'When Are Relations Neither True Nor False?
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From: Nam Nguyen on 13 Apr 2010 00:18 Marshall wrote: > On Apr 11, 7:51 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Apr 11, 5:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Marshall wrote: >>>>> On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>>> I reject your claim that intuition, of whatever kind, plays >>>>>>> any part in our thinking about the naturals. >>>>>> So is the naturals collectively a finite syntactical notion *to you*, >>>>>> since you'd reject the idea they're an intuition notion? >>>> Can you share with us your resounding, firm, answer to that simple >>>> question? >>> You mean, again? You want me to say "yes" again? Okay: >>> Yes. >> So your resounding answer is "yes" that the notion of the naturals >> is based on an intuition but is a notion of syntactical formal system. > > Not a "formal system" in this sense: > > http://en.wikipedia.org/wiki/Formal_system > > The naturals and their operators are a model, not a theory. Not only you're clueless as to exactly what the natural numbers be, you also don't use the technical terms correctly. First of all, if the naturals are collectively a model of a language then you should have not resoundingly said "yes" that they'd be "a finite syntactical notion", which refer to syntactical symbols, formulas, axioms, axiom-system, rules of inference: but not to interpretation, or truth values as when we talk about model truths. Secondly model would *require* intuition: if there's no intuition there's no interpretation, no truth values, hence no model! _You've contradicted yourself_ by first rejecting intuition "plays any part in our thinking about the naturals" and now stating "The naturals and their operators are a model". >> How would you know though such a formal system is syntactical >> consistent? > > The question of consistency is asked of theories, not of > models. I guess you are not really clear about the distinction. I wasn't asking you about the general consistency. I was asking you about *your contradictory claims* about the natural numbers. My hope here was that my question would help you to realize how wrong you were in rejecting intuition "plays any part in our thinking about the naturals". > Since the naturals and their operators are a model, > it makes no sense to ask about their consistency. Again it was about one of your contradictory claims that the naturals are of syntactical notion, which is something that we'd *only* refer to when talking about formal systems. Your statements flip-flop back and forth between the naturals being model theoretical and syntactical. Being model theoretical and being syntactical are 2 different (and opposing) notions. Can you now make up your mind and tell us once for all, which ways the natural numbers are: model theoretical, or syntactical? > >> and the 10-year old >> would most likely never claim he could use such an intuition >> to prove the consistency of PA, right? > > Sure. But we were talking about the natural numbers, not PA. You're wrong. My conversation here started when I responded to AK's statement: >> Anyone can intuit whatever they want. I said nothing about such >> matters, I merely noted I have, like any number of logic students >> over the decades, produced a proof of the consistency of PA in the >> course of my studies. Intuition has no more and no less to do with >> it than with any proof in mathematics. So AK did prove the consistency of PA. And his passage here seemed to deny PA's consistency proof would have to do anything with intuition [of the naturals used in the consistency proof of PA]. And that's why I later protested: Nam: >> Then, it also looks like a poor choice of using the _intuition_ about >> the naturals as a foundation of reasoning, as the school of thought AK, >> TF seem to have subscribed to, would suggest. And then you decided to join the conversation: Marshall: >> I reject your claim that intuition, of whatever kind, plays >> any part in our thinking about the naturals. So, Marshall. Let's not forget the conversation here is about the proof consistency of PA, as well as about intuition-or-not- intuition of the naturals. >> But AK would do >> so and you'd do the same or at least support such a claim. > > I am a poor judge of what AK would say. Then perhaps you should have stayed silent instead of trying to defend someone whose statement-validities you only have a "poor" judgment - at best! > For myself, I am > unclear enough about what you mean when you say "intuition" > to be able to answer you. Since you don't understand a very simple notion such as intuition [that a 10-year old student would understand], let me explain that to you. You yourself claimed that the natural numbers [collectively] is a model [I'm assuming of the language L(PA)]. But how would you know that for sure? The answer is you CAN NOT know that for sure, because you yourself will NEVER be able to list out all the necessary n-ary relations (sets) required by the definition of a language model. The only reason left for you to have a