I'm sure at some point we've all run into those most clever of theists who use "logic" to "prove" the existence of god. In this scenario, the demands of evidence for god's existence are usually met with some kind of explanation beginning with something generally accepted as truth which is then manipulated step-by-step resulting in some kind of "proof" that god must exist, based on the fact that the root argument is in some way irrefutable. They tend to rely on philosophical doublespeak, and are meant to confuse the person on the receiving end into thinking they've been provided with an argument for which there is no other answer, which in actuality is not the case. The following (taken from www.godlessgeeks.com. I thank them for this list, some of which is absolutely hilarious) are some prime (albeit skeletal: when presented during an actual argument, they are often made to look much deeper and more poetic, which is part of what makes them so confusing to some people) examples which I will break down, discredit, and reject:

1. ONTOLOGICAL ARGUMENT
(1) I define God to be X.
(2) Since I can conceive of X, X must exist.
(3) Therefore, God exists.

By virtue of this argument, yes god exists, however only as an idea. If it were otherwise, I could use this argument to "rationalize" the existence of anything really. Take the classic "invisible miniature unicorn" analogy. I could tell you I have an invisible miniature unicorn, and because I can conceive of this invisible miniature unicorn (even though I know that I simply made it up) it must certainly exist. That doesn't change the fact that I simply made it up. Rejected.

2. MODAL ONTOLOGICAL ARGUMENT
(1) God is either necessary or unnecessary.
(2) God is not unnecessary, therefore God must be necessary.
(3) Therefore, God exists.

This depends on a multitude of different criteria. Firstly, one needs to determine necessity. Then, what is it that god is necessary for? Is it actually necessary? Is god even really necessary for this to be so? The use of necessity in this argument is loose. It is based on the premise that god is required for certain things to work, a premise for which there is no evidence, only speculation. Until it is proven that god is actually a necessity, this argument is merely an opinion. Rejected.

3. ARGUMENT FROM BEAUTY, a.k.a. DESIGN/TELEOLOGICAL ARGUMENT
(1) Isn't that baby/sunset/flower/tree beautiful?
(2) Only God could have made them so beautiful.
(3) Therefore, God exists.

This is one of the more ridiculous ones out there. Since beauty is based on subjective opinion, it cannot be used as a basis for determining fact. In addition, as if it weren't already rejectable enough, there is a flip-side to this argument. If god made something so beautiful, who made something so ugly? This argument doesn't actually establish anything, it simply ignores the request for evidence and substitutes it with "Oh, that flower's quite nice. Must've been god." Rejected.

4. ARGUMENT FROM MIRACLES
(1) My aunt had cancer.
(2) The doctors gave her all these horrible treatments.
(3) My aunt prayed to God and now she doesn't have cancer.
(4) Therefore, God exists.

You hear it all the time. "My aunt/mother/dog is ill, we're praying for her recovery." The truth is, if it were only up to your praying, your loved one would most likely be a goner, unless for some reason their body is able to repair itself. The credit for healing is very often taken away from modern medicine and given to god. The truth is, if it weren't for modern medicine your aunt might not have lived long enough to get cancer, and perhaps (as radical as this may sound) it was all those horrible treatments that helped her recover. The fact that she prayed just gave you an excuse to thank god and not the doctors. Rejected.

5. MORAL ARGUMENT
(1) Person X, a well-known atheist, was morally inferior to the rest of us.
(2) Therefore, God exists

This is perhaps the most annoying argument of them all. It always seems to pop up despite actual, measurable evidence against the accusation that atheism promotes moral inferiority (in fact, there is evidence to support that the opposite is true for religion). What often adds to the absurdity of this argument is the subject in the position of "Person X." For the most part, the subject is Hitler. Hitler was not an atheist, nor was the Nazi regime an atheist organization. In fact, many fascist dictators were supported by the Vatican (since the big three in Europe - Mussolini, Hitler, and Franco, - were all Catholic) during their rise to power. Historical inaccuracies aside, there is nothing actually scientific about this argument. It's nothing more than a (rather bigoted) judgment of atheists based on the actions of one person. Rejected.

