Wittgenstein famously argued against this perspective, to which you could add the 'Cogito' as an example of supposedly directly accessing something both private and universal.
How do words refer to sensations? -- There doesn't seem to be any
problem here; don't we talk about sensations every day, and give them
names? But how is the connection between the name and the thing named
set up? This question is the same as: how does a human being learn the
meaning of the names of sensations? -- of the word "pain" for example.
Here is one possibility: words are connected with the primitive, the
natural, expressions of the sensation and used in their place. A child
has hurt himself and he cries; and then adults talk to him and teach
him exclamations and, later, sentences. They teach the child new pain
-- behavior. "So you are saying that the word 'pain' really means crying?" -- On the contrary: the verbal expression of pain replaces
crying and does not describe it.
For how can I go so far as to try to use language to get between pain
and its expression?
In what sense are my sensations private? -- Well, only I can know
whether I am really in pain; another person can only surmise it. -- In
one way this is wrong, and in another nonsense. If we are using the
word "to know" as it is normally used (and how else are we to use
it?), then other people very often know when I am in pain. -- Yes, but
all the same not with the certainty with which I know it myself! -- It
can't be said of me at all (except perhaps as a joke) that I now I am
in pain. What is it supposed to mean -- except perhaps that I am in
pain? Other people cannot be said to learn of my sensations only from
my behavior -- for I cannot be said to learn of them. I have them.
The truth is: it makes sense to say about other people that they doubt
whether I am in pain; but not to say it about myself.
"Only you can know if you had that intention." One might tell someone
this when one was explaining the meaning of the word "intention" to
him. For then it means: that is how we use it. (And here "know', means
that the expression of uncertainty is senseless
“When one says “He gave a name to his sensation” one forgets that a
great deal of stage-setting in the language is presupposed if the mere
act of naming is to make sense. And when we speak of someone’s having
given a name to pain, what is presupposed is the existence of the
grammar of the word pain; it shews the post where the new word is
““I can (inwardly) undertake to call THIS ‘pain’ in the future” – “But
is it certain that you have undertaken it? Are you sure that it was
enough for this purpose to concentrate your attention on your
feeling?” – A queer question. -”
These arguments and some others in his Philosophical Investigations, became the basis of the Private Language Argument.
You might also like to read about Munchausen's Trilemma, with a fundamental challenge to the pursuit of foundations of any kind to our knowledge.
It is not uncommon for people to respond with an 'argument by lack of imagination', that they feel if this is allowed to stand we cannot get anywhere with our thinking. So I will say I see the response as rooted in Hofstadter's Strange Loops as the hallmark of minds, where different modes of thought and experience are turned on each other in 'tangled hierarchies', allowing construction of a Coherentist fabric, rather than an Idealist's shining temple of knowledge built to a single architectural principle.