'When Are Relations Neither True Nor False?
in [Math]
|
From: Jesse F. Hughes on 6 Mar 2010 17:57 Newberry <newberryxy(a)gmail.com> writes: >> No to what? There is no notion of entailment? >> >> I haven't thought much about these things, but it seems to me that any >> semantics of a formal language should give me some way of determining >> entailment. Insofar as you have no entailment relation, it seems >> implausible that you have a semantics for your language. >> >> If I give you an arbitrary formula, can you specify the conditions >> under which that formula would be true? false? neither? > > The semantics is quite simple actually. For sentences with one > variable it is given by Venn diagrams. But when a set is empty (it > does not even appear on the Venn diagram) then the sentence that it > is a subset of an arbitrary set is neither true nor false. I suppose you find this perfectly clear and suitable. I'm afraid I find it vague. We've gone over a number of issues regarding your desired semantics in the past. I suppose I'll drop the issue here, since your answers are no more forthcoming now than they were in, say, 2007. -- "Just because you're ... in a Ph.d program it does not mean that you're up to the challenge of being a real mathematician. Only those who have a purity of mind and dedication to the truth as the highest ideal have a chance." --James Harris, as Sir Galahad the Pure.
From: Nam Nguyen on 7 Mar 2010 18:37 David Bernier wrote: > Nam Nguyen wrote: >> Alan Smaill wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> Jesse F. Hughes wrote: >>>> >>>>> So, you want to deny that Goedel's theorem is true. >>>> The "crank" tends to assert Goedel's theorem is false. >>>> The "standard theorist" would insist GIT is true. >>>> >>>> That leaves the "rebel" the only side who observes the >>>> method in Godel's work is invalid. >>>> >>>> Except for the relativists, why should we care about invalid >>>> truth or falsehood? >>> >>> >>> Because the notions of "truth" and "validity" are not beyond dispute, >>> and >>> maybe we/you/I have got those wrong. >> >> In a high level that makes sense, imho. But I'm not after the "absolute" >> truth, validity, or correctness. It's the _process and methods_ of >> reasoning that I'm really ... really after. >> >> My posting in threads after threads is a means to re-examine, evaluate, >> and revise the current methods of reasoning in FOL in general and in >> Incompleteness in particular. And my motivation of the re-examination >> is basically a following of what Shoenfield said about mathematical >> logic: >> >> "Logic is the study of reasoning; and mathematical logic is the study >> of reasoning done by the mathematicians. >> >> To _discover proper approach to mathematical logic_, we must therefore >> _examine the methods of the mathematicians_." >> >> [The highlights are mine]. >> >> For what it's worth, the long and short of it is what I've been doing for >> years is simply trying to evaluate, for possible shortcomings, of the >> current >> FOL regime and of anyone's reasonings I've come across in some way: >> myself, >> posters in this and other fora, Torkel Franzen, Godel, Hilbert, etc... >> >> I'm less interested for example whether or not, say, PA is consistent >> but I'm interested in based from what existing and historical reasoning >> backgrounds and by what methods one would logically conclude - with no >> emotion or "belief" - PA is consistent - or not. > > There's an ultrafinitist named Nelson, who rose > in professorial (USA) ranks to Associate Professor > of Mathematics or higher. That's a sign > of a definitely competent mathematician, generally > speaking. > For years, PA and/or ZFC were ok > for him. Now, his belief is that PA *might* > be inconsistent. Imho, it's best that we stay _neutral_ in term of _knowing_ in mathematical logic (FOL ). We can demonstrate some truths we know for certain; we can demonstrate for sure some truths we'd know in principle; but if there are truths we can *not* know then we can not know, and there isn't anything we can do to change it, however desired or "anguished" we might be. > > With mathematicians, there's a method to > their "madness" (English saying), and, hopefully, > no "madness" to their methods. For what it's worth I like this saying. I think it's a "madness" to try to balance it out between the methods based on what we can know (e.g. finite proof) and those based on what is impossible to know. New methods/governances regarding to, say, "absolute undecidability" might have to be devised and they might take some efforts. But imho there are some chances. > >> For instance, here's what MoeBlee said earlier: >> >> >> at least we do know how to formalize PRA. And if we cannot be >> >> confident that PRA is consistent then I don't know what substantive >> >> mathematical theory we could be confident as to consistency. >> >> This passage, imho, is a typical example of where one would forget that >> reasoning is a _process_ (as Shoenfield alluded to above), _not_ a >> mixture >> of intuitive and religious-like _beliefs_. If we agree what "formalize" >> means, what methods of reasoning are, what consistency and inconsistency >> means, then as a fact it's either PRA is or isn't inconsistent strictly >> based on the definitions and the methods, and it's either we do know >> *or* >> don't know about the fact. Period. "Confidence" isn't an issue nor does >> it have a role here, in the process of reasoning. >> >> The long and short of it is from what I've gathered, Godel's "proof" is >> invalid as a meta proof, because the methods used in reasoning failed >> in one of the few intuitive principles about the methods of logical >> reasonings. Let me cite the 2 obvious principles: >> >> (1) Principle of Consistency: >> >> No methods shall lead to contradictory conclusions. >> >> (2) Principle of (Method) Compatibility: >> >> If each of any 2 equivalent conclusions is expressed in an >> independent >> method then it shall not be the case that one method would lead to >> its perspective conclusion, while the other method wouldn not lead to >> the other counterpart conclusion. I'd like to add another Principle: (3) Principle of Symmetry (of Non-Logicality): Other than concepts that are tautologous or contradictory (in the underlying reasoning framework), a concept and its negation are totally _symmetrical_ with respect to methods of reasoning, in the sense that if there's a context that the concept is so-and-so then there's another context of the same category of contexts that its negation is also so-and-so. For instance, Let a concept be semantically reflected by a formula F, thenif F is provable in a system then there's another system that ~F is provable; if there's a language model that F is true, then the same could be said of ~F; if there's a "standard" model that F is true, then there's another model that ~F is true and this model could be equally considered as a "standard" model; etc... The symmetry of F and ~F is very much "absolute" as the symmetry of the elements in a 2-element set: there can't be _no preference_ between the 2 elements in term of set membership; if one of them could be used as the first element of an ordered pair, then so could be the other one, e.g. This Principle basically would absolve any "standard-ness" in reasoning and would pay the way for relativity in mathematical reasoning, imho. >> >> In Godel's work, the knowledge of the natural numbers as a sheer >> intuition >> or as a model of a FOL formal system is incompatible with rules of >> inference >> in so far as both of them are methods of reasonings (and they are). >> >> The reason being is "undecidability" of formal systems is purportedly >> "equivalently" defined by both methods, but the natural number would lead >> to proof of consistency while the rules of inference method would not!
From: Nam Nguyen on 7 Mar 2010 19:11 Nam Nguyen wrote: >>> The long and short of it is from what I've gathered, Godel's "proof" is >>> invalid as a meta proof, because the methods used in reasoning failed >>> in one of the few intuitive principles about the methods of logical >>> reasonings. Let me cite the 2 obvious principles: >>> >>> (1) Principle of Consistency: >>> >>> No methods shall lead to contradictory conclusions. >>> >>> (2) Principle of (Method) Compatibility: >>> >>> If each of any 2 equivalent conclusions is expressed in an >>> independent >>> method then it shall not be the case that one method would lead to >>> its perspective conclusion, while the other method wouldn not >>> lead to >>> the other counterpart conclusion. > > I'd like to add another Principle: > > (3) Principle of Symmetry (of Non-Logicality): > > Other than concepts that are tautologous or contradictory (in > the underlying reasoning framework), a concept and its negation > are totally _symmetrical_ with respect to methods of reasoning, Sorry. I meant "shall be totally _symmetrical_ ...". > in the sense that if there's a context that the concept is so-and-so > then there's another context of the same category of contexts that > its negation is also so-and-so. > > For instance, Let a concept be semantically reflected by a formula > F, thenif F is provable in a system then there's another system > that ~F is provable; if there's a language model that F is true, > then the same could be said of ~F; if there's a "standard" model > that F is true, then there's another model that ~F is true and > this model could be equally considered as a "standard" model; etc... > > The symmetry of F and ~F is very much "absolute" as the symmetry > of the elements in a 2-element set: there can't be _no preference_ > between the 2 elements in term of set membership; if one of them > could be used as the first element of an ordered pair, then so > could be the other one, e.g. > > This Principle basically would absolve any "standard-ness" in reasoning > and would pay the way for relativity in mathematical reasoning, imho.
