The End of an Era - That of The Natural Numbers
in [Math]
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From: William Hughes on 26 May 2010 07:32 On May 26, 2:28 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 26, 12:52 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Jesse F. Hughes wrote: > > >>> What else is possible, when the topic is logic, but one of the > >>> conversants is a blowhard incapable of realizing that, whenever P is > >>> false in a structure, ~P is true in that same structure? > >> Why not, if this is the _degenerated structure_ (the false structure) > >> of a language? Are you surprised that a tautology and a contradiction > >> are equivalent in a _degenerated formal system_ that's called inconsistent? > > > So in this *model* both > > > There exists an x such that blue(x) > > > and > > > There does not exist an x such that blue(x) > > > are false ?!?. > > Of course, in this false model where all relevant sets are empty. > > Tell me why you're surprised about this? Because you have an empty universe and you are claiming that both there is an element in this universe and there is not an element in this universe are false. <snip> > You're acting as if you were defending a thesis on the reasoning > values of inconsistent theories! Nope, there are inconsistent *theories*, there are no inconsistent *models*. - William Hughes
From: Jesse F. Hughes on 26 May 2010 07:32 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Aatu Koskensilta wrote: >>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>>> >>>>> Other than that, I'm afraid any conversation I might have with you >>>>> would be fruitless. >>>> Well, yes, as already noted I've at long last concluded it's totally >>>> pointless to try to discuss logic with you, my contributions thus >>>> reduced to general observations and cheap pot-shots. >>>> >>> I guess cheap shots will be flying around then. >> >> What else is possible, when the topic is logic, but one of the >> conversants is a blowhard incapable of realizing that, whenever P is >> false in a structure, ~P is true in that same structure? > > Why not, if this is the _degenerated structure_ (the false structure) > of a language? Because, by the definition of truth in a structure, if P is false in M, then ~P is true. That's why not. This definition is clearly stated on p. 19 of Shoenfield, but you're too addle-brained to understand it. > Are you surprised that a tautology and a contradiction are > equivalent in a _degenerated formal system_ that's called > inconsistent? Surely you're not incapable of understanding that, > are you? Of course. In an inconsistent theory, ~P <-> P is provable. That observation does *not* imply that both ~P and P are false in some mathematical structure. (Hint: inconsistent theories have no models. Duh.) I'm not sure why I'm actually in this discussion with Nam, who has shown an infuriating unwillingness to learn from others and a very sad inability to understand the authors he reads. I suppose I'll let it go at this time. -- "Clouds are always white and the sky is always blue, And houses it doesn't matter what color they are, And ours is made of brick." -- A new song by Quincy P. Hughes (age 4)
From: Marshall on 26 May 2010 08:06 On May 26, 4:09 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > So "There does not exist an x such that blue(x)" is true > for the empty model, while "There does not exist an x such > that blue(x)" is false. But both sentences are valid for > the empty model. This paragraph seems to have a typo in it. Are the two sentences quoted supposed to be identical? But you say "both sentences" so it seems like they are not supposed to be identical. Could you clarify this? Marshall
From: Marshall on 26 May 2010 08:18 On May 24, 11:58 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On May 24, 1:03 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> (iv) A universal statement "for all x A(x)" is true if and only if each > >> object satisfies "A(x)" > > >> Note that "each object" would presuppose or require that there exist > >> objects and that "if and if " means the statement is false when there > >> is NO object, or no object satisfying A(x). > > > You know, you often complain about how your "opponents" > > don't address technical definitions (although of course they > > do.) Well, here's a dead simple exampleof a > > technical issue that you just get butt-wrong. > > > I've commented before on your basic failure to > > understand this issue. It's called "vacuous truth" and > > you claim to understand it, but you also continue > > to make it clear that you don't, as above. > > You just don't seem to know how to argue here. Your opponent > has advanced an idea there [...] For the moment I am not addressing any of those various ideas. I am simply addressing a single, entirely seperable point, vacuous truth in FOL. You can write paragraph after paragraph about things *other* than that, but that will have no bearing on this particular, elementary point. > > From: > > >http://en.wikipedia.org/wiki/Vacuous_truth > > > quote: > > A vacuous truth is a truth that is devoid of content because it > > asserts something about all members of a class that is empty or > > because it says "If A then B" when in fact A is false. For example, > > the statement "all cell phones in the room are turned off" may be true > > simply because there are no cell phones in the room. > > But anyone could cite any technical source at any time, whether or not > it's actually relevant! A mere distraction. The question is whether the particular source here is relevant or not. I claim that the above entry on vacuous truth is in fact relevant to the meaning of vacuous truth. Do you claim it is not? > For example, how does the information in this link refute your opponent's > notion (which he used in his argument) that being true or being false is > just a meta mapping and that there are different kinds of truth mapping > allowed in FOL edifice? We could also ask how does it address the problem of global warming. However that is not what I am talking about at the moment. At the moment I am talking about vacuous truth. You said: > >> (iv) A universal statement "for all x A(x)" is true if and only if each > >> object satisfies "A(x)" > > >> Note that "each object" would presuppose or require that there exist > >> objects This is a very simple point, and you have it wrong. And everyone here knows it. And I've documented it directly. Marshall
From: Daryl McCullough on 26 May 2010 08:18
Marshall says... > >On May 26, 4:09=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> >> So "There does not exist an x such that blue(x)" is true >> for the empty model, while "There does not exist an x such >> that blue(x)" is false. But both sentences are valid for >> the empty model. > >This paragraph seems to have a typo in it. Doh! The second sentence was supposed to be "There exists an x such that blue(x)", the negation of the first sentence. A typo like that makes an entire post incomprehensible. -- Daryl McCullough Ithaca, NY |