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I'm trying to understand the generalization rule of inference better. Suppose we have a language $L$ consisting of some predicate symbols with the sizes corresponding to them. Now, we construct some theory $T$ in the following way. If $f$ is a valid closed formula over $L$ (i.e. $f$ is true for all interpretations of $L$), then $f\in T$. $T$ may also include some other closed formulas over $L$ (by closed I mean of course not containing free variables). Now, the system of inference rules usually includes the modus ponens (MP) and the generalization rule (GEN). Can we state that we don't need GEN for $T$, i.e. every theorem of $T$ that can be proved with MP+GEN can be also proved only with MP?

Bertrand Haskell
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    It depends on the axiom systems... We may have an axiom system (see Enderton) with suitable universal quantifier axioms and only MP as inference rule. – Mauro ALLEGRANZA Jul 17 '20 at 12:59
  • @MauroALLEGRANZA yes, I thought this may happen but didn't find a name for the "standard" (?) system I've met in Introduction to Mathematical Logic by Mendelson. Thanks for the link. The book seems very interesting. – Bertrand Haskell Jul 17 '20 at 13:10
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    See also this related post – Mauro ALLEGRANZA Jul 17 '20 at 13:13
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    In my own system of logic (more like that actually used by mathematicians AFAICT), I have no separate rule of universal generalization. That function is incorporated into my Conclusion Rule which discharges a premise. It also automatically does existential generalizations as required, though existential generalization is also available as a separate rule on the Logic menu. – Dan Christensen Jul 17 '20 at 14:56
  • @DanChristensen thanks for your message and my respect to those who invents something new. I've downloaded the software and will try to understand it. How long did it take you to write the program? Does your system allow to get all the theorems that can be proved by the common methods? – Bertrand Haskell Jul 17 '20 at 17:04
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    The time from when I first thought to implement my system as a software package to its initial release was at least 20 years in my spare time. It was designed to introduce students to the basic methods of proof using simple, even trivial exercises in formal proof. Formal proofs in general have their practical limitations for more complicated proofs. They are typically many times longer than their informal versions. You can see examples at my blog (Click on Author's Blog at my homepage.) – Dan Christensen Jul 17 '20 at 17:48

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