Final answer:
True, the believability of a syllogism does not affect judgment when using system 2 thinking, which is analytical and not influenced by the initial plausibility but rather focuses on logical structure and true premises.
Step-by-step explanation:
The statement 'the believability of a syllogism does not influence your judgment if you're using system 2 thinking' is generally true. System 2 thinking, as a form of reasoning, is characterized by effortful and fair consideration of the evidence rather than relying on initial plausibility or appeal. This type of cognitive processing is deliberate and analytical, thus, the believability of the premises in a syllogism does not automatically guarantee the truth of the conclusion. Instead, judgments should be based on the logical structure of the syllogism and whether the conclusion necessarily follows from true premises. According to Gilbert Harman's no false lemmas condition, a true belief should not be inferred from any false premises. Therefore, even if a syllogism appears believable, if it's analyzed using system 2 thinking and found to be based on false premises, the judgment would deem it invalid.