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."
lee@uhccux.uhcc.hawaii.edu (Greg Lee) (01/14/89)
From article <1883@buengc.BU.EDU>, by bph@buengc.BU.EDU (Blair P. Houghton): " ... " 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. Hurm. Where does Carnap say that?? " ... " S <=> S " " Such things are essential in developing the contextual meaning of words. Do you mean that when people say things that appear paradoxical, they are really inviting you to find an interpretation of their words that relieves the paradox? If so, I agree. Greg, lee@uhccux.uhcc.hawaii.edu
bph@buengc.BU.EDU (Blair P. Houghton) (01/15/89)
In article <2996@uhccux.uhcc.hawaii.edu> lee@uhccux.uhcc.hawaii.edu (Greg Lee) writes: >From article <1883@buengc.BU.EDU>, by bph@buengc.BU.EDU (Blair P. Houghton): >" ... >" 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. > >Hurm. Where does Carnap say that?? It wasn't a quotation. I pretty much remember that somewhere in there was a statement to the effect that "the logic of languange is in its syntax" and only in its syntax. --Blair
mirk@warwick.UUCP (Mike Taylor) (01/16/89)
Two people have individually stated in this debate that there are 5
truth values, these being True, False, Ambiguous, Undecidable and
Meaningless. It is intuitively clear to most people what is meant by
a sentence being either true or false. The latter three categories
are not so clear.
Am I right in thinking that an ambiguous sentence is one that can
consistently be either true or false and that an undecidable one
cannot be either? What, then is a meaningless sentence? Surely
either an ambiguous sentence or an inconsistent one is meaningless?
Or do people simply mean not well-formed?
To clarify what I mean above by my understanding of the terms
ambiguous and undecidable, I suspect that the sentence "This sentence
is true" is ambiguous, since it can consistently be either true or
false, and "This sentence is false" is undecidable, since it cannot be
either. Both sentences seem to me to be meaningless.
If this is correct, then Dave Peru's original assertion, that the mind
teeters between considering a paradox to be true or false, is correct
of ambiguous sentences, but not of undecidable ones - there is an
analogy here with unstable and unstable equilibrium points of a
physical system.
______________________________________________________________________________
Mike Taylor - {Christ,M{athemat,us}ic}ian ... Email to: mirk@uk.ac.warwick.cs
*** Unkle Mirk sez: "Em9 A7 Em9 A7 Em9 A7 Em9 A7 Cmaj7 Bm7 Am7 G Gdim7 Am" ***
------------------------------------------------------------------------------wsinkees@eutrc3.UUCP (Kees Huizing) (01/17/89)
From article <905@ubu.warwick.UUCP>, by mirk@warwick.UUCP (Mike Taylor): > If this is correct, then Dave Peru's original assertion, that the mind > teeters between considering a paradox to be true or false, is correct > of ambiguous sentences, but not of undecidable ones - there is an > analogy here with unstable and unstable equilibrium points of a > physical system. In my view, the analogon of an ambiguous sentence is an *indifferent* equilibrium. Kees Huizing Eindhoven Univ of Techn, Dept. of Math & Comp. Sc., Pb 513, 5600 MB Eindhoven email: mcvax!eutrc3!wsinkees.UUCP or wsdckees@heitue5.BITNET
smoliar@vaxa.isi.edu (Stephen Smoliar) (01/17/89)
In article <905@ubu.warwick.UUCP> mirk@uk.ac.warwick.cs (Mike Taylor) writes: >Two people have individually stated in this debate that there are 5 >truth values, these being True, False, Ambiguous, Undecidable and >Meaningless. It is intuitively clear to most people what is meant by >a sentence being either true or false. The latter three categories >are not so clear. > >Am I right in thinking that an ambiguous sentence is one that can >consistently be either true or false and that an undecidable one >cannot be either? What, then is a meaningless sentence? Surely >either an ambiguous sentence or an inconsistent one is meaningless? >Or do people simply mean not well-formed? > >To clarify what I mean above by my understanding of the terms >ambiguous and undecidable, I suspect that the sentence "This sentence >is true" is ambiguous, since it can consistently be either true or >false, and "This sentence is false" is undecidable, since it cannot be >either. Both sentences seem to me to be meaningless. > This has been giving me trouble, too. Another possible interpretation is that "meaningless" refers to sentences that are not well-formed; but then, of course, they are not really sentences, in which case it does not make sense to talk of their having truth values.
bph@buengc.BU.EDU (Blair P. Houghton) (01/18/89)
In article <905@ubu.warwick.UUCP> mirk@uk.ac.warwick.cs (Mike Taylor) writes: >Two people have individually stated in this debate that there are 5 >truth values, these being True, False, Ambiguous, Undecidable and >Meaningless. Wounds more like a communication problem, to me. Like my ol' anthro professor usedtasay: There are three levels of truth: 1. The Truth; 2. What People Believe is the Truth; 3. What People Tell You They Believe is the Truth. --Blair "...and whatever my urinalysis says..."
rjc@aipna.ed.ac.uk (Richard Caley) (01/20/89)
In article <905@ubu.warwick.UUCP> mirk@uk.ac.warwick.cs (Mike Taylor) writes: >Am I right in thinking that an ambiguous sentence is one that can >consistently be either true or false and that an undecidable one >cannot be either? What, then is a meaningless sentence? Surely >either an ambiguous sentence or an inconsistent one is meaningless? >Or do people simply mean not well-formed? A classic example of a meaningless sentence is "The current king of France is bald." ( Frege, I believe ) which is neither true nor false, since there is no king of France currently. -- rjc@uk.ac.ed.aipna AKA rjc%uk.ac.ed.aipna@nss.cs.ucl.ac.uk "Give me a beer and money sandwich: hold the bread" - Waldo 'DR' Dobbs
aam9n@uvaee.ee.virginia.EDU (Ali Minai) (01/20/89)
In article <7282@venera.isi.edu>, smoliar@vaxa.isi.edu (Stephen Smoliar) writes: > In article <905@ubu.warwick.UUCP> mirk@uk.ac.warwick.cs (Mike Taylor) writes: > > ........What, then is a meaningless sentence? Surely > >either an ambiguous sentence or an inconsistent one is meaningless? > >Or do people simply mean not well-formed? > > > This has been giving me trouble, too. Another possible interpretation is that > "meaningless" refers to sentences that are not well-formed; but then, of > course, they are not really sentences, in which case it does not make sense > to talk of their having truth values. At the risk of being called naive, let me suggest that meaningless sentences are those that obey all rules of syntax, but do not make sense in the context of mundane experience. "Mundane experience", of course, must be defined in subjective terms. For example, the sentence: The colour of her fear was a brilliant orange. would be considered meaningless by the vast majority of people in most contexts, though it is certainly well-formed (compare, "The colour of her skirt was a brilliant orange"). However, in a poem, or indeed, in any piece of literary writing, this sentence *could* make a great deal of sense. That this definition of meaninglessness requires a context should not put us off. After all, ambiguous and undecidable sentences too are so only in the context of the logic defined by us. One could convert an undecidable sentence into a decidable one by changing the nature of the logical substrate which provides the context, e.g. by allowing contradiction. That we do not is again because of mundane experience. Thus, logic is merely a subset (in a loose sense) of this experience. Sentences which are inconsistent within this logic are ambiguous or undecidable (since logic forces us to think in combinations of 'true' and 'false'). Sentences which are outside the domain of logic, and contradict perceptual/linguistic convention, can be considered meaningless. Meaninglessness cannot be defined in terms of "truth" and "falsehood", hence the problem. The statement (above) that "... either an ambiguous or an inconsistent sentence is meaningless", is, in my opinion, based on a very narrow and rigid notion of "meaning". As someone pointed out, some paradoxes can be seen as having two co-existent meanings, only one of which can be valid at any specific time---rather like a bistable system. Another issue here is that most paradox arises aout of a confusion between the language of an assertion and the meta-language by which we judge it. For example, the paradox in, "This sentence is false" goes away as soon as we distinguish between the two *levels* of meaning that---together--- generate it. I hope this makes sense. But then, what does! Ciao, Ali ----------------------------------------------------------------------- The opposite of one great truth is another great truth. Niels Bohr. -----------------------------------------------------------------------
