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I was reading some notes on propositional logic and I stumbled upon these related remarks:

"One way of measuring the strength of a logic is to ask whether it is decidable"

"One of the things we might mean when we say that a logic is trivial is that it is decidable"

"A logic L is decidable iff we could in principle program a computer to tell us of any given sentence of L in a finite period of time whether or not that sentence is a logical truth according to L"

"If logic is decidable then we could just grind out the consequences of any claim in an unthinking manner; in that sense, the logic took us nowhere new"

This is not the first time I read something like that, I often find propositional logic denoted as just a "toy".

Seems that this is all related, but I still don't have a grasp on the reasons.

Gabriele Scarlatti
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    I sure hope he doesn't think that finding a polytime (or any tractable) algorithm is trivial. – DanielV Feb 22 '19 at 17:36
  • In fact, the decision problem for classical propositional logic is NP-complete; and according to an answer to a question I posted over on CS stack exchange, the decision problem for intuitionistic propositional logic is PSPACE-complete. – Daniel Schepler Feb 22 '19 at 17:51
  • Note well that the term "trivial" is not well defined but heavily context dependent. – Somos Feb 22 '19 at 19:47
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    I'm not really sure how a more typical semidecidable proof theory takes us "somewhere new". We can "grind out the consequences of any claim in an unthinking manner" regardless of whether the logic is decidable or not. – Derek Elkins left SE Feb 23 '19 at 00:19

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Well, I certainly wouldn't say that a decidable logical system is trivial (well okay, it might be trivial, but not for that reason alone). The book The classical decision problem contains a lot of information about decidable logical systems, as does this survey paper, and I think it won't be hard to find a bunch which you find plenty interesting. So I'd consider the claim "decidability$\implies$triviality" to be unjustified colorful language.

However:

  • In light of Godel's incompleteness theorem, any "sufficiently rich" logical system is undecidable. So conversely, knowing that a logical system is decidable does put a limit on its strength; certainly it can't be used to "implement" all of mathematics, or even arithmetic.

  • Second, propositional logic in particular is indeed often considered just a toy, but not because it's decidable - rather, because it's (arguably) uninteresting. The semantics for (classical) propositional logic simply isn't very rich (just maps from the set of propositional atoms to $\{\top,\perp\}$) compared with other logics (e.g. first-order structures). This is related to the previous bulletpoint: any logic with a "sufficiently complicated" semantics will have to be undecidable.

Noah Schweber
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  • Seeing the title, I wondered if for this was maybe the author was suggesting that, due to GIT you mentioned, a "sufficiently rich" system is decidable only if it is inconsistent, and that is the sense in which he meant trivial. – DanielV Feb 22 '19 at 17:49