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I'm trying to better understand soundness, especially in contrast to semantic consistency. Here is what I've put together so far:

Soundness: A theory is sound if all theorems are true under all possible interpretations.

Semantic Consistency: A theory is semantically consistent if there exists an interpretation for which all theorems are true.

Note that I'm aware that consistency usually refers to syntactic consistency (namely that the theory cannot derive both $\varphi$ and $\lnot \varphi$, for any formula $\varphi$).

I'm primarily confused about the definition of soundness because it feels like a very hard property to satisfy. Take, for example, Peano Arithmetic. One axiom states that $0$ is not the successor of any number. I think I can find an interpretation that makes that false, for example, modular arithmetic. I'm fairly sure, though, that Peano Arithmetic is considered a sound theory so I suspect that I'm just confused about the definition of soundness.

Any help would be appreciated. Thanks.

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Your definition of soundness is incorrect. A theory is sound if it contains no sentences which are false with respect to a specific structure (or class of structures) of interest.

Now as an unfortunate matter of practice we generally only consider soundness when that specific context is understood in the background, so we omit it. For example, when we say

Peano arithmetic is sound

what we really mean is

Peano arithmetic is sound with respect to the structure $(\mathbb{N};+,\times)$; that is, for each axiom $\varphi$ of Peano arithmetic we have $(\mathbb{N};+,\times)\models\varphi$.

Meanwhile, Peano arithmetic is not sound with respect to the unique $1$-element $\{+,\times\}$-structure (basically, the trivial ring).


I think it's actually helpful to step a ways back and rephrase both soundness and consistency in a more general way. Specifically, given a class of structures $\mathbb{K}$ and a theory $T$, we say:

  • $T$ is sound with respect to $\mathbb{K}$ iff for every $\mathcal{A}\in\mathbb{K}$ we have $\mathcal{A}\models T$.

  • $T$ is semantically consistent with respect to $\mathbb{K}$ iff for some $\mathcal{A}\in\mathbb{K}$ we have $\mathcal{A}\models T$.

The fundamental difference then is the "every"/"some" distinction. There's a second difference too, however, which is down to practice: usually when we talk about soundness we're tacitly referring to a $\mathbb{K}$ consisting of a single structure, and almost always when we talk about semantic consistency we're tacitly referring to the class of all structures. So there are really two differences between soundness and semantic consistency at work here: soundness typically looks at a small context while semantic consistency typically looks at a large context, and with respect to that class soundness is a universal condition while consistency is an existential one.


That said, as I mentioned above we almost always work with reference to the class of all structures when thinking about semantic consistency. So in light of that your definition of semantic consistency is correct. Note that the completeness theorem shows that the "realist commitment" in the (with-respect-to-the-class-of-all-structures) notion of semantic consistency (namely, that mathematical structures are things which actually exist and about which sentences are true or false) can be removed; by contrast, there isn't really an analogous result for soundness(es).

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  • $\begingroup$ Thank you for your answer - that is very helpful and completely resolves my confusion around whether Peano arithmetic is sound. Now, though, the definitions for soundness and semantic consistency sound almost identical, right? Soundness refers to only true theorems w.r.t. a specific interpretation/structure while semantic consistency refers to only true theorems w.r.t at least one interpretation/structure? If that's the only difference, that's fine, I'm just trying to confirm. $\endgroup$ Apr 11, 2021 at 2:07
  • $\begingroup$ @user2869495 I mean, that's a difference between a specific and a general - that's a huge difference! As a relevant technical result from computability theory, per the completeness theorem the set of semantically consistent sentences is not too complicated (it's "co-recursively enumerable") but the set of sentences which are sound with respect to $\mathbb{N}$ is extremely complicated. $\endgroup$ Apr 11, 2021 at 2:10
  • $\begingroup$ Ok, that makes sense. I don't fully understand your last comment (that the set of sentences which are sound w.r.t. $\mathbb{N}$ is extremely complicated), but I'll save that question for when I'm farther along. Thanks! $\endgroup$ Apr 11, 2021 at 2:14
  • $\begingroup$ @user2869495 I've edited to go in a slightly different direction expositionally; feel free to revert if that made things worse. (Actually it occurs to me that you might not have enough reputation to revert, I don't know how that works; if you can't let me know and I'll roll it back myself.) $\endgroup$ Apr 11, 2021 at 2:16
  • $\begingroup$ Your edits definitely made things (even more) clear. Thanks. $\endgroup$ Apr 11, 2021 at 2:18

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