From: Nam Nguyen on 7 Mar 2010 23:09 Nam Nguyen wrote: >>> >>> The long and short of it is from what I've gathered, Godel's "proof" is >>> invalid as a meta proof, because the methods used in reasoning failed >>> in one of the few intuitive principles about the methods of logical >>> reasonings. Let me cite the 2 obvious principles: >>> >>> (1) Principle of Consistency: >>> >>> No methods shall lead to contradictory conclusions. >>> >>> (2) Principle of (Method) Compatibility: >>> >>> If each of any 2 equivalent conclusions is expressed in an >>> independent >>> method then it shall not be the case that one method would lead to >>> its perspective conclusion, while the other method wouldn not >>> lead to >>> the other counterpart conclusion. > > I'd like to add another Principle: > > (3) Principle of Symmetry (of Non-Logicality): > > Other than concepts that are tautologous or contradictory (in > the underlying reasoning framework), a concept and its negation > shall be totally _symmetrical_ with respect to methods of reasoning, > in the sense that if there's a context that the concept is so-and-so > then there's another context of the same category of contexts that > its negation is also so-and-so. > > For instance, Let a concept be semantically reflected by a formula > F, thenif F is provable in a system then there's another system > that ~F is provable; if there's a language model that F is true, > then the same could be said of ~F; if there's a "standard" model > that F is true, then there's another model that ~F is true and > this model could be equally considered as a "standard" model; etc... > > The symmetry of F and ~F is very much "absolute" as the symmetry > of the elements in a 2-element set: there can't be _no preference_ > between the 2 elements in term of set membership; if one of them > could be used as the first element of an ordered pair, then so > could be the other one, e.g. > > This Principle basically would absolve any "standard-ness" in reasoning > and would pay the way for relativity in mathematical reasoning, imho. > I have one more to add: (4) Principle of Humility: If there exists a concept C about non-finiteness, there exists a concept C', not necessarily identical to C, that no methods of reasoning shall be able to conclude whether or not it's a contradiction. It's my belief that an edifice of reasoning that conforms to these 4 principles (if not more) would be "logical", reasonable, humanistic, and conducive to our ability to face challenges in mathematical reasoning, moving forward. Thanks, in advance if you have any constructive comments and suggestions about these principles. Best Regards, Nam Nguyen
From: MoeBlee on 8 Mar 2010 11:22
On Mar 6, 10:44 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > I'm less interested for example whether or not, say, PA is consistent > but I'm interested in based from what existing and historical reasoning > backgrounds and by what methods one would logically conclude - with no > emotion or "belief" - PA is consistent - or not. > > For instance, here's what MoeBlee said earlier: > > >> at least we do know how to formalize PRA. And if we cannot be > >> confident that PRA is consistent then I don't know what substantive > >> mathematical theory we could be confident as to consistency. > > This passage, imho, is a typical example of where one would forget that > reasoning is a _process_ (as Shoenfield alluded to above), _not_ a mixture > of intuitive and religious-like _beliefs_. If we agree what "formalize" > means, what methods of reasoning are, what consistency and inconsistency > means, then as a fact it's either PRA is or isn't inconsistent strictly > based on the definitions and the methods, and it's either we do know *or* > don't know about the fact. Period. "Confidence" isn't an issue nor does > it have a role here, in the process of reasoning. There's nothing I wrote that indicates that I "forget" that reasoning is a "process". Nor did I mention nor rely on religious belief. Moreover, I did not dispute that PRA is consistent or it is not, regardless of what one believes about the matter. As to knowledge of consistency, unless one has a notion of 'knowledge' different from usual formulations such as 'true justfied belief' then one's knowledge or lack of knowledge of the consistency of PRA of course depends on at least three things: (1) whether PRA is consistent, (2) whether one believes PRA consistent, and (3) one's basis for believing PRA consistent. However, I am not adamant about such a view of knowledge, as I'm open to hearing alternative notions. Yet still, I would think that the process of reasoning is that of applying logic to premises. I don't know how one would reason in infinite regress without having some starting premises. So, in this context, some starting premises might be such things as "Modus ponens is truth preserving", "the law of identity obtains", etc. Of course, even these premises could be entailed by other premises, but still in such a process there will be certain other premises that were taken as starting points. As to what basis one should have for adopting such starting premises, I have not opined. I have not ruled out intuition, but I certainly have not endorsed that the basis should be religious. Now Nam's response is yet another in which he bypasses answering the question of what principles or system or ANYTHING he takes (whether what he takes is a matter of confidence or of his reasoning or whatever) as true or correct or acceptable or whatever. He has even disavowed that he can affirm that one display of a finite string of characters matches (or does not match, as the case may be) another display of a finite string of characters. So, I really don't know he expects there could be symbolic reasoning about ANYTHING. And, to the context of incompleteness, again, he hasn't said WHAT system of reasoning he would accept as proving the incompleteness theorem IF he doesn't already accept reasoning in the system that Koskensilta mentioned, which is an even MORE restricted version of PRA. MoeBlee |