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A Quadrilogue at the Edge of Logic


An Epistemological Improvisation on Chaos and Logic

in the spirit of Douglas Hofstatder

Auri: What do you mean, this project is false?

Ian: Maybe not the project, but the words you're using. In fact, your
sentence is false.

Matt: What? Are you saying this sentence is false?

Slide 1

Ian: Was that sentence false? Hmmm... Let's think about this.

Let's assume that the sentence, "This sentence is false" is true, then the sentence is false because it says so.
If, on the other hand, we say the sentence is false, then the sentence is false about being false and is therefore true. So we have a contradiction- the sentence is both false and true!

Covell: it seems like there must be away out of this.

Auri: Yeah, people must have thought about that sentence before. Hey, one of you tired overworked students out there- wanna take a stab at this?

Slide 2

Ian: You're not alone, dear participant. Yes, a whole lot of brilliant people like yourself, ancient and modern, have tackled this. Eubilides in the 4th century BC, a contemporary of Aristotle, put forth this so-called paradox. He described a Cretan prophet named Epimenides who said, "All Cretans are liars", which is essentially the same problem.

Matt: Couldn't we just summarily dismiss these paradoxes as meaningless by ruling out self-reference? How can a sentence meaningfully talk about itself?

Auri: Well, we don't seem to run into the same problem with the self-referential statement, "This sentence is true." Covell: So what? What does this true or false stuff matter when Cretans for many years might have to deal with the harsh consequences of people believing that they are liars due to this fool long ago Epimenides. I have a petition with many Cretans upset with this Epimenides fellow.

Ian: Anyway, if we want to set up any kind of system- like mathematics...

Matt: mmm... topology! 4-spheres! Hypercubes!...

Ian: .. Yes, all that good stuff... as I was saying, if we want a system to describe or model this mess we call reality, it'd be useful to distinguish whether things are true or false or somewhere in between dependent on the way we set up this system. After all, what use is a formal system in which we can't tell if 1=0 is a true statement or not? How could I tell you how many glasses of wine I drank last night?

Auri: Are you saying that casual conversation about your drug use is formally systematic?

Ian: Good point. No. I don't really know if spoken language is a formal system but I doubt it. I can say all kinds of inconsistent things but then again, Godel... proved that so can any sufficiently useful formal system.

Slide 3

Some people, like Laplacean determinists, might even argue that reality is a formal system in which all the particles forming everything are composed of assumed fundamental particles subject to basic laws, though I doubt this.

Auri: Are you getting at something? Ian: Fine. Anyway, ok, let's be a tad more careful. I'll describe my drug use and anything else quantitative using agreed-upon and seemingly obvious rules and symbols subject to straightforward reasoning processes. Adding, multiplying, generalizing, etc.

Covell: What's this have to do with truth?

Matt: Yeah, this conversation seems to have explosively bifurcated. We need a catastrophe to bring us back.

Slide 4

Auri: Well, if mathematical physics could have been used to create that maybe there's something applicable about all this (if not something scary as well.)

Matt: Well, luckily, logic seems nicely innocuous and abstract, if scary to certain students or postmodernists.

Ian: Yeah, well absolute truth is always offensive to most people. Maybe we should talk about accuracy instead, or shades of truth.

Covell: So, instead of saying absolutely true or false to a statement like, "Ralph is a spellbinding lecturer", we could assign some intermediate value that may be more meaningful.

Matt: And we could phrase those values with numbers.

Slide 5

Just for simplicity, why not say that 0 means false or totally inaccurate and 1 means true or totally accurate, with any real number in between representing an intermediate value.

Auri: So, "Santa Cruz is a nice place to live" might have a 0.9 value.

Matt: You might think so.

Covell: Wow, so truth-values are now numerical and continuous.. Could we translate truth statements into numerical expressions?

Ian: Yeah, actually, that Kurt Godel guy first came up with that thought when our grandparents were learning how to count.

Slide 6

He realized that logical expressions, which use notation to express thoughts that otherwise take awhile to say or write (just like all disciplines make shorthand) could be translated into statements about numbers and retain their essential meaning.

Covell: Wow, so you can then use techniques of number theory to play with the preserved thought and explore it in a totally new way.

Auri: Maybe this kind of connection could be used for exploring paradoxes. Maybe we could use math, or, I daresay, computers, to analyze paradoxical semantic "systems" like variations of the Liar.

Covell: They're self-referential about their truth values, so in effect they're recursive, like mathematical maps.

Ian: The next step came when Grim and Mar wondered what would happen if, instead of just evaluating your logical system once, you iterated it?

Covell: What do you mean?

Ian: Well, the traditional way of evaluating logical systems is pretty simple. You plug in what you know, and see what you can derive from that. It's similar to evaluating a mathematical function at a single point.

Matt: Grim and Mar wondered what would happen if they took the results from evaluating their logical system, and plugged them back in to the system. Essentially, they iterated the statements, and watched what happened.

Auri: How would you know when to stop?

Matt: Well, you could see if the truth values ever settled down to anything...

Auri: Like a point attractor?

Covell: I see... but what if your values never settled down?

Ian: Then you might have a strange attractor!

Auri: Waitaminute, waitaminute. How does the math actually work here? Say we had a logical statement...

Covell: like, say, "Chaos is valid mathematics."

Auri: And another...

Ian: How about, "What Matt says is true."

Auri: That works. Now, if you know whether "Chaos is valid mathematics," how do you determine the truth of what B has said?

Covell: Well, Ian is only telling the truth if I am.

Ian: Yeah. And if Covell is completely wrong, so am I. But aha! What if "Chaos is valid mathematics" is only partly true.

Auri: Most statements fall in to that category.

