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In the book "Logic" by Paul Tomassi, the author uses the term deductive apparatus to refer to the set of inference rules in propositional logic and first-order logic. The use of this term seems to suggest a formal system may contain another kind of deductive apparatus besides a set of inference rules. Is this the case, or is a set of inference rules the only type of deductive apparatus and the term "deductive apparatus" is simply synonymous with "inference rules?" If indeed other types of deductive apparatuses exist, could someone name them and point me in a direction where I could learn more? Thanks!

RyRy the Fly Guy
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  • I don't have the book. Can you clarify what you (and/or the book) mean by the term "inference rule"? There are typically many ways of presenting a logic: Hilbert-style, natural deduction-style, Gentzen-style sequent calculi of various kinds... . These may or may not all qualify as sets of inference rules, depending on what you mean by that. – Rob Arthan Jul 24 '23 at 22:11
  • Oddly enough, the textbook offers no formal definition of an inference rule. it only discusses the inference rules of propositional logic and first order logic and literally calls them "inference rules." – RyRy the Fly Guy Jul 25 '23 at 00:20
  • As far as I know, they are synonymous. I imagine that "deductive apparatus" is preferred to "set of inference rules" to a) avoid the misconception that there is a singular and universally-accepted collection of such rules and b) that perhaps they didn't wish to associate deduction too strongly with mathematics (specifically, set theory). The second point seems a little shaky to me, but the book is called "Logic"... – H. sapiens rex Jul 25 '23 at 03:45
  • You can see the post Formal System and Formal Logical System: a formal system (also called a logical calculus) consists of a formal language and a set of inference rules. The rules are needed in order to "produce" new formulas from axioms or assumed initial formulas. – Mauro ALLEGRANZA Jul 25 '23 at 05:44
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    See page 122: "Moreover, in our earlier discussion of inference and reasoning in PL, we developed our understanding of formulas and the logical connectives in terms of which inferences were licensed by formulas with a particular logical connective as the main connective. In the light of these considerations, we were able to construct a set of rules of inference for PL. In formal logical terms, when we constructed that set of rules of inference for PL we added a deductive apparatus to the formal language PL and in so doing we established a formal system." – Mauro ALLEGRANZA Jul 25 '23 at 13:25

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I am not sure what Tomassi has in mind when talking about 'formal systems' or 'deductive apparatus', but there are certainly ways to demonstrate that some claim or argument is deductively valid without the use of formal inference rules. For example: Truth-tables and truth trees (tableaux)

Bram28
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On page $7$ of Metalogic: An Introduction to the Metatheory of Standard First Order Logic by Geoffrey Hunter, the deductive apparatus of a formal system consists of at least one of the following:

$(a)$ a non-empty set of axioms

$(b)$ a non-empty set of inference rules

I have found this text to be an excellent bridge into Metalogic after reading Logic by Paul Tomassi.

RyRy the Fly Guy
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