From the development of the eye to the beauty of a waterfall to the exact value of the gravitational constant, theists may claim that anything natural with any quality to it whatsoever must have been deliberately crafted with humanity in mind. This is the Argument from Design.
Even if it were correct, it's a terribly egotistical way of looking at the world. And even if it were proven to be correct, no religion would have any basis upon which to claim that the designer or creator was its particular god or gods.
The basic answer to the argument from design is that there is no substantive evidence for it and therefore 1. to assume design in the presence of alternative theories supported by substantive evidence is putting one's head in the sand and 2. to assume design even in the complete absence of alternative theories is an argument from ignorance.
Beginning with evolution and the development of intelligent humans, there is a huge amount of geological, genetic and observed evidence to support the currently held view of the "tree of life". Evolution of subspecies is observed all the time, and contrary to a common objection whole new species have been seen to emerge, and recently. (This article on speciation has some examples.)
Contrary to another creationist talking point, there are tons of known transitional fossils. Contrary to Kirk Cameron, these don't look like half of one animal joined to half of another (like his famous Croco-duck). They're more like what you get if you morph a whole picture of one into a picture of the other, but stop halfway.
To dismiss evolution as a useless series of random changes is an argument from personal incredulity, which is a type of argument from ignorance. It's also wrong. The mutations are random, but only the beneficial mutations tend to be passed on by sheer survival and procreation skills. Evolution doesn't just try random things and get it right every time, it tries everything and goes with what works. It's like trying to hit a dartboard by spraying the whole wall with a machine gun. You'll miss a lot, but you'll hit it too.
Intelligence came about because at every stage in the development of primates, the ones who are just that little bit smarter than everyone else will always have the advantage. Over millions of years, it all adds up. Along with this comes morality (since good deeds are often rewarded), an appreciation of beauty (since it helps if what's pleasing to the eye is usually not diseased, poisonous or dirty) and emotions (to motivate us to do what's helpful to us and others).
Going back to the origin of life, abiogenesis as it was called could have occurred by a number of different chemical processes. So far scientists have used electricity (lightning) and a replica of the ancient atmosphere to create amino acids, which are pretty close. With a whole world full of chemicals being blown and washed into each other and billions of years to work, there was ample time and material for the components of the first replicating organism to slowly accumulate. The huge odds against this often given by folks like Hoyle generally assume that they all had to come together at once, which they didn't. Once one little bit of DNA was off and running, evolution and exponential growth took over.
Before we tackle the whole universe at once, let's consider Earth. Someone might claim that God put Earth exactly where it needed to be relative to the Sun so that liquid water and therefore life could form. We now know, however, that there are a lot more planets out there, and probably huge numbers of undiscovered ones. It's not that Earth was placed where liquid water could form. Rather, liquid water only forms on planets of the right temperature and Earth happened to fit the bill. Lots more planets might. This is called the anthropic principle: places aren't made for humans, humans just have a chance of turning up in hospitable places. Even on Earth there are many places we can't survive, like inside volcanoes and kilometres under the sea. So, we didn't emerge from there. Big surprise.
The largest design claim has to do with the fundamental constants of the universe. Six major ones are usually mentioned: those pertaining to gravity, electromagnetism, spatial dimensions and other less famous concepts. As is repeated endlessly, the slightest difference in any of them might result in matter being unable to form or stay together. This is the "fine-tuned universe" argument.
The problem is that even if this is true, there could still be other values of the constants which support matter. Perhaps instead of changing one or two slightly, you need to shift four of them by a huge amount. Considering that some of the constants could even be negative, you've got an infinite six-dimensional sample space in which to test hypothetical universes. We may never know whether our values are the only valid ones. Or, we may stumble upon another valid combination and that'll be the end of this argument.
Besides, the anthropic principle applies again if you consider the theory of a multiverse. If there are multiple (perhaps infinite) universes each with its own set of constants, of course we're going to turn up in the universe with a friendly combination. Other life forms may be thriving in universes where we wouldn't last for a second, and understanding how would require us to re-learn physics from scratch.
Contesting the argument from design is hard work, because to be most effective you need to know the going theories for whatever phenomenon is in question. I've tackled the most common ones, but be prepared for just about anything useful or pretty to be presented as direct evidence for gods. Then you need only find out where it really came from.
I've answered this argument a lot on this site, but I think it deserves a summary question. My aim is make this a reference for any subsequent "origin" questions.
The Cosmological Argument or Argument from First Cause is the proper form of the common argument that the universe must have been deliberately created, and you can't get "something from nothing". It predates Christianity, as Plato and Aristotle had their own versions.
That's the first issue with the argument: it only attempts to prove the existence of a Creator. It is therefore a deist argument, so when a theist uses it to prove a specific god with no further logic it's a step too far. Keep an eye out for this.
