Simple yet Profound Metatheorem
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From: Charlie-Boo on 22 Jan 2006 22:20 David C. Ullrich wrote: > On 9 Jan 2006 14:58:32 -0800, "Charlie-Boo" <chvol(a)aol.com> wrote: > >>Don't you see the parallel? If |- (|-Q == Q) is extremely bad > >>notation because the first |- means |- while the second |- refers to > >>some encoding of provability, then [] ( []Q == Q ) is extremely bad > >>notation because the first [] means |- while the second [] refers to > >>some encoding of provability. > > No, in standard modal logic people do not use the symbol [] > to mean two different things. Therefore your comment that my syntax does that is false, as my expression has the same meaning as the corresponding expression in Modal Logic - I changed only the syntax (to the less ambiguous |- instead of [] which has multiple interpretations), leaving the semantics the same. It is the same semantics as in Modal Logic - only the symbol is changed. If my semantics are wrong, then Modal Logic's semantics are wrong, as they are the same semantics. Got it?????? Once again you criticise a principle despite there being overwhelming evidence that it is in common use. > >> > A better > >> >counterexample for you (than meaningless propositional variables) would > >> >be P is any Godel sentence and Q is FALSE. > >> > >> A counterexample that depends on an _actual_ deep theorem is > >> "better" than a simple and totally elementary counterexample? > >> Fascinating. > > > >Your "proof" says, "suppose P (is an) atomic formula in the > >predicate calculus. Then P is not provable." This is not > >well-formed. (It is meaningless.) By that reasoning, PA is incomplete without regard to whether it is sound or consistent or not. Just pick an atomic formula and it and its negation are unprovable. Who needs Godel or Rosser then? > ************************ > > David C. Ullrich
From: David C. Ullrich on 23 Jan 2006 07:44 On 22 Jan 2006 19:20:08 -0800, "Charlie-Boo" <chvol(a)aol.com> wrote: >David C. Ullrich wrote: >> On 9 Jan 2006 14:58:32 -0800, "Charlie-Boo" <chvol(a)aol.com> wrote: > >> >>Don't you see the parallel? If |- (|-Q == Q) is extremely bad >> >>notation because the first |- means |- while the second |- refers to >> >>some encoding of provability, then [] ( []Q == Q ) is extremely bad >> >>notation because the first [] means |- while the second [] refers to >> >>some encoding of provability. >> >> No, in standard modal logic people do not use the symbol [] >> to mean two different things. > >Therefore your comment that my syntax does that is false, I made that comment because that's what you _said_ it meant. > as my >expression has the same meaning as the corresponding expression in >Modal Logic - I changed only the syntax (to the less ambiguous |- >instead of [] which has multiple interpretations), leaving the >semantics the same. > >It is the same semantics as in Modal Logic - only the symbol is >changed. If my semantics are wrong, then Modal Logic's semantics are >wrong, as they are the same semantics. > >Got it?????? > >Once again you criticise a principle despite there being overwhelming >evidence that it is in common use. > >> >> > A better >> >> >counterexample for you (than meaningless propositional variables) would >> >> >be P is any Godel sentence and Q is FALSE. >> >> >> >> A counterexample that depends on an _actual_ deep theorem is >> >> "better" than a simple and totally elementary counterexample? >> >> Fascinating. >> > >> >Your "proof" says, "suppose P (is an) atomic formula in the >> >predicate calculus. Then P is not provable." This is not >> >well-formed. (It is meaningless.) > >By that reasoning, PA is incomplete without regard to whether it is >sound or consistent or not. Just pick an atomic formula and it and its >negation are unprovable. Who needs Godel or Rosser then? > >> ************************ >> >> David C. Ullrich ************************ David C. Ullrich
From: Charlie-Boo on 23 Jan 2006 14:16
David C. Ullrich wrote: > On 22 Jan 2006 19:20:08 -0800, "Charlie-Boo" <chvol(a)aol.com> wrote: > >David C. Ullrich wrote: > >> On 9 Jan 2006 14:58:32 -0800, "Charlie-Boo" <chvol(a)aol.com> wrote: > >> >>Don't you see the parallel? If |- (|-Q == Q) is extremely bad > >> >>notation because the first |- means |- while the second |- refers to > >> >>some encoding of provability, then [] ( []Q == Q ) is extremely bad > >> >>notation because the first [] means |- while the second [] refers to > >> >>some encoding of provability. > >> > >> No, in standard modal logic people do not use the symbol [] > >> to mean two different things. > > > >Therefore your comment that my syntax does that is false, > > I made that comment because that's what you _said_ it meant. Too many pronouns. What did I say and what exactly did you interpret it to mean? > ************************ > > David C. Ullrich |