Why does soundness imply consistency?

Question

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:

1. 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)
2. 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 destination that are reach either $A$ or $\neg A$ (both not both). So every consistent system must include the law of excluded middle as a axiom, which of course I did not and then just included the negation of the only axiom as the only other axiom. So it doesn't feel I did anything too clever, but somehow something is wrong?

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I just realize it could be a problem because I am using Scott's informal definition. Even before I wrote the question I did check wikipedia but their definition didn't make sense to me. In particular the part that they say:

with respect to the semantics of the system

their full quote is:

every formula that can be proved in the system is logically valid with respect to the semantics of the system.

Answer 1

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.

Answer 2

Soundness and consistency are properties of deductive systems. Soundness can only be defined with respect to some semantics that is assumed to be given independently from the deductive system.

In the realm of semantics the two properties are related

Definition 1(Soundness[Semantics] -- borrowed from Wikipedia) Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based.

Definition 2(Consistency[Semantics]) A set of sentences $A$ in the language $\mathfrak{L}$ is consistent if and only if there exist a structure of the language $\mathfrak{L}$ that satisfies all sentences in $A$. A deductive system is consistent if there exist a structure that satisfies all formulas provable in it.

With the two definitions given above it is clear that soundness implies consistency. I.e. if the set of all provable sentences holds in all structures of the language then there exists at least one structure that satisfies them.

Answer 3

Your proof system is neither sound nor consistent, since $A$ is not a true proposition unless $A \equiv \top$, In which case $\lnot A \equiv \bot$ is not a true proposition. This argument shows that every sound proof system is also consistent.

Answer 4

Often when we come up with logical systems, they are motivated by an attempt to describe a pre-existing phenomenon. For example, Peano arithmetic is an attempt to axiomatize the natural numbers along with the operations of addition and multiplication.

Soundness can only be defined relative to the phenomenon you're attempting to describe, and essentially means that your axioms and inference rules really do describe the thing in question. So, for example, Peano arithmetic is sound because its axioms and inference rules really are true of the natural numbers.

This, of course, implies that you have a concept of "natural numbers" beyond Peano's definition of them, and some notion of what is true or false for the natural numbers without having derived these truths from any particular set of axioms. Trying to explain where those truths come from or how they can be verified can land you in philosophical hot waters. But if you take it as a given that there ARE natural numbers, and there is some collection of true facts about them, then you can view the axiomatization project as simply attempting to come up with a concise formal specification from which many of the most important truths can be derived. Then an axiomatization is sound if everything it can prove actually is in the pre-specified collection of truths, that is, if its properties and consequences match up with the interpretation you have in mind.

(Note in particular that your formal specification isn't going to prove everything that is true about the natural numbers, and moreover will not uniquely describe the natural numbers in that there are other structures, different from the natural numbers, in which Peano's axioms are also true.)

In first order logic, at least, a theory is consistent iff it has any models at all. Soundness means it has the specific model you wanted: the particular structure you were attempting to describe with your theory really is a model of your theory. From this perspective, it's clear why soundness implies consistency.

As an example of a theory that is consistent but not sound: Peano arithmetic, PA, is capable of encoding logical formulae as arithmetical constructions, and in particular you can encode the sentence "PA is consistent" ("there is no proof of falsehood from the axioms of PA"). Call this sentence Con(PA). You may also be aware that (since it is sound, and therefore consistent) PA can't prove Con(PA), by Gödel's first incompleteness theorem. This also means that the theory PA + $\neg$Con(PA) can't prove a contradiction, so it must be consistent. But it's not sound: it claims there exists a natural number encoding a proof of falsehood from the axioms of PA, but there can't possibly be such a number in the "real" natural numbers, since otherwise we'd be able to extract a genuine proof of the inconsistency of PA from it.

PA + $\neg$Con(PA) has models, but they're models which must include "extra" objects, "non-standard natural numbers", including one which it claims encodes the "proof" in question. The theory is simply not equipped with the necessary tools to distinguish these non-standard elements from the genuine bona-fide members of $\mathbb N$, or to demonstrate that the proof is not a legitimate proof.

You can alternatively interpret this as: PA + $\neg$Con(PA) is a perfectly legitimate logical system -- it just doesn't accurately describe the natural numbers, and the natural numbers aren't a model of it.

One more thing: we don't assume that axioms are true by definition. All axioms are by definition is just the basic building blocks of proofs. They're just claims: they're only true or false when applied to particular mathematical objects. You can have false axioms, it's just pretty silly, because your system will then necessarily and immediately not be sound.

Answer 5

To have a concise (and intuitive) answer I will paraphrase what Scott Aaronson said in his 6.045/18.400 MIT lecture. He said something like this:

Soundness means everything provable is true. Since consistency means there are no contradictions and soundness already involved the concept of truth and truth must be consistent (i.e. True != False), then its must mean Sound systems are also consistent. So Soundness implies consistency because (truly) true things don't have contradictions.

Now that I reflect I realize that I had some incorrect assumptions/ideas:

1. I didn't realize soundness was about semantics. Thus, I failed to realize that just using inference rules from the axioms is not enough to lead to true consequences (and that it doesn't guarantee it, which I thought was impossible to reach contradictory things as long as we started from the axioms and used valid inference rules).
2. I thought that as long as the axioms were true and the inference rules made sense everything that proceeded would be true. Which I now realize might not be true since if we just have a giant list of axioms and inference rules its hard to reason if everything that follows will be true. i.e. just starting from an axiom and using a valid inference rule is not enough to guarantee that the next step will be true.
3. The previous is essentially coupled with the fact that I didn't realize that there are two levels of complexity, 1) semantics 2) syntactics. Cranking the symbols crunching game may lead to contradictions.
4. I didn't realize I didn't know the proper characterization of truth, which derek made a great job in characterizing.