The basic premise that everything finite requires a cause is the least controversial part, but even this isn't rock solid. Possible exceptions are found in quantum mechanics, where particles move about in a probabilistic fashion. Until observed, a particle may be anywhere in a small area, and in a sense is everywhere in the area. When you observe it, it picks one spot and stays there. This is of course a gross oversimplification, but the point is that there's no known force moving the particles around. There may actually be no cause as such, and the universe may be far more spontaneous than we think.
Even "something from nothing" is plausible according to a related theory that "nothing" is really a quantum foam from which matter may emerge. This is purely theoretical at the moment (it makes mathematical sense, but there's not much physical evidence), but it's worth remembering that science is actually considering ways like this in which matter could just pop out of "nothing". It can't be dismissed entirely.
Causality may also be irrelevant if time wasn't linear at the beginning, if it had a beginning. An effect must follow its cause, but this is meaningless if chronological order hasn't settled down yet.
The universe is widely regarded by lay people (who aren't young-earth creationists) to have begun with the Big Bang. This may seem counterintuitive - how can something be created by an explosion instead of destroyed? - but it was no ordinary explosion. All the matter and energy in the universe was compressed into a singularity, a point so small it had no volume at all. (Absurd as this sounds, it happens today with large amounts of matter in the centres of black holes.) Then it expanded outwards, and it's still expanding to this day. Once the matter was in that singularity, nothing was created or destroyed, only distrubuted.
How did the matter get in there? Was the Big Bang the true beginning, or a continuation of something else? We haven't a clue. A god is one hypothesis. Other universes, with their own separate systems of time and space, are another. The quantum foam is an outside chance. Who knows what else we haven't thought of.
I like the idea of a multiverse, an eternal group or series of universes setting each other off. It's got one up on gods because it's multiple instances of a known object. We know there's at least one universe (this one), while we don't have a single example of an established god. If you see a huge cabbage patch where the whole crop's been eaten, and you find one fat little rabbit in the corner, do you assume that Bigfoot must have done most of the damage? No, you wonder where all the other rabbits are hiding.
The theory of expansion and contraction, of many Big Bangs and Big Crunches, has fallen apart recently with the discovery that the expansion of the universe is apparently accelerating. That means it will never return to the singularity, and it is not cyclical in the way we thought. That doesn't stop it from being cyclical in other ways, for example stretching until it tears a hole and then draining out to somewhere else.
The point is that if you do accept that everything finite must have a cause, something must be eternal. Either it's the universe/multiverse, or it's a god. There are many theories, and potentially many more, which allow for an eternal universe which needs no cause. Therefore an eternal god is not the only option, and anything which says so is a poor attempted proof of its existence.
Note: The argument that a god created the universe based on the universe's nature, order, awesomeness, etc. is not related to causality. It's the Argument from Design, which is next on the GBA hitlist.
The Ontological Argument strikes me as the equivalent of trying to win a lawsuit on a technicality. It's a full-blown a priori attempted proof which assumes only that a perfect being is conceivable. I won't argue this point, because although definitions may differ everyone gets some image in mind upon hearing the phrase "perfect being".
The thrust of the argument is that it's greater and more perfect to exist than not to exist. Since God in theory is the greatest and most perfect thing ever, He must exist.
The most obvious problem is that the argument is not the least bit specific about which God exists. Even if the argument were unassailable and the existence of a god were proven, we would still know absolutely nothing about the god's identity or nature. Jumping immediately from the existence of a god to the existence of your god is an unsupported assertion.
If you really wanted to be annoying, you could argue that since the argument can be used to prove the existence of multiple mutually exclusive gods (say, the God of Abraham and Ahura Mazda of the Zoroastrian faith) it's obviously a flawed argument. The theist reply is of course the above point that the argument makes no comment on the god's identity and most religions just have the wrong guy, but it's a good way to make people think.
The real problem is the premise that to exist in reality is greater and/or more perfect than to exist only in the mind. Something which doesn't exist isn't more perfect than something which does, but it isn't less perfect either. It has no qualities by which this can be judged. An apple which doesn't exist isn't red, but neither is it purple. Therefore it can't be redder or less red than a real red apple.
Existence isn't a property as such either. Even if it were, it wouldn't necessarily be a positive property, or something a perfect being must have. Something destructive like an earthquake might be better if it didn't exist.
There are plenty of objections along these lines by a great many people, the most famous being Bertrand Russell and Immanuel Kant. As stated in the question, there are also a great many rephrasings of the argument which try to circumvent these objections.
The net result is that major apologetic organisations have advised that the Ontological Argument in its current forms does not stand up to scrutiny, and other arguments like the Transcendental and Cosmological Arguments (the favourites) should be used instead.
That doesn't stop a lot of YouTubers from reciting obscure forms of the Ontological Argument and expecting them to be invincible. Look it up, and enjoy the logical knots both sides get themselves into when discussing it.
I always worry when someone uses this argument, because it may mean a few things. Maybe they don't think people have the intelligence to fully comprehend such a complex-sounding argument and will accept it by default. Maybe they haven't read the objections and don't expect anyone to look them up. From a big-picture perspective, they're using a less well known argument thinking it will take people by surprise, not considering that it's less well known for a reason. It just plain doesn't work.
