'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 30 Mar 2010 01:47 Jim Burns wrote: > I think it would be very useful to me in understanding > what you are trying to accomplish if you were > to give a summary of the best arguments AGAINST your > positions. As promised I'll summarize what I think as the best arguments against my positions. The caveats however are a) these are my own opinions and might not reflect what they might have actually thought and b) it seems to me a lot of arguments went back and forth and then became "dangling" in the sense that key questions weren't answered, terminologies were quite agreed, definitions weren't accepted, et...; consequently my discerning the nature of their arguments might not be 100% accurate. So, I could only do my best in the summarization, by extrapolating and triangulating from what were said in this thread or in the past ones that are directly relevant to the arguments. *** Imho, the 3 major and best arguments against my belief, that the nature of FOL reasoning is that of relativity or of being subjective, are the following objections: (O1) The Universality Objection: In this objection, the correctness in reasoning under one logical framework should be _universally constant_ and shouldn't be a function of individual subjective beliefs or knowledge. My claiming on the relativity nature of FOL reasoning seems to violate this natural and unobjectionable, say, "sanctity". (O2) The Philosophy Objection: In this objection, the ideas such that there are formulas written in the language of arithmetic that can be neither arithmetically true nor false are just philosophical ideas and thus can't be a basis to attack the current FOL reasoning. (O3) The Ordinary Mathematics Objection: This objection seems to be a cross-breed between O1 and O2. In this objection, FOL reasoning is build upon the ordinary mathematical knowledge that *in principle* should be universally _self evident_ to all who are trained or study mathematics. As such FOL reasoning should be universally the same and should _not_ be subjective or relativistic. On O1, perhaps the following conversation between DCU and me on Dec 22, 2005 would best reflect the objection (the argument) against my "reasoning relativity" position. It's in the thread "About Consistency in 1st Order Theories" where DCU gave some critique comments on my desire of changing FOL to make it conform to relativity of reasoning. NN: Specifically I'd like to change FOL in such a way that mathematical reasoning is no longer absolute: the reasoning agent's limited (finite) reasoning knowledge, and his/her freedom to make certain interpretation on certain levels of introspection, would all affect the results of the reasoning. DCU: If you want people to help you with this you might start by trying to convince people that there's a _need_ for this radical new version of logic. Exactly what the objective is is not clear to me. It seems possible that you might want to call it something other than "logic". Because whatever it is, it seems that in the thing you're looking for the "logic" is going to vary from person to person, and I suspect it's going to seem to a lot of people like the whole point to _logic_ is to study _correct_ reasoning, which will _not_ vary from person to person. Now, the class of mathematical facts that a given individual is actually able to prove certainly varies from person to person. If you want to study that somehow fine, but that seems more a topic in something like psychology than pure logic. If I'm correct in thinking that in the system you have in mind the _definition_ of correct reasoning is going to vary from person to person that seems even less like "logic". That's about 5 years ago and DCU isn't participating in the current thread and I don't know if he has changed his opinion on the subject of relativity in reasoning I've have been pursuing since. But apparent from his comments above are counter arguments against my position of mathematical relativity in reasoning. On O2, and O3, I believe some of AK's conversations in this thread and others in the past and some of TF's writings would reflect these 2 objections in various degrees. I have to admit though I don't have right off specifics in what AK said and might need more time to search for some samples. But let me excerpt some of what TF wrote about ordinary mathematical truth that should be universally accepted and "there is no need to assume that we are introducing any problematic philosophical notions". In TF's Chapter 2, "G�del�s Theorem" one would see: In a mathematical context, on the other hand, mathematicians easily speak of truth. If the generalized Riemann hypothesis is true..., There are strong grounds for believing that Goldbach�s conjecture is true..., If the twin prime conjecture is true, there are infinitely many counterexamples.... In such contexts, the assumption that an arithmetical statement is true is not an assumption about what can be proved in any formal system, or about what can be seen to be true, and nor is it an assumption presupposing any dubious metaphysics. Rather, the assumption that Goldbach�s conjecture is true is exactly equivalent to the assumption that every even number greater than 2 is the sum of two primes, the assumption that the twin prime conjecture is true means no more and no less than the assumption that there are infinitely many primes p such that p+2 is also a prime, and so on. In other words the twin prime conjecture is true is simply another way of saying exactly what the twin prime conjecture says. It is a mathematical statement, not a statement about what can be known or proved, or about any relation between language and a mathematical reality. Similarly, when we talk about arithmetical statements being true but undecidable in PA, there is no need to assume that we are introducing any problematic philosophical notions. That the twin prime conjecture may be true although undecidable in PA means simply that it may be the case that there are infinitely many primes p such that p+2 is also a prime, even though this is undecidable in PA. Again, these are only my thoughts of what the arguments against my position be. It would certainly be helpful if those who oppose my position could further clarify in technical clarity what they perceive are problems in my positions. I also don't mind in subsequent posts to further defend my position or to provide more counter-arguments. Cheers, -Nam Nguyen
