Re: CDR: Re: The End of the Golden Age of Crypto
Jim Choate wrote: On Wed, 20 Nov 2002, Peter Fairbrother wrote: Completeness has nothing to do with whether statements can or cannot be expressed within a system. A system is complete if every sentence that is valid within the system can be proved within that system. Introduction to Languages, Machines and Logic A.P. Parks ISBN 1-85233-464-9 pp 240 and 241 A non-mathematical easy to read primer (quotes from Springer-Verlag). I don't have a copy. If Alan Parkes says Godelian completeness is other than the definition above then he is wrong - possible, he is a multimedia studies teacher, and afaik is not a mathematician - but I suspect you misread him. FYI, I just googled completeness godel. First five results plus some quotes are at the bottom. Five minutes, which I could have spent better. RTFM. -- Peter Fairbrother ... Googling completeness and Godel, first five results: http://www.math.uiuc.edu/~mileti/complete.html No simple definition of completeness. Nice intro to models though. www.chaos.org.uk/~eddy/math/Godel.html Completeness is the desirable property of a logical system which says that it can prove, one way or the other, any statement that it knows how to address. www.uno.edu/~asoble/pages/1100gdl.htm Completeness = If an argument is valid, then it is provable http://www-cs-students.stanford.edu/~pdoyle/quail/questions/11_15_96.html A complete theory is one contains, for every sentence in the language, either that sentence or its negation. http://www.wikipedia.org/wiki/Kurt_Godel -- link to http://www.wikipedia.org/wiki/Goedels_completeness_theorem It states, in its most familiar form, that in first-order predicate calculus every universally valid formula can be proved.
Re: CDR: Re: The End of the Golden Age of Crypto
Jim Choate wrote: Para-consistent logic is the study of logical schemas or systems in which the fundamental paradigms are paradoxes. It's a way of dealing with logical situations in which true/false can't be determined even axiomatically. Most paraconsistent logics deal with paradoxes, but I know of none whose fundamental paradigms are paradoxes. That barely makes sense to me, and is certainly not true. Paraconsistent logics often* allow some but not all sentences within the logic to be both true and false. In paraconsistent logics that have simple notions of true and false** it is usually (at least sometimes) possible to axiomatically determine whether a sentence is true or/and false - they wouldn't be much use if you couldn't! (not that they are much use anyway). * Many logicians would say they all do, according to Vasiliev and Da Costa's original definition. Some would say only some do. And some logician somewhere will disagree with almost anything you say about paraconsistent logics... ** Not all do, eg some have multi-value truths. Some have conditional truths, or truths valid only in some worlds. Some have true, false, both and neither. And so on. As usual, some logicians will disagree with this. For those who might care, paraconsistent logics are usually defined as non-explosive* logics. Ha! There is some argument (lots!**) about that, but it's the generally accepted modern definition (or at least the one most often argued about). * logics in which ECQ does not hold. ECQ = Ex Contradictione Quodlibet, anything follows from a contradiction. In most normal logics, if any single sentence and it's negation can both be proved, then _every_ sentence can be proved both true and false. This property is known as explosiveness. ** For instance, it has recently been shown that some logics traditionally known as paraconsistent, eg Sette's atomic P1 logic, are explosive, contrary to that definition. There are arguments about the meaning of negation as well, all of which confuse the issue. BTW, the name doesn't have anything to do with paradoxes, at least according to the guy who invented it. The para bit is supposedly from an extinct word (I forget the language, Puppy-something, really) for arising out of, coming from. Some say it's from the Greek para- beyond; but I've never heard the paradox story before. I hope this at least interested some, and was not just troll-food. -- Peter Fairbrother
Re: CDR: Re: The End of the Golden Age of Crypto
Jim Choate wrote: Para-consistent logic is the study of logical schemas or systems in which the fundamental paradigms are paradoxes. It's a way of dealing with logical situations in which true/false can't be determined even axiomatically. Most paraconsistent logics deal with paradoxes, but I know of none whose fundamental paradigms are paradoxes. That barely makes sense to me, and is certainly not true. Paraconsistent logics often* allow some but not all sentences within the logic to be both true and false. In paraconsistent logics that have simple notions of true and false** it is usually (at least sometimes) possible to axiomatically determine whether a sentence is true or/and false - they wouldn't be much use if you couldn't! (not that they are much use anyway). * Many logicians would say they all do, according to Vasiliev and Da Costa's original definition. Some would say only some do. And some logician somewhere will disagree with almost anything you say about paraconsistent logics... ** Not all do, eg some have multi-value truths. Some have conditional truths, or truths valid only in some worlds. Some have true, false, both and neither. And so on. As usual, some logicians will disagree with this. For those who might care, paraconsistent logics are usually defined as non-explosive* logics. Ha! There is some argument (lots!**) about that, but it's the generally accepted modern definition (or at least the one most often argued about). * logics in which ECQ does not hold. ECQ = Ex Contradictione Quodlibet, anything follows from a contradiction. In most normal logics, if any single sentence and it's negation can both be proved, then _every_ sentence can be proved both true and false. This property is known as explosiveness. ** For instance, it has recently been shown that some logics traditionally known as paraconsistent, eg Sette's atomic P1 logic, are explosive, contrary to that definition. There are arguments about the meaning of negation as well, all of which confuse the issue. BTW, the name doesn't have anything to do with paradoxes, at least according to the guy who invented it. The para bit is supposedly from an extinct word (I forget the language, Puppy-something, really) for arising out of, coming from. Some say it's from the Greek para- beyond; but I've never heard the paradox story before. I hope this at least interested some, and was not just troll-food. -- Peter Fairbrother
Re: CDR: Re: The End of the Golden Age of Crypto
On Wed, 20 Nov 2002, Peter Fairbrother wrote: Completeness has nothing to do with whether statements can or cannot be expressed within a system. A system is complete if every sentence that is valid within the system can be proved within that system. Introduction to Languages, Machines and Logic A.P. Parks ISBN 1-85233-464-9 pp 240 and 241 -- We don't see things as they are, [EMAIL PROTECTED] we see them as we are. www.ssz.com [EMAIL PROTECTED] Anais Nin www.open-forge.org
Re: CDR: Re: The End of the Golden Age of Crypto
On Wed, 20 Nov 2002, Peter Fairbrother wrote: Completeness has nothing to do with whether statements can or cannot be expressed within a system. A system is complete if every sentence that is valid within the system can be proved within that system. Introduction to Languages, Machines and Logic A.P. Parks ISBN 1-85233-464-9 pp 240 and 241 -- We don't see things as they are, [EMAIL PROTECTED] we see them as we are. www.ssz.com [EMAIL PROTECTED] Anais Nin www.open-forge.org
