peru@soleil.UUCP (Dave Peru) (01/04/89)
When you view the meaning of a paradox, your brain is on a razor's edge.
Depending on what side you fall, the paradox is decidedly true or false.
Example: This statement is false.
Infinity is like a paradox. Consider the following number line:
+--+--+--+-->
0 1 2 3
From one point of view, sitting on the number line. You start running from
the origin to the right, 1, 2, 3, ... You go on to infinity. However, from
another point of view, this is always considered ONE number line. A single
defined entity. Defined, yet undefined (unbounded), by definition.
Getting back to paradox, here's a test of your understanding:
Paradox is not this sentence. True or false?shani@TAURUS.BITNET (01/04/89)
In article <551@soleil.UUCP>, peru@soleil.BITNET writes: > > From one point of view, sitting on the number line. You start running from > the origin to the right, 1, 2, 3, ... You go on to infinity. However, from > another point of view, this is always considered ONE number line. A single > defined entity. Defined, yet undefined (unbounded), by definition. > Yes. A great example. You see, the whole point here is, in which logic phase do you state your deffinition. If you have some knoladge in sets theory then you probebly know that the set of all natural numbers is the first infinite ordinal, an thus, it *is* a unit of the next higher ordninal. Infact, every ordinal is the set of all smaller ordinals, and thus, the priveus ordinal is his unit. Now. what does this has to do with paradoxes? because a paradox is meerly an indication for you that you are looking at things from a too low logic phase. you can't look at the natural numbers as a unit, from the point of view of the natural numbers! you have to do what is called a change of a secon degree, and to pass into a higher logical phase. Ofcourse, the only way to do that is to know that you are already there, i.e. to look at the set of natural numbers as a subset of that higher logical phase. O.S.
bwk@mbunix.mitre.org (Barry W. Kort) (01/06/89)
I continue to be stimulated by Dave Peru's roving mind. In article <551@soleil.UUCP> peru@soleil.UUCP (Dave Peru) writes about paradox: > When you view the meaning of a paradox, your brain is on a razor's edge. > Depending on what side you fall, the paradox is decidedly true or false. > Example: This statement is false. When sorting sentences into the two categories, TRUE and FALSE, paradoxes arise with sentences that defy such categorization. As in most paradoxes, this one reveals that two categories are insufficient to classify sentences. A third category is UNDECIDABLE. A fourth category is MEANINGLESS. A fifth category is AMBIGUOUS. > Getting back to paradox, here's a test of your understanding: > > Paradox is not this sentence. True or false? For now, I would categorize Dave's sentence as AMBIGOUS. Does it mean "This sentence is not an example of a paradoxical sentence."? Does it mean "This sentence is not the definition of the category of objects known as 'paradoxical sentences'."? Raymond Smullyan speaks of "meaningless sequences of words". Each word in the sequence may have unambiguous meaning, but the *sequence* may not have clear meaning. --Barry Kort Today's quote: "There are two kinds of people in this world: Those who divide the world into two kinds of people and those who don't."
geb@cadre.dsl.PITTSBURGH.EDU (Gordon E. Banks) (01/08/89)
In article <551@soleil.UUCP> peru@soleil.UUCP (Dave Peru) writes: >When you view the meaning of a paradox, your brain is on a razor's edge. >Depending on what side you fall, the paradox is decidedly true or false. >Example: This statement is false. > On the contrary, when presented with a paradox, one's mind tends to first call it true, then false, then true, then false as one considers it over and over. It is not resolvable. A good analogy is an optical illusion. When one looks at say, the classical optical illusion that can look like a vase or a couple kissing, one usually finds one's interpretation slowly oscillating between the two possibilities. We don't know exactly what is going on at the micro level, but it is likely that the intermediate level brain networks responsible for segmenting images are passing up interpretations which then are conciously perceived and accepted or rejected. It is a very interesting question, but certainly can be duplicated by artificial image processing systems.
