John M. Frame in The Doctrine of the Knowledge of God:
In important ways, the unbeliever's knowledge is like the believer's. Surveying the outline of the last section, we can say (1) that God is knowable but incomprehensible to believer and unbeliever alike and (2) that in both cases the knowledge can be described as covenant knowledge. Both believer and unbeliever know about God's control, authority, and presence. The knowledge of the unbeliever, like that of the believer, is a knowledge that God is Lord (cf. passages mentioned earlier). And both forms of knowledge are subject to God's control, authority, and presence. The unbeliever, like the believer, knows God only on God's initiative, though he refuses to obey that authority. His knowledge is not only a knowledge about God, but a knowledge of God himself (Rom. 1:21). Indeed, it is a confrontation with God as present, though he experiences the presence of God's wrath (Rom. 1:18), not His redemptive blessing (cf. Exod. 14:4, where the Egyptians' knowledge of God occurs in the midst of the experience of judgment).
... So we come to the analysis that I consider the most adequate. Let's take it in several steps. (1) All unbelievers know enough truths about God to be without excuse and may know many more, as many as are available to man. There is no limit to the number of true, revealed propositions about God that an unbeliever can know. (2) But unbelievers lack the obedience and friendship with God that is essential to "knowledge" in the fullest biblical sense-the knowledge of the believer. Yet at every moment, they are personally involved with God as an enemy. Thus their knowledge of Him is more than merely propositional. (3) The unbeliever's disobedience has intellectual implications. First, it is itself a stupid response to God's revelation. (4) Second, disobedience is a kind of lying. When we disobey God, we testify to others and to ourselves that God's Word is untrue. (5) Third, disobedience involves fighting the truth-fighting its dissemination, opposing its application to one's own life, to the lives of others, and to society. Sinners fight the truth in many ways. They (a) simply deny it (Gen. 3:4; John 5:38; Acts 19:9), (b) ignore it (2 Peter 3:5), (c) psychologically repress it, (d) acknowledge the truth with the lips but deny it in deed (Matt. 23:2f.), (e) put the truth into a misleading context (Gen. 3:5, 12, 13; Matt. 4:6), and (f) use the truth to oppose God. We should not fall into the trap of assuming that all sinners always use the same strategy. They do not always deny the truth in word or repress it into their subconscious. (6) Fourth, lying and fighting the truth involve affirmations of falsehoods. We must not assume that every sentence uttered by an unbeliever will be false; unbelievers can fight the truth in ways other than by uttering falsehoods. Yet disobedience always involves the acceptance of atheism, whether so stated in words or merely acted on in life (there is no significant difference between denying God's existence and acting as if God does not exist). (7) Fifth, these falsehoods may conflict with true beliefs that the sinner holds. At some level, every unbeliever holds conflicting beliefs, for example, God is Lord and God is not Lord. (8) Sixth, these falsehoods affect every area of life, including the epistemological. Thus the unbeliever has false notions even about how to reason-notions that may conflict with true notions that he also holds. (9) Seventh, the believer and the unbeliever differ epistemologically in that for the believer the truth is dominant over the lie, and for the unbeliever vice versa. It is not always clear which is dominant, which is to say that we do not have infallible knowledge of another's heart. (10) Finally, the unbeliever's goal is an impossible one-to destroy the truth entirely, to replace God with some alternative deity. Because the goal is impossible, the task is self-frustrating (see Ps. 5:10; Prov. 18:7; Jer. 2:19; Luke 19:22; Rom. 8:28; 9:15f.). The unbeliever is condemned out of his own mouth for he cannot help but affirm the truth that he opposes. And because the unbeliever's views are false, even his limited success is possible only because God allows it (see Job 1:12; Isa. 10:5-19). Adding to the fact that the unbeliever frustrates himself, God also frustrates him, restraining him from accomplishing his purposes (Gen. 11:7) and using him to accomplish God's purposes instead (Ps. 76:10; Isa. 45:1f.; Rom. 9:17). Thus the unbeliever's efforts accomplish good in spite of himself.
The last paragraph represents the most adequate view of the matter that I know of. Yet the question remains a very mysterious one. Scripture says that the unbeliever knows and that he does not know. Scripture does not give us an epistemological elucidation in as many terms; that elucidation must be drawn carefully out of what Scripture says about other matters. And much more work remains to be done before we will have a formulation that is credible to the church (even the Reformed churches) generally. Van Til is at his best in his Introduction to Systematic Theology (24-27) where he admits the difficulty of the questions (something he does not often do) and rests content with a description of the natural man as "a mixture of truth with error" (27). 1 will continue to assume the truth of the analysis under j above, but I would not advise anyone to be dogmatic about the details. Certainly they should not be used as tests of orthodoxy.