The phrase "given the axiom" is correct and usable in written English.
It is often used in mathematical or logical contexts to introduce a premise or assumption that serves as the basis for an argument or proof. Example: "Given the axiom that all triangles have 180 degrees, we can prove that this particular triangle also has 180 degrees."
If truth in a framework is explained in terms of truth given the axioms of the framework, we will want to know about the truth-value of the axioms themselves.
If the framework was mathematical, existence questions, such as whether there are prime numbers greater than 1000, were to be answered by means of proof, given the axioms and rules of logic introduced by the framework.
This takes care of the clause for universal quantifiers; but the argument uses the axiom of choice, and in fact it is not hard to see that the statement that Hintikka's and Tarski's definitions of truth are equivalent is itself equivalent to the axiom of choice (given the other axioms of Zermelo-Fraenkel set theory).
Given the other axioms of Zermelo–Fraenkel set theory, the existence of bases is equivalent to the axiom of choice.
In the paper [50] of Jakubík are given the following axioms for sequential convergence in a vector space Y: (B1) If x n ?
For instance, given the following axioms (illustrated in Manchester OWL syntax [23]): ZincAtom subclassOf MetalAtom (1) MetallicCompound equivalentTo Compound and hasAtom some MetalAtom (2) ZincOxide subclassOf Compound and hasAtom some ZincAtom (3).
As Gary Hatfield writes, "the problem is not to carry out proofs (which might well be assented to, given the definitions and axioms), but to discover the axioms themselves, (which are hopelessly obscured by the prejudices of the senses)" (1986, 71).
However, questions persist regarding how quarantine worked to control this disease, given the time-tested axiom that quarantine is most useful only when patients are infectious before becoming symptomatic, thus directly preventing secondary infections (6 ).
Although one can make room for types within set theory via equivalence classes or isomorphism types in general, the basic criterion of identity within that framework is given by the axiom of extensionality and thus, ultimately, reference is made to specific sets.
These include the instances of the equivalence schema (E) for propositions: The proposition that p is true iff p. Given the truth of such axioms, it follows that propositions exist and have the features attributed to them by our axioms.
