Higher-order logic is the union of first-, second-, third-, ..., nth-
order logic; i.e., higher-order logic admits quantification over sets
that are nested arbitrarily deeply.
https://en.wikipedia.org/wiki/Higher-order_logic
*All orders of logic in one formal system*

There are many ways to further extend second-order logic. The most
obvious is third, fourth, and so on order logic. The general principle,
already recognized by Tarski (1933 [1956]), is that in higher order
logic one can formalize the semantics—define truth—of lower order logic.
https://plato.stanford.edu/entries/logic-higher-order/

"Simple type theory, also known as higher-order logic"
The seven virtues of simple type theory
https://www.sciencedirect.com/science/article/pii/S157086830700081X
*All orders of logic in one formal system*

Thus a single formal system have every order of logic giving every
expression of language in this formal system its own Truth() predicate
at the next higher order of logic.

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 4/1/24 12:15 PM, olcott wrote:
> Higher-order logic is the union of first-, second-, third-, ..., nth-
> order logic; i.e., higher-order logic admits quantification over sets
> that are nested arbitrarily deeply.
> https://en.wikipedia.org/wiki/Higher-order_logic
> *All orders of logic in one formal system*
>
> There are many ways to further extend second-order logic. The most
> obvious is third, fourth, and so on order logic. The general principle,
> already recognized by Tarski (1933 [1956]), is that in higher order
> logic one can formalize the semantics—define truth—of lower order logic.
> https://plato.stanford.edu/entries/logic-higher-order/
>
> "Simple type theory, also known as higher-order logic"
> The seven virtues of simple type theory
> https://www.sciencedirect.com/science/article/pii/S157086830700081X
> *All orders of logic in one formal system*
>
> Thus a single formal system have every order of logic giving every
> expression of language in this formal system its own Truth() predicate
> at the next higher order of logic.
>

Note, Tarski doesn't talk about a "Higher Order" as in Higher Order
logic, showing the truths, but going to a Meta-Logic about the Logic system.