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https://en.wikipedia.org/wiki/Proof_calculus says

A proof system includes the components:

  • Language: The set of formulas admitted by the system, for example, propositional logic or first-order logic.
  • Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems.
  • Axioms: Formulas in L assumed to be valid. All theorems are derived from axioms.

https://en.wikipedia.org/wiki/Formal_system#Deductive_system says:

A deductive system, also called a deductive apparatus or a logic, consists of the axioms (or axiom schemata) and rules of inference that can be used to derive theorems of the system.

Are proof systems and deductive systems the same concept? What differences and relation are between them?

https://en.wikipedia.org/wiki/Formal_system#Proof_system mentions proof systems, but I don't find it says what relations and differences are between deductive systems and proof systems.

Thanks.

Tim
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  • @MauroALLEGRANZA If the two are the same concept, why do https://en.wikipedia.org/wiki/Formal_system#Deductive_system and https://en.wikipedia.org/wiki/Formal_system#Proof_system exist as different parts in the same article? – Tim Sep 06 '20 at 19:34
  • @MauroALLEGRANZA About "deductive system and proof system are the same concept". Now I think that a deductive system is a formal system with its consequence relation having some special properties, while a proof system is a rule-based system i.e. having a set of inference rules. These two are different concepts, but are equivalent (see Herre & Schroeder-Heister “Formal Languages and Systems” p7.) Also https://math.stackexchange.com/q/3802956/#comment7871288_3803045 – Tim Sep 07 '20 at 12:55

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