The phrase "and assert the existence" is correct and usable in written English.
It can be used in contexts where one is claiming or stating that something exists or is true.
Example: "The researchers aim to provide evidence and assert the existence of previously undocumented species in the region."
Alternatives: "and affirm the presence" or "and declare the existence".
Continuing in this fashion one arrives at stronger and stronger axioms that assert the existence of larger and larger levels of the universe of sets.
As we have shown above, (5), (8), and (20) are not to assert the existence of kèrén, fùqīn, and huì respectively.
However, it is evident that (8), (14), and (24) are not to assert the existence of fùqīn, liǎngsōu chuán, and kèrén.
Nor is it about some totality A: it asserts simply of each and every A that it is B. And although it does not directly assert the existence of A, it certainly presupposes it.
Second, it exhibits more clearly an essential point about the relation between the laws of motion and the inertial frames: that the laws assert the existence of at least one inertial frame.
On Hume's account, we are not entitled to assert the existence of both ideas and the minds, human or divine, that have them, but only the former.
A properly reformulated theory of 'logical' objects should have separate principles: (1) one or more principles which assert the existence of logical objects, and (2) a separate identity principle which asserts the conditions under which logical objects are identical.
Evaluating (5) is very difficult since there is no consensus on whether kinds have essences (Quine would say 'no') and there is no consensus among those who assert the existence of essences as to what they are.
They are not metaphysical propositions because they do not assert the existence of metaphysical kinds (mind and matter) but of the methodological assumptions that govern the study of mind and nature.
If the Axiom of Infinity is reformulated to assert the existence of Vω , then the ranks proved to exist as sets by Zermelo set theory are exactly those which appear in the natural model Vω+ω of this theory.
