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


andrewspencers@xxxxxxxxx wrote:
> george wrote:
> > As for the "same form" part, YES, OBVIOUSLY,
> > modus ponens and the propositional tautology
> > ((p->q)^p)->p) OBVIOUSLY DO have the same form.
> > Equally obviously, you may take ANY propositional
> > tautology whose main connectives are ^ and -> and
> > make a sound rule of inference out of it, just
> > by translating the ^'s to commas and the ->'s to
> > horizontal lines or whatever other symbol is being
> > used to divide premises from consequents.

> Ok, if the metalanguage uses the symbols "," and "/" instead of the "^"
> and "->" used in the object language (the object language here being
> the sentential calculus), then I can write modus ponens as "[(p->q),p]
> / q", but this just makes in clear that the sentence (it is a sentence,
> right?) is written in the metalanguage instead of the object language;
> if both languages used the same syntax (which had been my unstated
> assumption), then modus ponens would be written exactly the same as the
> propositional tautology.

That would be very confusing: inference rules are not about SC, but
about what can be done with SC.

As an analogy, think of SC as being like all the physical stuff
(engine, wiring, etc.) in the hood of your car; and inference as being
like all the events or processes that happen there when you turn the
key. Each process is equivalent to some of the stuff being there - the
spark plug will spark iff it's clean and properly in place, etc; but
none of the processes is the exact same thing as the stuff - the spark
isn't the same thing as the plug being there.

> But aside from the question about which syntax is used in which
> language, my original question was: why is the metalanguage even needed
> in the first place, i.e. why do inference rules have to be written in
> the metalanguage, and can't be written in sentential calculus?

The 'metalanguage' involves a different level of description; it isn't
about what wff are true or false in SC, but about what one can do using
SC.

> "[(p->q),p] / q" written in the metalanguage seems to mean exactly the
> same thing as "[(p->q) ^ p] -> q" written in sentential calculus,
> especially considering that in both cases the variables p and q range
> over the same things: sentences of the sentential calculus.

They correspond - for instance, they're logically equivalent (it has to
be that one's true iff the other is true) - but they don't say the same
thing. The second says that "If p only if q, and p, then q is a true
statement"; while the first says, "Given that p only if q, and p, are
true statements, q is a true statement".

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