Has any other mathematician defined mathematical sentences in this particular way? I have come up with a particular way to represent mathematical sentences, such that even distinct sentences with the same truth value are indeed distinct entities. I came up with this way while thinking of how to differentiate the sentences $4=4$ and $2+2=4$. I would define the first sentence as the ordered triple $(4,=,4)$, where the second component is the equality relation on the natural numbers. And, I would define the second sentence as the ordered triple $((2,+,2),=,4)$, where the first component is itself the ordered triple $(2,+,2)$, whose second component is the binary operation of addition on the naturals. And also, for example the sentence $2+(2 \times 1) = 4$, would be represented in my scheme as the ordered triple $((2, +, (2, \times, 1)), =, 4)$. I hope it is clear now how to go about doing this. Anyway, my real question is, has any other mathematician done this or some very similar thing? I would love to see some references for this.

The idea of defining formulas of finitary logics as tuples is quite old. See for instance R. Smullyan: First-Order Logic. New York, 1968, p.7, where formulas of classical propositional logic are defined as 1-, 2- or 3-tuples.
