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Soundness: $a \vdash b \implies a \vDash b$, i.e. if we can prove something, it will also be true. We don't want a system where we start out with something true and dedice something false. However it is conceivable that even if our system is sound, maybe it's quite incomplete/limited regarding what we can express, which is why we also would like...

Completeness: $a \vDash b \implies a \vdash b$, i.e. if we can show something is true, it's also provable. We want to be able to prove all true statements. However it is conceivable that even though we can prove all true statements, maybe it also proves false ones as well, which is why we'd also like the soundness property from before.

Do I have the right idea?

If so, how would I begin to prove soundness? If we are already given $a \vdash b$ I'm not sure what all we must iterate over in e.g. propositional logic to show that we always get true statements. Especially since it seems possible I could pick a false $b$ that contradicts.

user525966
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1 Answers1

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You have the right understanding of what soundness and completeness are. As to how to prove soundness, the right tool is induction - specifically, by induction on proof complexity, you show that there is no witness to "$a\vdash b$" unless in fact $a\models b$.

Exactly what this looks like will depend on the specific proof system you use. In the case of sequent calculus, it amounts to showing that the "basic sequent rules" are soundness-preserving: each rule only deduces sound sequents from sound sequents. Since every proof is well-founded, this means that no unsound sequent can every "creep in."

For example, consider the rule which takes in the sequents $\Gamma\vdash\varphi$ and $\Gamma\vdash\psi$ and outputs the sequent $\Gamma\vdash\varphi\wedge\psi$. Supposing that our "input sequents" are sound, in any model $\mathcal{M}$ of $\Gamma$, $\varphi$ is true (by the first sequent's soundness) and $\psi$ is true (by the second sequent's soundness). But then by the definition of satisfaction, $\varphi\wedge\psi$ is true in $\mathcal{M}$.

That is, from the assumption that $\Gamma\vdash\varphi$ and $\Gamma\vdash\psi$ are each sound, we've concluded "every model of $\Gamma$ satisfies $\varphi\wedge\psi$;" but this latter statement is exactly the soundness of the sequent $\Gamma\vdash\varphi\wedge\psi$!

Noah Schweber
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  • Can you show an example in classical propositional logic for Hilbert, and ND so I can try the rest out myself once I see the general idea? – user525966 Sep 09 '18 at 16:02
  • @user525966 I'm not sure what you mean. Everything I've done above is in classical propositional logic. – Noah Schweber Sep 09 '18 at 16:02
  • I assume sequent is different from Hilbert and ND? – user525966 Sep 09 '18 at 16:03
  • They're all classical propositional logic, they're just different proof systems for it. And they'll all follow the same basic pattern; e.g. the example I gave above is just $\wedge F$ in natural deduction. – Noah Schweber Sep 09 '18 at 16:05
  • So we basically need to show (true variable) (operator) (true variable) = true, for all operators we've defined? – user525966 Sep 09 '18 at 16:08
  • @user525966 Basically, yes. – Noah Schweber Sep 09 '18 at 16:11
  • True and true, true or true, true implies true... although, not true = false, true xor true = false, true nand true = false, so some operators are not truth preserving – user525966 Sep 09 '18 at 16:11
  • @user525966 No, you're looking at the wrong level (and I misunderstood your previous comment). "Operators" here are sequent rules (or similar), like "From $\Gamma\vdash\varphi$ and $\Gamma\vdash\psi$, deduce $\Gamma\vdash\varphi\wedge\psi$": methods for moving from a set of existing deductions to a new deduction. You need to show that each such operator preserves truth: from true (or rather, sound) sequents, we can only derive true sequents via those operations. – Noah Schweber Sep 09 '18 at 16:13
  • What? How do we know which operators we're working with here? – user525966 Sep 09 '18 at 16:14
  • They're the rules in your proof system. E.g. here. You need to show, again, that all the ("starting") rules in your proof system are soundness preserving. For example, I've shown above that "From $\Gamma\vdash\varphi$ and $\Gamma\vdash\psi$, deduce $\Gamma\vdash\varphi\wedge\psi$" is soundness preserving; now I just need to go through all the other rules in my specific sequent calculus system. – Noah Schweber Sep 09 '18 at 16:15
  • Let us continue this discussion in chat. – Noah Schweber Sep 09 '18 at 16:16
  • Oh so with Hilbert system our only rule would be modus ponens? Showing that true and (true implies true) proves true, via truth tables? – user525966 Sep 09 '18 at 16:17