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 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
