Torkel Franzen on truth
in [Logic]
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From: MoeBlee on 7 Jan 2008 13:01 On Jan 5, 1:14 pm, george <gree...(a)cs.unc.edu> wrote: > On Jan 4, 1:46 am, herbzet <herb...(a)gmail.com> wrote: > > MoeBlee wrote: > > > Authors such as Enderton take a theory to be any set of sentences > > > closed under entailment (which, thanks to the completeness theorem, is > > > a set of sentences closed under provability). > > That is the canonical definition. > "Canon" meaning what it means, that is the ONLY definition > you are going to get to use, without first explicitly attacking > that definition. It's the one I use, but I don't know that it is canonical. Many authors use other definitions. > > > I adopt Enderton's definition. > > > OK. > > It's NOT ok. I thought you just aid it is the canonical definition. So it's not okay for me to use what you yourself regard as the canonical definition? And even if it is not canonical, it is at least quite common. Moreover, adopting a definition that is so common and also available to refer to in such a widely referenced book as Enderton's seems quite okay. > It does violence to the word (theory). > If you can't tell what's an axiom and you therefore can't > tell even APPROXIMATELY *whether* some sentence is "in" > (declared "provable") by the theory OR NOT, then you don't > ACTUALLY HAVE any coherent *theory* (NON-local-technical sense) > of what makes sentences "in" the theory true! To express what you just mentioned, we also have <A T> where A is an axiomatization of T. Granted, for many authors, a theory is an ordered pair <A T> where T is the set of consequences of A. If you prefer that definition or some other arrangement, then fine for you. But Enderton's definition is also handy for certain reasons (as evidenced by the way he uses it in his book). Live and let live should be the principle when it comes to such differences in definitions. Sure, one can argue that certain definitions are better than others, but your continual TIRADES about such matters are foolish. > > > > I usually think of "a theory" as having a recursive (or at least r.e..) > > > > set of axioms. > > THAT *IS* the *CORRECT* definition. > As MoeBlee is explaining, it is NOT the standard one. > The point is that the standard is just broken. Your continual DECREES as to what is "correct" are tiresome. Fine that you may give reasons for preferring definition D to definition D' (which should allow for understanding also the advantages of definition D' over definition D). But your continual HARANGUES and FIATS on such things as definitions are foolish. And please make up your mind: Most of the time you're telling people that they are wrong not to follow standard conventions (or the "paradigm", a word you use), and here you're saying it is NOT okay to do so. MoeBlee
From: herbzet on 7 Jan 2008 18:06 MoeBlee wrote: > herbzet wrote: > > MoeBlee wrote: > > > For any structure S in a language L the structure assigns a truth-value > > to every sentence of L. For every sentence phi of L, one of phi and > > ~phi will be true and one will be false in S. On the assumption that > > not all sentences of L are validities and their negations, then S will > > be a model of some sentences that are not validities, and hence will > > be a model of some non-null theory of language L. > > > > Perhaps you would like to clean that up a bit. > > It occurred to me Friday night that I spaced out the obvious and > trivial: It happens. -- hz
From: herbzet on 7 Jan 2008 18:17 MoeBlee wrote: > herbzet wrote: > > MoeBlee wrote: [...] > > > > Yuh, I guess the argument turns on whether you want to affirm > > > > > > a) 2 = 1 + 1 is false in no model (of PA) > > > > > > or > > > > > > b) 2 = 1 + 1 is false in some models (of the language of PA). > > > > > I affirm both. > > > > But not, I hope, without the parenthetical qualifications. > > That would be very confusing. > > What is confusing? If one knows the definitions, both a) and b) are > clearly true. Without the parenthetical qualifiers, a) and b) are clearly contradictory. > model of a theory > > vs. > > moel for a language. > > Those are clearly distinct. Well, I guess context is important for an overloaded term. > > > > Actually, I find it even more annoying when someone is being a jerk > > > > while being right! > > > > > I guess that's a joke or something. Of course, it's subjective what > > > annoys a person. But it does seem disconnected to me to be more > > > annoyed by someone who is at least correct. > > > > I can assure you from long experience, I'd rather play chess with > > a poor loser than a swaggering, ungracious winner. > > Ah, yes, chess does have its own special poignant psychology. Yes, winning is unanswerable. No need to slam the pieces down. -- hz
From: herbzet on 7 Jan 2008 18:22 MoeBlee wrote: > To express what you just mentioned, we also have <A T> where A is an > axiomatization of T. Does "axiomatization of T" mean "recursive axiomatization of T"? Or can A be any old set of sentences? > Granted, for many authors, a theory is an ordered > pair <A T> where T is the set of consequences of A. -- hz
From: MoeBlee on 8 Jan 2008 13:47
On Jan 7, 3:17 pm, herbzet <herb...(a)gmail.com> wrote: > MoeBlee wrote: > > herbzet wrote: > > > MoeBlee wrote: > > [...] > > > > > > Yuh, I guess the argument turns on whether you want to affirm > > > > > > a) 2 = 1 + 1 is false in no model (of PA) > > > > > > or > > > > > > b) 2 = 1 + 1 is false in some models (of the language of PA).. > > > > > I affirm both. > > > > But not, I hope, without the parenthetical qualifications. > > > That would be very confusing. > > > What is confusing? If one knows the definitions, both a) and b) are > > clearly true. > > Without the parenthetical qualifiers, a) and b) are clearly contradictory. Sorry, I mistakenly misread you. I thought you meant some other parenthetical qualifier needed beyond those already present. That was plainly my lapse. So, right, a) and b) are compatible as long as the parentheticals you included are included. MoeBlee |