Question

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:

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.
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.
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?

Answer 1

In mathematical logic, sentences can be classified into different orders based on their complexity. The notation ℒ(ℒ(S)) is not well-formed, but ℒ(ℒ(S) = a) can be used to demonstrate an increase in the order of logic. This formulation is related to Tarski's reasoning but not directly applicable to his undefinability thesis.

Step-by-step explanation:

A sentence in mathematical logic can be classified into different orders based on the complexity of the sentence. For example, the sentence 'A is an element of B' is considered first-order logic, denoted as ℒ(A is an element of B) = 1. If we have a sentence like 'A is an element of an element of B', it would be second-order logic, denoted as ℒ(A is an element of an element of B) = 2, and so on.

Regarding the notation ℒ(ℒ(S)), it is not a well-formed expression or concept in logic. We cannot use ℒ(S) as an input for ℒ() because the function only takes a sentence S as its argument. However, we can use a modified notation ℒ(ℒ(S) = a), where a represents a fixed point. In this case, the value of ℒ(ℒ(S) = a) would be a + 1, which demonstrates an increase in the order of logic with each iteration.

This formulation does not directly correspond to Tarski's reasoning or his undefinability thesis, but it explores similar ideas in metalogic.