rwallace@vax1.tcd.ie (01/21/89)
In article <7282@venera.isi.edu>, smoliar@vaxa.isi.edu (Stephen Smoliar) writes: >>To clarify what I mean above by my understanding of the terms >>ambiguous and undecidable, I suspect that the sentence "This sentence >>is true" is ambiguous, since it can consistently be either true or >>false, and "This sentence is false" is undecidable, since it cannot be >>either. Both sentences seem to me to be meaningless. >> > This has been giving me trouble, too. Another possible interpretation is that > "meaningless" refers to sentences that are not well-formed; but then, of > course, they are not really sentences, in which case it does not make sense > to talk of their having truth values. I don't really see what the problem is about paradoxes like "This statement is false". Many formal systems contain expressions which cannot be evaluated within that system. Mathematics contains things like 1/0. Logic contains things like "This statement is false". Although a logical statement, it has no logical solution. So what? Why get worried about it? "To summarize the summary of the summary: people are a problem" Russell Wallace, Trinity College, Dublin rwallace@vax1.tcd.ie
bwk@mbunix.mitre.org (Barry W. Kort) (01/21/89)
In article <7282@venera.isi.edu> smoliar@vaxa.isi.edu.UUCP (Stephen Smoliar) writes about the distinction between "meaningless sentences" and "non-well-formed sentences": > This has been giving me trouble, too. Another possible interpretation > is that "meaningless" refers to sentences that are not well-formed; > but then, of course, they are not really sentences, in which case it > does not make sense to talk of their having truth values. I guess the semanticists can have a field day with this one. I tried to avoid the problem by referring to "locutions" as candidates for classification into the categories True, False, Undecided, Ambiguous, and Meaningless. I have no problem defining "sentence" to exclude meaningless sequences of words. But we still need a category in which to toss such junk. Ordinarily, a locution (would-be sentence) is granted the dignity of sentencehood (innocent until proven guilty). But such civility does not preclude the necessity of dealing with the occasional utterance of nonsense. --Barry Kort
hh5s@lucifer.acc.virginia.edu (Heiko Hecht) (01/22/89)
It seems to me that the question whether we need more than two truth-values
(true - false) depends on the extent to wich we want to make logic
paradox-proof. To remove paradoxes we basically have three choices:
1. We declare logic as not applicable to certain sentences:
e.g. "The king of France is bald" because it has no empirical reference,
"This sentence is false" because it is self-referencing,
"All people are liars" because it includes the person who writes it.
2. We introduce "new" truth values like undecidable or meaningless. The
question is whether this is a good idea, because it becomes very fuzzy.
My favorite is "uncomfortable":
e.g. "You are stupid" is not true, then again it may not be false and
it is probably undecidable even though it may be very meaningful.
3. Meta-logic to the rescue! If 1. and 2. don't work, we can always try to
claim that the sentence in question is actually (or includes) a
meta-logic sentence that refers to its own truth/falseness.
But what if choices 1. to 3. don't seem to work, does anyone have suggestions
as to how to resolve the following paradox:
"The following sentence is true"
"The preceeding sentence is false" ?
Heiko Hecht (hh5s@virgina.edu) "Say yes to paradoxes"lee@uhccux.uhcc.hawaii.edu (Greg Lee) (01/22/89)
From article <479@aipna.ed.ac.uk>, by rjc@aipna.ed.ac.uk (Richard Caley): " ... " "The current king of France is bald." " " ( Frege, I believe ) which is neither true nor false, since there is no " king of France currently. Russell, but you're following the Strawson line. Russell said it was false. I agree with Russell. 'Meaningless' is a loaded term. There's no telling its intended reference until you know what theory is being pushed. (This should not be taken to imply that it is different from other terms.) Greg, lee@uhccux.uhcc.hawaii.edu
fsjl@pnet12.cts.com (Fragano Ledgister) (01/24/89)
lee@uhccux.uhcc.hawaii.edu (Greg Lee) writes: >From article <479@aipna.ed.ac.uk>, by rjc@aipna.ed.ac.uk (Richard Caley): >" ... >" "The current king of France is bald." >" >" ( Frege, I believe ) which is neither true nor false, since there is no >" king of France currently. > >Russell, but you're following the Strawson line. Russell said it was >false. I agree with Russell. > >'Meaningless' is a loaded term. There's no telling its intended >reference until you know what theory is being pushed. (This should >not be taken to imply that it is different from other terms.) > > Greg, lee@uhccux.uhcc.hawaii.edu Back in the days when I was an undergraduate studying philosophy (among other things), my teacher used this example when teaching Russell's epistemology. He then went on to say that when he was a graduate student at the Sorbonne and this phrase came up he argued that it was true. De Gaulle, he said, was the king of France, and de Gaulle was bald -- hence the 'present king of France' was bald. Since Mitterand is also bald (and since I've been hearing descrip- tions of him as 'a de Gaulle-like figure'), perhaps we might consider it to be true today. UUCP: uunet!serene!pnet12!fsjl ARPA: crash!pnet12!fsjl@nosc.mil INET: fsjl@pnet12.cts.com
kerber@uklirb.UUCP (Manfred Kerber) (01/25/89)
Heiko Hecht writes: >> But what if choices 1. to 3. don't seem to work, does anyone have suggestions >> as to how to resolve the following paradox: >> >> "The following sentence is true" >> "The preceeding sentence is false" ? This can be excluded by Russell's "Theory of Types" as described in "Principia Mathematica" or in the American Journal of Mathematics p.222 ff, Vol.XXX, 1908. In order to avoid paradoxies Russell introduces a strict hierarchy of types. The first sentence of the above example is of type "sentence about sentence". Then the second must be of type "sentence". On the other hand in order to make a statement about the first, the second must be of type "sentence about sentence about sentence", both is impossible. So such a self-reference, direct or indirect, is excluded. Manfred Kerber
bwk@mbunix.mitre.org (Barry W. Kort) (01/25/89)
Let's play with the assertion, "The current king of France is bald." If we put this into symbolic logic notation, we get For all x, if x is the current king of France, then x is bald. Or in slightly more melifluous English, Every person who happens to be the current king of France also happens to be bald. Now in Aristotelian Logic, the above is equivalent to the denial of its negative: It is not the case that there is a non-bald individual who is the current king of France. Or again, in clearer English, There is no one who is both non-bald and the current king of France. I think we would all agree that the above statement is a meaningful and accurate description of the French state of affairs. That is, the assertion is True. So if you believe in the Law of the Excluded Middle, the original assertion and the denial of its negation have equivalent truth values (namely True). If this line of reasoning leaves you uneasy, consider the possibility of throwing away the Law of the Excluded Middle. Then we can no longer transform an assertion into the denial of its negation, and we can pleasantly argue over the category into which the "current bald king of France" can be tossed. --Barry Kort
