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


andrewspencers@xxxxxxxxx wrote:
> George Dance wrote:
> > > "[(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.

> BTW you didn't dispute my claim that in both cases "p" and "q" range
> over the sentences of the sentential calculus. Is my claim really
> correct? I thought that any variable "x" which ranged over the
> sentences of some language L could not exist in L, but must exist in
> some language M which is a metalanguage of L.

You may be confusing me with another 'george' - one who tends to shout
a lot for emphasis. I've said nothing about 'metalanguages.'
What I mentioned were two different 'levels of description'.

'Metalanguage' was a Tarski invention for one such case of different
levels, which does not require them even being two different languages
- if the two of us were talking about cars or women, for instance, and
then began talking about what a statement one of us had said earlier
(whether it was really true or false), we'd have shifted
from using English as an object language to using it as a metalanguage
without even making a distinction. Metalanguages exist only
relatively; the same language can be an OL or a ML in two different
contexts, or even in two different applications of the same context.

To make all that relevant to your 'claim', I'd say that I agree with
it; it looks quite clear that, in PC, formulas involving propositional
variables like p and q (plus the operators) are 'metalanguage'
statements relative to formulas involving constants like P and Q (plus
the same variables). Which may contradict what 'george' et al have
been telling you (I haven't paid that much attention to their posts
here), but that's not a problem; he and I are not singing using the
same hymn book anyway.

.

Relevant Pages

  • Re: Torkel Franzen on truth
    ... language that M is a structure for. ... In such contexts as I mentioned, what is the harm of using 'model' and ... Except we don't have an example of anyone disagreeing with George ... simply because he is otherwise a royal jerk. ...
    (sci.logic)
  • Re: Computability and logic
    ... George, I apologize to you for all errors and unsubstantiated ... then this is a VERY simple language; ... is translate it into a string. ... program, the result being a single NUMERIC entity (both explicit, if I ...
    (sci.logic)
  • Re: Basic Axioms
    ... in these natural language assertions, but George Greene's ... the listings and read the snippits in the message listing. ...
    (sci.logic)
  • Re: Torkel Franzen on truth
    ... language that M is a structure for. ... To which George objected: ... which might seem like a rather pedantic distinction (or, ... simply because he is otherwise a royal jerk. ...
    (sci.logic)
  • Tarskis theory of truth is rubbish
    ... Mathematicians cant tell us what makes a mathematical statement true ... the australian philosopher colin dean points out tarskis theory of truth ... if a grammar of a language must be in its metalanguage, ... If the language under discussion is L, ...
    (sci.logic)