Matt: So if what Covell says is only partly true, then what Ian says is also partly true. In fact, Ian statement's truth value can be thought of in terms of error. The error is the difference between the claimed value...

Ian: "What Covell says is true." (emphasis on true)

Matt: And the original statement...

Covell: "Chaos is valid mathematics."

Matt: So if you write the "error" as the difference between the truth values of those two statements.. And take the absolute value, so things stay positive... then you have a formula for the error.

Slide 7

Covell: You know the error. How then do you figure out the actual truth of what Ian said?

Auri: Well, if the error is zero, then what you said is completely true. And if the error is one, then it's completely false.

Ian: And if the error is somewhere in between, then so is the truth value, right.?

Covell: That sounds good. But what if Ian said, "What Covell says is false?"

Matt: Then you are claiming a truth value of zero, so the equation comes out like this ( writes Ian = 1 - | 1 - (Ian)| ).

Auri: (Flipping through the book) Now, what about saying something like

"It is very true that chaos is valid mathematics?"

Matt: What do they do in the book?

Auri: Here they square the whole statement.

Ian: That can't be right. Do they explain why?

Auri: Apparently it's a convention from Fuzzy Logic. (writes Very = squared? on board)

Covell: I don't think so. That's one of the ways that this mathematical approach fails. It's hard to justify operations like squaring and square roots coming from simple logical statements.

Auri: (Joking) I'd love to see a logical system with a sine function. Or how about a log?

Slide 8
Slide 9
Slide 10
Slide 11
Slide 12

Matt: A Note: Grim and Mar didn't do any mathematical analysis as to whether these attractors are mathematically chaotic. Although it looks pretty chaotic, I am not completely sure that that is going on here.

Auri: So do these pictures really "solve" any contradictions? Can we extract their truth value from the pictures?

Covell: I'm not so sure. (show a picture of the origami) For example, can you really say anything about the truth value a logical system that ends up on this attractor?

Ian: Well, we can't really see where the answer is, but (pointing to empty area) we can see where it is not. We are only a little closer to any comprehensible answer.

Matt: And if we look at the simple liar, it fills up half of our square. What kind of answer is that? Not much. But all this talk about the liar's paradox and self-referentiality makes me think again of Godel's numbering and his incompleteness theorem.

Auri: What's that?

Ian: That's what Godel stumbled upon when he translated logic into arithmetical statements. He found a proof that any logical or mathematical system is either inconsistent or incomplete, or both. And the crux of the proof is basically that all of these systems contain the Liar's Paradox.

Auri: Inconsistent? Incomplete? Both? What's that supposed to mean?

Matt: Inconsistent means that two statements are contradictory. And incomplete means that a statement cannot be proven.

Covell: So Godel proved that there are some things you cannot prove?

Auri: (swooning) How incredibly postmodern!

Matt: Exactly. And he did it in 1939. Ahead of his time, eh?

Ian: Yeah. And it had deep repercussions for logic and math. It meant that a rigorous foundation for mathematics was impossible.

Auri: Aye caramba! (still swooning) It seems that we can never know anything for sure these days!

Slide 13

Matt: Anyway, could this Dynamical Semantics have any implication on Godel's Theorem? They are both based on the Liar's Paradox.

Covell: Well, let's think about this. Dynamical Semantics allows for iteration, or re-evaluation of your system, and it allows for statements to be somewhere between true and false.

Matt: Right. Both of those things are done in everyday life, and usually not done in math. Dynamical Semantics might give us a way out of the existential gridlock of Godel's paradox, at least in daily life, if not in mathematics.

Covell: Whoa there, big fella. Don't go on any intellectual magic carpet rides. What about the down-to-earth applications of Chaos Logic? Does this have anything to do with reality?

M and Ian: huh, reality?

Auri: I've got an idea. We live in a culture swamped with information. We, as students especially, are constantly asked to question the validity and reliability of the information we find. Perhaps we could use dynamical semantics to process and judge the accuracy of a wide spectrum of information on a given topic.

Matt: yes, since some of our information sources might comment on other sources , we have self referentiality, an essential aspect of dynamical semantics.

Ian: Grim and Mar, in the Philosophical Computer, are also concerned with information reliability; they call it epistemic dynamics. In terms of Chaos, they used the example of a surveilance institution using dynmamical semantics theory to calculate the reliability of their spy, 007.

Covell: Very useful; new technology for the CIA.

Auri: Well lets look at a grassroots way to use it. Say for example we read in the New York times about an air-strike in Iraq and in the article we find:

Slide 14

Covell: Doesn't the New York times have its own contextually positioned biases

that influence their reporting.

Auri: Exactly. So our task would be to gather many news reports about

this air strike as well as the criticisms and opinions on the very

reliability of these sources. Then, perform a dynamical semantics

analysis of the data and see what kind of graph we get.

Matt: So the graph would help us judge the reliability of the New York Times report.

Auri: However, the very process of judging reliability is dangerous.

Covell: The parameters we insert into our program would reflect our very own biases based on our situational knowledge, say, for example, we've all lived as white males. The type of information we chose to look at would affect our answers.

Ian: Well, that seems unavoidable. Just think about Quantum Physics..

Auri: So, uh, is the Liar's Paradox resolved?

Matt: Kind of yes and kind of no. We have the neat pictures.

Ian: Not resolved in Aristotelian or even continuous logic, but it doesn't seem to be paradoxical in this new system. So maybe there is some sort of resolution.

Slide 15

Covell: So is this project false?

Auri: What do you mean, this project is false?

Ian: Maybe not the project, but the words you're using. In fact, your

sentence is false.

Matt: What? Are you saying this sentence is false?

Ian: Was that sentence false? Hmmm... Let's think about this.

Slide 16

Last modified: Mon Mar 8 07:47:49 PST 1999