'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 11 Apr 2010 11:46 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> If you don't remember or have any doubt about my position, please >> allow me to clearly reiterate my position: >> >> It's impossible to have logically acceptable methods for proving a >> syntactical consistency. > > But there's nothing clear about this reiteration. What is a "logically > acceptable method"? > A method which would conform to definition of syntactical (in)consistency, to definition of model of formal systems, to definition of rules of inferercne.
From: Nam Nguyen on 11 Apr 2010 11:53 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> So what you're saying is you just _intuit_ PA system be consistent, no >> more no less. Of course anyone else could intuit the other way too! > > Anyone can intuit whatever they want. I said nothing about such matters, > I merely noted I have, like any number of logic students over the > decades, produced a proof of the consistency of PA in the course of my > studies. Intuition has no more and no less to do with it than with any > proof in mathematics. > So, is your proof of the consistency of PA a syntactical proof of a _consistent_ formal system? (I mean, anyone could syntactically prove such proof in an inconsistent formal system). If not then in what sense would your proof not require an _intuition about the natural numbers_?
From: Nam Nguyen on 11 Apr 2010 12:15 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> So what you're saying is you just _intuit_ PA system be consistent, no >>> more no less. Of course anyone else could intuit the other way too! >> >> Anyone can intuit whatever they want. I said nothing about such matters, >> I merely noted I have, like any number of logic students over the >> decades, produced a proof of the consistency of PA in the course of my >> studies. Intuition has no more and no less to do with it than with any >> proof in mathematics. >> > > So, is your proof of the consistency of PA a syntactical proof of a > _consistent_ formal system? (I mean, anyone could syntactically prove > such proof in an inconsistent formal system). If not then in what sense > would your proof not require an _intuition about the natural numbers_? Iow, your notion of "proof" is intuitive (and vague) which is why I mentioned you "just _intuit_ PA system be consistent".
From: Nam Nguyen on 11 Apr 2010 12:52 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> If you don't remember or have any doubt about my position, please >>> allow me to clearly reiterate my position: >>> >>> It's impossible to have logically acceptable methods for proving a >>> syntactical consistency. >> >> But there's nothing clear about this reiteration. What is a "logically >> acceptable method"? >> > > A method which would conform to definition of syntactical (in)consistency, > to definition of model of formal systems, to definition of rules of > inferercne. For example, one can _actually prove_ this T = {(a=b) /\ ~(a=b)) is inconsistent, even though one wouldn't be able - or wouldn't care - to intuit the semantic of each symbol in the axiom. That's a "logically acceptable" proof.
From: Alan Smaill on 11 Apr 2010 13:58
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > David Bernier wrote: >> Do you see problems with starting with a false premise, not(P) ? > > Yes. I'd break "the Principle of Symmetry": if we could start with not(P), > we could start with P. And so you could, so that's not a problem. But what would you prove? -- Alan Smaill |