Why does soundness imply consistency?

I was reading the question Consistency and completeness imply soundness? and the first statement in it says:


I understand that soundness implies consistency.

Which I was quite puzzled about because I thought soundness was a weaker statement than consistency (i.e. I thought consistent systems had to be sound but I guess its not true). I was using the informal definition Scott Aaronson was using in his 6.045/18.400 course at MIT for consistency and Soundness:

• Soundness = A proof system is sound if all the statements it proves are actually true (everything provable is True). i.e. IF ( $\phi$ is provable) $\implies$ ($\phi$ is True). So IF (there is a path to a formula) THEN (that formula is True)

• Consistency = a consistent system never proves A and NOT(A). So only one A or its negation can be True.

Using those (perhaps informal) definitions in mind I constructed the following example to demonstrate that there is a system that is sound but not consistent:

$$ CharlieSystem \triangleq \{ Axioms=\{A, \neg A \}, InferenceRules=\{NOT(\cdot) \} \}$$

The reason it's I thought it was it was a sound system is because by assumption the axioms are true. So A and not A are both true (yes I know the law of excluded middle is not included). Since the only inference rule is negation we get that we can reach both A and not A from the axioms and reach each other. Thus, we only reach True statements with respect to this system. However, of course the system is not consistent because we can prove the negation of the only statement in the system. Therefore, I have demonstrated that a sound system might not be consistent. Why is this example incorrect? What did I do wrong?

In my head this makes sense intuitively because soundness just says that once we start from and axiom and crank the inference rules we only reach at destinations (i.e statements) that are True. However, it does not really say which destination we arrive. However, consistency says that we can only reach dest

I recommend looking into formal logic beyond vague, hand-wavy descriptions. It's interesting and highly relevant to computer science. Unfortunately, the terminology and narrow focus of even textbooks specifically about formal logic can present a warped picture of what logic is. The issue is that most of the time when mathematicians talk about "logic", they (often implicitly) mean classical propositional logic or classical first-order logic. While these are extremely important logical systems, they are nowhere near the breadth of logic. At any rate, what I'm going to say largely takes place in that narrow context, but I want to make it clear that it is happening in a particular context and need not be true outside of it.

First, if consistency is defined as not proving both $A$ and $\neg A$, what happens if our logic doesn't even have negation or if $\neg$ means something else? Clearly, this notion of consistency makes some assumptions about the logical context within which it operates. Typically, this is that we are working in classical propositional logic or some extension of it such as classical first-order logic. There are multiple presentations, i.e. lists of axioms and rules, that could be called classical propositional/first-order logic but, for our purposes, which doesn't really matter. They are equivalent in some suitable sense. Typically, when we are talking about a logical system we mean a (classical) first-order theory. This starts with the rules and (logical) axioms of classical first-order logic, to which you add given function symbols, predicate symbols, and axioms (called non-logical axioms). These first-order theories are usually what we're talking about as being consistent or inconsistent.

Next, soundness usually means soundness with respect to a semantics. Consistency is a syntactic property having to do with what formal proofs we can make. Soundness is a semantic property that has to do with how we interpret the formulas, function symbols, and predicate symbols into mathematical objects and statements. To even begin to talk about soundness, you need to give a semantics, i.e. an interpretation of the aforementioned things. Again, we have a separation between the logical connectives and logical axioms, and the function symbols, predicate symbols, and non-logical axioms. What makes connectives connectives and logical axioms logical axioms from the semantic point of view is that they get treated specially by semantics while function symbols, predicate symbols, and non-logical axioms do not. The typical semantics for classical first-order logic is to interpret formulas as set-theoretic relations on some (power of a) set called the "domain" given as part of a particular semantics, and the connectives as the more or less obvious set-theoretic analogues, e.g. $[\![\varphi\land\psi]\!]=[\![\varphi]\!]\cap[\![\psi]\!]$ where I use $[\![\varphi]\!]$ as the interpretation of the formula $\varphi$. In particular, $[\![\neg\varphi]\!]=D\setminus[\![\varphi]\!]$ where $D$ is the domain set. The idea is a formula is interpreted as the set of (tuples of) domain elements that satisfy the formula. A closed formula (i.e. one with no free variables) is interpreted as a nullary relation which is to say a subset of a singleton set which can only be that singleton or the empty set. A closed formula is "true" if it isn't interpreted as the empty set. Soundness is then the statement that every provable (closed) formula is "true" in the above sense.

It is easy from here, even from the sketch I've given, to prove that soundness implies consistency (in the context of classical first-order logics and the semantics I've sketched). $$[\![\varphi\land\neg\varphi]\!]=[\![\varphi]\!]\cap(D\setminus[\![\varphi]\!])=\varnothing$$ If your logic is sound, then every provable formula interprets as a non-empty set, but $[\![\varphi\land\neg\varphi]\!]$ is always interpreted as the empty set no matter what formula $\varphi$ is, and so it can't be provable, i.e. your logic is consistent.

StackExchange Computer Science Q#88274, answer score: 17