Trying to get all this terminology straight in logic

Question

Really confused on the terminology here.

According to a few resources as-linked:

Valid is defined as a logical form where it is impossible to have true premises leading to a false conclusion. It doesn't mean the premises are necessarily true. Just that if they were true, the conclusion would need to be true as well.

Sound is defined as a valid logical form where the premises are indeed true.

But then we also have "semantic consequence" denoted $\Gamma \vDash \varphi$ which normally means "if everything on the lefthand side is true, then the stuff on the righthand side is true." Sometimes this is also referred to as "validity" or "tautology."

But this also seems like it is clashing with the definition of "sound" as well.

So I'm really quite confused how we're supposed to be defining and separating these terms out. I'm seeing different answers on Wiki, different answers in books, different answers in other Math StackExchange posts, etc.

I'm going nuts over here trying to separate all these concepts (metalogic vs. logic, soundness vs. validity, semantic vs. syntactic, etc) when every other resource is constantly merging stuff together in vague ways.

Once and for all what is the definition of validity? Soundness? Completeness? Syntactic consequence? Semantic consequence? Logical consequence? Tautology? Theorem? Axiom? Inference? Formula? Sentence? Expression? Proposition? Statement? Relation? Connective? Operator? Are any of these actually synonyms of each other? How are they different?

Answer 1

It is important to stick to mathematical logic here as opposed to the definitions from philosophy. The first sentence of the wikipedia

In mathematical logic, a logical system has the soundness property if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.

while not fully precise, is the sense in which 'sound' is generally used in mathematical logic. This corresponds to a system having valid axioms and validity-preserving rules of inference, so you can see how it corresponds loosely to the philosophical notion of true premises and correct (i.e. truth-preserving) argumentation.

As for validity, the wikipedia sentence

A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language. In propositional logic, they are tautologies.

uses 'valid' in the way it is used in mathematical logic. Here the connection to the philosophical notion of a valid argument that you cite is a little less direct. Let $A$ and $B$ be your premises and $C$ your conclusion. And say $$ (A\land B) \to C $$ is valid in the mathematical sense. That means that $A\land B \to C$ is true in any interpretation, which means that it is impossible for $A$ and $B$ to be true and for $C$ to be false. So the two notions of validity connected, but even more loosely than was the case for soundness.

Note that the validity of $(A\land B)\to C$ is the same thing as $C$ being a semantic consequence of $A$ and $B,$ i.e. $$ A,B\models C$$ which has more of the flavor of "a valid argument from $A$ and $B$ to $C$," though strictly speaking, there's no 'argument' here. (However, the syntactic variation $A,B\vdash C,$ which is equivalent in the presence of a completeness theorem, means there's a proof of $C$ from assumptions $A$ and $B$... that's more of an 'argument'.)

I've tried to be conciliatory to the philosophical definitions here, but in math it's important to get comfortable with the fact that precise definitions vary from treatment to treatment. (Heck, the other day I learned that not all books have the same definition of 'compact set' in a general topological space, which was a definition that I thought was more or less sacrosanct.) It should also go without saying that the definitions don't need to conform to the colloquial meaning of the words, though it's nice when they're at least not wildly misleading.

In a field with as many moving parts in mathematical logic we will often need to adjust and adapt our terminology as we develop new ideas and try to apply mathematical logic to more exotic situations. Even in the beginning of my answer where I gave some 'general purpose' definitions and remarks, I had some particular contexts in mind, and we will need to make these things fully precise in any detailed treatment. For instance, when we move to the specific context of classical propositional logic, "valid" becomes synonymous with "tautology" and we make precise the ideas of "interpretation" and "truth" in a given interpretation. When we flesh things out in a given context (especially when working in more exotic situations than, say, classical propositional or predicate logic) everything I said is subject to revision.

Answer 2

The notation $\Gamma \models \varphi$ means $\varphi$ is true in every structure in which all statements in $\Gamma$ are true.

The notation $\Gamma \vdash \varphi$ means $\varphi$ can be proved by using the statements in $\varphi$.

The latter depends on some notion of proof. One wants such a notion to satisfy three desiderata:

• Soundness, i.e. if $\Gamma\vdash\varphi$ then $\Gamma\models\varphi.$
• Completeness, i.e. if $\Gamma\models\varphi$ then $\Gamma\vdash\varphi.$
• Effectiveness, i.e. there is a proof-checking algorithm, which will correctly identify its input as either a valid proof or not.

Note that $\Gamma$ is allowed to be a set of statements whereas $\varphi$ is just one statement. So why not conjoin all the statements in $\Gamma$ into one? The problem here is that if one is allowed to join infinitely many statements into one, then putting such an infinite conjunction in the role of $\varphi$ has the result that one cannot satisfy all three desiderata simultaneously. Any proof of that takes a lot of work.