weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) (04/01/88)
I've just come across what looks like an extremely fascinating book.
_The Liar: An Essay in Truth and Circularity_, by Jon Barwise and
John Etchemendy, Oxford University Press, 1987. It only costs $20;
its ISBN is 0-19-505072-X. Buy this book.
I will summarize as best I can as to what is in it; it's definitely
on my summer reading list. The level is that of standard first-year
logic. (My first impressions include, inter alia, that it gives pre-
cisely what I was asking for in earlier postings about infinite re-
gressions and self-justifying in my disagreements with Paul Torek: viz,
while *some* infinite regressions are harmful, not *all* of them are.)
Saul Kripke, in "Outline of a theory of truth" _The Journal of Phil-
osophy_ 72 (1975), pp 690-716, re-opened the question that had suppos-
edly been answered by Tarski: "what is truth?" He proposed a theory
that permitted circular reference. It has not been generally accept-
ed, but the realization that simple schemes in the spirit of Zermelo
for banishing of the Liar paradox are fundamentally unacceptable be-
came widespread. Variants have been proferred; what B&E present is
a model-theoretic approach to analyzing these theories that permit
circular reference, based on a version of set theory that permits
non-well-founded sets. The axioms they use, in an intuitive form due
to Aczel, lead to unique sets with any given pattern of abstract cir-
cular reference, eg, there is only one x such that x={x}, only one y
such that y={0,y}, etc.
B&E develop, first a Russellian inspired interpretation of truth,
where propositions are definitive statements about the "real world"
out there, and then second an Austinian inspired interpretation,
where propositions always have context-sensitive features.
An example they use to illustrate this difference concerns "Claire
has the 3 of clubs". Start by assuming that a fixed moment of time
and a fixed set of unambiguously identified people are being referred
to--this kind of understanding is for ease of speaking, and not a
significant feature of any interpretation. In their Russellian view,
this statement has an absolute truth value, determined by identifying
Claire and the 3 of clubs. In their Austinian view, the statement's
truth value can depend on its speaker's understanding of the situa-
tion: if the speaker mistook Emily for Claire, or the deuce for the
three-spot, then the statement is false, even if it turns out that
the real Claire had the real 3 of clubs all along.
In both cases B&E work out a non-well-founded model theory that per-
mits liar-type paradoxes in both situations to be resolved--moreover,
it permits an understanding of *why* the resolution works as it did.
In the Russellian case, the liar paradox becomes false, while the
truth-teller puzzle ("This sentence is true.") becomes model-depen-
dent. Note that this is not the same as context-sensitive: once
one has identified the "real world" correctly in their Russellian
analysis, one gets a rigid true/false determination. They find,
however, that the Russellian resolution is unnatural, and can even
define semantically the notion of a proposition being "paradoxical".
In the Austinian case, they give explicit constructions of Liars,
ie, utterers of "This sentence is false", some of whom are telling
the truth, and some of whom are telling a falsehood! And the same
can be done for other utterers of paradoxical/puzzling sentences.
In particular, they show the relevance of the difference between
negation and denial, ie, between asserting a falsehood and denying
a truth.
They contrast the two views by saying that their Russellian solution
is analogous to the introduction of the set/class distinction into
naive set theory and the proof that the Russell class of sets that
don't contain themselves is the universal class, while their Austin-
ian solution compares with relativizing the Russell class to within
a fixed (not necessarily well-founded) set. The first just shows
one has left ZFC, while the latter is the standard diagonalizing
technique of construction entirely within ZFC. In the Austinian
case, they derive Reflection Principles which allow for arbitrary
great generality in scope of reference.
I hope to understand this book by the time Raymond Smullyan's version
comes out!
"Corollary 23: A sentence "phi" is intrinsically paradox-
ical in the Russellian semantics just in case "phi" is
intrinsically deniable, while "not phi" is necessarily
false.
"The difference between denial and negation, once pointed
out, is easy enough to acknowledge, but even easier to
forget. This is especially true in logic, where the em-
phasis is place on truth as a property of sentences, and
denials are relegated to the pragmatic wastebasket. And
in general, ignoring this distinction does little serious
harm, any more than ignoring relativistic effects causes
problems in trips to the supermarket. But at speeds ap-
proaching that of light, ignoring relativistic effects
gives paradoxical results. Similarly, in the realm of
semantics. Corollary 23 points out that, when approach-
ing sentences like the Liar, we risk paradox if we ig-
nore the difference between negation and denial."
