The End of an Era - That of The Natural Numbers
in [Math]
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From: Daryl McCullough on 22 May 2010 23:28 Nam Nguyen says... >How could any formula be true in a model in which its universe >U and any n-ary predicate are empty? In the empty model, any formula of the form forall x Phi(x) is true, and any formula of the form exists x Phi(x) is false. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 23 May 2010 02:14 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> It's now your turn (as well as WH's) to answer my question: if you >> think my answer is incorrect, then where and why (in some technical >> details)? > > Not my turn. I don't bother with substantive discussions with you. > It is obvious that you don't really understand logic at all and these > discussions never seem to have a positive effect on your ignorance. > > I just wanted to see if you'd be prodded to actually answer William's > question. I'll leave the replies to him. > If you "don't bother with substantive discussions" with me on technical matters then you don't ask or prod me. You, Jesse F. Hughes, are an intellectual dishonest and coward.
From: Nam Nguyen on 23 May 2010 02:38 William Hughes wrote: > On May 22, 9:41 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> On May 22, 7:37 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> there's no formula is true in M3 >>> Nonsense The formula >>> (not(exits x) or blue(x)) >>> is true in M3 and provable in T3. >> Did I already say "by definition of model it's false" > > Many times. You are under the delusion that by saying > something many times you can change it from incorrect to > correct. Oh, I offered explanations with technical details and all that anyone could search or see from the ng server so that is never an issue with me. In the same post and on the same subject I had: >> _By definition of model_ do you not agree the formula >> P(x1, x2, ..., xn) would be false in an empty predicate >> P? And you _refused_ to engage in the technical dialog. I also pointed out you were making a technical mistake because your "(not(exits x) or blue(x))" isn't a well formed formula in L(T3), and you _refused_ to acknowledge the mistake or offer a correction. Your attacking utterance here is indeed very idiotic!
From: Nam Nguyen on 23 May 2010 02:43 Daryl McCullough wrote: > Nam Nguyen says... >> How could any formula be true in a model in which its universe >> U and any n-ary predicate are empty? > > In the empty model, any formula of the form forall x Phi(x) > is true, Can you explain in some details why? > and any formula of the form exists x Phi(x) is false.
From: Daryl McCullough on 23 May 2010 07:45
Nam Nguyen says... >Daryl McCullough wrote: >> Nam Nguyen says... >>> How could any formula be true in a model in which its universe >>> U and any n-ary predicate are empty? >> >> In the empty model, any formula of the form forall x Phi(x) >> is true, > >Can you explain in some details why? Well, in classical logic, "forall x, Phi(x)" is interpreted to mean the same thing as "not (exists x, not Phi(x))". I guess you could consider it just a convention, but it's a very useful one for reasoning (which is what logic is for, after all). For example, suppose I tell you that all the coins in my pocket are quarters. Then you see me draw a quarter out of my pocket and spend it. Then afterward, it is *still* true that all the coins in my pocket are quarters. The only way that it could change is if I add a new coin to my pocket that is *not* a quarter. It can never become false by removing quarters. So saying that all the coins are quarters does not tell you that there are any coins at all. If you want to say that, you can say "all coins in my pocket are quarters, and there exists a coin in my pocket". You might say that the statement "all coins in my pocket are quarters" is *meaningless* in the case where there are no coins in my pocket, but it isn't. It's a definite bit of information that could be useful to know in some circumstances. For example, suppose that it's discovered that the copper in pennies causes cancer. You know that I sometimes carry change in my pocket, and are worried about me. But then, using classical logic, you could reason 1. All the coins in his pocket are quarters. 2. Therefore, there are no pennies in his pocket. You don't need to know how many quarters there are in order to make your conclusion. The convention that every universal statement is true in the empty domain gives us a way to *prove* that our domain really is empty: If you can prove that for your domain, "forall x, Phi(x)" and you can also prove for your domain that "forall x, not Phi(x)" then you are justified in concluding that your domain is empty. -- Daryl McCullough Ithaca, NY |