engelson@cs.yale.edu (Sean Philip Engelson) (01/26/89)
>Heiko Hecht writes: >>> But what if choices 1. to 3. don't seem to work, does anyone have suggestions >>> as to how to resolve the following paradox: >>> >>> "The following sentence is true" >>> "The preceeding sentence is false" ? > In article <3715@uklirb.UUCP>, Manfred Kerber resolves the paradox by introducing Russell's hierarchy of types, saying that the first is of types "Sentence about Sentence", thus the second must thus be of type "Sentence", but it's also "S about S". However, if you allow infinite types, the paradox remains, as both sentences can be of type T, where T is defined as the fixed point of T', as follows: T' = "Sentence" | "Sentence about T'" Each sentence is of type T and can thus refer to the other. Is there any a priori reason to exclude infinite types? -Sean- ---------------------------------------------------------------------- Sean Philip Engelson, Gradual Student Who is he that desires life, Yale Department of Computer Science Wishing many happy days? Box 2158 Yale Station Curb your tongue from evil, New Haven, CT 06520 And your lips from speaking (203) 432-0677 falsehood. ---------------------------------------------------------------------- I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones. -- Albert Einstein
bph@buengc.BU.EDU (Blair P. Houghton) (01/27/89)
In article <3715@uklirb.UUCP> kerber@uklirb.UUCP (Manfred Kerber) writes: >Heiko Hecht writes: >>> But what if choices 1. to 3. don't seem to work, does anyone have suggestions >>> as to how to resolve the following paradox: >>> >>> "The following sentence is true" >>> "The preceeding sentence is false" ? > >This can be excluded by Russell's "Theory of Types" as described in "Principia >Mathematica" or in the American Journal of Mathematics p.222 ff, Vol.XXX, 1908. >In order to avoid paradoxies Russell introduces a strict hierarchy of types. >The first sentence of the above example is of type "sentence about sentence". >Then the second must be of type "sentence". On the other hand in order to make >a statement about the first, the second must be of type "sentence about >sentence about sentence", both is impossible. So such a self-reference, >direct or indirect, is excluded. How bout a few symbols? 1. S1 ==> S2 __ 2. S2 ==> S1 note the precise notation. "==>" says that if S1 is true, then S2 is true. It says nothing about what happens if S1 is false. Thus, since S2 claims S1 is false, then we know nothing of the validity of S2. It's not a paradox, it's incomplete. --Blair "Any ideas? Any at all. About anything. C'mon, just a _little_ thought. Don't be such a politician. C'mon, you can do it..."
diana@coracle.cis.ohio-state.edu (Diana Smetters) (01/27/89)
In article <43843@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes:
< Let's play with the assertion,
<
< "The current king of France is bald."
<
< If we put this into symbolic logic notation, we get
<
< For all x, if x is the current king of France, then x is bald.
<
I think that some of the controversy may stem from the alternate translation,
where:
"The current king of France is bald."
is rendered into logic as:
"There exists an x, such that x is the current king of France and x is bald."
due to a different theoretical analysis of definite descriptions. This
statement is false, without causing any problems with the law of the
excluded middle.
--Diana Smettersap1i+@andrew.cmu.edu (Andrew C. Plotkin) (01/27/89)
/>> "The following sentence is true" />> "The preceeding sentence is false" ? / / In order to avoid paradoxies Russell introduces a strict hierarchy of types. / The first sentence of the above example is of type "sentence about sentence". / Then the second must be of type "sentence". On the other hand in order to make / a statement about the first, the second must be of type "sentence about sentence / about sentence", both is impossible. So such a self-reference, direct or / indirect, is excluded. I'm not positive about this, but it was my understanding that that produces a weaker system. Godel showed that a mathematical system *can* talk about the truth (at least, about provability within itself) without self-contradiction. That is, the above sentences can be allowed, but are given the status "unprovable". Now we humans think we can do better than that; we keep saying that we can define truth without that sort of cop-out, which is why we get so tangled by these paradoxes... --Z
jrk@s1.sys.uea.ac.uk (Richard Kennaway CMP RA) (01/28/89)
In article <48717@yale-celray.yale.UUCP> engelson@cs.yale.edu (Sean Philip Engelson) writes: [refers to the paradox: "The following sentence is true"/"The preceding sentence is false"] >In article <3715@uklirb.UUCP>, Manfred Kerber resolves the paradox by >introducing Russell's hierarchy of types, saying that the first is of >types "Sentence about Sentence", thus the second must thus be of type >"Sentence", but it's also "S about S". However, if you allow infinite >types, the paradox remains, as both sentences can be of type T, where >T is defined as the fixed point of T', as follows: > T' = "Sentence" | "Sentence about T'" >Each sentence is of type T and can thus refer to the other. Is there >any a priori reason to exclude infinite types? You said it yourself: "the paradox remains". (Well, an a posteriori reason, I suppose...) Ever since Russell showed that the system of logic Frege had just published was self-contradictory, people have been trying to find ways of avoiding the paradoxes of self-reference. However, all the solutions so far seem (IMHO) pretty ad hoc. -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. uucp: ...mcvax!ukc!uea-sys!jrk Janet: kennaway@uk.ac.uea.sys
jrk@s1.sys.uea.ac.uk (Richard Kennaway CMP RA) (01/28/89)
In article <43843@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes: >Let's play with the assertion, > > "The current king of France is bald." > >If we put this into symbolic logic notation, we get > > For all x, if x is the current king of France, then x is bald. [further discussion deleted] I would part company with you here. Your rephrasing in logic doesnt seem to me to mean the same. If someone unaware of the French system of government asked you "Is the current king of France bald?" would you reply "No"? I would reply "France is not a monarchy". The statement "The current king of France is bald" makes a false presupposition, and if I were trying to formalise the English language, I would not want to assign any truth-value to that sentence. -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. uucp: ...mcvax!ukc!uea-sys!jrk Janet: kennaway@uk.ac.uea.sys
weltyc@cs.rpi.edu (Christopher A. Welty) (01/28/89)
>In article <43843@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes: > Let's play with the assertion, > > "The current king of France is bald." > > If we put this into symbolic logic notation, we get > > For all x, if x is the current king of France, then x is bald. > And this statement, according to first order logic, is equivalent to : For all x, x is not the current king of france or x is bald Which is true (when the or is understood not to be exclusive). In article <32698@tut.cis.ohio-state.edu> Diana Smetters <diana@cis.ohio-state.edu> writes:> > > "There exists an x, such that x is the current king of France and x is bald." > [which is false] Actually I think this the correct (logic) interpretation of the statement. The problem is that the statement implies indirectly that there exists a current king of france, in fact the statemnet assumes this. When this assumption is shown to be false, the statement becomes false (according to logic). But this is missing the point. Two-valued logics can not deal with the information in this sentence. The truthness or falsness is not the point, it's the meaninglessness. The english statement does not say `there is a current king of france', nor does it say `all people are either the current king of france or bald'. The point is that we, as humans, are able to form sentences like this and understand that there is no `truth value' associated with it - it's meaningless. of course, if you had a many-valued logic you might be able to have values for things like meaningless (see the first posting on this subject). Christopher Welty --- Asst. Director, RPI CS Labs weltyc@cs.rpi.edu ...!njin!nyser!weltyc
