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This paper addresses Frege's discussion of illogical thought in the introduction to Basic Laws of Arithmetic. After a brief introduction, I discuss Frege's claims that logic is normative vis-à-vis thought, and not descriptive, and his opposition to the idea that logical laws express psychological necessities. I argue that these two strands of Frege's polemic against psychologism constitute two motivating factors behind his allowing for the possibility of illogical thought. I then explore a line of thought—originally advanced by Joan Weiner—according to which Frege should have rejected illogical thought as not constituting a genuine possibility. I argue that, once developed, this line of thought constitutes an important correction (moreover, one that is consistent with his two aforementioned anti-psychologistic strands) to Frege's own response to the possibility of illogical thought.