Re: Why are rules of inference not laws of sentential calculus?

andrewspencers@xxxxxxxxx wrote:
> Tarski says on p. 47 of "Introduction to Logic and to the Methodology
> of Deductive Sciences",
> "Besides the rules of definition, of which we hve already spoken, we
> have other rules of a somewhat similar character, namely, the RULES OF
> INFERENCE or RULES OF PROOF. These rules, which must not be mistaken
> for logical laws, amount to directions as to how sentences already
> known as true may be transformed so as to yield new true sentences. In
> the proof carried out above, two rules of demonstration have been made
> use of: the RULE OF SUBSTITUTION and the RULE OF DETACHMENT (also known
> as the MODUS PONENS RULE)."
>
> He had previously already stated several laws of sentential calculus in
> both natural and formal language, including for example the law of
> hypothetical syllogism: "if p implies q and q implies r, then p implies
> r", i.e. "[(p->q) ^ (q->r)] -> (p -> r)" (prefixed by universal
> quantifiers for p, q, and r). He then says on p. 48, only in natural
> language,
> "The rule of detachment states that, if two sentences are accepted as
> true, of which one has the form of an implication while the other is
> the antecedent of this implication, then that sentence may also be
> recognized as true which forms the consequent of the implication."
> He doesn't state this rule in formal language, but wouldn't it be
> "[(p->q) ^ p] -> q"?

Nope. Rules of proof are not stated in the formal language, but
generally given in English. Formulas in a language are just collections
of symbols put together according to particular rules. They can't tell
you how to derive formulas from other formulas, which is why you need
rules of proof.

>
> So, if something like "[(p->q) ^ (q->r)] -> (p -> r)" is a logical law
> (i.e. law of sentential calculus), then why is "[(p->q) ^ p] -> q" not?

((p->q) ^ p) -> q) is a theorem of the sentential calculus, but this
theorem is not the rule of detachment.

.

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