ucbvax!garnet!weemba Matthew P Wiener/Brahms Gang/Berkeley CA 94720
"Logicians, it is said, abhor ambiguity but love paradox."g-rh@cca.CCA.COM (Richard Harter) (04/03/88)
In article <8224@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >I've just come across what looks like an extremely fascinating book. >_The Liar: An Essay in Truth and Circularity_, by Jon Barwise and >John Etchemendy, Oxford University Press, 1987. It only costs $20; >its ISBN is 0-19-505072-X. Buy this book. I will put it on my buy list... Sounds very interesting. Re the example of claire and the three of clubs. It sounds at first hearing that the difference between the Austinian version and the Russellian version is that the Russellian version reduces to the Austinian version if we add terms that make statements about context, i.e. "Claire has the three of clubs and this person is Claire and this person holds this card and this card is the three of clubs." This does not sound so interesting. However I have my doubts about this "adding terms that make statements about context"; I shall be interested in seeing what is actually done. Re the liars paradox. The following (in loose form) seems satisfactory to me: We make a distinction between statements and data. Statments about data are either true or false. We can also make statements about statements. Statements about statements do not have to be either true or false. However we can classify statements as reducible (founded) or irreducible (unfounded). Reducible statements can be resolved in terms of true and false. True and false do not directly apply to irreducible statements; instead the relevant question is "Can truth or falsity be consistently assigned to the statement". There are four possibilities (both T and F, T only, F only, and neither.) The reducible statements satisfy a two valued logic; the irreducible ones a four valued logic. Moreover, truth of a reducible statement is not the same thing as 'T only' for an irreducible statement. The latter says that you can consistently treat it as being true, but not as false; it has no 'truth' value per se. An immediate corollary is that it is not permissable to quantify over irreducible statements in the usual form of truth and falsity, because true and false are not correct categories for irreducible statements. Whether this in fact works, is quite another matter. But, if it does, it seems satisfactory to me. I rather suspect it doesn't -- it all sounds too simple minded not to have been analyzed and shown wanting a long time ago. -- In the fields of Hell where the grass grows high Are the graves of dreams allowed to die. Richard Harter, SMDS Inc.
ma261aai@sdcc3.ucsd.EDU (Stephen Bloch) (04/03/88)
In article <8227@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >The axioms they use, in an intuitive form due >to Aczel, lead to unique sets with any given pattern of abstract cir- >cular reference, eg, there is only one x such that x={x}, only one y >such that y={0,y}, etc. I'm reminded of Alain Colmerauer's handling of self-referential trees [see, for example, Proceedings of the Int'l Conf. on 5th Gen. Computer Systems, 1984] in which the task is to tell whether two data structures (let's say, trees) with unbound variables at some of the nodes can be unified by a suitable assignment of variables, and if so what this assignment is in a least Herbrand model (the "least" allows for unique solutions -- it's not actually true that there is only one x such that x={x}, but there's a unique minimal solution. Somebody correct me if I'm taking the name of Herbrand in vain.) And in fact most of his examples look like "find y such that y = (0 y)". Colmerauer gives algorithms for solving systems of simultaneous equations (and "inequations", i.e. "x CANNOT be unified with y" as opposed to "x is not NECESSARILY unified with y", which is the usual understanding of "not" in a Prolog system) of this sort, and I did an undergrad independent study implementing (in Prolog) a Prolog interpreter using said algorithms as a unifier. The rest of the book review is fascinating, too, and I'll try to find the book. Another Obnoxious Math Grad Student Steve Bloch
vespa@ssyx.ucsc.edu (Adam Alexander Margulies) (04/04/88)
In article <8224@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >I've just come across what looks like an extremely fascinating book. >_The Liar: An Essay in Truth and Circularity_, by Jon Barwise and >John Etchemendy, Oxford University Press, 1987. It only costs $20; ^^^^^^^^^^^^^^^^^ !!!!!!!!!!!!!!!!! Obviously no longer a student. ;-) I said, type it NOW, Adam! || ||Adam Margulies | \ ||_ /| ||ARPA: vespa@ucscb.ucsc.edu | ||\`o_O' ||BITNET: vespa@ucsci.BITNET | || ( ) ||UUCP: ...!ucbvax!ucscc!ssyx!vespa| -----------------------------------||--mU-m-|| | |DISCLAIMER: || ATT: (408)429-8868 | | These are NOT my opinions. They are my dog's. |
weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) (04/04/88)