jrk@s1.sys.uea.ac.uk (Richard Kennaway CMP RA) (01/28/89)
In article <416@s1.sys.uea.ac.uk> jrk@uea-sys.UUCP (me) writes: ... >you reply "No"? ... Sorry, I meant "Yes". -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. uucp: ...mcvax!ukc!uea-sys!jrk Janet: kennaway@uk.ac.uea.sys
sp299-ad@violet.berkeley.edu (Celso Alvarez) (01/28/89)
From an absolutely non-technical, lay perspective, here are some ideas about the truth value of "The current king of France is bald". My intuition is that the logic paraphrasis of the sentence is >> "There exists an x, such that x is the current king of France and x is >> bald." [which is false] (Diana Smetters <diana@cis.ohio-state.edu> in article <32698@tut.cis.ohio-state.edu>). Compare: "The king of France [Louis XIV] was bald" is true; but both "The current king of France is bald", and "The current king of France WAS (used to be) bald" contain a linguistic presupposition (rather than an assumption) that a king exists in France that is 'current'. Since this is false, the falsehood of the entire proposition lies on the fact of the non- existence of a king in France, not on whether he is or used to be bald (verb tense makes no difference). (In fact, aren't we dealing with several, embedded logical propositions?). And since the statement "The current king of France" (uttered while pointing at a portrait or photograph) is false in itself, why claim that the whole sentence is "meaningless"? (cf. Christopher A. Welty (weltyc@cs.rpi.edu) in article <374@rpi.edu>): CW>The point is that we, as humans, are able to form sentences like this and CW>understand that there is no `truth value' associated with it - it's CW>meaningless. In fact, I have the feeling that people understand that the whole statement is false (fallacious) precisely because there is no king in France, that is, precisely due to powerful mechanisms of linguistic pressuposition. While the statement does not *assert* that there is a current king of France, it *presupposes* it linguistically, in very much the same way as the statement "The sun shines" presupposes `The sun is' (for clarity, I hope, compare the meaningless of The current kangkt of France is bald , where the presupposed proposition `The kangkt is' cannot be true nor false. Or not?). Celso Alvarez sp299-ad@violet.berkeley.edu
lee@uhccux.uhcc.hawaii.edu (Greg Lee) (01/29/89)
From article <19625@agate.BERKELEY.EDU>, by sp299-ad@violet.berkeley.edu (Celso Alvarez): " ... (In fact, aren't we dealing with several, embedded " logical propositions?). Yes, and together with the fact that only a non-embedded clause can be straightforwardly denied, I think that's all there is to it. If in a conversation the problem sentence is stated, and one replies 'No', this would be taken as denying that the guy is bald. That's because 'is bald' is predicate of the matrix sentence -- it has to do with the *syntactic* relation between assertion and response. If one wishes to deny that there is a king of France, which is also part of what was stated, a simple 'No' won't do. Although we haven't discussed here the definiteness of the description 'the king of France', the above is roughly along the lines of Russell's analysis. I see no warrant here for distinguishing between assertion and presupposition if this means any more than matrix versus embedded. I think the intuition that the sentence is meaningless in the event there is no king of France is founded on a confusion between a sentence's being false, on the one hand, and the syntactic resources offered by natural language for conveying that it is false, on the other. Mind the syntax, and the semantics will take care of itself. Greg, lee@uhccux.uhcc.hawaii.edu
bwk@mbunix.mitre.org (Barry W. Kort) (01/29/89)
In an article long, long ago, someone asked how to resolve the following paradox: > "The following sentence is true." > "The preceeding sentence is false." If one adopts Aristotle's Law of the Excluded Middle, then one has that the above pair of sentences is mutually inconsistent. But remember our discussion about Intuitionist Logic, where we threw away the Law of the Excluded Middle, and invented a few middle possibilities besides True and False. Consider, if you will, the following pair of sentences: "The following sentence is provable." "The preceding sentence is unprovable." The paradox seems to have vanished. The first statement can be both True and Unprovable. The second sentence can be both True and Provable. (But please don't ask me to supply the proof. I didn't say they were provable by *me*!) The point is twofold: Not all True sentences are provable and not all unprovable sentences are False. Thus we need a third category: Undecidable. We can then resolve the paradox by chastising both sentences for overstating the case. They could have gotten along very nicely if they had scaled back their dogmatic assertions along the lines of the second, more harmonious pair. --Barry Kort
bwk@mbunix.mitre.org (Barry W. Kort) (01/29/89)
In article <43843@linus.UUCP>, I took the assertion, "The current king of France is bald." and put it into symbolic logic notation to get For all x, if x is the current king of France, then x is bald. In article <32698@tut.cis.ohio-state.edu> Diana Smetters <diana@cis.ohio-state.edu> writes: > I think that some of the controversy may stem from the > alternate translation, where: > > "The current king of France is bald." > > is rendered into logic as: > > "There exists an x, such that x is the current king of France > and x is bald." > > due to a different theoretical analysis of definite descriptions. This > statement is false, without causing any problems with the law of the > excluded middle. Ain't it amazing how English is so ambiguous, that even educated people can reasonably disagree on the semantics of a simple word like "the"? --Barry Kort
kerber@uklirb.UUCP (Manfred Kerber) (01/30/89)
In answer to Blair P. Houghton: >>>> >>>> "The following sentence is true" >>>> "The preceeding sentence is false" ? >> >How bout a few symbols? > >1. S1 ==> S2 > __ >2. S2 ==> S1 No, this notion is not sufficient, because one has also 1.' NOT S1 ==> NOT S2, analogous for 2. So in both cases one has a ``<==>'' instead of a ``==>''. One gets S1 <==> NOT S1. It is a real paradox. In answer to Sean Philip Engelson: >In article <3715@uklirb.UUCP>, Manfred Kerber resolves the paradox by >introducing Russell's hierarchy of types, saying that the first is of >types "Sentence about Sentence", thus the second must thus be of type >"Sentence", but it's also "S about S". However, if you allow infinite >types, the paradox remains, as both sentences can be of type T, where > >T is defined as the fixed point of T', as follows: > T' = "Sentence" | "Sentence about T'" >Each sentence is of type T and can thus refer to the other. Is there >any a priori reason to exclude infinite types? No, there is no a priori reason to exclude infinite types, but the paradoxies of Russell can be a reason to avoid self-refential assertions. One method is to use a strict hierarchy of types. This does not exclude to use infinitely many types, but your fix point construction is not okay. I see no problem to use types associated to each ``ordinal number'' inclusive omega ordinals, as e.g. by: T(1) = "Sentence", T(i) = "Sentence about Sentence of Type T(k) with k<i" So one can have types: T(1),T(2),T(3),...,T(omega),T(omega+1),T(omega+2),... and so on, where omega is the ordinal associated to N, the set of natural number. Indeed one gets a weaker system as Andrew C.Plotkin remarks. An assertion as "This sentence is true" is excluded without being paradox. Manfred Kerber
weltyc@cs.rpi.edu (Christopher A. Welty) (01/31/89)
