XPost: sci.logic
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.
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