The End of an Era - That of The Natural Numbers
in [Math]
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From: Jesse F. Hughes on 1 Jun 2010 21:19 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Whatever the reason you have when saying x=x "is not true" is a > direct contradiction with Marshall's belief (if not with Aatu's > and Jesse's as well), in the original challenge I brought up above > about showing any "absolute (formula) truth". I see no contradiction with anything I've said. x=x is not a closed formula and so nothing from p. 19 of Shoenfield applies to it. Note: Nam's misreading of Shoenfield is very easy to understand. On p. 19, he gives a definition of truth value in a structure by induction on the length of the formula. Unfortunately, Nam thinks that the clause for non-equation atomic formulas (A is of the form pa_1,...,a_n) applies for arbitrary formulas. He completely ignores the clause for, say, negation, which says that ~B is false in M iff B is true in M. Not that Nam will either understand or admit this. -- Jesse F. Hughes. Me: It's very sad when one's husband or wife dies. Quincy (Age 4 1/2): Yeah. You might want to tell them something and you just can't. [Long pause] Like "Take out the trash."
From: herbzet on 1 Jun 2010 23:57 Aatu Koskensilta wrote: > Daryl McCullough writes: > > > If we understand the logical form of this syllogism, we know > > that it is valid, even if we have no idea what "mortal" means, > > or what "men" are, or who Socrates is. In particular, we don't > > need to know whether there *are* any men. > > I seem to recall universal statements have existential import in > Aristotle's logic. It appears that view basically stems from a nineteenth-century view of Aristotle's logic. Another view of his logic is: From http://plato.stanford.edu/entries/square/#AriForOFor : [...]On this view affirmatives have existential import, and negatives do not � a point that became elevated to a general principle in late medieval times. The ancients thus did not see the incoherence of the square as formulated by Aristotle because there was no incoherence to see. 1) In the syllogism cited by Daryl, the major premise is both universal *and* affirmative, so by the above view your observation would hold: the premise would be false if there are no men. 2) The modern view, of course, is that universal terms do not have existential import, so the first premise would be true on that view if there are no men. 3) The syllogism cited by Daryl is not a categorical syllogism, but has a singular minor premise -- which would be false if there are no men. The rules for syllogisms with singular terms vary from writer to writer -- Aristotle did not not treat of it. 4) So, on what view is the syllogism valid or invalid, if there are no men? Beats me. The syllogism seems valid to me in any case -- were the premises true, so would the conclusion be. -- hz
From: Marshall on 2 Jun 2010 02:10 On Jun 1, 12:56 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marshall <marshall.spi...(a)gmail.com> writes: > > In fact, it seems to me that if there is a simple frontrunner reason > > for studying logic at all, it is just exactly to *avoid* doing this > > sort of thing. But perhaps that's just my imperialist, narrowly > > utilitarian view. I am sure Jesse or Aatu or someone like that will > > correct me if I'm wrong here. > > I'll correct you even when you're right. My generosity knows no bounds! But this is a central feature of your charm! That, along with general observations and cheap potshots. Not to mention a wonderful sense of language. Marshall
From: Nam Nguyen on 4 Jun 2010 02:30 Alan Smaill wrote: > > You claim that your notion of model is equivalent to Shoenfield's > notion. Yet Shoenfield follows Tarski's truth definition: > the negation of a formula is true in a structure if and only if > the formula is false in the structure. You either didn't read it carefully, or did too carefully to the point of being pedantic and missed what he had said there. That's all. For example, take the condition iii he had in defining (true) model that I mentioned a few times. Nam (responding to William Hughes), May 19 > iii) For each n-ary predicate symbol p of L other than =, an n-ary > predicate pM in |M|. > > (|M| means the universe of M). > > ... explain why that condition has the phrase "other than =". Do you > understand what that phrase mean in the relationship with logical > and non-logical predicates, with the definition of being true and > being false? Nam, (responding to Jim Burns), May 19 > I cited _text book_ definition of model (e.g. condition iii pg 18, > phrase "other than =", Shoendfield, and other quotes), and > nobody _including you_ gave a slight reflection on them? Nam, May 22 > The implication of condition iii is that despite all theories T's > must extend _the_ logical theory T0 = {x=x}, a [true] model of > T0 will NOT exist until we have in our mind a [true] model of > a consistent T. Allan, May 28 > (iii) For each n-ary predicate symbol p of L ..... " Your last post on the same subject (condition iii) is about 10 days older than when I first posted it and not only you didn't respond to my analysis on the phrase "other than =" but you also snipped it in your last mentioning. Why? But whether or not the snipping was intentional, it has cost your reasoning here and prevented you from seeing that I actually followed Shoenfield's definition even though (and I did say that many times) he was using _true_ model in his definition, and I'm talking about false model. > > By all means don't do the usual thing -- > but don't then claim that what you have is faithful to > Shoenfield and Tarski. Again, true model and false model both are defined in term of set-hood and set membership, which would reflect Tarski's concept of "concrete and factual" truth in the realm of abstraction and which both Schoenfield and I used for 2 different cases: the typical case and the atypical one. Let's revisit the definition again this time we'd use strictly notation and hopefully you'd see the issue better. The language: L = L(c1,c2,blue,non-blue) be a language with 2 individual constant symbols: c1, c2; and 2 unary predicate ones: blue, non-blue. Lets define the following: U = {1,2}, 1 = {}, 2 = {{}}, pBlue = {1}, pNon-Blue = {2} M = { <'A',U>, <'c1', 1>, <'c2', 2> <'blue', pBlue = {1}>, <'non-blue',pNon-Blue = {2}> } Let me now present to you very short but technical questions so that your answers would illustrate the nature of being true/false in a model-set, whether or not U is empty. Q1. Is the _set_ M a complete structure of L? (I.e. Does M miss any element?) Q2. Suppose M is a structure of L, is blue(c1) true? Why? Q3. Suppose M is a structure of L, is c1 = c2 true? Why? Q4. Let R be a 2-ary "defined symbols" (Shoenfield, pg 6) in L and be defined as R(x,y) df= (blue(x) \/ non-blue(y)). Is R(c1,c2) false in M, if M is a structure of L? Why?
From: Daryl McCullough on 4 Jun 2010 06:35
Nam Nguyen says... > >Alan Smaill wrote: > >> >> You claim that your notion of model is equivalent to Shoenfield's >> notion. Yet Shoenfield follows Tarski's truth definition: >> the negation of a formula is true in a structure if and only if >> the formula is false in the structure. > >You either didn't read it carefully, or did too carefully to the >point of being pedantic and missed what he had said there. That's all. >For example, take the condition iii he had in defining (true) model >that I mentioned a few times Yes, you clearly are very confused by it, but I really don't understand your confusion. What's really weird about crackpots, is that they are not content to just have their own alternative theory. They insist that they understand the *standard* theory better than the non-crackpots. -- Daryl McCullough Ithaca, NY |