In article <19625@agate.BERKELEY.EDU> sp299-ad@violet.berkeley.edu (Celso Alvarez) writes: > [...] since the statement > > "The current king of France" > >(uttered while pointing at a portrait or photograph) is false in itself, >why claim that the whole sentence is "meaningless"? You are adding here a context to the statement, implying that the more formal meaning of the statement is `This is the current king of France'. But that, I would claim, is a COMPLETELY different statement than just "The current king of France", because this statement is NOT false. It is based upon the presupposition (which, as you point out, is the correct term) that there IS a current king of France. But the statement itself DOES NOT make this claim. I claim that the understanding of this statement requires a notion beyond that of truth or falsehood. Christopher Welty --- Asst. Director, RPI CS Labs weltyc@cs.rpi.edu ...!njin!nyser!weltyc
bwk@mbunix.mitre.org (Barry W. Kort) (01/31/89)
In article <416@s1.sys.uea.ac.uk> jrk@uea-sys.UUCP (Richard Kennaway) takes exception to my translation of "The current king of France is bald." into symbolic logic notation Richard writes: > I would part company with you here. Your rephrasing in logic doesnt > seem to me to mean the same. I am glad you see it that way, Richard, for that was the point I was attempting to make. The conventional symbolic logic notation does not have the expressive power to handle pathological sentences such as the cited example. There are more recent advances in symbolic logic which are worth investigating if one wishes to decide how to dispose of the current king of France. In addition to answering "True" or "False", a modern thinker is permitted to respond with "Huh?" or "Mu" among other possibilities. --Barry Kort
sp299-ad@violet.berkeley.edu (Celso Alvarez) (02/01/89)
In article <429@rpi.edu> weltyc@cs.rpi.edu (Christopher A. Welty) writes: CAW> In article <19625@agate.BERKELEY.EDU> sp299-ad@violet.berkeley.edu CAW> (Celso Alvarez) writes: CA> the statement "The current king of France" (uttered while pointing at a CA> portrait or photograph) is false in itself, CAW> You are adding here a context to the statement, implying that the more CAW> formal meaning of the statement is `This is the current king of CAW> France'. But that, I would claim, is a COMPLETELY different statement CAW> than just "The current king of France", because this statement is NOT CAW> false. If for you "The current king of France" is NOT false (your emphasis), then it is true. Its meaning is `There is a king in France at present'. But "The current king of France is bald" must be false, because, while there happens to be a king, there happens to be no bald person who is the king of France. Without adding a (temporal and spatial) context, however, "The current king of France" is true. It just depends on when the statement was uttered. My point is, the context affects the truth-value of "The current king of France" as much as it affects that of "The current king of France is bald". CAW> I claim that the understanding of this statement requires a notion CAW> beyond that of truth or falsehood. I don't see any basic difference between "The current king of France" and "There is a king in France at present" in terms of propositional content. The problem is that truth/falsehood is linguistically built in presuppositions. The typical example of linguistic presupposition (profusely employed by lawyers and other experts of interrogation) runs along the lines of "Did John stop drinking heavily?"; the fact that John didn't use to drink heavily cannot be denied just with a simple "yes". I prefer to see the truth-value of utterances in terms of native (e.g. speakers') categories. If the categories employed in philosophy of language or logic contradict those of natural conversationists, speech act theory is futile. While most people probably recognize the difference in the falsehood of "The current king..." and that of "The Earth is square", if presented with the statement about the king, people would probably say 'It's false', 'It's absurd' or 'It makes no sense'. If forced to choose between true-false, some might say 'It's false, because it makes no sense'. But I doubt anyone would reason 'It's TRUE, because it makes no sense'. And how about the logicity of these non-sequitors (I'll translate)?: It was Thursday, yet it was raining ("Era jueves, y sin embargo llovia") Although Camoes was Portuguese, he was one-eyed ("Camoes, aunque era portugues, era tuerto"). or... It's half overcrowded in here. I'm mildly/fairly exhausted. The prisoner was sort of riddled with bullets. Celso Alvarez sp299-ad@violet.berkeley.edu
laverman@prismab.prl.philips.nl (Bert Laverman) (02/01/89)
In article <429@rpi.edu> weltyc@cs.rpi.edu (Christopher A. Welty) writes: (As well as many others) >> >> "The current king of France" >> >> [.. many statements about him and those statements deleted ..] This morning in the newspaper I found an article telling that the current claimer of the French Throne died in a skiing accident. Fate let the (supposed) descendant of `guillotined' Louis meet a steel cable at appropriate height. So anyway, whatever statement about the current King of France will alas be invalid. :-/ BTW I saw his photo: He had enough hair! # Disclaimer: # I don't post # If I do, certainly not on behalf of my employer
chalmer@silver.bacs.indiana.edu (david chalmers) (02/02/89)
In article <44071@linus.UUCP> bwk@mbunix.mitre (Barry Kort) writes: >Consider, if you will, the following pair of sentences: > > "The following sentence is provable." > "The preceding sentence is unprovable." > >The paradox seems to have vanished. The first statement >can be both True and Unprovable. The second sentence >can be both True and Provable. (But please don't ask >me to supply the proof. I didn't say they were provable >by *me*!) > >The point is twofold: Not all True sentences are provable >and not all unprovable sentences are False. Thus we need >a third category: Undecidable. > >We can then resolve the paradox by chastising both sentences >for overstating the case. They could have gotten along >very nicely if they had scaled back their dogmatic assertions >along the lines of the second, more harmonious pair. Sorry. It's kind of obvious that if a sentence is provable, then it's PROVABLE that it's provable. Because if a proof exists, a PROOF that the proof exists just consists in displaying the original proof. Got that? (There may be weird logics in which this is untrue - for instance where a proof could 'exist' but be unable to be constructed - but this kind of thing isn't relevant here.) Here are your two statements again: 1: Sentence 2 is provable. 2: Sentence 1 is unprovable. So: if Sentence 1 above is True, then (by reading what it says) Sentence 2 is provable. But as we've just said, this entails that the statement "Sentence 2 is provable" is itself provable. But this statement is just Sentence 1! So now we see that Sentence 1 is provable, CONTRADICTING Sentence 2. So, oops, now Sentence 2 is FALSE, so it can't be provable. Continuing on the merry-go-round of paradox...this means that Sentence 1 has to be false. But if that's the case, then of course it's unprovable, so Statement 2 is TRUE after all! Aaargh! Stop! I want to get off. But in fact this yields the only resolution of this 'paradox.' That is: Sentence 1 is false (and thus unprovable), and Sentence 2 is true but unprovable. Check it out - it's consistent. (Now someone might say to me: haven't you just PROVED that Sentence 1 is unprovable, so doesn't that make Sentence 2 provable after all...but I'm going to quit while I'm ahead.) Never underestimate the power of a paradox to bite its own tail. Dave Chalmers (dave@cogsci.indiana.edu) Center for Research on Concepts and Cognition Indiana University
bph@buengc.BU.EDU (Blair P. Houghton) (02/02/89)
In article <418@s1.sys.uea.ac.uk> jrk@uea-sys.UUCP (Richard Kennaway) writes: >In article <416@s1.sys.uea.ac.uk> jrk@uea-sys.UUCP (me) writes: >... >>you reply "No"? >... > >Sorry, I meant "Yes". Please take this discussion over to rec.arts.dichotomy --Blair "Sorry, it's just that We here on comp.ai are meditating, and our machines have better mantras than we do, and we're grumpy over it."
