C.S. Lewis' argument is basically that there exists no basis for rejecting of the supernatural. All such attempts to do so always involve unjustified presupposition. However, his arguments are much more interesting than that, so I'll let him speak for himself:
"This point of scientific method merely shows (what no one to my
knowledge ever denied) that if miracles did occur, science, as
science, could not prove, or disprove, their occurrence. What cannot
be trusted to recur is not material for science: that is why history
is not one of the sciences. You cannot find out what Napoleon did at
the battle of Austerlitz by asking him to come and fight it again in a
laboratory with the same combatants, the same terrain, the same
weather, and in the same age. You have to go to the records. We have
not, in fact, proved that science excludes miracles: we have only
proved that the question of miracles, like innumerable other
questions, excludes laboratory treatment." (God in the Dock, p.140)"
"If anything extraordinary seems to have happened, we can always say
that we have been victims of an illusion. If we hold a philosophy
which excludes the supernatural, this is what we always shall say.
What we learn from experience depends on the philosophy we bring to
experience before we have settled, as well as we can, the
philosophical question." (Miracles, p.1-2)
"When a thing professes from the very outset to be a unique invasion
of Nature by something from outside, increasing knowledge of Nature
can never make it more or less credible than it was at the beginning.
In this sense it is mere confusion of thought to suppose that
advancing science has made it harder to accept miracles. We always
knew they were contrary to the natural course of events; we know still
that if there is something beyond Nature, they are possible."
"The whole mass of seemingly irregular experience could never have
been turned into scientific knowledge at all unless from the very
start we had brought to it a faith in uniformity which almost no
number of disappointments can shake." (Miracles, p.109)
Lewis also argues that the supernatural has, in fact, entered into the natural realm as evidenced of by man's faculty of reason:
"[We have] already discovered that in every human being a more than
natural activity (the act of reasoning) and therefore presumably a
more than natural agent is thus united with a part of Nature: so
united that the composite creature calls itself 'I' and 'Me'."
The epistemology of Cornelius Van Til is based on the principle that true knowledge cannot be had in isolation from the whole. Since God alone has access to all knowledge, His revealed truth is the only truth in which we can fully trust. According to my understanding of this, concepts do not consist of isolated instances alone but rather they are the product of the sum total of our understanding of the subject in question. Knowledge cannot be had in isolation from one's overall worldview, so true knowledge cannot be had presupposing a false worldview. If it is untrue that the natural can exist in complete independence from the supernatural, then it is impossible to keep any given concept completely isolated from this false presupposition. For that reason, it doesn't make sense to presuppose the non-existence of the supernatural. He says the following:
"The antitheist maintains that the term existence may be applied as a
predicate to any 'fact' even if the 'fact' of God’s existence is not a
fact. On the other hand the theist maintains that the term 'existence'
cannot be applied intelligently to any 'fact' unless the 'fact' of
God’s existence is a fact. In other words, the antitheist assumes that
we can begin by reasoning univocally, while the theist maintains that
we cannot begin otherwise than by reasoning analogically, i.e., on the
presupposition of the truth of that which the Scripture says of God."
(Survey of Christian Epistemology)
"So C. C. J. Webb, in his book Problems in the Relation of God and
Man, clearly indicates his agreement with the Idealistic theory of
the judgment which contends that parts apart from the whole have no
meaning, and synthesis can have no meaning apart from an equally
ultimate analysis. At the same time Webb thinks it quite possible to
investigate the phenomenon of the moral consciousness according to the
ordinary method of scientific empiricism. We may say then that on the
one hand Webb thinks it impossible to think intelligibly of the
non-existence of either God or the universe, and still wants to study
the universe as though totally new things were appearing in it, while
on the other hand he thinks it quite possible to start with the
antitheistic method of ordinary empiricism and come at last to a
theistic position." (Survey of Christian Epistemology)