# Does a function assigning any sentence to some 𝘢th-order logic exist?

I feel like I'm just reinventing Tarski's wheel with this idea, or maybe I'm even remembering what I've looked over with respect to Tarski's undefinability thesis and phrasing it in a way that resonates with me. Anyway, this is what I'm curious about:

1. Let ℒ(S) be a function that takes a sentence S and outputs which ath-order logic that S "belongs" to. For example, "A is an element of B," would (presumably) be first-order, so ℒ(A is an element of B) = 1. Then (I suppose) ℒ(A is an element of an element of B) = 2, and so on.
2. What happens, then, if we take S schematically and have ℒ(ℒ(S))? But this seems ill-formed, or deficient, or something along that line. Perhaps it would be better to write ℒ(ℒ(S) = a); then a (Lawverean?) fixed point of this is such that ℒ(ℒ(S) = a) = a.
3. The Tarskian maneuver: but alternatively, maybe it's not possible to use ℒ(S) so schematically, but we must always defer to ℒ(ℒ(S) = a) specifically and get ℒ(ℒ(S) = a) = a + 1 (I suppose this is of a piece with how Russell had type indexes increase).

Is the above a way to formulate Tarski's reasoning, or does it apply to a separate (if similar/related) issue in metalogic?

• I don't see any reason why such a function would be impossible to define for the Tarskian hierarchy. I think it would be better to use the word 'level' rather than 'order'. Usually in logic, 'order' refers to what you are quantifying over. First order means quantifying over things, second order is quantifying over classes or predicates, etc. What you are referring to here is the distinction between object language, metalanguage, metametalanguage, etc. Jul 8 at 21:49
• @Bumble yeah I'm not sure how it all works, the SEP article on Tarski's truth definitions said the object-/meta-language distinction is usually handled by a division of labor between the logic and some set theory, but was originally meant for the lower-/higher-order logic distinction itself. And then I don't know that "element of" would go with "things/properties" followed by "element of an element of" with "sets of things/properties of properties," since I've seen it said on the MathSE/OF that flipping the ∈ symbol converts all element-of sentences into set-of sentences. Jul 9 at 0:14
• I was wondering elsewise, then, if the lower-/higher-order sequence was schematically more useful from the perspective of logicality than reference to specific orders (like first- and second-). But I don't have much of a handle on the specifics, so trying to fly off into a system of logic whose order-signature was kept as a variable more than a constant is beyond me for now. Jul 9 at 0:18
• Sideways-speaking, are some people overblowing the importance of Lawvere's characterization of undefinability/incompleteness/halting/etc.? I've seen some commentary on Reddit that seemed to go either way, as if Lawvere's fixed-point theorem is either supposed to be really important and "demystifying" or is much less important ("trivially important," if that phrase makes any sense, maybe) and not such as takes away from the apparent epistemic impact of incompleteness and related phenomena. Jul 9 at 0:24
• If you are interested in hierarchies of logics and metalogics and non-Tarskian logics, there are some interesting papers by Eduardo Barrio on this subject. philpapers.org/rec/BARILO-2 and philpapers.org/rec/BARAHO-17 Jul 9 at 0:38