feeling that you know the naturals is a model at all is your INTUITION. Do you understand now that you were wrong before when you resoundingly claimed: >> I reject your claim that intuition, of whatever kind, plays >> any part in our thinking about the naturals. > My best guess would be: would > we consider the hunch of a ten year old to be a proof that > PA is consistent? Speaking for myself, no. If you meant > something else, please clarify. Above I already explained to you the simple notion of intuition that a 10-year old would understand! > If you do decide to clarify, > please recall that PA is not the natural numbers, that PA is > a theory and the naturals+their operators are a model, > and that consistency is a question that only applies to > theories. Where did I claim otherwise? My questions to you were meant for you to realize you didn't know what you were talking about when rejecting that intuition "plays any part in our thinking about the naturals". You should read the conversation more carefully. > Also recall that the fact that all the axioms of > PA are true for the naturals is a model-theoretic proof > that PA is consistent. Marshall, how do you prove that - WITHOUT INTUITION? You're clearly clueless about what you are asserting. >> Oh I learnt the naturals numbers the same way mathematicians >> would learn: it's an intuitive notion! > > The above sentence does not signify anything clearly enough > for me either to agree nor to disagree. > > Every time I engage with you in any least substantive > way, I end up feeling like I have wasted my time. > Not sure why I do it. I really would be better served > just to laugh at your silly ideas and be done with it. What silly ideas? You meant when I alluded that the notion of the naturals is a notion of intuition? If so, then you'd be also laughing of, say, Shoenfield!
From: Nam Nguyen on 13 Apr 2010 00:34 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Alan Smaill wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> To claim some formula being >>>> absolutely true (or false) is to destroy the notion of such symmetry. >>> Do you agree that symmetry is only >>> broken if the cases "P" and "not P" are treated differently? >> Of course I'd not agree. Even in relativity, given an appropriate context, >> P and ~P must necessarily be treated differently because of LEM. > > Now you've really lost me. > LEM *is* symmetric itself, isn't it? I have no idea what you meant here by "LEM *is* symmetric itself". You have to explain what you meant by that. The word "symmetry" I've used here is in the context of the Principle of Symmetry I articulated before with some clear examples and are about concepts, formulas - but not about LEM]. > >> The Principle addresses a different issue: P or ~P can't be uniformly >> treated as true or false in _all_ contexts. That's all the Principle of >> Symmetry would stipulate, and all I've really said (as above). > > So how do you know it is relevant to the particular context at hand, > that of assuming "not P" hypothetically (ie for sake of argument), deriving > a contradiction, and concluding "P". And also in the context where "P" > and "not P" are switched systematically in such an argument? I'm not sure what you're trying to argue here. My argument only concerns the alleged "absolute" truths that DB tried to suggest, and in which I counter claimed any such absolute truth would break the Principle of Symmetry. That's all I've really argued in this conversation!
From: Nam Nguyen on 13 Apr 2010 02:09 Marshall wrote: > On Apr 11, 10:43 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Apr 11, 9:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> To be fair the "standard theorists" and I don't seem to fight on >>>> the relativity of mathematical truth in general. They know >>>> ~(1+1=0) is true or false, relative to what kind of arithmetic >>>> we've chosen as the underlying one (e.g. arithmetic modulo 2 or >>>> arithmetic modulo 3). >>> In other words, what a sentence means is "relative" to >>> what meaning we choose for the symbols it is composed of. >> NO. > > No? You don't think the meaning of a sentence depends > on the meaning of the symbols it contains? Really? You misinterpreted my "No". It was supposed to mean "No, there was no need for you to introduce semantic into the issue with your phrase 'In other words'". I was talking about the relativity of truth, not semantic. There's also the relativity of semantic but that was _not_ what I was referring to when mentioning about my not-fighting with the "standard theorists". Also, your remark "what a sentence means..." is borderline being irrelevant: I was talking about truth, not semantic! >> Semantics and truth are 2 _different notions_! > > Is there any relationship between them at all? Is there? Why do you ask? Suppose there is, semantics and truth are still _2 different notions_, aren't they? >> And I'm taliking about "the relativity of mathematical truth". > > Yes you are. That's what's so funny. Are you taking a cheap shot at my typo "taliking"? If not, what's so funny about "the relativity of mathematical truth"?
