At root, subjectivism, intrinsicism and Objectivism are theories of the nature of concepts or “universals.” Here, I summarize them in regard to their view of the nature of truth and knowledge. Since truth is an attribute of statements composed of concepts, each school’s view of truth is a direct outgrowth of its view of concepts.
Subjectivism holds that truth, in effect, resides only in the mind. For a subjectivist, a particular statement can be true for one person and false for another, based solely on one’s mental choices, subjective processing, or emotions. (Kant (by implication), Wittgenstein, James, Sartre, etc.) “Truth” amounts to whatever one believes, and there is no such thing as “knowledge” of reality; only some sort of “experience” inside one’s own mind.
Intrinsicism holds that truth resides disembodied out in the world. Typically, intrinsicists hold that all people have to do is somehow “open their hearts to God,” or “pay attention to their intuitions,” or “open their minds to the light of truth,” and the “external truth” will infallibly push its way in. If the truth is already “out there,” then there’s no reason to think that any special processing is required to reach it; one merely has to absorb it. (Plato, Aristotle (partially, in regard to essences), Apostle Paul, Augustine, etc.) For an intrinsicist, conceptual knowledge is whatever external truths one happens to have absorbed. A particular statement is “true” for everyone, whether they have any evidence or not. (And it’s an arbitrarily answerable question whether various people can be held responsible for not grasping all the “floating truth” out there.) (1)
Objectivism holds that truth and falsehood are aspects of conceptual knowledge. Truth (and perceptual knowledge) is a relationship between a consciousness and reality. Truth is reality, as conceptually processed by a consciousness. Truths do not exist disembodied in external reality. Only physical entities (and their aspects–including other consciousnesses) exist in external reality. I can only reach a truth when I choose to conceptually process percepts by reasoning (by the method of logic.) For an Objectivist, a particular statement can be true for one person and false for another, only when there is a radical difference in the relevant perceptual evidence available to the two people. It does not depend on mental choices, subjective processing, emotions, or whims. (2) A statement can also be arbitrary for one person and either true or false for another: People can have different levels of evidence that change how the statement ranks on their “epistemological determinacy” scale. (From arbitrary, to possibly true or false, to probably true or false, to certainly true or false.)
There is much more to be said about this topic, and I recommend Objectivism: The Philosophy of Ayn Rand, by Leonard Peikoff, for more.
(1) To be clear, most modern, intellectual intrinsicists (and many such subjectivists) go to great pains to cloak their theory of knowledge in the appearance of reasoning from observation. They use the language of natural science and the formalism of deductive arguments. But this is all rationalization or inconsistency, because, for intrinsicists, the ultimate basis of “knowledge” is just to “feel the [allegedly external] truth.” For subjectivists, whatever their pretenses about subjectivism being necessitated by objective science, that self-contradiction wipes out objectivity on their part, and they thus imply that there’s no such thing as knowledge of reality. (What distinguishes knowledge of reality from fantasy is that knowledge is objective.)
(2) Some clarification on mental choices and truth: When I say that truth does not depend on mental choices, I am referring to what one might call “normative truth”: what the person ought to consider true. This depends solely on what parts of reality the person has observed, (i.e. what evidence he has.) What the person will actually claim as truth does depend on a mental choice: the choice to think about the evidence or not. This is why I say, “I can only reach a truth when I choose to conceptually process percepts by reasoning…” The normative truth is a potential I have, based on my current context of perceptual evidence. Reaching this truth is exercising the choice to fulfill (recognize) the normative truth.
Note also that the position of one’s body can be affected by prior mental choices. Only in this narrow sense can what one perceives–and thus one’s normative truth–be affected by mental choices. Once one has actually perceived something, mental choices are irrelevant to the normative truth.
[Substantial Edit: 2/28/15: My statement that, “For an Objectivist, a particular statement cannot be true for one person and false for another, (2) but it can be arbitrary for one person and either true or false for another,” in the fourth paragraph was altered to say that it is possible for something to be true for one person and false for another, based on evidence. I no longer agree with the old statement, and I don’t think Objectivism supports it. Footnote (2) was also altered to explain the current view.]