Validity of Deductive Arguments

Question

I’ve been reading some mathematical logic texts and encountered differences in the definition of the validity of arguments, as presented below

An argument is valid iff it is impossible for all the premises to be true and the conclusion to be false.
An argument is semantically valid iff when all the premises are true, then the conclusion must also be true. An argument is syntactically valid iff the premises under some Proof System deduce the conclusion (that is, the conclusion has to be written whenever the premises are written).

It seems like the definition of validity in Definition 1 is the same as the one for semantically validity, in which the truth of each premise is evaluated by some truth assignment. However, the syntactical validity is only concerned with the form of the argument.

Now my Rhetoric professor defined the validity of an argument with Definition 1. However, it seems to me that Definition 1 is some more simple way of understanding a more complex issue presented in Definition 2, and when we talk about the validity of an argument (say in a mathematical logic context), we are in generally referring to the definition for syntactical validity, the one that only cares about the form of the argument rather than semantical validity.

Is my understanding correct? If so, consider the following deduction $$ \begin{align*} &p\\ &q\\ \hline &\therefore r \end{align*} $$ where $p, q$, and $r$ are all false statements and there is no logical connection between each of the three. Would this argument be semantically (vacuously) valid (since the premises are all false) but syntactically invalid (since there is no deduction going on because there is no logical connection between the statements)?

Syntactically speaking, assigning 'false' to variables 'p', 'q', and 'r' is not consistent with their form. Thus, asking about it's syntactically validity doesn't work, because variables are not constants, and the form of a propositional variables works out such that it can either hold as true or false. You can have an argument of the form 0, 0 $\vdash$ 0, where '0' means 'falsum' which would be both a syntactically valid and semantically valid argument. — Doug Spoonwood, Sep 12 '23 at 02:52
Last point: "from p and q, then r" is NOT valid. For premises p and q both TRUE and conclusion r FALSE, the definition of valid argument is not satisfied. — Mauro ALLEGRANZA, Sep 12 '23 at 05:44
Yes, as you have noted, the two definitions above of "valid" and "semantically valid" are the same. Usually, we specify "semantically" in the context of a formalized system with interpretation, in which case validity can be defined semantically, i.e. by way of the interpretation, and syntactically, i.e. by way of the deductive apparatus of the system (axioms+rules). In most case, like classical propositional logic, we can prove mathematically that the two definitions are equivalent. — Mauro ALLEGRANZA, Sep 12 '23 at 05:47
See also the post Is an argument valid simply if its form is valid? as well as Logically, deductively, tautologically, semantically, syntactically valid arguments. — Mauro ALLEGRANZA, Sep 12 '23 at 06:48
See also Valid vs. sound (and if these differences matter in syntax vs. semantics, metalogic vs. logic) as well as Logic terminology. — Mauro ALLEGRANZA, Sep 12 '23 at 06:50
Do you mean that $p$ and $q$ are contradictions rather than that they are false? False premises are potentially true in an alternate interpretation, so don't necessarily produce a valid argument. Anyway, I guess, the logical connection between contradictions is their logical equivalence to one another. — ryang, Sep 12 '23 at 11:34

Validity of Deductive Arguments

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