Logically, Objectivism starts with a set of axioms, which are the self-evident preconditions of all knowledge. The axioms must be accepted in any attempt to prove them either true or false, so they are below the ability or the necessity of proof. One can show that the axioms are axioms by showing how all claims of knowledge of any kind presuppose them. This validates them, but one cannot prove them from more fundamental premises, since there are no premises more fundamental. (1)
For Objectivism, the axioms are preconditions of all knowledge, but they are not the starting points of a deductive chain that defines the rest of the philosophy. The rest of the philosophy–its (non-axiomatic) epistemology, ethics, politics and esthetics–are all proved essentially through induction. The principles are arrived at through observation of reality and integration of observed instances into general principles, with the axioms and more fundamental principles serving as reference points for the derivation of narrower, less fundamental principles.
The structure of Objectivism is like a skyscraper shaped like an X. The axioms form the foundation on which everything else rests. Metaphysics and epistemology are the lower legs of the X, ethics is the center of the X, politics and esthetics are the upper arms of the X. At every step beyond the axioms and their corollaries, the structure of the philosophy is built out of new observations of reality, as the skyscraper would be built out of new material from its surroundings. But the upper levels are dependent on the lower levels. If we remove load-bearing members (principles) from the base of the philosophy, the structure above them comes crashing down. (2)
So, if someone says, “I don’t see how you can get from “Existence exists” (the basic axiom) to “Rational egoism is the proper ethics for man.” The answer is that you can’t, deductively. But deduction is not a method of generalization. You cannot get general principles that are based on reality strictly from deduction. If we want to reach general statements that correspond to reality, the method we must use is induction. This is the method Ayn Rand employed to reach Objectivist principles. (3)
(1) The axioms are identified conceptually by the broadest possible generalization from observation, but they can’t be proven using any principles more fundamental. See Objectivism: The Philosophy of Ayn Rand by Leonard Peikoff.
(2) See YouTube Intro to Objectivism and Understanding Objectivism by Leonard Peikoff.
(3) Yes, I am aware of David Hume’s alleged disproof of induction. The refutation of his view essentially consists of referring to the Objectivist axioms and their validation, (specifically, the Law of Identity and its corollary, the Law of Causality) along with identification of the contextual nature of inductive generalizations.