Are inference rules just tautologically (or syntactically) valid arguments?

Question

"An argument is valid iff the following implication is a tautology: $h_1∧h_2∧...∧h_n⇒C$ where $h_1∧h_2∧...∧h_n$ are the hypothesis and $C$ the conclusion."

A classic inference rule is modus ponens for example: $A⇒B$, $A$, therefore $B$. This is a valid argument, because $(A∧(A⇒B))⇒B$ is a tautology. This works for every inference rule.

So are inference rules just tautologically valid arguments (true in virtue of their form)? is this all they are?

One more doubt, if inference rule are valid arguments can i just "symbolize" modus ponens for example as: $A⇒B,A⊧B$ ? or there's another symbol for that? One of my books uses the symbol "$⇒$" (and $→$ for implications) but i think it can be a little confusing..

Thank you!

Answer 1

The intent of syntactical inference rules is to reflect semantically valid inferences/arguments, yes.

However, there is no requirement that inference rules be valid. You can define inference rules any which way you want. My favorite invalid inference rule is:

$$\frac{}{\therefore P}\qquad \text{(hokus ponens)}$$

because this will allow me to complete any formal proof in 1 step :)

A second thing to notice is that inference rules typically reflect baby inferences, i.e. inferences that are obviously valid. We typically do not like to have very complicated (but valid) arguments as our inference rules; the whole conceptual idea of formal proofs is to break things down into those baby inferences: if we see that each of the individual inferences are valid, then we can be confident that the whole argument is valid as well. We can then treat that whole argument as a Lemma, maybe, but treating the whole argument as an inference rule defeats this purpose.

Answer 2

• Arguments appear at the language level; an argument is a string of formulas.

• Any talk aboud validity ( semantic or syntactic) belongs to the metalanguage level. You cannot say " this formula or this inference is valid" inside the logical language.

• Inference rules belong to the metalanguage level: when I state an inference rule, I am not performing an inference, I am stating what one is allowed or even should to do while performing an inference : " from $(\phi \rightarrow \psi)$ and $\phi$, infer $\psi$".

Analogy : when I state the rule " Sentence --> Noun Phrase + Verb Phrase", I am not using english language; I am talking about english language ( namely, about its syntax).

• Most inference rules correspond to a parallel metalogical statement according to which a given formula is a logical law/ truth ( i.e. a tautology).

Example : corresponding to the metalogical statement above is the metalogical statemment " the formula $ [(\phi \rightarrow \psi) \land \phi\rightarrow \psi]$ is a tautology".

• But this is not always the case : the conditional proof rule has no corresponding tautology.