'When Are Relations Neither True Nor False?
in [Math]
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From: David Bernier on 10 Apr 2010 05:01 Nam Nguyen wrote: > Nam Nguyen wrote: >> 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. > >> *** >> >> 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. >> >> 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. > > I'm still hoping to hear some responses, from Jim, my opponents, or from > anyone. In a proof by contradiction, we assume the negation of what we want to prove (i.e. not(P) ). Then, with that assumption, we arrive at "G .and. not(G)" . This is done in an "imaginary world" where P is false. Because we followed logic, and arrived at "G .and. not(G)", we conclude that not(P) is false. So P is true. Do you see problems with starting with a false premise, not(P) ? By the way, I don't remember hearing of the philosopher P. F. Strawson before this or a related thread. Wikipedia on philosopher P. F. Strawson < http://en.wikipedia.org/wiki/P._F._Strawson > . David Bernier
From: John Jones on 10 Apr 2010 05:18 Newberry wrote: > I had claimed that if for all a in the range of x > > (y)Aay (1) > > is vacuously true then > > (x)(y)Aay (2) > > is vacuously true. I have got an objection that this is scope fallacy > and that the inference is incorrect. I do not know, but I no longer > claim that (2) is vacuously true. I do claim that if for all a in the > range of x > > (y)Aay (1) > > is neither true nor false then > > (x)(y)Aay (2) > > is neither true nor false. A new version of my paper titled 'When Are > Relations Neither True Nor False?' is posted here > http://www.scribd.com/doc/26833131/RelationsAndPresuppositions-2010-02-13. > > Note that given > > ~(Ex)(Ey)[(x + y < 6) & (y = 8)] > > in Strawson's logic of presuppositions the formula is neither true nor > false for any choice of y. Let y be 8: > > ~(Ex)[(x + 8 < 6) & (8 = 8)] > i.e. > (x)[(x + 8 < 6) -> ~(8 = 8)] > > the subject class is empty and hence the sentence is neither true nor > false. > > Let y be anything but 8, say 9: > > ~(Ex)[(x + 9 < 6) & (9 = 8)] > i.e. > (x)[(9 = 8) -> ~(x + 9 < 6)] > > the subject class is empty and hence the sentence is neither true nor > false. > > Comments appreciated. Why don't you give an example? An example of a relationship that is neither true nor false is "Encore, Encore!" or "I promise", etc. If we are real mathematicians, and mathematics represents reality, how would you express these phrases in mathematical syntax? Other examples which aren't performatives are any statement whatever: It would first help to lay out what it is for something to be true or false. For a proposition to be true or false, we must show that there is an alternative proposition to that stated. But we can't assume that there is an alternative.
From: Nam Nguyen on 10 Apr 2010 11:38 David Bernier wrote: > Nam Nguyen wrote: >> Nam Nguyen wrote: >>> 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. >> >>> *** >>> >>> 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. >>> >>> 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. >> >> I'm still hoping to hear some responses, from Jim, my opponents, or >> from anyone. > > In a proof by contradiction, we assume the negation of > what we want to prove (i.e. not(P) ). Then, with that assumption, we > arrive > at "G .and. not(G)" . This is done in an "imaginary world" where P is > false. > Because we followed logic, and arrived at "G .and. not(G)", we conclude > that not(P) is false. So P is true. But what about the situation where P and not(P) can't be assigned to a truth value, in an absolute sense (i.e. it must be clear to _all_ that it's true, or false)? > > Do you see problems with starting with a false premise, not(P) ? Yes. I'd break "the Principle of Symmetry": if we could start with not(P), we could start with P. > > By the way, I don't remember hearing of the philosopher P. F. Strawson > before this or a related thread. > > Wikipedia on philosopher P. F. Strawson > < http://en.wikipedia.org/wiki/P._F._Strawson > . > > David Bernier
From: Aatu Koskensilta on 11 Apr 2010 11:05 Newberry <newberryxy(a)gmail.com> writes: > You proved PA consistent? Sure, as have innumerable others in the course of their logical studies. But perhaps you can now comment on how you feel about the more interesting mathematical result, that for any formal theory T extending Robinson arithmetic, either directly or through an interpretation, there are infinitely many true statements of the form "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" which are unprovable in T if T is consistent? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 11 Apr 2010 11:12
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > So what you're saying is you just _intuit_ PA system be consistent, no > more no less. Of course anyone else could intuit the other way too! 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. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |