Atheists cannot know anything
Apologists such as Sye Ten Bruggencate claim that the possibility that a person is wrong about any or all of their beliefs is equivalent to "knowing nothing".  Related claims include Atheists Can't Logically Hold Their Opinion and Atheism Cannot Justify Reason and Truth. These criticisms are sometimes advanced together with presuppositional apologetics. This argument is a red herring since they may destroy their opponent's belief system but they still have not proved the validity of their own view. It is also a tactic to keep skeptics on a defensive footing in debates and prevents proper analysis of the apologist's views. 
- "By demonstrating that unbelievers can not argue, think, or live without presupposing God, presuppositionalists try to show unbelievers that their own worldview is inadequate to explain their experience of the world and to get unbelievers to see that Christianity alone can (make) [sic] sense of their experience. "
By questioning the basis of atheist's views, presuppositionalists sometimes claim they may ignore all logical arguments and questions placed before them on the grounds that they presuppose God. For example, Sye accuses people of being solipsists, which is poisoning the well.  This is a fallacy because any person who expresses a positive belief may be questioned about their belief without the questioner automatically incurring any fallacies.
Unreasonable burden for justification
"If you know something, it is impossible to be wrong about it"
- — Sye Ten Bruggencate 
Sye uses Plato's definition of knowledge which is "justified true belief" which equates "knowledge" with "fact" (which suffers from the Gettier problem). This jargon definition should be stated up front in a debate since, in modern discourse, it is an unusual use of the word "knowledge". Sye uses an unreasonable standard for knowledge to be considered justified: anything less than perfect justification is dismissed (this is the Nirvana fallacy). This view ignores the possibility of tentative knowledge of synthetic propositions, which accounts for the vast majority of all knowledge about reality (or arguably all knowledge), which is absurd.
Based on their equivocated concept of "knowledge" and unreasonable standard of justification, apologists arguing that God or the Bible is the only possible foundation of certain knowledge. However, they have not yet demonstrated that certain knowledge is possible for synthetic propositions. Dues to philosophical ideas such as Descartes' evil demon and brain in a vat, it is difficult to demonstrate certain knowledge of reality is even possible. It is rather odd to insist that atheists must justify something that might not even exist!
The apologist may claim that any attempt to justify our reliance on our senses must not rely on more sense experience, as this would be circular. In a sense, the origin of logic faces the problem of the Münchhausen trilemma. Apologists are trying to solve the conundrum using an argument from ignorance with the claim "God did it". Empiricists and skeptics may answer that our memory, reason and sensory experience corroborate each other (which is a form of Psychologism) and this justifies our tentative belief in their (limited) reliability, but this is not "knowledge" in the Platonic sense. Some philosophers simply take these to be foundational knowledge, for instance Karl Popper founded scientific knowledge use an intermediate approach incorporating some dogmatism, some infinite regress and some perceptual experience.  To reject our experience of our senses and reason as insufficient justification, as some apologists appear to do, is a form of skeptical extremism, which is often rejected by philosophers as absurd.
Logic does not require God
One main claim made by presuppositional apologists is that atheists can not account for their use of logic and, therefore, the theistic accounting for it is the best one. Apologists repeatedly claim skeptics use circular reasoning to justify logic or reason:
- "A philosophical rationalist, for instance, one who believes that human reason is the supreme arbiter of truth, must, ultimately, prove his point by appealing to human reason. "
This is simply incorrect, logic and reason are and always has been accounted for without invoking a creator or logic itself. Many conceptions of logic exists but one accounting based on psychologism (pioneered by John Stuart Mill and criticised by Gottlob Frege and others ) is as follows:
- Physical reality, experienced a posteriori through our senses and cognition, appears to fit certain patterns. Reality requires some subjective interpretation of phenomena for it fit into the schema of logical propositions. This enables us to seek for patterns in reality.
- We descriptively call the most fundamental patterns in reality as "the logical absolutes" of logic (the law of identity, the law of non-contradiction, the law of excluded middle). The logical absolutes exist only in our minds. The logical absolutes are called this because they are useful in our reality. As abstract concepts, they are true in all possible universes but may be inaccurate or irrelevant in some universes that are unlike our own.
- The logical absolutes provide the basis for the formulation of other laws of formal logic. Certain usages of propositions are not consistent with the logical absolutes and are called logical fallacies. Logical laws may be formalised in an abstract system similar to mathematics. These laws of logic are indirectly related to our experience of reality but ultimately derive from it.
Alternatively, we can approach the problem in another direction and devise a system of logic a priori and attempt to validate it against our experience of reality. However, an arbitrary system of thought does not always fit our observation of reality. Testing a machine to see if the machine works is not circular but is common sense and, furthermore, I do not need to know where the machine came from in order for it to work; the same goes for logic. Either way, the established laws of logic are intrinsically tied to our perception of reality and validated against our experience.
Isn't logic objective?
- "If this were true, any group of people could, at any time, change the laws of logic within their group. [...] If logic were indeed just a set of conventions, it would not be testable. Life would be arbitrary, and there would be no universal rules of logic. "
The laws of logic are often considered to be objective and axiomatically true. If they are dogmatically used as a premise, they can be treated as abstract and objectively true (this is allowed under the rules of logic). However, this dogmatic basis of argumentation does not automatically apply to conclusions about the real world, nor does it attempt to justify why that specific formulation was adopted rather than some other system of thought, which requires a subjective justification using psychologism. In short, as an abstract system of rules, logic is objective; but as a description of reality, logic is not objective.
Apologists sometimes try to equate the "objective laws of reality" with the laws of logic (which effectively reduces to the Natural-law argument). Even if the laws of reality exist objectively, they are distinct from our descriptive formulation of logic. The formulation used by humans may be specific to humans and possibly did not exist prior to humans.
Some argue that the abstract formulation exists independently of humans, rather like the Platonic concept of ideals. If this is the case, there are an infinite number of other formulations that are just as real (but do not happen to describe our world). To claim the laws of logic pre-exist and are some how significant is the Texas sharpshooter fallacy.
- -Sigh- Oh Sye - Can you be Wrong About Everything you Claim to Know?
- The Refining Reason Debate, May 31st
- Theopedia, Presuppositional apologetics, retrieved 4th July 2014
- Karl Popper, "The Logic of Scientific Discovery", p. 87
- Stanford Encyclopedia of Philosophy, Psychologism, substantive revision Mon Nov 7, 2011