In article <26505@cca.CCA.COM>, g-rh@cca (Richard Harter) writes: [but first, a word from our sponsor:] >>_The Liar: An Essay in Truth and Circularity_, by Jon Barwise and >>John Etchemendy, Oxford University Press, 1987. It only costs $20; >>its ISBN is 0-19-505072-X. Buy this book. >Re the example of claire and the three of clubs. It sounds at first >hearing that the difference [is merely adding contextual indentifying >clauses, which isn't interesting; it can't be that simple.] You are correct, it isn't that simple. For example, you can't "add" con- text to a *self-referential* sentence. >Re the liars paradox. The following (in loose form) seems satisfactory >to me: We make a distinction between statements and data. Statments >about data are either true or false. We can also make statements about >statements. Statements about statements do not have to be either true >or false. That this is unsatisfactory has been felt for a long time by many. Our intuition very strongly says that statements about statements "refer", and it is only after some analysis that we discover limits on this. We can say, so much for intuition, but no one has ever made good, convinc- ing *models* for these self-referential sentences in the first place be- fore, with or without funny logics. Kripke first made concrete part of this feeling with the following example: (A) Most of Nixon's assertions about Watergate are false. (B) Everything Jones says about Watergate is true. Pretty harmless looking, no? But what if A is Jones' *only* assertion about Watergate, and B was asserted by Nixon, who also said, coinciden- tally, 2k other things about Watergate, k of them clearly true, and k of them just as clearly false? As Kripke said, "there can be no syn- tactic or semantic `sieve' that will winnow out the `bad' cases while preserving the `good' ones." > However we can classify statements as reducible (founded) >or irreducible (unfounded). Reducible statements can be resolved in >terms of true and false. True and false do not directly apply to >irreducible statements; instead the relevant question is "Can truth >or falsity be consistently assigned to the statement". There are >four possibilities (both T and F, T only, F only, and neither.) What about (L*) "This sentence is either false or irreducible." ? If true, then it isn't false, and it isn't irreducible. Nope. If false, then true. Nope. So it's irreducible, hence true. Whoops. >An immediate corollary is that it is not permissable to quantify over >irreducible statements in the usual form of truth and falsity, because >true and false are not correct categories for irreducible statements. So how do you *identify* if a statement is reducible or not? You need to know *before* you attempt to permissibly analyze a statement, accord- ing to your above comments, and yet the reducibility may hinge on the internals of the sentence. Unless you come up with an *internal* categorization of reducible/irredu- cible statements--and Kripke's example or variants thereof makes this very unlikely--any such theory will always be vulnerable to just strengthened liars. You as might as well stick to identifying truth. >Whether this in fact works, is quite another matter. But, if it does, >it seems satisfactory to me. Ignoring the L* problem, it is unsatisfactory in that it doesn't *explain* anything. Just saying, "paradox go home" doesn't mean anything. Indeed, rejecting "vicious circles" on the grounds that they are "vicious circles" is--you guessed it--a viciously circular argument. In contrast, and this was the point of view that bugged so many people when I raised it so many months ago, *justifying* vicious circles in a circular manner is self-con- sistent. And that is the beauty of B&E's book. They give an intuitive explanation for what happens with these self-referential sentences. No hokey rules about what is allowed or isn't. As an example of a non-hokey rule, con- sider the following pseudoproposition p: p := Max has the 3 of clubs OR p. Its negation is q: q := Max doesn't have the 3 of clubs AND q. It works out that *both* are true under their natural truth predicates. But the following self-reference *is* acceptable: p := Max has the 3 of clubs OR True(p). Its negation is q: q := Max doesn't have the 3 of clubs AND False(p). B&E give a simple *internal* grounding rule that eliminates the former pair but not the latter. And they point out how natural this rule is by comparing the situation with English: "Max has the 3 of clubs or this sentence is true", is something we try to analyze, but "Max has the 3 of clubs or this sentence", is ungrammatical. In contrast, a reducible/irreducible criterion is an *external* distinc- tion, and cannot be justified before the fact. ucbvax!garnet!weemba Matthew P Wiener/Brahms Gang/Berkeley CA 94720
g-rh@cca.CCA.COM (Richard Harter) (04/04/88)
This is one of several articles in response to Matthew's comments. In this article I clarify (?) my remarks on the liars paradox and reducibility and irreducibility. In article <8307@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >>Re the liars paradox. The following (in loose form) seems satisfactory >>to me: We make a distinction between statements and data. Statments >>about data are either true or false. We can also make statements about >>statements. Statements about statements do not have to be either true >That this is unsatisfactory has been felt for a long time by many. Our >intuition very strongly says that statements about statements "refer", >and it is only after some analysis that we discover limits on this. We >can say, so much for intuition, but no one has ever made good, convinc- >ing *models* for these self-referential sentences in the first place be- >fore, with or without funny logics. Well, my intuition doesn't say this at all, but that's probably the fault of my intuition. Let me see if I can make this clear. Suppose I have a rather long list of statements. Suppose I want to go through the list and determine which ones are true and which are false. What happens? I find that there are statements about data