Disproving Odin - An Experiment in arguing with a theist with Theist logic

[polymath257]

A couple of things. One, an inductive argument is going to have, within its premises, mention of the subject (or in this case, it's negation) it is arguing for. There is no inherent logical problem nor is there a fallacy in this fact alone. I am not sure if you have a logical reason for thinking this important in this case or that you just think it should not happen in general. I would point you back to the All Men are Mortal...Socrates is Mortal argument. 

Two, not that it matters for the argument but a Platonist would say that a number of abstract objects would be included in the set of things that do not being to exists. Many mathematicians are Platonists. Another is people like Polymath and Grandizer think the universe exists infinitely in the past so did not begin to exist. It could also be another argument that there is nothing in that set. It's really not the question the argument was designed to analyze.

Syllogisms cannot be used to prove the existence of things.  There's nothing wrong with the Socrates is mortal syllogism because it does not attempt to prove the existence of immortals.  

All men born of women are mortal
Socrates was not born of a women
Therefore Socrates is immortal

Anytime a syllogism provides new facts about the world instead of sorting out the facts we have, it involves a fallacy of some kind.

I'm not sure whether you are limiting yourself to the old Aristotelian syllogisms, but the more modern versions absolutely can prove the existence of things.

So, The statement 'there is an x such that P(x)' is equivalent to 'it is false that for all x, P(x) is false'. So, to prove an existence, we simply need to negate a universal claim. So,

All A are B.
C is not B
-----------
There exists something that is not A.

is a correct deductive form.

Your claim that syllogisms do not provide 'new' facts about the world is, while technically true, very misleading. As an example, take any non-trivial result in mathematics. That result is proven via syllogisms from previous results. Hence, in a sense, it is not 'new knowledge'. But that is clearly not a useful way to talk about things. The new, non-trivial, result is definitely new knowledge that was not clear from the previously known material and in which the syllogism produced a new perspective.

More specifically, suppose we start out in geometry class with the basic axioms for lines and angles. Later, the Pythagoren theorem is proven from those assumptions. To say that is not 'new knowledge' even though it is 'contained' in the axioms seems to be a very strange way to use the language.