From: Newberry on 30 Mar 2010 01:55 On Mar 29, 10:17 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > But if we leave out all the vacuous sentences we can still do all the > > useful arithmetic as we know it. Although all the people on this board > > believe that such sentences are true nobody argued that they were > > useful. Aatu even said that they did not belong in ordinary > > mathematical reasoning. > > You're imagining things. > > > Furthermore there is a reason to think that they are neither true nor > > false. I cannot think of any good reason for claiming that 1 + 1 = 2 > > is not true. > > Well, perhaps you might be moved to answer the following query, which I > have now presented on several occasions: > > How are we to apply your ideas about vacuity, meaningfulness, truth, > proof, what not, in context of the following mathematical observation: > for any consistent theory T extending Robinson arithmetic, either > directly or through an interpretation, in which statements of the form > "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" can be > expressed, there are infinitely many Diophantine equations D(x1, ..., > xn) = 0 that have no solutions but for which "the Diophantine equation > D(x1, ..., xn) = 0 has no solutions" is not provable in T. > > On an ordinary understanding, a statement of the form "the Diophantine > equation D(x1, ..., xn) = 0 has no solutions" is true just in case D(x1, > ..., xn) = 0 has no solutions. According to your account some such > statements are neither true nor false No. > -- owing, so I gather, to your > eager zeal for equating formal provability with truth, contra G del and > his lackwit lackeys -- You never answered my question what you ment by "Goedel." a) Every system capable of arithmetic is syntactically incomplete b) In every system capable of arithmetic there are true but unprovable formulae. > regardless of whether the corresponding > Diophantine equations are soluble or not. What are we to make of this? > In what way, and just how, do your musing connect to our mathematical > experience or reasoning? You may of course introduce whatever technical > terminology you wish, defining truth and falsity whichever way you see > fit, to suit your whim and fancy, but unless answers to questions like I > have presented are forthcoming your fiddling will be of no apparent > interest in the wider scheme of things in mathematics, in the philosophy > of mathematics, in our everyday arithmetical reasoning, thinking, > reflection. > > It's presumptuous of me, but I nonetheless surmise it is at least in > part a hope of yours, in these your endless mumblings Usenetical on > matters logical, the liar, vacuity, what not, that others come to > recognise the wonderful clarity of your theory of meaning, your take on > logic, your this and that. On this surmise, I can only suggest you make > some attempt to demonstrate the value of your insight to the skeptic > masses by showing how some conundrum can be exorcised, some problematic > enigma eradicated, by way of transparent and compelling reasoning > involving your novel insights, some traditional baffler dissected with > particular cunning made possible by the notions you champion, and so > on. This will be impossible if the skeptic is not willing to discuss anything specific I have written and only utters general derogatory epithets. > You could do well to emulate Dummett or Kreisel, neither of whom I'm > an unreserved fan, but who never fail to stimulate and inspire, even if > it is only to cry out in philosophical exasperation. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Daryl McCullough on 30 Mar 2010 06:30 Newberry says... > >On Mar 29, 10:17=A0pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> How are we to apply your ideas about vacuity, meaningfulness, truth, >> proof, what not, in context of the following mathematical observation: >> for any consistent theory T extending Robinson arithmetic, either >> directly or through an interpretation, in which statements of the form >> "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" can >> be expressed, there are infinitely many Diophantine equations >> D(x1, ...,xn) = 0 that have no solutions but for which >> "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" >> is not provable in T. >> On an ordinary understanding, a statement of the form "the Diophantine >> equation D(x1, ..., xn) = 0 has no solutions" is true just in case >> D(x1 ..., xn) = 0 has no solutions. According to your account some >> such statements are neither true nor false > >No. What do you mean, no? You are proposing to equate truth and provability. Some sentences of the form "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" are neither provable nor refutable. It follows from your equating of truth and provability that they are neither true nor false. The reasoning that there are statements that are true, but unprovable goes like this: 1. Let D(x1,...,xn) be a polynomial in the variables x1, ..., xn. 2. If m1, ..., mn are n integers such that D(m1,...,mn) = 0, then the statement "There exists x1, ..., xn such that D(x1,...,xn) = 0" is provable. We can easily prove this by plugging in m1, ..., mn and checking to see if the result is 0. 3. If the statement "There exists x1, ..., xn such that D(x1,...,xn) = 0" is not provable, then there are no integers m1, ..., mn such that D(m1, ..., mn) = 0. This follows immediately from 2. 4. Note that if Phi is the formula "There exists x1, ..., xn such that D(x1,...,xn) = 0", then 3. has the form: "If Phi is not provable, then ~Phi". In other words, if Phi is not provable, then the negation of Phi holds. 5. Therefore, if Phi is neither provable nor refutable, then the negation of Phi holds. So if Phi is neither provable nor refutable, then Phi is false. ("Phi is false" means the same thing as "The negation of Phi holds"). 6. Therefore, if Phi is neither provable nor refutable, then there is a statement, Phi, that is false, but not provably false. There is another statement, ~Phi that is true, but not provable. 7. Therefore, if there is a statement Phi (of the appropriate form) that is neither provable nor refutable, then provability and truth are not the same. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 30 Mar 2010 06:34 Newberry says... >Indeed. But if we leave out all the vacuous sentences we can still do >all the useful arithmetic as we know it. The undecidable statements of a theory are *not* vacuous. If you have a statement of the form "there is no solution to the polynomial equation D(x1, x2, ..., xn) = 0", it's not vacuous. It tells us something very specific, namely that a search for solutions to the polynomial equation will never succeed. Such a statement may be undecidable in a particular theory, but it's not vacuous. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 30 Mar 2010 06:51
Newberry says... >There are two issues here. >a) The two tokens have the same subject and the same predicate. >b) The resolution can be seemingly defeated by forcing all tokens into >one type. > >Not sure why you think they are related. Any theory of truth that is worth considering, if two sentence tokens have the same subject and same predicate, then they have the same truth value. Otherwise, your notion of truth is unconnected with the meaning of sentences. >Let's take a) first. Gaifman's evaluation procedure is such that if >two tokens have the same subjects and predicates one can nevertheless >be true and the other neither true nor false. > >Now b): > This sentence is not truthy. > "This sentence is not truthy" is not truthy. > >These two sentences have the same subjects and predicates. The former >is self-referential the latter is not. Using Godel coding, you can eliminate direct self-reference and thereby make the two sentences identical. Then it is a contradiction to say that one is truthy and the other is not. -- Daryl McCullough Ithaca, NY |