The Transcendental Argument for the existence of God (TAG for short) demands a certain sardonic respect due to its sheer ambition. In its full form, it claims that logic (and by extension rationality, sense, morality and any argumentation at all) can only exist if the Christian God does.
In simplified form it's not Christian-specific, but it can be used at any point in an argument to override the whole thing and declare that the argument is only possible (or evidence is only understandable as a concept, or our senses are only reliable) if there's a god, so one must exist.
This approach does not convince many atheists as far as we know. It seems like the equivalent of winning at chess by knocking over all the pieces. Nevertheless it's difficult to find a clear hole in it which Christians in particular haven't already closed with an addendum (see how much longer CARM's list of defenses is than the argument itself).
One good way to make it a lot less convincing, strangely, is to temporarily presuppose the existence of God. If God exists, He still isn't guaranteed to be the source of logic, because how could we check? We can't go to a universe without God to see whether logic fails there, either because we're stuck in this universe or because God's omnipresence extends beyond it. In other words, we can't remove God to see whether logic is independent of Him.
Therefore even if God existed and we all knew it, that logic is dependent on Him could only ever be an assertion and the Transcendental Argument is still not self-evident. If He doesn't exist, of course, then the TAG is not only moot but flat out false.
I have two other major objections which CARM's pre-emptive defenses don't fully cover. Firstly, logical absolutes, rather than being conventions, eternal or anything else CARM mentions, may not really exist at all but instead may only be apparent.
Secondly, if logical absolutes do exist, saying that they must be the product of an absolute transcendent mind is an argument from ignorance. (Likewise is the assertion that if the physical universe were to disappear they would still be true. Again, how would you check?) Even if they're not the product of the physical universe or human minds, there may be any number of unknown alternatives besides a transcendent mind, or any mind at all.
I realise that most objections to the TAG are simply alternative hypotheses and doubts as to its basic assertions, but that's really all you need. If there is any possible alternative, an argument presented as the only possible state of affairs cannot be a proof until it clearly dismisses all competition. Once the possibility of an a priori proof is gone, the TAG loses its power and is just another thing theists say.
The short answer is no, because "theism is false" is actually further than most atheists are willing to go.
Even Richard Dawkins would not unequivocally say there are no gods, which must be true in order to conclude that theism is false. The chapter of The God Delusion directly concerned with God's existence is titled, "Why there *almost* certainly is no God".
Would you say that because Dawkins felt he needed the "almost", Richard Dawkins is not an atheist? Of course not. Atheism is not a stone cold denial, it's not a positive belief that there are no gods (that's "strong atheism"). It's a lack of belief in any god or equivalent and the associated conclusion that there probably aren't any. To invite atheists to argue that theism is definitely false is like inviting an astronomer to argue that aliens are impossible.
Here it is from the standpoint of statistics. It's impossible to disprove a null hypothesis; you can only find sufficient evidence to reject it and accept an alternate hypothesis. The absence of any gods is the null hypothesis, because it requires no evidence to even suggest.
This is proven by the existence of "implicit atheists", those few with no exposure to religion and no self-taught religious beliefs. Many young children and some remote tribespeople are examples. Atheism is the default position for a human being, at least to begin with. Any god is therefore an alternate hypothesis which you may accept on the basis of whatever you regard as evidence. You can never deny the possibility outright. I don't try.
Moving from the disputed truth of theism to the pros and cons of professed theism: With the possibility of a god intact in my mind, this atheist is ostensibly at risk of paying dearly in hell while believers are rewarded. This is only likely to be the case if there is only one possible god. There are 20,000 or so invented gods and an infinite number of other possible gods. Whatever the probability of SOME god existing, the probability of a particular god is one in infinity, or near to zero as darn it. Live as though only your god or gods exist and, infinity to one, you're backing the wrong horse. The vast majority of invented gods are very unforgiving of that. Better to approach the worst-case scenario of a foreign god with no other gods in hand, I say. Besides, wouldn't any god see through a fake self-serving belief professed only to escape punishment?
As you can see, theism invites criticism even when you admit it is not definitely false.
The following is an attempt to reason against theism as strongly as an honest atheist should dare.
Premise 1: There is no apparent empirical evidence for the existence of any god or equivalent supernatural being.
(Break this premise if you like by presenting some. The value and definition of "apparent empirical evidence" will need hammering out first of course, and I won't do that here.)
Premise 2: For the truth of theism to be at all likely, some empirical evidence for the existence of a god or equivalent supernatural being would need to be apparent.
(You'd need a very good reason why some other type of evidence is acceptable, or even evidence at all.)
Conclusion: Theism is likely false.
I invite readers to try writing their own step-by-step logical arguments in a similar vein. Have fun.
(As a sidenote, Penn Jillette once wrote an article called "There is no God." You might like it. - Brian Sapient)