bwk@mbunix.mitre.org (Barry W. Kort) (01/11/89)
In article <1975@cadre.dsl.PITTSBURGH.EDU> geb@cadre.dsl.pittsburgh.edu (Gordon E. Banks) enters the fray on Dave's paradoxical sentence: >In article <551@soleil.UUCP> peru@soleil.UUCP (Dave Peru) writes: >>When you view the meaning of a paradox, your brain is on a razor's edge. >>Depending on what side you fall, the paradox is decidedly true or false. >>Example: This statement is false. >On the contrary, when presented with a paradox, one's mind tends to first >call it true, then false, then true, then false as one considers it >over and over. It is not resolvable. I disagree. I suggest that we consider the law of logic that trips us up here: Aristotle's Law of the Excluded Middle. This law says that a sentence must be either True or False. There are no other possibilities. We now know better. A sentence may be formally undecidable. A sentence may be ambiguous, admitting multiple meanings. A sentence may be a meaningless sequence of words, admitting no meaning whatsoever. Now, let us consider the pathological locution, "This sentence is false." If we abandon the Law of the Excluded Middle, we are left with the problem of categorizing the locution in question as one of: 1) True, 2) False, 3) Undecidable, 4) Ambiguous, or 5) Meaningless. We have already tried to categorize the sentence as either True or False, and come to a contradiction in either case. So we discard those two possibilities. The locution is apparently neither Meaningless nor Ambiguous. This leaves only Undecidable, which seems consistent with everthing else we know about the pathological sentence. This example illustrates the main idea of paradoxes: they reveal the incompleteness of our thinking. The paradox is not unresolvable. We resolve it by inventing the previously excluded middle. Interestingly enough, Saul Kripke has developed several new branches of logic (Modal Logic and Intuitionist Logic being the two that seem most interesting here). Like non-Euclidean geometry, new logics arise by throwing away unnecessary restrictions in the Axioms of the formal system. Not all thinking is deductive. Some of the most fascinating thinking is creative. Cantor and Conway created transfinite numbers in different ways. What new possibilities can you imagine if you throw off the yoke of unnecessarily restrictive rules? --Barry Kort and unambiguous.
bph@buengc.BU.EDU (Blair P. Houghton) (01/13/89)
In article <43519@linus.UUCP> bwk@mbunix (Barry Kort) writes: >In article <1975@cadre.dsl.PITTSBURGH.EDU> geb@cadre.dsl.pittsburgh.edu >(Gordon E. Banks) enters the fray on Dave's paradoxical sentence: > >>In article <551@soleil.UUCP> peru@soleil.UUCP (Dave Peru) writes: > >>>When you view the meaning of a paradox, your brain is on a razor's edge. >>>Depending on what side you fall, the paradox is decidedly true or false. >>>Example: This statement is false. > >>On the contrary, when presented with a paradox, one's mind tends to first >>call it true, then false, then true, then false as one considers it >>over and over. It is not resolvable. > >I disagree. I suggest that we consider the law of logic that trips >us up here: Aristotle's Law of the Excluded Middle. This law says >that a sentence must be either True or False. There are no other >possibilities. We now know better. A sentence may be formally >undecidable. A sentence may be ambiguous, admitting multiple meanings. >A sentence may be a meaningless sequence of words, admitting no meaning >whatsoever. Hurm. Astirottle rears his ponderous head. He'd never read Rudy Carnap's eminently unreadable "The Logical Syntax of Language." All sentences (ansatzen) can be reduced logically to their syntax; semantics are irrelevant to logic. Paradox is no different. "I am not me" and "this ansatz is untrue" both reduce to _ S => S or, actually _ S <=> S Such things are essential in developing the contextual meaning of words. I.e., while syntax (logic) is not dependent on semantics, semantics is dependent on syntax. In other (invisible) words: Sy => Se Hand me a paradox of any sort, give me twelve years to finish Carnap's book, and I shall decompose your paradox into it's syntactic contradictions. It will have no semantic ones. --Blair __ "2B + 2B, es ist die frage."
