Is an argument valid simply if its form is valid?

Question

Can I conclude that an argument is valid if its argument form is valid?

I realize that a false premise may lead to an incorrect conclusion (which is not what I'm asking). I see a lot of questions about whether specific arguments are valid, but I'm asking in general. I just want to confirm that I'm understanding this concept.

Definition from Rosen's book:

An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true.

An argument form in propositional logic is a sequence of compound propositions involving propositional variables. An argument form is valid no matter which particular propositions are substituted for the propositional variables in its premises, the conclusion is true if the premises are all true.

Argument form looks like (i can't seem to get perfect formatting):

$p \to q$

$p$

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$\therefore q$

Perhaps another way to ask this is, if the argument form is a tautology (I'm verifying if tautology means valid argument form), then can I say that the argument is valid? (again, I'm not asking about the conclusion of the argument but about its validity)

This question gives an example of a valid argument form as: $((p \wedge (p \to q)) \to q$ or Modus ponens. There are other examples too.

Thanks,
Paul

Answer 1

An argument : "if $\varphi_1$ and $\varphi_2$, therefore $\psi$" is valid, in symbols :

$\varphi_1, \varphi_2 \vDash \psi$

if and only if : $(\varphi_1 \land \varphi_2) \to \psi$ is a tautology, in symbols :

$\vDash (\varphi_1 \land \varphi_2) \to \psi$.

See this post and this post for details.

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I think that, according to Rosen's definition :

an argument in propositional logic is a sequence of propositions

we have to understand a specific argument with "natural language" expressions, like the following :

"if $n$ is even and $n$ is greater than $2$, then $n$ is even"

which is an instance of the "argument form" :

"if $p$ and $q$, then $p$"

where an "argument form" is a sequence of compound propositions involving propositional variables.

Thus, Rosen's definition of valid argument form :

an argument form is valid no matter which particular propositions are substituted for the propositional variables in its premises, the conclusion is true if the premises are all true

is formalized in propositional logic with the definition of the relation of of tautological implication (or consequence) between a set of propositional formulae : the premises, and a formula: the conclusion :

$\Sigma$ tautologically implies $\tau$ (written : $\Sigma \vDash \tau$) iff every truth assignment for the sentence symbols in $\Sigma$ and $τ$ that satisfies every member of $\Sigma$ also satisfies $τ$.

We have that, if $\Sigma$ is a finite set of formulae, i.e. $\Sigma = \{ \varphi_1, \varphi_2,\ldots,\varphi_n \}$ :

$\varphi_1, \varphi_2,\ldots,\varphi_n \vDash \tau$ iff $\vDash \varphi_1 \land \varphi_2 \land \ldots \land \varphi_n \to \tau$

i.e. iff $\varphi_1 \land \varphi_2 \land \ldots \land \varphi_n \to \tau$ is a tautology.

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In conclusion, we can say that a (specific) argument (like the one above) is valid if it is an "example" of a valid argument form; in the example above : $(p \land q) \to p$.

Answer 2

The answer is "yes." An argument's validity/invalidity depends only its logical form. The form of the argument you gave is valid. (And it's called "modus ponens")