6. ARGUMENT FROM CREATION, a.k.a. ARGUMENT FROM PERSONAL INCREDULITY
(1) If evolution is false, then creationism is true, and therefore God exists.
(2) Evolution can't be true, since I lack the mental capacity to understand it; moreover, to accept its truth would cause me to be uncomfortable.
(3) Therefore, God exists.

This argument is amongst the weakest arguments for the existence of god, and I've seen it many times. In fact it makes an even better argument against god, especially the god of the person making the argument. Firstly, it is a statement of ignorance on the part of the believer, since although you may not understand evolution, plenty of other people do. On that notion it can be rejected immediately, but let's go a bit further shall we? If we were to turn this argument back from the atheist standpoint, it would look something like this:

(1) If evolution is false, then creationism is true, and therefore God exists.
(2) You believe evolution can't be true, since you lack the mental capacity to understand it; moreover, to accept its truth would cause you to be uncomfortable.
(3) Therefore, you would rather believe in something archaic and unverifiable than take the time to educate yourself on something for which there is far more evidence, simply because it makes you "uncomfortable." Rejected.

7. ARGUMENT FROM THE BIBLE
(1) [arbitrary passage from OT]
(2) [arbitrary passage from NT]
(3) Therefore, God exists.

This one is easy. It really goes for any sacred text, but in this case we'll use the bible. Keeping in mind we've asked for evidence, we've now been provided with biblical verse. We've asked for proof of certain claims, and we've been given the very text from which those claims are made. You cannot use the bible as proof of the bible, it's circular and redundant. Rejected.

8. ARGUMENT FROM LACK OF DISPROOF, a.k.a. ARGUMENT FROM SHIFTED BURDEN OF PROOF
(1) You can't prove God doesn't exist!
(2) Therefore, God exists.

This one is as old as the debate itself. Many theists know atheists like to go on about the burden of proof, but they often see it as the atheists' way of trying to get away without having to argue for their side. The truth is that the burden of proof is a scientific principle, not a cop-out. Science's default position is typically the lack of something's presence/effect, and it is presence/effect that needs to be proven. Of course, this refusal to follow the scientific method is often followed by a refusal of science, which is a basis for instant rejection.

9. ARGUMENT FROM ANECDOTAL EXPERIENCE
(1) I have experienced feelings of God's presence in my mind.
(2) Therefore, God exists.

This argument has got to be the most vacuous attempt at proving the existence of god I've ever been given. It proposes that I am basically just supposed to take your word for it, which is about as reasonable as me saying "the Earth is flat and orbited by the Sun, take my word for it." It provides no source of real information other than emotional testimony, and it's about as reliable as a telephone psychic. In what way can I measure this feeling? How can I test it repeatedly, verify it's consistency, confirm its truth? The answer is, obviously, that there is no way. So, in the spirit of your argument, I've got a feeling you're wrong! Rejected.

10. ARGUMENT FROM LACK OF EYEWITNESS
(1) You weren't there to witness abiogenesis/evolution/Big Bang/etc.
(2) Therefore, God exists.

This final argument is the mantra of creationists everywhere. They attest that one could not possibly believe something one wasn't present for. Aside from this being the epitome of the proverbial pot calling the kettle black, it's more a matter of evidence than of personal witness. We have not witnessed abiogenesis, evolution, the Big Bang etc. but we have evidence that they occurred. They've left a trace that we can examine, their results have been tested and duplicated in laboratories. Our understanding of them has grown with our investigation. On the other hand, the purveyors of religion have got a book. There is no solid, verifiable evidence of the claims it makes, and the questioning or investigation of its contents are often greatly discouraged. No, we weren't there to witness the beginning of the universe, but neither were you, and the evidence is certainly not in your favor. Rejected.

There are many, many other arguments in the same vein as these, but I think I've made my point. As "logical" as these can look (especially with the verbose, intellectually deep-seeming writing commonly involved), they are more or less excuses. They hold no more weight as proof than a straight up "it's true because I say so." Perhaps most simply put, it is a very weak attempt at intellectual intimidation and a very clever way of lying to oneself.