long@hpsgrt1.HP.COM (Shyh-Lai LONG) (02/02/89)
/ hpsgrt1:comp.ai / weltyc@cs.rpi.edu (Christopher A. Welty) / 6:32 am Jan 31, 1989 / In article <19625@agate.BERKELEY.EDU> sp299-ad@violet.berkeley.edu (Celso Alvarez) writes: > [...] since the statement > > "The current king of France" > >(uttered while pointing at a portrait or photograph) is false in itself, >why claim that the whole sentence is "meaningless"? You are adding here a context to the statement, implying that the more formal meaning of the statement is `This is the current king of France'. But that, I would claim, is a COMPLETELY different statement than just "The current king of France", because this statement is NOT false. It is based upon the presupposition (which, as you point out, is the correct term) that there IS a current king of France. But the statement itself DOES NOT make this claim. I claim that the understanding of this statement requires a notion beyond that of truth or falsehood. Christopher Welty --- Asst. Director, RPI CS Labs weltyc@cs.rpi.edu ...!njin!nyser!weltyc ----------
rjc@aipna.ed.ac.uk (Richard Caley) (02/02/89)
In article <3038@uhccux.uhcc.hawaii.edu> lee@uhccux.uhcc.hawaii.edu (Greg Lee) writes: >From article <479@aipna.ed.ac.uk>, by rjc@aipna.ed.ac.uk (Richard Caley): >" ... >" "The current king of France is bald." >Russell said it was >false. I agree with Russell. Obviously this is a matter of opinion, but I can't agree it is false since I would not say that "No he isn't" Is the natural responce. I don't actually think it is meaningless either. -- rjc@uk.ac.ed.aipna AKA rjc%uk.ac.ed.aipna@nss.cs.ucl.ac.uk "Give me a beer and money sandwich: hold the bread" - Waldo 'DR' Dobbs
bwk@mbunix.mitre.org (Barry W. Kort) (02/02/89)
In article <3781@uklirb.UUCP> kerber@uklirb.UUCP (Manfred Kerber) writes: > One gets S1 <==> NOT S1. It is a real paradox. The paradox goes away if you admit the possibility that S1 is unprovable (or undecidable). Then we merely have that S1 is unprovable if and only if NOT S1 is unprovable. Therefore I have proven that S1 is unprovable. (I have also proven that NOT S1 is unprovable.) Any questions? --Barry Kort
bwk@mbunix.mitre.org (Barry W. Kort) (02/02/89)
In article <3091@silver.bacs.indiana.edu> dave@cogsci.indiana.edu (David Chalmers) brilliantly analyzes the 2-sentence paradox: "The following sentence is provable." "The preceding sentence is unprovable." David, a bit dizzy from his ride, steps off the merry-go-round and concludes: > Sentence 1 is false (and thus unprovable), > and Sentence 2 is true but unprovable. > Check it out - it's consistent. Actually, I think we have to clarify the meaning of "provable". We have a Goedel sentence here which is formally unprovable (underivable) using deduction. But we can nevertheless see that it is true if we permit ourselves to transcend the rules of formal derivation. What would happen if I edited the sentences to read: "The following sentence is formally unprovable but informally provable." "The preceding sentence is formally unprovable but informally provable." We can see that the two sentences have now become identical: "This sentence is formally unprovable but informally provable." Or, if you prefer, "It is evidently the case that this sentence is not formally derivable." And here we have the seeds of intuitionist logic. --Barry Kort
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
jwm@stdc.jhuapl.edu (Jim Meritt) (02/04/89)
This sentence is false. (IMHO, a paradox is a function of the language it is expressed in, and not of the universe. YOU may not understand it, but so what?) Disclaimer: "It's mine! All mine!!!" - D. Duck
dave@cogsci.indiana.edu (David Chalmers) (02/04/89)
In article <44270@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes: > >What would happen if I edited the sentences to read: > > "The following sentence is formally unprovable but informally provable." > "The preceding sentence is formally unprovable but informally provable." > >We can see that the two sentences have now become identical: > > "This sentence is formally unprovable but informally provable." Yep, I'm with you Barry. This sentence is almost like Godel's sentence "This sentence is formally unprovable", with the extra twist if recognizing it's provability at "one level up" (the level of meta-logic, if you like). One minor problem: the sentence need not now be paradoxical or cause any difficulities - it's quite OK for THIS sentence simply to be false. One major problem: it would be difficult to make such a statement cause any problems a la Godel. To do this it would have to be grounded in a given system (to make it really meaningful, as opposed to word play), where it talked about itself in a concrete way. But if you grounded it in "Level 0" (normal) logic, then any claims it makes about "Level 1" ('informal') logic won't have the viciously circular property that makes it bite it's own tail. If you ground it in Level 1, informal logic (as I think you'd prefer), then there are no longer any problems, the statement is simply true, and provable within the system. (Though its truth might vary with the formulation.) >And here we have the seeds of intuitionist logic. Why? I guess the intuitionists were happy that unprovable statements exist, because this was what they had always thought, but for different reasons. I think that an intuitionist would deny the validity of "informal provability," though. Dave Chalmers
dave@cogsci.indiana.edu (David Chalmers) (02/04/89)
In article <44268@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes: >In article <3781@uklirb.UUCP> kerber@uklirb.UUCP (Manfred Kerber) writes: > > > One gets S1 <==> NOT S1. It is a real paradox. > >The paradox goes away if you admit the possibility that S1 >is unprovable (or undecidable). Then we merely have >that S1 is unprovable if and only if NOT S1 is unprovable. > Sure, you've shown that S1 is unprovable. But the paradox is still there. It still seems that S1 can be neither true nor false. According to the views of most mathematicians, meaningful mathematical statements must have some truth-value, irrespective of whether or not they are provable. Even Godel's result never argued with this (his famous unprovable sentence G, say, was in fact true, although the proof had to be outside its particular system.) To abandon the notion that a meaningful mathematical statement must be either true or false leads one straight into the arms of the intuitionists. (Well, I guess there are a few of them about.) All the problems with S1 stem from the fact that it is not meaningful - it talks about nothing apart from it's own truth-value, so it is not grounded in reality. So any 'paradox' shouldn't worry us. Dave Chalmers P.S. Will people please shut up about the French royal family. This is really semantic games with absolutely no interesting conceptual problems behind it all.
lee@uhccux.uhcc.hawaii.edu (Greg Lee) (02/04/89)
From article <505@aipna.ed.ac.uk>, by rjc@aipna.ed.ac.uk (Richard Caley):
" In article <3038@uhccux.uhcc.hawaii.edu> lee@uhccux.uhcc.hawaii.edu (Greg Lee) writes:
" >From article <479@aipna.ed.ac.uk>, by rjc@aipna.ed.ac.uk (Richard Caley):
" >" ...
" >" "The current king of France is bald."
" >Russell said it was
" >false. I agree with Russell.
"
" Obviously this is a matter of opinion, but I can't agree it is false
" since I would not say that
"
" "No he isn't"
"
" Is the natural responce.
I agree that that is not the natural response. Neither is "No, there
isn't." This has to do with syntactic constraints between assertion and
reply in conversations -- to a first rough approximation, a reply
must correspond to the main clause in what it is a reply to. If we thus
distinguish between the sentence *being* false and the circumstances
that determine whether we can give a *reply* to the effect that it is
false, we can avoid resorting to a peculiar third truth value (in this
specific case, anyway).
So, I say, the sentence means the same as "There is a (unique) current
king of France, and he is bald," but it has a different structure and so
is not syntactically equivalent in that the two sentences admit of
different replies.
The description of such examples in terms of presuppositions and third
truth values is a pre-theoretical taxonomy. It doesn't tell us what is
really going on, but just supplies a terminology for enumerating the
facts.
