Torkel Franzen on truth
in [Logic]
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From: george on 22 Dec 2007 12:41 > "G. Frege" wrote: > > Hence isn't it the /interpretation/ that assigns > > truth to an axiom? This is what is known as "a stupid question". Maybe FF should look in the dictionary under "Axiom". Axioms have to be provable because THAT'S THE DEFINITION OF "Axiom". The Axioms are the things that you just presume true FROM THE BEGINNING, the things to which you PRE-assign a definitional proof-length of 0 (or of 1 if takes some effort to show that some string is recognizable as an axiom). On Dec 21, 3:30 pm, herbzet <herb...(a)gmail.com> wrote: > Well, they're true in all their models, by definition. But axioms are NOT UNIQUE in that regard! That is NOT a special property Of Axioms! EVERY last wff is true in all of ITS models! MODELS have to be models OF something! It is, as I was trying to beat through MoeBlee's head a minute ago, "structures" and "interpretations" that get to be unattached. If a structure is a model of a wff then the the wff is true in the structure. MoeBlee's point was that people were in a very generalized habit of using "model" withOUT requiring it to be a model of anything in PARTICULAR. That is an observable fact about the brute weight of usage; neither I nor anyone else GETS to disagree with it, so MoeBlee probably thought it would therefore be safe, or at least defensible, to assert it. He was entirely wrong about that. People's general habit of doing this IS BAD. It is sloppy. IT NEEDS reform. People NEED to clean up their (usage)ACT. Instead, MoeBlee chose to DEFEND the fact that people generically tend to talk this way AS ACCEPTABLE because it is allowed by definitions that he can quote from Enderton. That is using a good work to confuse people and make discourse in general LESS accurate and it is not intellectually acceptable behavior. > Last time George and I spoke on this point, he was of the opinion, > if I understood him correctly, that interpreting the axioms in > structures which falsify (one or more of) them doesn't count -- > axioms are always taken as true regardless of how they are > interpreted; interpretation is otiose. Close, yeah. It amazes me that some people can just tune into my wavelength (if they feel like it) while others must insist that I'm just evil. > I'll admit it seems like a rather pointless exercise to interpret > axioms in structures in which they are false, it just happens > to be involved in the Tarskian conception of logical consequence: > in those structures too the axioms imply their theorems /and > nothing else/. The verb "imply" is the key word in that sentence. Material implication is convenient in some ways and misleading in others; please let us NOT have another thread about "vacuously true".
From: G. Frege on 22 Dec 2007 12:39 On Sat, 22 Dec 2007 09:27:14 -0800 (PST), george <greeneg(a)cs.unc.edu> wrote: > > FF [...] HE was WRONG about the original reply to > apoorv's use of unrestricted comprehension! > Actually, I didn't encourage apoorv in using "unrestricted comprehension". > > What WAS RELEVANT was ZFC, and that ZFC DOES NOT HAVE > unrestricted comprehension! > Right. Though Paul R. Halmos mentions the convention to write {x : phi(x)} _even in ZFC_, if we can show ExAy(y e x <-> phi(y)). Hence we may actually formulate the definition 0 =df {x : x =/= x} (using this convention). Of course, FIRST we have to show ExAy(y e x <-> y =/= y). And you KNOW that this _can_ be shown (in ZFC). F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 22 Dec 2007 12:41 On Sat, 22 Dec 2007 09:30:47 -0800 (PST), george <greeneg(a)cs.unc.edu> wrote: >> >> Wait a second. Aren't those axioms /true/ only when /interpreted/ >> (i.e. in a model)? Hence isn't it the /interpretation/ that assigns >> truth to an axiom? >> > It is the designation of an axiom AS an axiom that assigns > PROVABILITY to the axiom. > That's right. But that was not my question. :-) (I asked for TRUTH, not PROVABILITY.) F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 22 Dec 2007 12:43 On Sat, 22 Dec 2007 09:17:22 -0800 (PST), george <greeneg(a)cs.unc.edu> wrote: > > Nam had said one thing wrong and FF had (as usual) > corrected it INcorrectly. > :-) Keep up the good worke, george. F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 22 Dec 2007 12:46
On Sat, 22 Dec 2007 09:41:57 -0800 (PST), george <greeneg(a)cs.unc.edu> wrote: >> >> Hence isn't it the /interpretation/ that assigns >> truth to an axiom? >> > Maybe FF should look in the dictionary under "Axiom". > No, really not. > > Axioms have to be provable because THAT'S THE DEFINITION OF "Axiom". > Right. > > The Axioms are the things that you just presume true FROM THE > BEGINNING, ... > Oh, wait a second. Aren't you mixing up TRUTH with PROVABILITY here? :-o F. -- E-mail: info<at>simple-line<dot>de |