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"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!

cekami7844
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  • Having said that, every inference rule produces a valid argument (both syntactically and semantically) because the conclusion is a logical consequence of the premises. Thus, using the MP example: $A \to B, A \vDash B$ and $A \to B, A \vdash B$. – Mauro ALLEGRANZA May 12 '20 at 11:04
  • @MauroALLEGRANZA but $∀xPx→Pa$ is a tautology, https://www.umsu.de/trees/#(%E2%88%80xPx%E2%86%92Pa), you can't prove it with truth table, but if you construct a truth tree of the negation, and the tree closes (so there is no way that the negation is true) the sentence is a tautology – cekami7844 May 12 '20 at 11:09
  • Rules of inference are part of the formal system and they are the "building blocks" of formal proofs (see the main answer to your previous post). Thus: $A \to B, A \vdash B$. – Mauro ALLEGRANZA May 12 '20 at 12:40
  • And rules of inference are valid with respect to the corresponding semantics (rules preserve truth). Thus: $A \to B, A \vDash B$. – Mauro ALLEGRANZA May 12 '20 at 12:41
  • See also another previous post by you. – Mauro ALLEGRANZA May 12 '20 at 13:35
  • "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 isn't correct. A tautology is a formula in the system, which follows formation rules. A∧(A⇒B)⇒B is not a formula (not at the same level as a tautology), and thus isn't a tautology. Also, modus ponens has two premises, NOT one as (A∧(A⇒B)) suggests. Modus ponens is a valid argument, because when (A⇒B) holds true, when A holds true also, then B holds true also. – Doug Spoonwood May 12 '20 at 17:01
  • @DougSpoonwood That formula was meant with the parenthesis (A∧(A⇒B))⇒B IS a tautology – cekami7844 May 12 '20 at 19:15
  • @cekami7844 Sorry. I misread your statement. There does exist the tautology ((A∧(A⇒B))⇒B). However, the rule of inference isn't that. It's A, (A⇒B) $\vdash$ B. It doesn't have a conjunction in it, such that A, (A⇒B) is well-formed in the object logic. In other words, the rule of inference is not a tautology. The rule of inference also doesn't work out as valid, because of the tautology. It's because when each premise holds true, so does the conclusion. – Doug Spoonwood May 12 '20 at 22:24

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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.

Bram28
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  • 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.
  • what is the "conditional proof rule"? – cekami7844 May 16 '20 at 22:01
  • Conditonal prrof rule says : " if , after having admitted A as hypothesis, you dérive B under this hypothesis, you may infer ( unhypothetically) : (A --> B) ". –  May 17 '20 at 14:30
  • https://www.iep.utm.edu/prop-log/#SH5e , §5 e) –  May 17 '20 at 14:32