reductio ad falsum versus reductio ad absurdum - sci.logic

Message from discussion reductio ad falsum versus reductio ad absurdum

futurist

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More options Sep 6 2005, 2:30 pm

Newsgroups: sci.logic

From: "futurist" <adamgold...@adamgolding.com>

Date: 6 Sep 2005 11:30:27 -0700

Local: Tues, Sep 6 2005 2:30 pm

Subject: Re: reductio ad falsum versus reductio ad absurdum

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> > > Given
> > > |- P
> > > |- P -> Q
> > > infer
> > > |- Q

> > > You must use meta language "infer" and not object language "->" because
> > > |-P is not a statement of the object language.

> > modus ponens does NOT conclude a statement in the metalanguage, so no
> > rule which ends in "infer |- Q" could be modus ponens. you must mean:

> > Given
> > P
> > P->Q
> > infer
> > Q

> That is modus ponens. That is different that what you were trying
> which is what I explained to you above which is a meta-theorem
> that's an immediate result of MP and the definition of |-.

i may have mistakenly given the impression that i was interested in the
meta-theorem but i am not. i am interested in precisely describing the
rules of the object language that is ND.

> > which is expressed as P, P->Q |- Q

> Correct.

> > > You have to express the rule in the
> > > metalanguage.

> > yes, which is why I'm trying to express the rule using sequents,
> > because sequents are in the meta language. however you seem to keep
> > trying to push the rule into the meta-meta language (English statements
> > about sequents), placing it out of reach for application in the object
> > language...

> The object language doesn't have the capacity to discuss itself!

of course not. which is why the rule OF a language cannot be expressed
IN that language. thus i am using sequents which are NOT part of ND,
and are thus ONE level up, and are in the *meta*-language, as opposed
to the meta-meta-language, which is where you place us when you connect
sequents with english. the rules should be expressed in the
meta-langage. the meta-meta-language would, i suppose, be better for
'rules about rules' but i don't see any need for that here.

> > actually i never used '->' for infer, i used '|-' -- although i think
> > you're right that it is/was causing me confusion--it happens to be in
> > Allen and Hand that many of the 'rules' about what you can infer also
> > happen to be truths about what literal wffs follow from others (i.e. P
> > |- P v Q) but this appears to break down for CP and RAA... (although
> > i dont' totally understand why yet...)

> Yes, |- doesn't mean infer, it means there's a sequence of wffs
> such that ...

> > > Or to use words
> > > If you assume A and can derive Q and ~Q, then you can infer ~A
> > > To restate the outline. From
> > > A |- Q, ie assuming A and deriving Q
> > > A |- ~Q, ie assuming A and deriving ~Q
> > > conclude
> > > |- ~A

> > the last thing you wrote is a weaker form, i think, (what another
> > poster called the 'constructive reductio', because it says that both Q
> > and ~Q have to follow from A *alone*--so it doesn't provide for the
> > following case:

> > Q
> > ---
> > A (Assumption)
> > ...
> > ~Q
> > ~A

> You haven't defined and explained the semantics and syntax of
> ---

it's nothing but the horiztonal line that separates the premises from
their consequences in every version of ND i've ever seen... at any rate
the sketch immediately above was not an attempt at a precise
formulation of the rule--it was just to discuss an example--an argument
which wouldn't fall under your description of the rule, thus showing
your description of the rule is not general enough, or at least that a
second rule is required.

> > since A |- Q is never demonstrated here (and may in fact be false)

> > i believe i saw that equivalence awhile back--and I'm sure ill
> > encounter it again when i take metalogic, although ND is definitely the
> > system for me right now, since i want to be applying it heavily to
> > natural language in some of my philosophy courses.

> Spare me, what's that? Metalogic is to logic like metaphysics is to
> physics?

no. the study of metaphysics was so named partially because aristotle's
metaphysics, which had no title from the author, came 'after' ('meta')
aristotle's physics in traditional compilations of his work.
metaphysics is not 'physics about physics', it is the study of
principles about causation, existence, truth, etc. which are intended
to have greater generality than topics discussed in real-world physics.
so the sense of 'meta' is similar but not the same.

metalogic is to logic as metamathematics is to mathematics. thus
metalogic and metamathematics are essentially the same discipline.
Godel's theorem is a metalogical/metamathematical theorm. statments
like 'ND is complete' are statements of metalogic.

> > > and in philosophical logic, the essential precision of
> > > mathematical logic is likely neglected.

> > i cannot decode this sentence--do you mean that PC is more 'precise'
> > than ND?

> Before getting into the detains of logical precision, first be more
> precise about your use of language using capital I and I'm instead of i
> and i'm. Then with demostrated attention to details, attention to
> precise mathematical and logical details will be easier.

I cannot decode the sentence mentioned above--do you mean that PC is
more 'precise' than ND?

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