Is FOL a formal system or a formal axiomatic system? That is, is there anything in FOL apart from syntax, semantics and inference rules? Or do the syntactic rules count as axioms?
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2I disagree with the decision to close - I think this question is sufficiently focused and admits a good answer (and incidentally stems from the regrettable over-loading of the term "formal system"). I've voted to reopen. – Noah Schweber Feb 26 '20 at 19:53
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See FOL. – Mauro ALLEGRANZA Feb 26 '20 at 20:30
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1FOL is a language (syntax rules) + semantics (definition of interpretation) + proof calculus. There are many different way to set up a proof calculus: someone with axioms and rules and some (like Natural Deduction a Truth Trees) with only rules. – Mauro ALLEGRANZA Feb 27 '20 at 07:36
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1Thus, the presence of axioms is not strictly necessary. – Mauro ALLEGRANZA Feb 27 '20 at 07:37
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Maybe useful the post Axiom Systems and Formal Systems as well as Are axioms assumed to be true in a formal system? – Mauro ALLEGRANZA Feb 27 '20 at 07:46
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@MauroALLEGRANZA thank you for your response! this is very helpful – FlappyBoy123 Feb 27 '20 at 22:30