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“If p, then q” is taken in logic texts to be identically equivalent to “Not (p and not-q).” “Not (there is a naturally evolved bird with talons, and it is not a bird of prey)” is identically equivalent to “If there is a naturally evolved bird with talons, then it is a bird of prey.” It seems that we know up front that this “identically equivalent” relation holds however much our knowledge of birds increases; it cannot be found false. Whether there are presently unknown conditions under which this particular “If p, then q” can be found false is open, though until specific prima facie plausible conditions of that sort are proposed (at least in a sketchy way), that open possibility is a vacuous possibility, a degenerate, impotent sort of possibility, whether the if-then concerns nature or mathematics. The nature of birds is a matter of identity, but it seems a wider sort of identity than that in the “identically equivalent” relation. And the latter would seem to be something one learns about later than the former, although maybe the latter is already present in a precursor way in prelinguistic action schemata (eg. there’s more than one way to get attention, more than one form under the schema get attention). In his book How We Know, Harry Binswanger takes syllogistic inference to be a case in which what is already implicit in the premises is drawn out and made explicit in the conclusion. That is a common perspective on deductive inference. The syllogism is a form of “If p, then q” in which p is a conjunction of two propositions: “If r and s, then q.” For r and s to be true and to bear implicit truths, of course, r and s both have to express awareness of facts (254–55). This viewpoint is smooth with the views of Rand that logic is a form of identification and that existence is identity. In his book Objectivism: The Philosophy of Ayn Rand, Leonard Peikoff remarked: “The method of logic . . . does reflect the nature and needs of consciousness. It also reflects the other factor essential to a proper method: the facts of external reality. The principle which logic provides to guide man’s mental steps is the fundamental law of reality” (120–21). There are no contradictory facts in reality, I should add, to be thought in conjunction if thought is aimed at fact. To put forth without evidence or design for evidence the thesis that there are naturally evolved birds with talons that are not birds of prey contradicts evident facts without resolving the purported contradiction with other (not-adduced) evident facts. I suggest that denials of modus ponens should be understood as that sort of denial under the basic conception of logic in Objectivism. Logical validities are never independent of all facts of reality. Some excerpts from Nathaniel Branden’s lectures The Basic Principles of Objectivism: “Logic is the tool of reason. Logic is based on facts, on the fact that that which is, is; but it is not a science of facts. It is a science of method (75).” “One proves a proposition by demonstrating that it is logically necessitated, that its denial would contradict facts already known to exist. . . . . “Until one has grasped that A is A, and that contradictions cannot exist, there can be neither proof nor the concept of ‘proof’. . . . “The Law of Identity is a genetic root of the concept of ‘proof’. . . . (73, transcription in The Vision of Ayn Rand)

People who say this are usually people who didn't get past undergrad (or have animosity towards people who do get up to the PhD or graduate level). It's a kind of anti-intellectualism. Have you taken graduate courses in philosophy? Have you taken time to understand really complex philosophy, even if you didn't agree? I'm not trying to demean you here - I'm asking that you check your premises that you can and should badmouth people who dare to say something formally in philosophy as would be expected of people in universities who do philosophy of math or logic. What SK is saying isn't really so scary in all. Oist epistemology is not simple or easy. Sometimes people think it might be, but this is because Rand only wrote an introduction. Adding some formalism doesn't destroy anything. If Oist epistemology can't survive some formal treatment, then it would be a trash epistemology. But the cool thing is that it can. Type theory is probably the absolute best avenue to follow to improve or fairly criticize Oist epistemology from a formal perspective. If you don't like philosophy from a very formal perspective (which this thread is about), that's fine. If you want to participate with it though, you should take the time to understand before criticizing. Rand was generally informal about her philosophy. That doesn't mean it can't get a fair formal treatment.