The End of an Era - That of The Natural Numbers
in [Math]
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From: William Hughes on 26 May 2010 00:25 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 ?!?. - William Hughes
From: Nam Nguyen on 26 May 2010 01:28 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? Let me guess: because it's a revulsion in reasoning right? But isn't a formula and its negation being equivalent _also_ a revulsion? FOL reasoning allows the latter to happen; why are you surprised when there's case for the former? Technically speaking, out of infinitely many numbers of theories of a language how many theories has such a "strange" property that _all formulas_ are equivalent? Just one right? Similarly, how many structures out of infinitely many of them that has a "weird" property that _all formulas_ have the same truth value? Just one right? Birds of the same feathers flock together, don't they? For crying out loud, what can one expect from an inconsistency besides "strangeness" and "weirdness"? You're acting as if you were defending a thesis on the reasoning values of inconsistent theories!
From: Daryl McCullough on 26 May 2010 06:55 Nam Nguyen says... >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? You are confusing two different things: truth in a structure, and provability in a theory. If S is a structure for a language L, then truth in a structure is defined in such a way that every closed sentence of L (a sentence without free variables) is assigned a value "true" or "false", and the set of true sentences is disjoint from the set of false sentences. In the case of the empty structure (no elements), this assignment is pretty simple: (1) Every sentence of the form "Ax Phi(x)" is assigned "true". (2) Every sentence of the form "Ex Phi(x)" is assigned "false". (3) There are no quantifier-free closed sentences. What about sentences that are *not* closed? Well, to interpret open sentences (ones with free variables), we also have to have an assignment function (one that assigns an element of the domain to each variable). In the case of the empty domain, there are no assignment functions, so the empty domain cannot be extended to give a truth value to open sentences. There is a subtle distinction between "true" and "valid" for a structure. If Phi is an open sentence, then it is considered "valid" for a structure if every assignment results in a sentence that is true. Under this definition, since there are no assignments for the empty structure, it follows vacuously that *every* formula (and its negation) is valid for the empty structure. So, the set of *true* formulas for the empty domain is a consistent set. The set of *valid* formulas for the empty domain is inconsistent. In any case, the open formula "x=x" is valid in every structure, including the empty structure. The open formula "~(x=x)" is only valid in the empty structure. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 26 May 2010 07:09 William Hughes says... > >On May 26, 12:52=A0am, 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 inconsisten= >t? > >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 ?!?. This is a place where the subtle distinction between "true" and "valid" is important. A model only assigns "true" or "false" to *closed* sentences (ones without free variables). To give a truth value to open sentences, you have to supplement the model with a function assigning values to each variable. An open sentence is said to be "valid" for a model if *every* assignment makes it true. For *non-empty* models, nobody really cares much about the distinction between "true" and "valid", but for the empty model, there is a big difference. For the empty model: 1. Every universally quantified sentence is true. 2. Every existentially quantified sentence is false. 3. Every sentence is valid. The reason for the last fact is that there *are* no assignments possible for the empty model, so it vacuously follows that every assignment makes every sentence true. 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. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 26 May 2010 07:12
Nam Nguyen says... >> 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. That's wrong. >Tell me why you're surprised about this? Because it's not true. >You're acting as if you were defending a thesis on the reasoning >values of inconsistent theories! We're not talking about inconsistent theories, we're talking about *structures* for a language. Every structure for a language has an associated theory (the set of closed sentences true in that structure), and that theory is always consistent. -- Daryl McCullough Ithaca, NY Ithaca, NY |