When using natural language, most people will shift their interpretation of a statement to the nearest neighbor that makes sense to them, or they will keep the meaningful part of the interpretation and disregard the part that makes no sense. In dealing with the Barber's Paradox, we can still count on the fact that the Barber shaves all those *other than himself* that do not shave themselves, and we leave the question as to whether he shaves himself open for further evidence.
In logic and mathematics, however, these sorts of partial interpretations are disallowed, not because we're intellectually unable to resolve them, but because the way we resolve them might vary from person to person, and the logical or mathematical language is designed to minimize ambiguities and miscommunications at the expense of brevity. So while in natural language, we can still assign a (partial) meaning to the statement, in logic we must (in most systems) throw out the whole thing. Statements in any language are "things", which can be described by other statements in the same or another language, provided it is sufficiently general. The statement "P(X)" is not the same as the statement "P(X) is true". The meaning of "P(X) is meaningless" is distinct from the (potential absent) meaning of the statement "P(X)" that is being referred to. So while "P(X)" as a statement may play a role in determining the truth value of referring statements such as "P(X) is true" and "P(X) is meaningless", it still does not have meaning in and of itself. Nonsense is nonsense, even when we identify it as such. Logic systems are typically constructed using a simplified metalogic, which is somewhat circular, but unavoidable because we must describe the new language with the only tool we have for describing things: language. I'm pretty familiar with this idea of using logical statements that assign truth values to other logical statements. There is definitely utility in this syntactic/formal approach to logic, since it helps us design languages that are highly regular and analytic in their structure. However, the purpose of a language is not to follow rules, but to describe things, and this applies to logic just as well as any other language. The strict syntactical rules are an additional constraint we place on formal, as opposed to natural, languages. Meaning, which is universal to language and is fundamental to its purpose, is a mapping from language terms onto reality (or at least our internal representation of it), allowing us to describe those things. Without meaning, language is useless, and so I see the arbitrary assignment of a truth value to a meaningless statement as an exercise in futility, unless that truth value is the Meaningless truth value I already described, which serves as a label to tell us not to invest further time trying to assign meaning to it (and not to apply the principle of the excluded middle to it). In natural language, we can take an alternate route, and assign meaning and truth values to individual cases the statement maps to which happen to be consistent. In formal logic, we are not permitted to do this because it violates the regularity constraints we place on logic as a regular/analytic alternative to natural language, and so we throw out the entire statement. The problem I am pointing out is not about whether the statement should be thrown out, but rather *how*. In "standard" logic, this is done by assigning a value of False to the statement. But for a Meaningless statement, this act of assigning False works together with the principle of the excluded middle to produce an error: claiming that the negation of the Meaningless statement is in True, when in fact it is also Meaningless. Most incarnations of logic fail to distinguish between the cases where an un-True statement can be negated to get a True one, and cases where the negation of the un-True statement is also un-True. So either we don't get to use the power of the excluded middle where it is appropriate, or we are at risk of using it where it is inappropriate. On Thu, Nov 8, 2012 at 8:45 AM, Jim Bromer <[email protected]> wrote: > Aaron Hosford <[email protected]> wrote: > 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. > ---------------------------------------------------- > > Well, I wasn't really thinking of pure "Boolean Logic" when I said that > Logic could be pointed at meaninglessness. I was saying that a logical > assessment could be made of situations which could include > meaninglessness. However, that application of logic does not entail or > demand that the domain of X be defined according to some kind of doctrine > as long as the logical values could be computed when the referents of the > domain are defined or the statement could be defined as meaningless or as > being complex. > > The illustration of a paradox exemplifies why logic can be such a valuable > asset when analyzing meaning. So while the assignment of a man who might > be the 'barber who shaves everyone in town who does not shave themself' > might be impossible due to a hidden paradox, or it might simply imply that > the barber does not shave, the sentence is meaningful. The fact that (the > situation can exist where) no X can be said to logically fill the > definition of an element of the situation does not mean that the sentence > is meaningless (unless you are a logical positivist or something.) > > The assignment of the the truth value of a logical statement and the > assignment of the references of the statement are not the same. So you can > have a statement which assigns the truth value of a paradoxical logical > statement. I don't accept the Goedel thing because I no longer believe in > Kantor's diagonalization theory. > > One problem with logic is that when you create logically > compound sentences without using the constructions of the logical > method the statements are not going to be completely logical. Then when > you use those compound sentences in an applied logical statement the hidden > complexities of the sentence are going to interfere with the logical > assessment of the sentences. > > Since the compound sentences that we typically use do contain hidden > complexities they cannot be assigned logical values very easily. To make > them logical we usually have to refine them in some way but this can change > their meaning. > > However, if logical perfection was the goal one could imagine that a > referent (or is it predicate) sentence can refer to a finite number of > logical sitatituations which could then be resolved. So if a natural > sentence is meaningless in the natural way it simply would not have a > logical value but another logical statement which refered to sentence could > designate it as meaningless. Incidentally, the "meaningless" statement > would then have some meaning in the greater context of the discussion. > This seems like it is itself a paradox but its meaning is only relevant to > the fact that it had been assessed as meaningless! The meaning then is > derived from the statement that refers to the meaningless statement. > > Most of care very much about statements which are false. > > Jim Bromer > > > On Wed, Nov 7, 2012 at 1:17 PM, Aaron Hosford <[email protected]> wrote: > >> 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 <[email protected]> wrote: >> >>> On Mon, Nov 5, 2012 at 11:38 PM, Aaron Hosford <[email protected]>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 >>> >> *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
