful that they are truth apt. We do form "Assume that ..." with some sentences that are not truth apt, e.g. "Assume that Mona Lisa is beautiful" or "Assume that Gandalf is a wizard", but that draws on truth-like predicates other than the truth predicate proper ("true" according to popular opinion, according to fiction, etc.). There is no meaningful such surrogate for paradoxical sentences. On those theories that do assign truth value to the Liar at all, it is false.
    – Conifold
    Mar 12 '21 at 18:28
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    Isn't saying "This sentence is false" only defining the token "false" to refer to the sentence? It's only because we also use the word "false" to refer to a truth value that it is confusing. You cannot evaluate the truth value of the sentence before it is complete, so stating "This sentence has truth value <some truth value>" requires being able to resolve "This sentence has truth value <the truth value of "This sentence has truth value <....>>". and so on, ad infinitum which is an absurdity.
    – JohnRC
    Mar 13 '21 at 10:06

It's not totally clear to me what you're asking here, but one approach to self-referential statements that might be what you're looking for is Barwise and Etchemedy's use of non-well-founded sets in their book The Liar. A more book based on the same approach is Roy Cook's The Yablo Paradox, which is basically an infinitary version of the Liar; the Yablo paradox uses multiple sentences that (each) refers to sets of sentences.

The need for non-well-founded sets (for these purpose) somewhat controversial though, because e.g. McLarty argues that one can do pretty much the same thing as in Liar but without AFA by using fixed points e.g.

For example, we define [Fa_] as a function taking propositions to propositions. Then a liar proposition is a fixed point for [Fa_], a solution to p = [Fap].

He then proceeds to build an algebra for the semantics of such proportions. The ultimate point of it being that:

[Barwise and Etchemedy] show the graph-theoretic ideas Aczel used to motivate AFA are valuable in analyzing circular reference. These graphs may serve other ends besides, but so do lots of structures and we would do as well not to worry about enshrining them all in the membership relations of set theories. [My] algebras apply the graph theory directly to semantics.

So in some sense it's all about modelling such prepositions as graphs (with loops), but the dispute is at what level are these graphs encoded.

In general, one can get much of the results (e.g. final coalgebras) based on non-well-founded sets in a "classic" ZF context; see "Doing without AFA" in SEP.

As Cook mentions, more strictly speaking, in the finite case, sentences like " "L: Assume L is true," are dealt with in Gaifman's pointer semantics (which is quite inspired from computer science, I think.) Gaifman semantics deal with languages like

line 1: The sentence on line 1 is not true.

Or in a simplified presentation

(1) (1) is not true.

Essentially, after some graph-based inference, sentences get assigned values in a Kleene (strong) 3-valued logic, with the middle/GAP value standing for unresolved self-referential statements: For a slinghy more complicated example.

as Gaifman says, Pointer Semantics implies strong Kleene’s 3-valued truth table. A sentence has the form “A → A” might not be true. For example,

(3) If (3) is true, then (3) is true.

where “if … then …” is interpreted by strong Kleene’s 3-valued logic. The token (3) forms a closed loop and is assigned the value GAP. And other sentence tokens saying “If (3) is true, then (3) is true” get the value T.

He also discussed adding the "is gappy" predicate to the language.

It gets a bit more complicated if one wants to add other connectives to the logic, as Cook does; basically one ends up solving systems of equations in some logic. E.g. just the liar paradox is x = not x which has no Boolean solutions, but e.g. has a solution in Kleene's K3 (namely the x = GAP). In a more general context of multiple equations, the problem is basically satisfiability and (thus) related to graph coloring.

Since Cook is dealing with an infinite number of sentences, things get a bit more complicated as applying Rado's selection principle and what not. Aside: to give a bit of a spoiler as to a (main) result in Cook's book: he shows that the Yablo paradox has the same characteristic set as the Liar under Finzler–Aczel set theory and "below" (i.e. with AFA, SAFA, or FAFA) but that these two problems have distinct characteristic sets under BAFA (i.e. with Boffa's anti-foundation axiom.) So, whether the Yablo paradox is "circular" or not depends on that choice of axiom.

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