which are true or false and can be marked accordingly. I winnow out these. Call this set of statements S_0. Then I notice that there are statements that refer to statements in S_0. I can unambiguously mark these as true or false. And so on. After I have marked S_0, S_1, and so on I may have some left. These I call irreducible. Paranthetically, I'm using the term irreducible, because they don't have to form a closed referential loop, e.g. (1) Statement 2 is true. (2) Statement 3 is true. ..... Now, it's possible that I can tease out the truth of these statements in the manner of a logic puzzle. I.e. I would mark statement x as true and then see what other statements would then be true and vice versa, mark it as false and see what happens. Now if it happened to be the case that all statements could be assigned truth values in this manner, I might be satisfied. In fact, I can't. There is no unique assignment. Even if I could, however, I am actually doing two different things. The truth value of the reducible statements follows directly from the data. In the case of the irreducible statements I am really saying "The only consistent assignment is to mark X as true, etc." I am not really saying that X is true; I am only saying that it will be consistent the rest if it is treated as being true. If this be truth it is a different kind of truth. Now it simply doesn't bother me that there is no unique list of assignments, that there is no unique model that fits all cases. What I can actually DO is to say and determine things such as "If I mark statement X as being true then it is consistent with all of the reducible statements, but it is not if I mark it false." and "I cannot consistently mark statement X as being either true or false." Now these are statements of fact -- I can either do the markings as claimed or not. These statements have truth value. In short, irreducible statements have no intrinsic truth value at all. This is not the same as saying that they are meaningless or that they have no referents; it is simply saying that they don't (despite appearances) fall into the category of things with truth values. It is the statements about how they can be marked that have truth value. -- In the fields of Hell where the grass grows high Are the graves of dreams allowed to die. Richard Harter, SMDS Inc.
g-rh@cca.CCA.COM (Richard Harter) (04/04/88)
_The Liar: An Essay in Truth and Circularity_, by Jon Barwise and John Etchemendy, Oxford University Press, 1987. It only costs $20; its ISBN is 0-19-505072-X. Buy this book. In article <8307@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >In article <26505@cca.CCA.COM>, g-rh@cca (Richard Harter) writes: ... Proposing the following counterexample. >What about (L*) "This sentence is either false or irreducible." ? >If true, then it isn't false, and it isn't irreducible. Nope. If false, >then true. Nope. So it's irreducible, hence true. Whoops. No soap. (L*) is irreducible. It has no truth value. Can I consistently mark it as true? No. Can I consistently mark it as false. No. It is not (L*) that is a problem. Consider the following statement (L**) (L*) is irreducible. Is this a reducible statement? If it is, is there something wrong with the definition of reducibility that I gave? -- In the fields of Hell where the grass grows high Are the graves of dreams allowed to die. Richard Harter, SMDS Inc.
g-rh@cca.CCA.COM (Richard Harter) (04/05/88)
_The Liar: An Essay in Truth and Circularity_, by Jon Barwise and John Etchemendy, Oxford University Press, 1987. It only costs $20; its ISBN is 0-19-505072-X. Buy this book. Okay, alright already, I will. In article <8307@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >Kripke first made concrete part of this feeling with the following example: > (A) Most of Nixon's assertions about Watergate are false. > (B) Everything Jones says about Watergate is true. >Pretty harmless looking, no? But what if A is Jones' *only* assertion >about Watergate, and B was asserted by Nixon, who also said, coinciden- >tally, 2k other things about Watergate, k of them clearly true, and k >of them just as clearly false? As Kripke said, "there can be no syn- >tactic or semantic `sieve' that will winnow out the `bad' cases while >preserving the `good' ones." I would accept Kripke's statement, on the grounds that he knows much more about these things than I do. (A true reducible statement :-)). However I do not see that this is a good example. If we eliminate the 2k statements as window dressing, we are left with, in schematic form (A) 'B' is false (B) 'A' is true Now I am content to say that neither (A) nor (B) are true or false, that they are not, despite appearances, statements having truth value. But that is beside the point -- what I do not see is that this example justifies or illuminates Kripke's assertion "there can be no ...". Perhaps there cannot be such a sieve. But in this instance we have no problem in separating the sheep from the goats. Now I would expect that the problem of deciding whether statements are reducible or irreducible is undecidable, i.e. there is no decision procedure. I don't know this, but it sounds like the sort of thing that is undecidable. What this example does show that is that statements cannot be classified one by one. And it follows simply enough that there are infinite sets of statements that can't be classified in a finite procedure. -- In the fields of Hell where the grass grows high Are the graves of dreams allowed to die. Richard Harter, SMDS Inc.
