I am not asking for a defense of or pro/con of the existence of an omnipotent (or multiple omni-x) being, or for the existence of square-circles or any other similar thing. These arguments are well documented within this site, for example Is the definition of God consistent?
My question concerns the terminology associated with a logically inconsistent definition or an argument flowing from it, and whether or not assigning a truth value to the conclusion of such argument is a named logical fallacy.v
The basic laws of logic indicate:
- a valid argument is one such that if all propositions are true, then the conclusion is true.
- if any proposition to a valid argument is false, then we cannot determine whether the conclusion is true or false. It may remain true even though a proposition is false.
- with an invalid argument, it doesn't matter whether the propositions are true or false - we can never determine the truth of the conclusion.
Given the above, what happens if I define something in a way which is logically inconsistent and use that definition in an initial premise/proposition for an argument?
An illogical/incoherent thing is not able to be addressed by logic, other than perhaps to assign it to the set of objects which are incoherent. So how does an incoherent definition flow in an argument?
If a definition used in a premise/axiom/proposition of a logical argument is illogical/incoherent/paradoxical, then do we say that the proposition itself is incoherent or paradoxical as a result?
Continuing on to the logical argument that flows out of such a proposition,**
- Do we say that such an argument is also incoherent or paradoxical because one of its propositions is?
- Or is it more correct to say that such an argument is simply invalid? Which is to say we cannot establish the validity of such an argument (it is outside the realm of logic to determine it's validity)
- Or something else?
Is there a name for the fallacy of attempting to determine the logical truth value for the conclusion of such an argument?
Note: One may also be able to discuss this in mathematical terms, with the concept of infinity, division by zero and similar concepts which can be used to show impossible things (i.e. 2 + 2 = 5, etc.) by using improper or illogical definitions at the start of the proof
Note 2: I don't think this requires going to a formal system of symbolic logic - if it does, please help me understand why