http://en.wikipedia.org/wiki/Barber_paradox
Traditional First Order Boolean logic doesn't provide a way out of this paradox, because it doesn't provide for the possibility that P(X) = "X shaves those who don't shave themselves" is inherently impossible to assign a single truth value to, without making special restrictions on the domain of X. First Order Boolean logic is itself a model, and the model is broken because it can't handle these sorts of cases. It is not that complicated to add an additional truth value, Meaningless, in addition to True and False. It destroys the principle of the excluded middle, but this was the source of the problem in the first place. With Meaningless as a truth value, the principle of the excluded middle is transformed from "Precisely one of P(X) and ~P(X) is True, and the other is False," to "Provided P(X) is not Meaningless, precisely one of P(X) and ~P(X) is True, and the other is False." This also has some interesting implications when looking at Godel's incompleteness, since we can take the route of assigning the Meaningless truth value to the diagonalization statement, which is equivalent to "This statement cannot be proven," instead of giving it a value of True, and the proof falls apart because it depends on the strict True/False dichotomy. (What does it really mean to say, "This statement cannot be proven," anyway? Who cares? How does it affect the rest of the universe of discourse? Meaningless is the appropriate truth value.) Ultimately, what we really care about is not False or Meaningless statements, but the True ones. These are the statements that directly map to reality (or the universe of discourse, at least), and actually *tell *us something about it. False and Meaningless are only important in the sense that a backdrop is important. They are counterfactuals. In other words, the primary meta-logical language in which we as humans use to define and explain the behavior of various logical and mathematical systems only cares about True statements, not un-True ones. The principle of the excluded middle is a convenient way to sometimes find True statements from un-True ones, but it fails if the un-True statement is Meaningless as opposed to False (which is the negational opposite of True). This is equivalent to saying that un-True statements are only useful if we can get additional True statements by applying an operation (negation) to them, and that we can do so for False -- but not Meaningless -- statements. Negating a Meaningless statement just gets you another Meaningless statement. It deeply disturbs me that logicians, mathematicians, and their "customers" in the sciences so often fail to make this simple observation. On Wed, Nov 7, 2012 at 11:25 AM, Jim Bromer <jimbro...@gmail.com> wrote: > On Mon, Nov 5, 2012 at 11:38 PM, Aaron Hosford <hosfor...@gmail.com>wrote: > ch of the confusion and failure generated by Boolean logic, the most > commonly used and least versatile form of "fully functional" logic, arises > from the failure of Boolean logic to recognize the distinction between > different kinds of false statements (pieces of language). Boolean logic > only recognizes statements that are false because their opposites are true. > But it completely disregards the existence of statements which cannot be > mapped to reality to verify their truth. These sorts of statements are not > false because their opposites are true, but because they are meaningless. > This lack of distinction (a bias towards perceiving truth and falsehood but > not meaninglessness) is the source of many mathematical and logical > paradoxes. > > > > > I do not agree with that. Logic can be, uh, pointed at meaningless and > therefore incorporate that as a possible situation that can be found in a > model. The real problem, as I see it, is that the model would have to be > too complicated to be truly useful as a complete model of every > possibility. So instead we have to incorporate models of variations of > unknowns or undefineds into our models and then rely on bounded models so > that the logical 'discussion' (so to speak) might be used effectively. > This turns deductive logic into inductive logic and makes the effective use > of logic conjectural. > > Jim Bromer > Jim Bromer > > > On Mon, Nov 5, 2012 at 11:38 PM, Aaron Hosford <hosfor...@gmail.com>wrote: > >> Mike, here's something for you to chew on: >> >> I like to think of the term "random" as meaning the absence of a * >> detectable* pattern, due to biases and/or blind spots of the observer or >> perspective. Under this definition, it is implicit that anything can be >> seen as random or non-random given an appropriate observer or perspective. >> >> I think our minds are simply filters for reality, designed to pluck out >> the most common and useful patterns through appropriate biases. We look for >> the patterns we do because they help us accomplish our evolutionary >> purpose. When something looks random to us, it simply means we can't >> identify a pattern, not that there isn't one. >> >> Mathematics is a language for representing the sort of patterns the human >> mind tends to perceive. It is, in other words, a language for representing >> reality in terms of our intrinsic human perceptual biases. We have integer >> arithmetic because we perceive discrete, countable units in the universe. >> We have calculus because we perceive continuous phenomena such as curves, >> areas, volumes, and flows in the universe. We have geometry and topology >> because we perceive shapes and invariances of shape in the universe. We >> have logic because we perceive a mapping between situations and their >> descriptions (language, including math) in the universe. And we find >> commonalities and relationships between the various branches of >> mathematics, described in the terms of logic because logic is the language >> we use to talk about languages, including those of mathematics. >> >> Much of the confusion and failure generated by Boolean logic, the most >> commonly used and least versatile form of "fully functional" logic, arises >> from the failure of Boolean logic to recognize the distinction between >> different kinds of false statements (pieces of language). Boolean logic >> only recognizes statements that are false because their opposites are true. >> But it completely disregards the existence of statements which cannot be >> mapped to reality to verify their truth. These sorts of statements are not >> false because their opposites are true, but because they are meaningless. >> This lack of distinction (a bias towards perceiving truth and falsehood but >> not meaninglessness) is the source of many mathematical and logical >> paradoxes. >> >> As for your "irregular forms", Mike, what all this boils down to is that >> if the regularity of some aspect of the world isn't visible to mathematics, >> it isn't visible to *people*. Mathematics simply codifies what our >> brains already do. When we notice a regularity in reality, we create a >> branch of mathematics to describe it. If we don't see a regularity in a >> portion of reality, we call it random and mentally disregard it. I am >> identifying concepts themselves as human-perceivable regularities in the >> world, in case that isn't readily apparent. Thus if we can conceptualize >> something, it must be regular in our minds, and either a branch of >> mathematics exists to describe those regularities, or we can create one. >> (These new branches of math always start out as human language and become >> steadily more formalized as we become more certain about what the >> regularities are and how to more effectively describe them in language.) >> >> So, in summary: >> 1) A concept is a human-perceivable regularity. >> 2) Mathematics is a highly refined language for describing >> human-perceivable regularities. >> Therefore: >> 3) Mathematics is a highly refined language for describing concepts. >> >> Mathematics (and language in general) is how we tell each other about the >> world. If something can't be described in terms of mathematics, it's >> because it either can't be perceived, or we have identified a new branch of >> mathematics that needs to be created. Eventually, we will have covered all >> the filtering biases evolution has built into our minds, and mathematics >> will be sufficient to describe all concepts the human mind can perceive >> without further additions to the language. >> >> >> >> >> >> >> >> *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/303/23050605-bcb45fb4> | > 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