peru@soleil.UUCP (Dave Peru) (02/02/89)
From the original posting: >When you view the meaning of a paradox, your brain is on a razor's edge. >Depending on what side you fall, the paradox is decidedly true or false. >Example: This statement is false. > > [bunch of stuff deleted] > >Paradox is not this sentence. True or false? The meaning of the sentence "Paradox is not this sentence" I think is kind of neat. Consider the sentence "This statement is false", which I think is the same in meaning as the following two sentences: The next sentence is false. The previous sentence is true. Anyway, the meaning of the sentence "This statement is false" creates something we call a paradox. The sentence "Paradox is not this sentence" I think goes one step further. In the first case, it's like a snake eating its own tail. In the second case, it's like the snake had finished eating.
gary@cgdra.ucar.edu (Gary Strand) (02/03/89)
Article <583@soleil.UUCP> peru@soleil.UUCP (Dave Peru) says : >When you view the meaning of a paradox, your brain is on a razor's edge. >Depending on what side you fall, the paradox is decidedly true or false. >Example: This statement is false. What I think is happening is that people are assuming that a given English sentence must have some kind of logical truth/falsehood to it, merely because we can state it as a sentence. I can think of literally thousands of perfectly good sentences that are in fact total nonsense, to wit: "Bananas are elephants." "All good men are Buicks." "Truth is defined to be that which is sugar." "For something to be false means that it is wavy like a reed in a gale." All these sentences are perfectly good from a purely syntactical viewpoint, ie they are gramatically correct, but that says nothing about whether or not they actually MEAN anything. I think this also applies to such things as: "This sentence is paradox." "The following sentence is false." "The previous sentence is true." (or whatever the doublet is) My point is that English allows the generation of thousands of sentences that need not have any meaning. Thus, there is more to a 'correct' sentence than just following grammatical rules. Does thus make sense? =============================================================================== Great spirits have always encountered | First, it was Nuclear Winter. Then it violent opposition from mediocre minds. | became the Ozone Hole. The beast has - Albert Einstein | risen again -- The Greenhouse Effect! ------------------------------------------------------------------------------- Gary Strand Eigennutz geht vor Gemeinnutz (303) 497-1398
dave@cogsci.indiana.edu (David Chalmers) (02/03/89)
In article <1361@ncar.ucar.edu> gary@cgdra.ucar.edu (Gary Strand) writes: > What I think is happening is that people are assuming that a given English >sentence must have some kind of logical truth/falsehood to it, merely because >we can state it as a sentence. > > I can think of literally thousands of perfectly good sentences that are in >fact total nonsense, to wit: > > "Bananas are elephants." > "All good men are Buicks." > "Truth is defined to be that which is sugar." > "For something to be false means that it is wavy like a reed in a gale." > > All these sentences are perfectly good from a purely syntactical viewpoint, >ie they are gramatically correct, but that says nothing about whether or not >they actually MEAN anything. Sure these sentences mean something. They're quite coherent to me. The only trouble with them is that they're FALSE. This is a far cry from being meaningless. > I think this also applies to such things as: > > "This sentence is paradox." > > "The following sentence is false." > "The previous sentence is true." (or whatever the doublet is) Now these sentences (at least the last pair, anyway) are different. The last pair cannot be assigned truth-values (not even false ones), which means that something different is going on. This leads a lot of people to the conclusion that they're meaningless. For a lot of people, the reason they draw this conclusion is that (for them) meaning is DEFINED in terms of truth-value. Personally, I think this is putting the cart before the horse. Truth-value or no truth-value, these sentences are meaningless because they are CONTENT-FREE. They say nothing about the world; they say nothing interesting at all about anything but their own truth-value. And as truth-values have to be ultimately grounded in reality, this is equivalent to saying nothing at all. Someone might say to me: 'in Godel's theorem, doesn't he construct some silly sentence equivalent to "This statement is false", and use it to draw powerful conclusions about mathematics? So how can you say that this sentence is meaningless?'. But the beauty of Godel's construction is that at the SAME TIME as the sentence is talking about itself, it is also talking about a complex mathematical proposition (because the statement can be interpreted on two levels). So this seemingly "meaningless" sentence is in fact grounded in hard reality. So that's the lesson: meaning comes first, and truth-value second. And I should set one thing straight. There is absolutely no paradox about the statement "This sentence is paradox." The statement is simply false. Dave Chalmers (dave@cogsci.indiana.edu) Center for Research on Concepts and Cognition Indiana University