The originality and importance of Newman's philosophical thought has generally been under-appreciated.  But in spite of such general under-appreciation, some important twentieth century philosophers (e.g., Bernard Lonergan in Understanding and Being, and Alasdair MacIntyre in Whose Justice? Which Rationality) have explicitly acknowledged their "massive debt" (MacIntyre's own words) to Newman.  And even in cases where there is no traceable influence  from one thinker to another, it can be fairly stated that Newman's thought anticipated important themes made famous by later thinkers, such as Ludwig Wittgenstein, Hans-Georg Gadamer, Thomas Kuhn, and Michael Polanyi.

In many matters, to think correctly is to think like Aristotle
"While we are men, we cannot help, to a great extent, being Aristotelians, for the great Master does but analyze the thoughts, feelings, views, and opinions of human kind. He has told us the meaning of our own words and ideas, before we were born. In many subject-matters, to think correctly, is to think like Aristotle; and we are his disciples whether we will or no, though we may not know it."
From The Idea of a University, Discourse V

Human reasoning is largely a matter of practice,  not rule-following.
"The mind ranges to and fro, and spreads out and advances forward with a quickness which has become a proverb, and a subtlety and versatility which baffle investigation. It passes on from point to point, gaining one by some indication; another by probability; then availing itself of an association; then falling back upon some received law; next seizing on some inward instinct, or some obscure memory; and thus it makes progress not unlike a clamberer on a steep cliff, who, by quick eye, prompt hand, and firm foot, ascends, how, he knows not how himself, by personal endowments and by practice, rather than by rule, leaving no track behind him, and unable to teach another. It is not too much to say that the stepping by which great geniuses scale the mountain of truth is as unsafe and precarious to men in general as the ascent of a skillful mountaineer up a literal crag. It is a way which they alone can take; and its justification lies alone in their success. And such mainly is the way in which all men, gifted or not gifted, commonly reason — not by rule, but by an inward faculty. Reasoning, then, or the exercise of reason, is a living, spontaneous energy within us, not an art."
From "Implicit and Explicit Reason," Sermon 13 in Fifteen Sermons Preached Before the University of Oxford (Notre Dame: University of Notre Dame Press, 1998).

Merely deductive, secular knowledge cannot be a principle of action.
"Science gives us the grounds or premisses from which religious truths are to be inferred; but it does not set about inferring them, much less does it reach the inference;—that is not its province. It brings before us phenomena, and it leaves us, if we will, to call them works of design, wisdom, or benevolence; and further still, if we will, to proceed to confess an Intelligent Creator. We have to take its facts, and to give them a meaning, and to draw our own conclusions from them. First comes Knowledge, then a view, then reasoning, and then belief. This is why Science has so little of a religious tendency; deductions have no power of persuasion. The heart is commonly reached, not through the reason, but through the imagination, by means of direct impressions, by the testimony of facts and events, by history, by description. Persons influence us, voices melt us, looks subdue us, deeds inflame us. Many a man will live and die upon a dogma: no man will be a martyr for a conclusion."
From Section 6 of "The Tamworth Reading Room," Discussions and Arguments

We are often possessed of 'unperceived impressions' in the sense that we are often ruled by views and prejudices which we do not even recognize as our own.
"... the impression made upon the mind [by an idea] need not even be recognized by the parties possessing it.  It is no proof that persons are not possessed, because they are not conscious, of an idea.  Nothing is of more frequent occurrence, whether in things sensible or intellectual, than the existence of such unperceived impressions.  What do we mean when we say that certain persons do not know themselves, but that they are ruled by views, feelings, prejudices, objects which they do not recognize?"
From "The Theory of Developments in Religious Doctrine," Sermon 15 in Fifteen Sermons Preached before the University of Oxford (Notre Dame: University of Notre Dame Press, 1998)

Life is for action, and action presupposes faith in things that cannot be proved.
"Life is for action.  If we insist on proofs for every thing, we shall never come to action: to act you must assume, and that assumption is faith."