Greg, lee@uhccux.uhcc.hawaii.edubph@buengc.BU.EDU (Blair P. Houghton) (02/05/89)
In article <3145@aplcomm.jhuapl.edu> jwm@aplvax.UUCP (Jim Meritt) writes: > >This sentence is false. > >(IMHO, a paradox is a function of the language it is expressed in, and >not of the universe. YOU may not understand it, but so what?) 1. Read R. Carnap, _The_Logical_Syntax_of_Language_, and discover that logic is independent of the language. _ 2. S = S is in symbolic-logic language, but has a depiction in every language I know of, and is paradoxical regardless of the words or their arrangement, so long as the syntax can be reduced to its logic, and, extrapolating from Carnap, the language would be entirely useless if it couldn't. 3. No comment on what I think 'IMHO' could possibly mean to you :-) --Blair "Everyone out of the building. Walk, don't run."
bwk@mbunix.mitre.org (Barry W. Kort) (02/07/89)
In article <17219@iuvax.cs.indiana.edu> dave@duckie.cogsci.indiana.edu (David Chalmers) writes: > In article <44270@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes: > > And here we have the seeds of intuitionist logic. > > Why? I guess the intuitionists were happy that unprovable statements > exist, because this was what they had always thought, but for different > reasons. I think that an intuitionist would deny the validity of > "informal provability," though. I was hoping to propel us into a journey of discovery of intuitionist logic, a subject which intrigues me (mainly because I barely comprehend it). If I understand Kripke's ideas, intuitionist logic admits a powerful new method of proof based on a formalization of analogy. I suspect that intuitionists are reasoning by analogy when they turn up those delicious true but underivable theorems. --Barry Kort
smoliar@vaxa.isi.edu (Stephen Smoliar) (02/08/89)
In article <43843@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes: >Let's play with the assertion, > > "The current king of France is bald." > >If we put this into symbolic logic notation, we get > > For all x, if x is the current king of France, then x is bald. > >Or in slightly more melifluous English, > > Every person who happens to be the current king of France > also happens to be bald. > This is fine as far as it goes; but, at least for the sake of argument, we should acknowledge that this is NOT the approach which Bertrand Russell took in his paper "On Denoting" (MIND, 1905). There, Russell introduced a specific denotation clause, that is, a form which explicitly stood for the denotation of some entity; so he was concerned with the interpretation of expressions which used this form, such as using "C(the father of Charles II.)" to represent a sentence about the father of Charles II. Quoting from his paper: Observe that, according to the above interpretation, whatever statement C may be, "C(the father of Charles II.)" implies:-- "It is not always flase of x that 'if y begat Charles II., y is identical with x' is always true of y," which is what is expressed in common language by "Charles II. had one father and no more." Consequently if this condition fails, EVERY proposition of the form "C(the father of Charles II.)" is false. Thus e.g. every proposition of the form "C(the present King of France)" is false. Most of the paper is concerned with justifying this rather convoluted interpretation. In a nutshell, the point is tries to make is the following: Thus "the present King of France," "the round square," etc. are supposed to be genuine objects. It is admitted that such objects do not SUBSIST, but nevertheless they are supposed to be objects. This is in itself a difficult view; but the chief objection is that such objects, admittedly, are apt to infringe the law of contradiction.
rjc@aipna.ed.ac.uk (Richard Caley) (02/08/89)
In article <43843@linus.UUCP> bwk@mbunix.mitre.org (Barry Kort) writes: >Let's play with the assertion, > "The current king of France is bald." >If we put this into symbolic logic notation, we get > For all x, if x is the current king of France, then x is bald. Says who? I certainly would not take this as a valid translation. Apart from anything else, the second sentence contains all the problems of the first, it is of the form \/x (x=King) -> bald(x) Which is, of course equivalent to bald(King) However This is just hiding the problem ( the definite reference to an non existent entity ) behind a layer of verbage. "The Current King of France" Still does not refer, so, in the above FOPC, the constant 'King' has a somewhat strange status. Now, if you are takeing "the current king of France" to be a description ( on a par with 'blue' ) which I can just about see, then your translation does not mean the same thing as the original - since it now describes a property of a set ( the set of entities which are "the current king of France" ) rather than an individual. This is less problematical since an empty set is a perfectly straght forward object. If one really doesn't want to bring in possible worlds, then I believe that the only alternative is to say the sentence is 'odd' in some way ( meaningless being another way of saying 'odd' ). The sentence does not have a meaning in its own right since if it was used by someone the natural responce by a well informed English speaker would not have any relationship to the assertion made by this sentence, one would deny the presupposition. >--Barry Kort -- rjc@uk.ac.ed.aipna AKA rjc%uk.ac.ed.aipna@nss.cs.ucl.ac.uk "Give me a beer and money sandwich: hold the bread" - Waldo 'DR' Dobbs
rjc@aipna.ed.ac.uk (Richard Caley) (02/09/89)
In article <3201@uhccux.uhcc.hawaii.edu> lee@uhccux.uhcc.hawaii.edu (Greg Lee) writes: >From article <505@aipna.ed.ac.uk>, by rjc@aipna.ed.ac.uk (Richard Caley): >" "The current king of France is bald." (1) >" I would not say that >" "No he isn't" (2) >" Is the natural responce. >I agree that that is not the natural response. Neither is "No, there >isn't." This has to do with syntactic constraints between assertion and >reply in conversations -- to a first rough approximation, a reply >must correspond to the main clause in what it is a reply to. But there _is_ no subordinate clause, unless we introduce one as a crowbar to get the facts to fit the theory. A perfectly respectable responce ( not a reply, however ) would be "but there _is_ no king of France" (3) I would never say "No, there isn't" (4) would even be a candidate, since (1) makes no _assertion_ of existence, it just presumes it. >If we thus >distinguish between the sentence *being* false and the circumstances >that determine whether we can give a *reply* to the effect that it is >false, we can avoid resorting to a peculiar third truth value (in this >specific case, anyway). Ok, I don't like third truth values either, however I don't think that the way to go is to add more and more complexity to the 'meaning' we ascribe to an utterance. I don't think that (1) has a meaning in isolation any more than " This is black " (5) does. Without knowing what I am pointing at you can neither agree nor disagree with (5), you can only decide that I am foolish for saying it. >So, I say, [ (1) ] means the same as "There is a (unique) current >king of France, and he is bald," but it has a different structure and so >is not syntactically equivalent in that the two sentences admit of >different replies. Here is a shot at trying to change your mind. ( not original by any means ). If (1) is false ( as it must be if it has the meaning which you give it ) then, assuming we do not throw out the law of the excluded middle, its negation must be true - ie either " The current king of France is not bald " (6) or, more conservativly, " It is not true that the current king of (7) France is bald " Certainly I would not assert (6). If (7) is true but (6) isn't then we must explain why they are not logically equivalent - I think most people would say that they were. On your reading, (6) means " There is a unique king of France and he is (6') not bald " whereas (7) means " Anyone who is king of France is either (7') not bald or shares the throne " ( I think I have done my rewriting correctly, no doubt someone will correct me if not ). Now, I can't give a cast iron proof that (7') is an incorrect reading, certainly it seems rather farfetched to me. >The description of such examples in terms of presuppositions and third >truth values is a pre-theoretical taxonomy. It doesn't tell us what is >really going on, but just supplies a terminology for enumerating the >facts. Why is this so for presuppositions and third truth values and not for syntactic constraints on replies? Certainly 3-valued logics can be defined just as formally as syntactic constraints, as can rules for deriving presuposition sets from sentences. > Greg, lee@uhccux.uhcc.hawaii.edu -- rjc@uk.ac.ed.aipna AKA rjc%uk.ac.ed.aipna@nss.cs.ucl.ac.uk "Give me a beer and money sandwich: hold the bread" - Waldo 'DR' Dobbs
kerry@bcsaic.UUCP (Kerry Strand) (02/11/89)
"The current king for France" has certainly gotten his PR. Consider the statement "The bald king is bald." Obviously, this is a true statement. Does it matter that there is no bald king as to whether this statement is true or not? Some would say that the statement is false if the presupposition that a bald king exists is false. Neither syntax nor semantics is sufficient to determine the truth value of the statement; it takes knowledge of the context of the statement to determine that. -- Kerry Strand kerry@atc.boeing.com uw-beaver!ssc-vax!bcsaic!kerry Boeing Advanced Technology Center (206)865-3412 P.O. Box 24346 MS 7L-64 . Seattle, WA 98124-0346 .
