The End of an Era - That of The Natural Numbers
in [Math]
|
From: Alan Smaill on 23 May 2010 17:33 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Let me give you an example step by step so you could > understand the concept of the false model of an inconsistent > theory. > > Let "blue" be an unary predicate symbol of a language that also > has an individual constant "e", and let: > > T1 = {P(e)} > > be consistent with this model M1: > > M1 = { > <'A',U>, <'e',(e0}>, <'=',{<e0,e0>}>, <'~=',{}>, > <'blue',{e0}>, <'~blue',{}> > } This is a strange way to give a model -- e.g. the interpretation of the negation of the predicate "blue" is determined by the general definition of satisfaction, so it is confusing to specify it here. > where e0 = {}, U = {e0}, '~=' and '~blue' are a (FOL) defined symbols > for the unary predicates != and ~blue(x), respectively. What is A? > Let now: > > T2 = {~P(e)} > > M2 = { > <'A',U>, <'e',(e0}>, <'=',{<e0,e0>}>, <'~=',{}>, > <'blue',{}>, <'~blue',{e0}> > } > > Now let T3 be our intended theory > > T3 = {P(e) /\ ~P(e)} > > The false model for T3 is: > > M3 = { > <'A',{}>, <'e',(}>, <'=',{}>, <'~=',{}>, > <'blue',{}>, <'~blue',{}> > } Stop right there. You have <'e',(}> (I assume you mean <'e',{}>). But the interpretation of 'e' has to be a *member* of the domain of interpretation, in the usual version of model, as in Shoenfield. You have broken that part of the definition. > where now U = {}. Note that both the negating predicates 'blue' > and '~blue', as well as the predicate '=' are empty sets, but > despite that M3 can never be empty! And what entity does "e0" refer to? > So, by definition of being true, being false in FOL model, > all formulas are are defined as being false in M3, and that's > the only model for T3 - the false model. Not by the usual definition of being true or false in FOL model. -- Alan Smaill
From: Alan Smaill on 23 May 2010 17:39 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> .... >>> But where is the word "true" in all that syntactical, >>> rules-of-inference-based, proof? >> >> Nowhere. >> >> Let's try this one step at a time. >> >> Do you agree that >> >> all x. (( x = 0 & x =/= 0 ) -> P(x)) >> >> is provable in FOL with equality, whenever P is a unary predicate >> in the language under consideration? > > If by that you mean the meta statement {( x = 0 & x =/= 0 )} |- P(x) is > true, then yes I agree. ("Provable" means provable in some formal system). Well, good to get that small amount of agreement. But, no, as I said, I'm asking about provability "in FOL with equality" in particular. Is any such statement provable in that logic? (There are many equivalent proof systems known for that logic.) -- Alan Smaill
From: Nam Nguyen on 24 May 2010 04:03 Daryl McCullough wrote: > 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))". Accordingly to Shoenfield, "we may define A in term of E [and vice versa]". So yes the 2 formulas are equivalent and if by convention we'd like to assign to one of them a semantics, the other should have the same assignment. > > 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). As alluded to above it's virtually a necessary convention. A relevant point I'd like to add here though is formula, its semantic, and its truth, are _different_ components of FOL edifice each of which would have different relevant definitions and/or rules of reasoning using them. > For example, suppose I tell you that all the coins in my pocket > are quarters. So, per Tarski's concept of truth, _factually_, the set of coins in your pocket is non-empty _and_ all of them are quarters. As put in some details by the URL: http://en.wikipedia.org/wiki/Semantic_theory_of_truth we'd have this definition of truth: (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). As we could see from the definition (iv), model theoretically truths are based on knowledge/ intuited fact or, in Shoenfield's terminology, "observation" and NOT on inference or reasoning. If the fact about something is so-and-so then and only then the statement about it being so-and-so is true. If there's no such fact the the statement is false. > Then you see me draw a quarter out of my pocket > and spend it. Now the set of coins in your pocket could be empty but nobody would know. So, the statement "all the coins in my pocket are quarters" can't be assigned a Tarski's truth value > Then afterward, it is *still* true that all the > coins in my pocket are quarters. According to Tarski's truth definition you can't map the statement to a Tarski's truth value. (Of course there are other truth mappings but in this context we wouldn't interested in them!) > 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". But that's exactly what Tarski's definition (iv) would stipulate. > > 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. But iirc I never said the statement is meaningless. Otoh, I did mention above that formula, semantic, and truth are different, and we shouldn't have them "mixed" up, so to speak. > 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. I'm sorry I don't know for sure if any of this has anything to do with model truths as defined by Tarski.
From: Frederick Williams on 24 May 2010 05:56 "Jesse F. Hughes" wrote: > > Frederick Williams <frederick.williams2(a)tesco.net> writes: > > > "Jesse F. Hughes" wrote: > > > >> Hence, a model must have non-empty support, > > > > Free logic may have empty models. > > Yes, but I believe the context here is classical FOL. You can't expect me to jump into a month old discussion and know what the context is :-) -- I can't go on, I'll go on.
From: Marshall on 24 May 2010 08:45
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. 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. So your earlier claim: > Note that "each object" would presuppose or require that > there exist objects is false on its face. This is a really, really elementary fact about how universal quantification works in first order logic. It is really clear that you've gotten it wrong, and there are vast numbers of references to document this. Now, you wanna make up your own logic, go ahead! If you want to insist on your: > Note that "each object" would presuppose or require that > there exist objects then you are necessarily NOT talking about FOL. Marshall |