An Essay in Aid of a Grammar of Assent, Chapter 4

No single human mind can fully understand any one truth, for appreheding any one proposition requires the apprehension of a large multitude of other propositions.
"No mind, however large, however penetrating, can directly and fully by one act understand any one truth, however simple.  What can be more intelligible than that 'Alexander conquered Asia', or that 'Veracity is a duty'? but what a multitude of propositions is included under either of these theses! still, if we profess either, we profess all that it includes."
An Essay in Aid of a Grammar of Assent, Chapter 5

There can be no general criterion for determining the accuracy of inferences in concrete matters.
"Every one who reasons, is his own centre; and no expedient for attaining a common measure of minds can reverse this truth; -- but then the question follows, is there any criterion of the accuracy of an inference, such as may be our warrant that certitude is rightly elicited in favour of the proposition inferred, since our warrant cannot, as I have said, be scientific?  I have already said that the sole and final judgment on the validity of an inference in concrete matter is committed to the personal action of the ratiocinative faculty, the perfection or virtue of which I have called the Illative Sense, a use of the word 'sense' parallel to our use of it in 'good sense', 'common sense', a 'sense of beauty', &c.; and I own I do not see any way to go farther than this in answer to the question."

An Essay in Aid of a Grammar of Assent, Chapter 9

The nature of inferior animals is complete from the moment such animals come into existence, but the human being begins with 'nothing realized' and must progress gradually in an activity of self-perfection and self-making.
"What is the peculiarity of our nature, in contrast with the inferior animals around us?  It is that, though man cannot change what he is born with, he is a being of progress with relation to his perfection and characteristic good.  Other beings are complete from their first existence, in that line of excellence which is allotted to them; but man begins with nothing realized (to use the word), and he has to make capital for himself by the exercise of those faculties which are his natural inheritance.  Thus he gradually advances to the fullness of his original destiny.  Nor is this progress mechanical, nor is it of necessity; it is committed to the personal efforts of each individual of the species; each of us has the prerogative of completing his inchoate and rudimental nature, and of developing his own perfection out of the living elements with which his mind began to be.  It is his gift to be the creator of his own sufficiency; and to be emphatically self-made."

An Essay in Aid of a Grammar of Assent, Chapter 9

In matters of duty, social intercourse, and taste, the individual must rely on his or her own judgment, or on what Aristotle called phronesis.
"... how does the mind fulfil its function of supreme direction and control, in matters of duty, social intercourse, and taste?  In all of these separate actions of the intellect, the individual is supreme, and responsible to himself, nay, under circumstances, may be justified in opposing himself to the judgment of the whole world; though he uses rules to his great advantage, as far as they go, and is in consequence bound to use them.  As regards moral duty, the subject is fully considered in the well-known ethical treatises of Aristotle.  He calls the faculty which guides the mind in matters of conduct, by the name of phronesis, or judgment.  This is the directing, controlling, and determining principle in such matters, personal and social.  What it is to be virtuous, how we are to gain the just idea and standard of virtue, how we are to approximate in practice to our own standard, what is right and wrong in a particular case, for the answers in fulness and accuracy to these and similar questions, the philosopher refers us to no code of laws, to no moral treatise, because no science of life, applicable to the case of an individual, has been or can be written.  Such is Aristotle's doctrine, and it is undoubtedly true."

An Essay in Aid of a Grammar of Assent, Chapter 9

Even if general moral codes are available to us, it remains necessary for individuals to apply such general codes, and thus to make their own judgments in matters of personal duty.
"An ethical system may supply laws, general rules, guiding principles, a number of examples, suggestions, landmarks, limitations, cautions, distinctions, solutions of critical or anxious difficulties; but who is to apply them to a particular case? whither can we go, except to the living intellect, our own, or another's?  What is written is too vague, too negative for our need. It bids us avoid extremes; but it cannot ascertain for us, according to our personal need, the golden mean.  The authoritative oracle, which is to decide our path, is something more searching and manifold than such jejune generalizations as treatises can give, which are most distinct and clear when we least need them.  It is seated in the mind of the individual, who is thus his own law, his own teacher, and his own judge in those special cases of duty which are personal to him."
An Essay in Aid of a Grammar of Assent, Chapter 9

Click here for an online version of Newman's Grammar of Assent