Any theory that propounds an opposition between the logical and empirical, represents a failure to grasp the nature of logic and its role in human cognition. [IOTE, 112]
Do advocates of the ASD really propound an "opposition" between logic and experience? Perhaps some do; but without giving examples, Peikoff is merely issuing an unsubstantiated assertion. The ASD grew out of distinctions generated by Hume and Kant. These philosophers were attacking rationalistic speculation (what Kant called "pure" reason). They were not, however, banishing logic from human cognition.
Peikoff goes on the present a brief one-paragraph digest of the Objectivist theory of knowledge:
Man is born tabula rasa; all his knowledge is based on and derived from the evidence of the senses. To reach the distinctively human level of cognition, man must conceptualize his perceptual data --- and conceptualization is a process which is neither automatic nor infallible. Man needs to discover a method to guide this process, if it is to yield conclusions which correspond to the facts of reality --- i.e., which represent knowledge. The principle at the base of the proper method is the fundamental principle of metaphysics: the Law of Identity. In reality, contradiction is the proof of an error. Hence the method man must follow: to identify the facts he observes, in a non-contradictory manner. This method is logic --- "the art of non-contradictory identification." Logic must be employed at every step of a man's conceptual development, from the formulation of his first concepts to the discovery of the most complex scientific laws and theories. Only when a conclusion is based on a noncontradictory identification and integration of all the evidence at a given time, can it qualify as knowledge. [IOTE, 112-113]
Let's examine this paragraph sentence by sentence.