g-rh@cca.CCA.COM (Richard Harter) (04/06/88)
_The Liar: An Essay in Truth and Circularity_, by Jon Barwise and John Etchemendy, Oxford University Press, 1987. It only costs $20; its ISBN is 0-19-505072-X. Buy this book. In article <8307@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >In article <26505@cca.CCA.COM>, g-rh@cca (Richard Harter) writes: >>Re the example of claire and the three of clubs. It sounds at first >>hearing that the difference [is merely adding contextual indentifying >>clauses, which isn't interesting; it can't be that simple.] >You are correct, it isn't that simple. For example, you can't "add" con- >text to a *self-referential* sentence. That isn't what I had in mind -- the example addressed the issue of a sentence being true by accident under a Russellian interpretation. If I say "Claire has the three of clubs", I may be mistaken in several particulars, e.g. the person whom I thought was Claire was actually Emily, or what I thought was the three of clubs was actually the deuce, and so. The example claimed that the Austinian interpretation took into account context, i.e. not only does Claire have the three of clubs, but also my grounds for believing this are correct. What I was asking was whether, in a non self-referential sentence, an Austinian intepretation can be converted into a Russellian interpretation by adding clauses adding context. One might argue or even establish that this is not possible. One might even show that the attempt to add full context will convert a non self-referential statement into a self-referential statement!! One might be able to show that this is true, but that the self referential character is resolvable or innocuous. I haven't read the book yet, so I don't know. However, at first appearance, the mention of adding context to a self-referential sentence is a red herring. > >>An immediate corollary is that it is not permissable to quantify over >>irreducible statements in the usual form of truth and falsity, because >>true and false are not correct categories for irreducible statements. >So how do you *identify* if a statement is reducible or not? You need >to know *before* you attempt to permissibly analyze a statement, accord- >ing to your above comments, and yet the reducibility may hinge on the >internals of the sentence. In general, you can't. So? Some statements can be immediately identified as being reducible -- others can be immediately identified as being irreducible. If you have a finite set of statements and all referents to statements can be eliminated, the statements are reducible. If you have a finite set in which all external references to statements can be eliminated, but there are internal references which cannot be eliminated, the set of statements are not all reducible. >Unless you come up with an *internal* categorization of reducible/irredu- >cible statements--and Kripke's example or variants thereof makes this very >unlikely--any such theory will always be vulnerable to just strengthened >liars. You as might as well stick to identifying truth. There is no effective procedure for categorization. This should be clear. But I have yet to see any justification for the claim that this makes for vulnerability for strengthened liars paradoxes. Indeed, the situation is the converse, a liars paradox cannot arise because the definition of truth is being restricted to that which you can actually determine. That is not the problem -- the issue is whether such an approach is useful. If you take a restricted view of 'truth' do you eliminate essential parts of mathematics and logic? -- In the fields of Hell where the grass grows high Are the graves of dreams allowed to die. Richard Harter, SMDS Inc.
barwise@csli.STANFORD.EDU (Jon Barwise) (04/10/88)
John Etchemendy and I have been enjoying the discussion of The Liar, so I thought I would send some pointers to further work. Paul King, Computer Science, U. of Manchester, has done some work following up on our book. In particular, he has solved the 3rd open problem there. His treatment provides a much imporved version of our reflection theorem. Aczel's book on AFA is due off the press any day now. You can find out how to order it by sending a message to Dikran@csli.stanford.edu. I have a paper applying AFA to common knowledge in the TARK II volume (reference below), and a more mathematical version with proofs that I can send out by US Mail. There are a number of interesting papers on paradoxes in the TARK II volume. I would especially call attention to the ones by Gaifman and Koons as being relevant to the earlier discussions on this distribution list. There is a new book out by Sainsbury, Cambridge Univ. Press, called Paradoxes. It goes into some depth on a wide range of paradoxes, from Zeno, thorugh the Prisoner's Dilemma, to the Liar. Finally, Etchemendy and I have written a PS for the next printing of our book (next year) that might interest you. I can send it by US Mail, or a latex file by email. It provides a somewhat different slant on our work in the book, a slant that some people find much more attractive. TARK II: Reasoning about KNowledge, Ed by Moshe Vardi, March 1988, Morgan-Kaufmann (Los Altos, CA). This is the proceedings of the second TARK conference. Jon Barwise barwise@csli.stanford.edu