Jim,
No, not really. The use of a statement in classical logic was not based on > verification that the statement was meaningful according to the test that > it could be assigned a value of True or False. I don't remember much from > the little Greek philosophy that I have actually read, but, while I have > some difficulty imagining Greeks like Aristotle using formal logical > statements that could not be determined as being True or False, I can, very > easily, imagine them discussing statements that were not easily evaluable. > By "classical logic" I meant the logic of Frege, Russel, Whitehead, and the like. Universal statements (For all x, ...) are falsifiable, but not verifiable. >> > > Is that a universal statement and therefore not verifiable (as True)? > That means that it is not verifiable (in the sense that it cannot be > assigned a truth value) and would not be a meaningful statement. To make > it meaningful it must be an Existential statement (I assume from the > previous statement that you meant to say that) which would say that there > exists some universal statements (of some domains) which are not > verifiable. > Yea, that's what I meant. But I still need to assume a few meta-logical axioms to make that assertion. :) On Sun, Jun 24, 2012 at 10:44 AM, Jim Bromer <[email protected]> wrote: > On Thu, Jun 21, 2012 at 5:47 PM, Abram Demski <[email protected]>wrote: > > >> Right. So, if we follow the framework of classical logic, "verifiable" >> and "falsifiable" are just two (overlapping) special cases of a much wider >> notion of "meaningful". >> > > No, not really. The use of a statement in classical logic was not based > on verification that the statement was meaningful according to the test > that it could be assigned a value of True or False. I don't remember > much from the little Greek philosophy that I have actually read, but, while > I have some difficulty imagining Greeks like Aristotle using formal logical > statements that could not be determined as being True or False, I can, very > easily, imagine them discussing statements that were not easily > evaluable. Isn't that a thread that is thoroughly woven into the fabric of > the Socratic method? > > >> Universal statements (For all x, ...) are falsifiable, but not >> verifiable. >> > > Is that a universal statement and therefore not verifiable (as True)? > That means that it is not verifiable (in the sense that it cannot be > assigned a truth value) and would not be a meaningful statement. To make > it meaningful it must be an Existential statement (I assume from the > previous statement that you meant to say that) which would say that there > exists some universal statements (of some domains) which are not > verifiable. You might say, well, I was saying that it was not verifiable > as a meaningful statement but if that were true it would be verified that > it was both meaningful and falsifiable. (In other words, the > statement,"The Statement that, 'Universal Statements are meaningful', is a > falsfiiable but not verifiable statement," is also a contradiction > - according to the doctrine that says that a statement has to be verifiable > (its truth value can be determined) to make it meaningful.) > > Jim > On Thu, Jun 21, 2012 at 5:47 PM, Abram Demski <[email protected]>wrote: > >> Right. So, if we follow the framework of classical logic, "verifiable" >> and "falsifiable" are just two (overlapping) special cases of a much wider >> notion of "meaningful". Atomic propositions are both verifiable and >> falsifiable, because they can be checked directly ("in principle" that >> is... we may not be able to visit the core of the Earth to check its >> temperature on a given day, but we can treat this as an atomic fact >> anyway... if we like.) Existential statements (There exists x such that...) >> are verifiable, but not always falsifiable. If the Flying Spaghetti Monster >> exists, then we could prove it by finding him/her. But if not, we may never >> know. Universal statements (For all x, ...) are falsifiable, but not >> verifiable. If we find one counterexample, it is disproven; however, we can >> find all the examples we like, yet never prove it for certain. Popper >> concentrated on these kinds of statements, because they are the kind most >> relevant to scientists; scientists typically want universal laws. >> >> If we stack quantifiers ("For all x, there exists y such that..." "There >> exists x such that for all y..."), we get apparently meaningful statements >> which *might* never be verified or falsified. Many of the important >> theorems or open conjectures in mathematics have this form. (For all >> integers, there exists a unique prime factorization. For all maps, there >> exists a 4-coloring. For all even whole numbers, there exists a >> representation as the sum of two primes.) >> >> Of course, the positivists thought that all mathematical statements would >> turn out to be verifiable! But, alas, it's not the case! >> >> But, these days, Positivist ideas and Popper's "opposite" ideas get >> rolled together. My impression is that modern-day self-styled positivists >> are mostly falsificationist, but their core idea is more that scientific >> theories are only meaningfully different insofar as they make different >> predictions (so, in particular, a theory which doesn't make specific >> predictions is not meaningfully different from no theory at all). >> >> My perspective is, AGI systems should be able to address as meaningful >> any concept which humans find to be meaningful. The goal, then, is >> different from these kinds of theories. With humans, we need to limit the >> scope to keep people "scientific"... and that is the goal of positivist or >> popperian theories. With AGI, we need to first understand the scope in >> order to have a starting point... so we require a rather more ambitious >> epistemology! >> >> Best, >> >> Abram >> >> On Thu, Jun 21, 2012 at 1:32 PM, Jim Bromer <[email protected]> wrote: >> >>> This issue of the verifiability of a proposition is related to AGI (of >>> course.) In order to be able to use a proposition effectively one has to >>> verify that it is meaningful in the first place. Furthermore, you have to >>> make sure that it is not dull witted (like proving a statement by inferring >>> it from a direct contradiction.) So a proposition has to have some kind of >>> interpretation that would make it meaningful and not completely insipid or >>> directly contradictory. Next it needs to be related to the subject matter >>> that it is supposed to be supporting or be integrated with. The concept of >>> "meaningful" does seem to coincide with "relevant". But how do we write a >>> program to create meaningful and relevant propositions when the very fabric >>> of knowledge is being generated. There is always the possibility that if >>> the knowledge that had been generated previously was meaningful then there >>> would be no way out of the program just generating a lot more meaningless >>> propositions. >>> >>> Finally the system has to be able to generate or find some propositions >>> that will help it achieve some goal. This aspect of meaningful and >>> relevant is another complexity. >>> Jim >>> *AGI* | Archives <https://www.listbox.com/member/archive/303/=now> >>> <https://www.listbox.com/member/archive/rss/303/7190161-766c6f07> | >>> Modify <https://www.listbox.com/member/?&;> Your Subscription >>> <http://www.listbox.com> >>> >> >> >> >> -- >> Abram Demski >> http://lo-tho.blogspot.com/ >> >> *AGI* | Archives <https://www.listbox.com/member/archive/303/=now> >> <https://www.listbox.com/member/archive/rss/303/10561250-164650b2> | >> Modify <https://www.listbox.com/member/?&;> Your Subscription >> <http://www.listbox.com> >> > > *AGI* | Archives <https://www.listbox.com/member/archive/303/=now> > <https://www.listbox.com/member/archive/rss/303/7190161-766c6f07> | > Modify<https://www.listbox.com/member/?&;>Your Subscription > <http://www.listbox.com> > -- Abram Demski http://lo-tho.blogspot.com/ ------------------------------------------- AGI Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/21088071-c97d2393 Modify Your Subscription: https://www.listbox.com/member/?member_id=21088071&id_secret=21088071-2484a968 Powered by Listbox: http://www.listbox.com