From: David Bernier on 13 Apr 2010 04:37 Nam Nguyen wrote: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> [...] >>> The Principle addresses a different issue: P or ~P can't be uniformly >>> treated as true or false in _all_ contexts. That's all the Principle of >>> Symmetry would stipulate, and all I've really said (as above). >> >> So how do you know it is relevant to the particular context at hand, >> that of assuming "not P" hypothetically (ie for sake of argument), >> deriving >> a contradiction, and concluding "P". And also in the context where "P" >> and "not P" are switched systematically in such an argument? > > I'm not sure what you're trying to argue here. My argument only concerns > the alleged "absolute" truths that DB tried to suggest, and in which I > counter > claimed any such absolute truth would break the Principle of Symmetry. > That's all I've really argued in this conversation! If I understand some proof that uses a "reductio ad absurdum" argument, then it makes sense to me, or in other words the argument is valid. My mind is sort of stuck inside me, so the best I can do is to try to put my thoughts into words, hoping that what I write makes sense to others. Strictly speaking, what I arrive at are my conclusions. So I'm not saying that I arrive at absolute truths. Also, I know I can make mistakes. So my conclusions can be revised. If others don't accept my conclusions of logic, then I try not to get too upset. If it's not conclusive for them, then that's the way things are. Somebody referred to Strawson, the logician/philosopher. Would you know if Strawson thought of a "Principle of Symmetry" similar to yours? Presumably, this "Principle of Symmetry" you mention has been discussed in some book or article. Would you know of some reference for this? David Bernier
From: Alan Smaill on 13 Apr 2010 06:30
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Alan Smaill wrote: >>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>>> >>>>> To claim some formula being >>>>> absolutely true (or false) is to destroy the notion of such symmetry. >>>> Do you agree that symmetry is only >>>> broken if the cases "P" and "not P" are treated differently? >>> Of course I'd not agree. Even in relativity, given an appropriate context, >>> P and ~P must necessarily be treated differently because of LEM. >> >> Now you've really lost me. >> LEM *is* symmetric itself, isn't it? > > I have no idea what you meant here by "LEM *is* symmetric itself". > You have to explain what you meant by that. > > The word "symmetry" I've used here is in the context of the Principle > of Symmetry I articulated before with some clear examples and are about > concepts, formulas - but not about LEM]. LEM can be expressed by all formulae of a certain syntactic shape (P v not P) in the language to hand. Whether you take P as true or P as false makes no difference to LEM. In that sense, swapping P with not P makes no difference, LEM will apply in either case. Perhaps you can explain why "P and ~P must necessarily be treated differently because of LEM"? (Are you maybe thinking of what is called the law of non-contradiction, rather than the excluded middle?) >>> The Principle addresses a different issue: P or ~P can't be uniformly >>> treated as true or false in _all_ contexts. That's all the Principle of >>> Symmetry would stipulate, and all I've really said (as above). >> >> So how do you know it is relevant to the particular context at hand, >> that of assuming "not P" hypothetically (ie for sake of argument), deriving >> a contradiction, and concluding "P". And also in the context where "P" >> and "not P" are switched systematically in such an argument? > > I'm not sure what you're trying to argue here. My argument only concerns > the alleged "absolute" truths that DB tried to suggest, and in which I > counter claimed any such absolute truth would break the Principle of > Symmetry. That's all I've really argued in this conversation! Still not convinced on my side. I think we've gone as far as we can productively go, however. -- Alan Smaill email: A.Smaill at ed.ac.uk School of Informatics tel: 44-131-650-2710 University of Edinburgh |