lee@uhccux.uhcc.hawaii.edu (Greg Lee) (02/11/89)
From article <529@aipna.ed.ac.uk>, by rjc@aipna.ed.ac.uk (Richard Caley): \>" "The current king of France is bald." (1) \ \>reply in conversations -- to a first rough approximation, a reply \>must correspond to the main clause in what it is a reply to. \ \But there _is_ no subordinate clause, unless we introduce one as a \crowbar to get the facts to fit the theory. Did I say there was? (I might resort to a crowbar, but I haven't yet.) \... \Here is a shot at trying to change your mind. ( not original by any \means ). If (1) is false ( as it must be if it has the meaning which you \give it ) then, assuming we do not throw out the law of the excluded \middle, its negation must be true - ie either Correct. Its logical negation must be true. Neither (6) nor (7) below is the logical negation of (1), however. Your (7'), or something along those lines, is the logical negation of (1). \ " The current king of France is not bald " (6) \ \ " It is not true that the current king of (7) \ France is bald " \ \Certainly I would not assert (6). If (7) is true but (6) isn't then we \must explain why they are not logically equivalent - I think most people \would say that they were. Yes, I'd say that, too. \ On your reading, (6) means \ \ " There is a unique king of France and he is (6') \ not bald " Yes. The negative in the main clause of (6) goes with the corresponding clause in (6'). \whereas (7) means \ \ " Anyone who is king of France is either (7') \ not bald or shares the throne " No, of course it doesn't. You've made a theory here that putting 'it is not true that' in front of a sentence (of English) gives its logical negation, then attributed that theory to me, apparently. Straw man. \>The description of such examples in terms of presuppositions and third \>truth values is a pre-theoretical taxonomy. It doesn't tell us what is \>really going on, but just supplies a terminology for enumerating the \>facts. \ \Why is this so for presuppositions and third truth values and not for \syntactic constraints on replies? Syntactic constraints on replies are there anyhow -- that's just a fact. Describing one set of facts in terms of others -- what I proposed -- is an explanation. Devising a terminology for some facts in a way that does not do this -- what you propose -- is not an explanation. That's the difference. \Certainly 3-valued logics can be defined just as formally as syntactic \constraints, as can rules for deriving presuposition sets from sentences. I don't see what formalization has to with it. You don't think that bad theories are incapable of being formalized, do you? Greg, lee@uhccux.uhcc.hawaii.edu
jack@cs.glasgow.ac.uk (Jack Campin) (02/15/89)
> I was hoping to propel us into a journey of discovery of intuitionist > logic [...] If I understand Kripke's ideas, intuitionist logic admits a > powerful new method of proof based on a formalization of analogy. Firstly: intuitionistic logic is not Kripke's invention, as this seems to imply. He made important contributions to it in the mid-sixties but is hardly the dominating figure in it. It's a formalization, done by Arend Heyting before Kripke was born, of an approach to the foundations of mathematics due to L. E. J. Brouwer, dating back to 1920 or so. Secondly: it has NOTHING WHATEVER to do with analogy. It is a logic designed for an ontology of potentially infinite formal constructions. It even has something surprising to say about Rolle's Theorem, but I'm not going to say what. Read it up. The net is no place for detailed tutorials. Here are some places to look to find out more about it; they get tougher, and more recent, as you go down the list. There is a VAST literature on it, with connections to everything from the theory of meaning to polymorphically typed lambda calculus. Almost any book on the philosophy of mathematics will have something about it (usually borrowed from Heyting). A. Heyting: "Intuitionism - an introduction", North-Holland (short and zippy but predates the modern semantics of Beth and Kripke) M. A. E. Dummett: "Elements of Intuitionism", Oxford University Press (for the philosophical issues) M. C. Fitting: "Intuitionistic Logic Model Theory and Forcing", North- Holland (good for the Beth-Kripke semantics) D. van Dalen: article in volume II of the "Handbook of Philosophical Logic", D. Reidel (maybe the best introduction for logicians) D. Bridges and F. Richman: "Varieties of Constructive Mathematics", Cambridge University Press (more specifically mathematical) M. Beeson: "Foundations of Constructive Mathematics", Springer-Verlag (huge encyclopaedia) J. Lambek and P. J. Scott: "Introduction to Higher Order Categorical Logic", Cambridge UP (for the link with lambda calculus and the most recent semantics for intuitionistic logic) A. S. Troelstra & D. van Dalen: "Constructivism in Mathematics", North-Holland (a new and even huger encyclopaedia; I haven't seen this yet) J. L. Bell: "Toposes and Local Set Theories", Oxford UP (I haven't seen this yet; covers the same ground as Lambek & Scott, but Bell can write much more intelligibly than Lambek) > I suspect that intuitionists are reasoning by analogy when they turn up > those delicious true but underivable theorems. From an intuitionistic standpoint, this is gibberish. Intuitionists identify the notions of truth and proof; there is no transcendent arbiter of truth like a set-theoretic model. As there are many intuitionistic formalisms of varying degrees of power, naturally they don't all prove the same things; this also occurs in classical foundational approaches, but has different implications for an intuitionist, as they can't measure the success of a formalism by its degree of approximation to a divine cribsheet of mathematical truths. Intuitionists take two different attitudes to this problem. Brouwer thought there was no way to close off the limits of provability - mathematical ingenuity could always come up with new and stronger principles of proof, so statements are never definitively underivable. On the other hand people like Martin-Lof think there can be a final, definitive theory of all mathematics, and what it can't prove is meaningless (Martin-Lof said he'd found it, about five times :-). On either account, "true but unprovable" is a vacuous concept. There is one place where analogy comes into (classical) undecidability, but not *proof* by analogy. I posted Friedman's example of a PA-undecidable sentence to sci.logic two weeks ago; it is in a precise sense a finite analogue of a theorem about infinite objects. But Friedman's proof of its PA- undecidability - a proof using principles that would make any intuitionist's toes curl, incidentally - is not remotely an analogue of the infinitistic original, still less a "proof by analogy", and the analogy isn't what makes Friedman's theorem PA-undecidable. I don't believe there *are* any proofs by analogy, anyway. The whole point of logic is to undercut "obvious" analogies, like that between words of identical grammatical category, to show why you can't reason about them in the same way; aren't "proofs by analogy" all like: Nobody is taller than Jim Jim is taller than Mary so: Nobody is taller than Mary??? -- Jack Campin * Computing Science Department, Glasgow University, 17 Lilybank Gardens, Glasgow G12 8QQ, SCOTLAND. 041 339 8855 x6045 wk 041 556 1878 ho INTERNET: jack%cs.glasgow.ac.uk@nss.cs.ucl.ac.uk USENET: jack@glasgow.uucp JANET: jack@uk.ac.glasgow.cs PLINGnet: ...mcvax!ukc!cs.glasgow.ac.uk!jack