An often unstressted connotation of the word "logical" and "logic" is the non-classicist notion of logic. For example, it makes sense to speak of a "Marxist logic", a "psychoanalytic logic", or a "Republican logic". However, the definitions offered above do not stress even the possibility of these phrases making sense, as does the very common practice in our society of confusing logic with an Aristotelian or classical conception of logic as the specific system of dialectical reasoning in which argumentative forms such as modus tollens and modus ponens serve as rhetorical exemplars.
For example, modus ponens is represented in the following familiar rule of inference: if P then Q, P, therefore Q. According to the classical conception of logic, any system of reasoning that does not progress according to this rule can safely be said to be illogical.
This, however, isn't exactly fair given that logicality is not defined as any particular system of reasoning, but rather the broad and vague concept of a system of reasoning as in the Greek word logos, which could be translated as logic or reason or rational system of thought, from which our English "logical" is derived.
If we read Freud, for example, we can see that he stresses a great variety of progressions of thought in which a trope such as modus ponens is not exemplified. For example, Freud's notions of sublimation and repression make it possible to think ratioanlly (scientifically) in a form that could be represented as: if P then Q, P, not Q. As in the case of the familiar Freudian trick of arguing from the denial of a statement to the assertion of its content. For example: the patient recounts a dream, the analyst asks the patient if the dream means "Q", the patient denies it, and the analyst takes this as proof that it does. According to Freud, this is not an irrational movement of thought. It was part of Freud's claim that such a system of thought is systematically present in human psychical activity and is logical.
To say that Freud (or Marx, or Hegel, or Derrida, or postmodern thinkers) is illogical is to ambiguously state that Freud proferred systems of thought that are not covered by the classical logic (as described by Aristotle, Plato, analytic philosophers, et. al.). This does not mean that he proferred illogicality. What Freud offered was certainly a systematic representation of thinking with its own assumptions, axioms, and rules of inference. It is not, then, by definition illogical.