The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 24 May 2010 14:58 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 (any formula is false in the false model in of a language) and he has cited a few supporting technical notions to back him up: - Exhibits E1 and E2 where he stipulated being true/false is just a meta mapping which is non-syntactical. - Tarski's concept of truth which is not a notion about logical inference - Textbook definition of model making using of set membership (even in the case of the empty set), and following Tarski's concept of truth. Iow, he just didn't cite one thing and that was it! He chained the above (if not some more) in some logical patterns. So, unless you have concrete counter argument any of the above is incorrect your keep saying something like "your basic failure to understand this issue" is just an idiotic attack. I meant, was I wrong when I said being true/false is just a meta-mapping? Why? Was Shoenfield's definition not following Tarski' concept of truth? Why? etc... > > 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! 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? How does what we understand in this link would constitute that textbook definition is not following Tarski's concept of truth? etc... etc..., and etc... > > So your earlier claim: > >> Note that "each object" would presuppose or require that >> there exist objects > > is false on its face. Well yes, a crank could say that too and NOT have anything to explain why! > > 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. Well yes, a crank could say that too and NOT have anything to explain why! > > 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. Well yes, a crank could say that too and NOT have anything to explain why!
From: Alan Smaill on 24 May 2010 15:04 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> 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? > > 'A' is, irrc, a notation from Herbert B. Enderton indicating the > universal quantification over the universe U of discourse, another > poster (MoeBlee) in a past thread mentioned and used the notation. > Shoenfield didn't' use it iirc, but it won't matter much in the > definition of model (a la Tarski). > > The general definition of structure/model M (of a language), which > a model of a theory is, would be: > > M = {<>, <>, <>, <>, <>, <>, ....} > > or in some details noting we're in FOL=: > > M = {<'A',U>, <'=',p0>, <'s1',p1>, <'s2',p2>, <'s3', p3>, ...} > > Where 'A' is the Universal Quantification symbol, and each 'si' is > an n-ary predicate symbol of the language L, and each pi is the > predicate (a set) corresponding to si. Note that an individual constant > symbol, such as 'e' of L(T3), is a 0-ary symbol. Also, e0 is just > 0-ary function value for 'e', meaning e0 is an element of U that's > named 'e'. So far so strange. > Note also that for the 2nd element of an ordered pair <>, sometimes I > close them in brackets not at other times: e.g. <'s2',p2> and <'s2',p2> > have been used interchangeably. ???? > But that'd be just notation. > > The key feature of this definition I think my opponents have overlooked > is ultimately M is a _set_ and as such any language will have to have a > model: _M might be empty but M will exist as a set_ ! Well, no doubt. But when you say that you have a "model" for an inconsistent theory T, then there are some conditions relating the axioms of T to the structure defined so that M is in fact a model (that eaxh axiom of T is true in M). Do you agree? > Note also the fact > that per each language: there's only one inconsistent theory (as a collection > of theorems) that's a normal outcome of eg Shoenfield's approach. > and there's is only one model that all the the n-ary predicates > are empty. That one model (a set) is the model for that one theory. Do you think gthis agrees with Shoenfield's definitions? > What my opponents seem to get confused is that the meta expression > "There's no model for an inconsistent theory T" is actual wrong: there's > always that peculiar model (which again _is a set_): it might be the only > useless model; but that's why it's a model for the only useless theory > in that language! > >>> 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. > > The textbook definition of model (Shoenfield's included) follows > Tarski's concept of truth and makes use of the property "being in, > being a member of" with respect to a predicate, an ordered pair of > elements in the universe U. But since both the predicate p and U > can be empty set, there's nothing with <'e',{}>: by definition of a > structure, you have to _map_ a symbol ('e' in this case) to a predicate > ({}an empty set in this case). You should not map a constant or function symbol to a "predicate" or relation. Read what Shoenfield says about the interpretation of constants (function symbols); closed terms in the language should denote elements of the domain. >>> 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? > > As mentioned above that's a named element of U that gets map to > the language symbol 'e' So, U is *not* empty. >>> 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. > > That's not correct. Definition of a formula being true or false in > M requires ONLY the notion of the n-tuples of elements of U _factually_ > be in or not in M. (And M always exists as a model-set!). And that's a > text-book definition. Definition of truth of a formula involving a constant "e" requires a non-empty domain to interpret e. That's a text-book definition. > If it would help, one doesn't have to like the word the "the false > model", one just have to acknowledge that by definition of a structure > (a set) there's such a model that using the membership property and > following Tarski's concept of truth all formulas are interpreted as > false. Well, no. If ever a discussion leads to the conclusion that there are no yes/no questions, this one is it ..... -- Alan Smaill
From: Nam Nguyen on 24 May 2010 17:10 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> If it would help, one doesn't have to like the word the "the false >> model", one just have to acknowledge that by definition of a structure >> (a set) there's such a model that using the membership property and >> following Tarski's concept of truth all formulas are interpreted as >> false. > > Were you perhaps led to this important discovery by vigorously > reflecting on Shoenfield? > I don't mind, Aatu, having conversations, arguments, with you to really, really settle this whole issue, knowing that there's a risk I could be technically wrong. But you got to have an _intention to really resolve_ it even if it means to spend of some effort, details to bring it to _some closure_. Other than that, I'm afraid any conversation I might have with you would be fruitless.
From: William Hughes on 24 May 2010 19:56 On May 24, 5:03 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Note that "each object" would presuppose > or require that there exist objects Nope, this is simply incorrect. - William Hughes
From: William Hughes on 24 May 2010 20:03
On May 23, 3:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: <snip> > "(not(exits x) or blue(x))" isn't a well formed > formula in L(T3), Ok if you do not want to use existential qualifiers as predicates use the equivalent form For All x :(blue(x)) The important point is > How could any formula > be true in a model in which its universe U and any > n-ary predicate are empty? Note that by FOL any statement about the property of the elements of an empty set is true. So use the quantifier For All. Eg. For All x:(blue(x)) - William Hughes |