Final answer:
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.