In a Consistent System, Can a True Sentence Imply a False One?
in [Logic]
|
Prev: when math defines the boundary between finite versus infinite at 10^500 #696 Correcting Math
Next: FLT like 4Color Mapping, Poincare C. and Kepler Packing #697 Correcting Math
From: Nam Nguyen on 30 Jul 2010 19:14 Daryl McCullough wrote: > Nam Nguyen says... > >> If you clarified so. Now then, as I asked before, if e.g. I tell you I have >> a T that has a disprovable formula in it, would you be able to tell if that >> T is consistent, or not? > > No, because *every* theory (inconsistent or not) has disprovable formulas > in it. Right. So, would you see why it's odd, not making a lot of sense, not to restrict the definition only to the case of consistent theories? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 30 Jul 2010 20:06 Daryl McCullough wrote: > Nam Nguyen says... > >> You have not yet illuminated why such definition (of "disprovable) would >> make any technical sense in the case of an inconsistent theory. > > What kind of sense do you want it to make? What's wrong with the > conclusion that, if T is inconsistent, then every sentence is both > provable and disprovable in T? > > It's a technical claim which follows from the technical definitions > of "provable" and "disprovable". What more do you want? I'm very sure that defining a prime as having 1 and itself as its divisors has _some_ merits too because the well known 3 is a prime and because as you've just alluded to it's just a definition! But let me cut the chase here and say that a genuine technical definition has a few purposes but one important of which is it should not lead to contradictions, even if accidentally (via definition) by ambiguities. For example if I defined a theory as T "para-consistent" iff there's a disprovable formula F in it, would anybody here really know if T is consistent or not, given that it must be one way or another, and that that's a _clear cut_ technical definition hence can't be wrong? I mean: what is a useful definition if it's actually useless? In the case of "disprovable" definition, because most of the times when talking about disprovable formulas we in some ways already assume (even silently) the T be consistent, and so it's a perceived ambiguity when mentioning them. But that's only half of the story. The other half comprises 2 parts. A) we do have a very simple formal definition of an inconsistent theory that we can use and B) in the case of a consistent theory a "disprovable" formula is also unprovable and unprovability is a characteristics of a consistent theory. Given all that, if we restrict the definition only to consistent theories (or just re-confirm that is the case because there's still a likelihood it's actually the case that people do assume consistency) then it would make clear sense to the semantic of the word. Of course, as I alluded to before, what would be _the technical points_ to define something like "Bush-provable", "Obama-disprovable" or the like even when we technically could? -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: MoeBlee on 30 Jul 2010 20:16 On Jul 30, 6:06 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 30, 5:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > >> I'm sorry: I'm not interested in your cooking knowledge. When you have > >> anything technical to say then get back to the subject. > > > I wrote you a LIST of technical definitions and theorems!!! > > Where in that list would it contain your explanation why the given > definition of "disprovable" would make sense in the case of an > inconsistent theory? This is TOO delicious! Why would ANY definition make sense when it doesn't make clauses for ALL KINDS of special circumstances? You could say of ANY definition, "well, what about this situation and what about that situation?". Now, rather than guide you (who could possibly presume to guide Nam Nguyen in anything?) with this, I'm going to leave it at this: You only need to refer to a good discussion (such as in Suppes's 'Introduction To Logic') on the subject of definitions. And with that, I SWEAR TO GOD, I'm done. MoeBlee
From: Nam Nguyen on 30 Jul 2010 20:32 MoeBlee wrote: > On Jul 30, 6:06 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>> On Jul 30, 5:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> I'm sorry: I'm not interested in your cooking knowledge. When you have >>>> anything technical to say then get back to the subject. >>> I wrote you a LIST of technical definitions and theorems!!! >> Where in that list would it contain your explanation why the given >> definition of "disprovable" would make sense in the case of an >> inconsistent theory? > > This is TOO delicious! > > Why would ANY definition make sense when it doesn't make clauses for > ALL KINDS of special circumstances? Rhetorical, MoeBlee. We're only talking about 2 cases: consistent and inconsistent theories. NOT "ALL KINDS of special circumstances". Besides let be be what it be: if at some points we need clarity then so be it. Is that a "shock" or a surprise to you. > > You could say of ANY definition, "well, what about this situation and > what about that situation?". Can you give another example? > > Now, rather than guide you (who could possibly presume to guide Nam > Nguyen in anything?) with this, I'm going to leave it at this: You > only need to refer to a good discussion (such as in Suppes's > 'Introduction To Logic') on the subject of definitions. > > And with that, I SWEAR TO GOD, I'm done. I don't know about the other parts but it's good that you're done here. There doesn't seem much anyone would miss. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: MoeBlee on 30 Jul 2010 20:50
Oh, lordy lordy, I SWORE to you...please, God in heaven, make me strong to resist temptation... Your mortal servant, MoeBlee |