John Henry Newman, answered from the texts and cited to the page.
The illative sense is simply the mind reasoning — but reasoning as it actually operates, not as logicians have tidied it up for display. It is the power by which a trained or naturally gifted intellect moves from antecedents to consequents, weighing converging probabilities, detecting the hidden assumptions that lie beneath every argument, and arriving at a conclusion with a sureness that no formal apparatus of words and propositions could have delivered.1
Consider what it is not. It is not a trick of art, however elegant; no diagram, no syllogism, no algebraic notation does the work that the living mind does when it determines where proof ends and where mere probability falls short.2 The logician's machinery is useful — indispensable, even, for communication — but it is serviceable rather than sufficient, and imperfect at that.1
The mind must already be at work before the first premiss is set down, sorting and combining what is to count as evidence, and it remains at work after the last step, judging whether the conclusion actually follows from the whole. The closest analogy I know is Aristotle's phronesis — practical wisdom, the faculty by which a man judges what is right in a particular case.3
No ethical treatise can do that for him. It may supply rules, cautions, landmarks, distinctions; but the application to this case, for this person, in these circumstances, requires something the treatise cannot contain, something seated in the mind of the individual himself.3 The illative sense stands to reasoning generally as phronesis stands to moral conduct: it is the directing and determining principle, personal in its operation, answerable finally to itself.
That last phrase wants unpacking, because it sounds like a licence for arbitrariness, and it is not. The illative sense is a rule to itself not because it is exempt from correction, but because there is no higher court of the same kind to which it can appeal.4 A faculty that can be trained, that operates differently in different subject matters, that is exercised better by the gifted or the long-practised than by the novice — such a faculty is judged as any skilled judgement is judged, by its fruits across a range of cases, not by its conformity to a formula.
Bacon separating the physical from the theological system, Butler connecting the moral with the religious — logical formulas could never have sustained those investigations; it was the mind itself, working without explicit words, by a process that cannot be fully analysed, that detected the principles lying obscure and brought them into the light.1
There is a further distinction worth pressing, because the objection of fideism almost always rests on collapsing it. The illative sense does not produce mere opinion, nor does it produce the conditional conclusion of a formal inference. What it produces, when the convergence of probabilities is sufficient and the mind is rightly disposed, is real assent — unconditional, not hypothetical, carrying with it a sense of intellectual rest.5
The man who holds that Great Britain is an island, or that he will one day die, holds these things without qualification and without being able to demonstrate them in the logician's sense; the illative sense has done its work, and done it honestly. To call that fideism is to demand of all human knowledge a standard that mathematics alone can meet, and then to declare everything else arbitrary — which is itself a conclusion no demonstration has established.
It is a rule to itself, and appeals to no judgment beyond its own; and attends upon the whole course of thought from antecedents to consequents, with a minute diligence and unwearied presence, which is impossible to a cumbrous apparatus of verbal reasoning, though, in communicating with others, words are the only instrument we possess, and a serviceable, though imperfect instrument.An Essay in Aid of a Grammar of Assent
Nor, again, is it by any diagram that we are able to scrutinize, sort, and combine the many premisses which must be first run together before we answer duly a given question.An Essay in Aid of a Grammar of Assent
He calls the faculty which guides the mind in matters of conduct, by the name of phronesis, or judgment... no science of life, applicable to the case of an individual, has been or can be written... The authoritative oracle, which is to decide our path, is something more searching and manifold than such jejune generalizations as treatises can give.An Essay in Aid of a Grammar of Assent
It determines what science cannot determine, the limit of converging probabilities and the reasons sufficient for a proof. It is the ratiocinative mind itself, and no trick of art, however simple in its form and sure in operation.An Essay in Aid of a Grammar of Assent
In the first way of thinking we discover what we know to be the truth for which we have been seeking, and readily give our assent to it; but we cannot explain, either to others or even to ourselves, the reasons which underlie the certainty we have about the truth to which we assent.The Philosophical Notebook Vol 1, pp. 116–117