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I restudied deeply logic (from scratch) and set theory (ZFC) but I came across 2 issues:

  1. In propositional logic, we prove the Deduction theorem using induction, but induction itself is proved using logical principles based on deduction, doesn't that make it a circular proof ?
  2. In the development of logic we use sets, but in the development of set theory we use logic, so again is it a circular proof ?
Predator Monarch
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jul 31 '23 at 14:29
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    Versions of this question are asked frequently on this site. I linked one duplicate, but you can find more by searching for terms like "circularity" or "circular reasoning" and "set theory" or "foundations". – Alex Kruckman Jul 31 '23 at 15:34
  • it doesnt answer my questions, my question is mainly about deduction theorem not ZFC – Predator Monarch Jul 31 '23 at 15:42
  • Is there a step in the proof that you believe is circular in particular? – Michael Carey Jul 31 '23 at 15:51
  • The answer to your question lies in the distinction between the object theory and the meta theory, which is clearly explained in some of the answers to the linked duplicate. If you want an answer that addresses the deduction theorem specifically, search on this site for "deduction theorem circular reasoning". Again, variants of your question have been asked and answered many times here. – Alex Kruckman Jul 31 '23 at 17:10
  • I added a bunch of related questions to the duplicate list in the closure notification. I recommend you read all of the answers and comments there thoroughly. In particular, the second one (Why can we use induction when studying metamathematics?) may come closest to directly answering your question 1 about induction and the deduction theorem. – Alex Kruckman Jul 31 '23 at 17:18

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