Logic

Logic is the art of non-contradictory identification. Objectivism treats it not as a set of formal rules imposed on thought but as the method by which a volitional consciousness keeps its conclusions tied to the facts, which is why Ayn Rand defined it as an art — something a person practises, and can fail to practise.

“Logic is the art of non-contradictory identification.”
~ Ayn Rand, "Galt's Speech", For The New Intellectual

Reading the definition

Both halves of the definition carry weight.

Identification places logic in the world rather than in a symbol system. The purpose of a chain of reasoning is to say what something is. A valid argument is not a permitted rearrangement of tokens; it is a route from things already identified to something further about the same reality. This is why Objectivism denies that logic is a matter of convention, and why it holds that formal validity divorced from truth is only half the subject. See Identity and Truth.

Non-contradictory is the law of identity restated as a rule of method. A thing is what it is, so it cannot have contradictory attributes; therefore a contradiction in one's thinking is a signal of an error in it, not a discovery about the world. Rand's practical formulation was that in the presence of a contradiction one should check one's premises, because one of them is wrong.

Art means that logic is exercised by choice and can be practised well or badly. Man's consciousness is not automatically correct: it is capable of error, evasion and drift, and logic is the discipline that prevents them. See Reason and Volition.

Its foundation

Objectivism grounds logic in metaphysics rather than treating the two as independent. The law of non-contradiction is not an axiom accepted because the system needs one; it is a corollary of the axioms of existence, consciousness and identity.

This makes the laws of logic neither arbitrary conventions we could have chosen differently, nor Platonic entities in a realm of their own, nor innate structures of the mind imposed on an unknowable reality. They are identifications of the most general facts about anything that exists. See Primacy of Existence, Axioms and Law of Causality.

Logic also governs the formation of the concepts that arguments are made of. A conclusion drawn validly from terms that have been severed from their referents is a floating abstraction — the form of reasoning without its content. See Concept formation and Rationalism.

Fallacies

A fallacy is a recurring way of breaking the discipline, and naming them is useful because they repeat. Among those given particular attention in Objectivism:

Fallacy of the stolen concept: Using a concept while denying something it depends on — as in an argument against reason that must be reasoned.
Floating Abstraction: Employing a concept detached from the concretes that give it meaning.
Straw Man: Refuting a position the opponent does not hold.
Non sequitur: A conclusion that does not follow from the premises offered.
Ad hominem: Answering the person rather than the argument.
Package-Deal: Bundling incompatible things under one term so that accepting one drags in the rest.

See Fallacies and Logical Terms for the wider list.

Common misunderstandings

That logic guarantees truth. It guarantees only that a conclusion follows from its premises. Truth also requires that the premises be true and the concepts be validly formed. This is the error Objectivism labels rationalism: deduction cut loose from observation.
That being logical means ignoring emotion. Emotions are responses to one's value-judgments, and Objectivism treats them as information about those judgments rather than as noise. What it denies is that emotion is a means of knowledge. See Reason-Emotion Dichotomy.
That the laws of logic are arbitrary because alternative formal systems exist. Different formalisms answer different questions. None of them permits a thing to both be and not be what it is in the same respect at the same time, because nothing does.
That a paradox refutes logic. A genuine paradox is a signal that a premise or a definition is wrong. Rand's advice to check premises is the working procedure, not a rhetorical flourish.

Discussion

References

Rand, Ayn. "Galt's Speech", For The New Intellectual. Random House, 1961.
Rand, Ayn. Introduction to Objectivist Epistemology.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 4.

External links

Logic at the Ayn Rand Lexicon

Epistemology Topics
Senses \| Consciousness \| Volition \| Concepts: Unit, Concept-Formation
Objectivity \| Knowledge: Context, Hierarchy \| Reason: Certainty, Truth, the Arbitrary \| Emotions