One might deny that laws of logic exist, but not without presupposing the laws of logic (i.e. the law of non-contradiction). Since the affirmation of a proposition implies the falsehood of its contradictory, the denial of the laws of logic is self-refuting.
The possibility of rational inference presupposes the laws of logic (i.e. identity; non-contradiction), but the laws of logic entail that nonphysical, nonspatial, nontemporal reality of some sort be accepted. The laws of logic are not physical laws as is evidenced by the fact that they are applicable to possible worlds in which there are no physical objects. 
Given naturalism, if one is to know the laws of logic then a physical relationship with those laws along the lines of causal interaction between the brain and the laws of logic must obtain. However, it is impossible for the brain to be in such a physical causal relationship with the laws of logic if they are as described above.
Victor Reppert makes the following argument:
1. If naturalism is true, then logical laws either do not exist or are irrelevant to the formation of beliefs.
2. But logical laws are relevant to the formation of beliefs. (Implied by the existence of rational inference.)
3. Therefore, naturalism is false. 
Agreus has stated that, “Logic is self-evident so there is no reason to justify logic.”  There are a host of general skeptical worries with this response as well as an opportunity to reply with the same statement substituting “God” for “Logic”. For the sake of argument these possible responses will be set aside. Agreus has clearly affirmed that there are such things as laws of logic and has at the very least implied that they are knowable and that they are relevant to the formation of beliefs.
Agreus asked, “What do you mean when you say an atheist does not use logic as a result of their worldview?”  While my intention is not to answer in place of the intended recipient of this question I have provided one possible response to the concern Agreus has with respect to logic and atheism. The reason that Agreus cannot use logic as a result of his worldview is because his worldview does not give rise to the beliefs about logic that are assumed in the principles and respective operational features Agreus clearly accepts; these principles and operational features are in fact precluded by his worldview.
To restate what Ryft Braeloch wrote:
The point is that the non-Christian can make use of logic; however, it’s in spite of his world view, not as a result of it. Things like the laws of logic and properly functioning brains, etc., are what relevant Christian theological doctrines predict of mankind (e.g., imago Dei), whereas other views, like metaphysical naturalism, offer no such predictions. So when an atheist like Stein makes use of such terms and concepts, it is in spite of his world view, not because of it; i.e., he is borrowing intellectual capital from without. 
The things that the naturalist wishes to say about logic entail that naturalism is false. This is not a skeptical argument, nor a transcendental argument, but it might be argued that it has a transcendental thrust about it. To be sure, other details and sub-arguments will need to be provided in order to complete the argument for – say – Christian Theism. Agreus asks some excellent questions to this end:
For instance, how are the “laws of logic” like things as you state? How exactly does Christian theological doctrine make the predictions of “laws of logic” and “properly functioning brains” as you claim it does? 
Nevertheless, Agreus has a serious problem. What he has written entails that his own position is false.
 The use of “possible worlds” here is modal language akin to that typically used by philosophers. It should not be confused with Middle Knowledge or Molinism (though there is a relation), Star Trek, or Multiverse cosmologies. If you do not believe that God had to necessarily create you or save you but could have done otherwise then you are employing possible worlds semantics in layperson’s (as opposed to philosophical) terms. For more on possible worlds semantics as well as their use in theology see Jay Richards Untamed God, Ron Nash Ultimate Questions, and K. Scott Oliphint Reasons For Faith. Since the argument I am using proceeds from the hypothetical truth of a naturalistic atheism possible worlds semantics may be employed even if it turns out that they should be rejected within Christianity.
 Reppert, Victor. C.S. Lewis’s Dangerous Idea: In Defense of the Argument from Reason. Downers Grove, IL: InterVarsity Press, 2003. Pgs. 81-82. Explicit statement of the argument provided in syllogistic form and the preceding paragraphs concerning the laws of logic are quoted and paraphrased from this source (respectively).
 Ibid. The question was directed toward Ryft Braeloch.