You've all seen it. Sentences like "this sentence is false": if they're false, they're true, and vice versa, so they can't be either true or false. Some people solve this problem by doing something really complicated: they introduce infinite type hierarchies wherein every sentence you can express is given a "type", which is an ordinal number, and every sentence can only refer to sentences of lower type. "This sentence is false" is not a valid sentence there, because it refers to itself, but no ordinal number is less than itself. Eliezer Yudkowsky mentions but says little about such things. What he does say, I agree with: *ick!*

In addition to the sheer icky factor involved in this complicated method of making sure sentences can't refer to themselves, we have deeper problems. In English, sentences *can* refer to themselves. Heck, *this* sentence refers to itself. And this is not a flaw in English, but something useful: sentences ought to be able to refer to themselves. I want to be able to write stuff like "All complete sentences written in English contain at least one vowel" without having to write it in Spanish or as an incomplete sentence.^{1} How can we have self-referential sentences without having paradoxes that result in the universe doing what cheese does at the bottom of the oven? Easy: use fuzzy logic.

Now, take a nice look at the sentence "this sentence is false". If your intuition is like mine, this sentence seems false. (If your intuition is unlike mine, it doesn't matter.) But obviously, it isn't false. At least, it's not *completely* false. Of course, it's not true, either. So it's not true or false. Nor is it the mythical third truth value, *clem*^{2}, as clem is not false, making the sentence indeed false, which is a paradox again. Rather, it's something in between true and false--"of medium truth", if you will.

So, how do we represent "of medium truth" formally? Well, the obvious way to do that is using a real number. Say that a completely false sentence has a truth value of 0, a completely true sentence has a truth value of 1, and the things in between have truth values in between.^{3} Will this work? Why, yes, and I can prove it! Well, no, I actually can't. Still, the following, trust me, *is* a theorem:

Suppose there is a set of sentences, and there are N of them, where N is some (possibly infinite) cardinal number, and each sentence's truth value is a continuous function of the other sentences' truth values. Then there is a consistent assignment of a truth value to every sentence. (More tersely, every continuous function [0,1]^N -> [0,1]^N for every cardinal number N has at least one fixed point.)

So for every set of sentences, no matter how wonky their self- and cross-references are, there is some consistent assignment of truth values to them. At least, this is the case *if* all their truth values vary continuously with each other. This won't happen under strict interpretations of sentences such as "this sentence's truth value is less than 0.5": this sentence, interpreted as black and white, has a truth value of 1 when its truth value is below 0.5 and a truth value of 0 when it's not. This *is* inconsistent. So, we'll ban such sentences. No, I don't mean ban sentences that refer to themselves; that would just put us back where we started. I mean we should ban sentences whose truth values have "jumps", or discontinuities. The sentence "this sentence's truth value is less than 0.5" has a sharp jump in truth value at 0.5, but the sentence "this sentence's truth value is significantly less than 0.5" does not: as its truth value goes down from 0.5 down to 0.4 or so, it also goes up from 0.0 up to 1.0, leaving us a consistent truth value for that sentence around 0.49.

*Edit: I accidentally said "So, we'll not ban such sentences." That's almost the opposite of what I wanted to say.*

Now, at this point, you probably have some ideas. I'll get to those one at a time. First, is all this truth value stuff really necessary? To that, I say yes. Take the sentence "the Leaning Tower of Pisa is short". This sentence is certainly not completely true; if it were, the Tower would have to have a height of zero. It's not completely false, either; if it were, the Tower would have to be infinitely tall. If you tried to come up with any binary assignment of "true" and "false" to sentences such as these, you'd run into the Sorites paradox: how tall would the Tower be if any taller tower were "tall" and any shorter tower were "short"? A tower a millimeter higher than what you say would be "tall", and a tower a millimeter shorter would be "short", which we find absurd. It would make a lot more sense if a change of height of one millimeter simply changed the truth value of "it's short" by about 0.00001.

Second, isn't this just probability, which we already know and love? No, it isn't. If I say that "the Leaning Tower of Pisa is extremely short", I don't mean that I'm very, very sure that it's short. If I say "my mother was half Irish", I don't mean that I have no idea whether she was Irish or not, and might find evidence later on that she was completely Irish. Truth values are separate from probabilities.

Third and finally, how can this be treated formally? I say, to heck with it. Saying that truth values are real numbers from 0 to 1 is sufficient; regardless of whether you say that "X and Y" is as true as the product of the truth values of X and Y or that it's as true as the less true of the two, you have an operation that behaves like "and". If two people have different interpretations of truth values, you can feel free to just add more functions that convert between the two. I don't know of any "laws of truth values" that fuzzy logic ought to conform to. If you come up with a set of laws that happen to work particularly well or be particularly elegant (percentiles? decibels of evidence?), feel free to make it known.

1. ^ The term "sentence fragment" is considered politically incorrect nowadays due to protests by incomplete sentences. "Only a fragment? Not us! One of us standing alone? Nothing wrong with that!"

2. ^ I made this word up. I'm so proud of it. Don't you think it's cute?

3. ^ Sorry, Eliezer, but this *cannot *be consistently interpreted such that 0 and 1 are not valid truth values: if you did that, then the *modest sentence* "this sentence is at least somewhat true" would always be truer than itself, whereas if 1 is a valid truth value, it is a consistent truth value of that sentence.

The crisp portion of such a self-reference system will be equivalent to a Kripke fixed-point theory of truth, which I like. It won't be the least fixed point, however, which is the one I prefer; still, that should not interfere with the normal mathematical reasoning process in any way.

In particular, the crisp subset which contains only statements that could safely occur at some level of a Tarski hierarchy will have the truth values we'd want them to have. So, there should be no complaints about the system coming to wrong conclusions, except where problematically self-referential sentences are concerned (sentences which are assigned no truth value in the least fixed point).

So; the question is: do the sentences which are assigned no truth value in Kripke's construction, but are assigned real-numbered truth values in the fuzzy construction, play any useful role? Do they add mathematical power to the system?

For those not familiar with Kripke's fixed points: basically, they allow us to use self-reference, but to say that any sentence whose truth value depends eventually on its own truth value might be truth-value-less (ie, meaningless). The least fixed point takes this to be the case wh... (read more)

YKY,

The problem with Kripke's solution to the paradoxes, and with any solution really, is that it still contains reference holes. If I strictly adhere to Kripke's system, then I can't actually explain to you the idea of meaningless sentences, because it's always either false or meaningless to claim that a sentence is meaningless. (False when we claim it of a meaningful sentence; meaningless when we claim it of a meaningless one.)

With the fuzzy way out, the reference gap is that we can't have discontinuous functions. This means we can't actually talk about the fuzzy value of a statement: any claim "This statement has value X" is a discontinuous claim, with value 1 at X and value 0 everywhere else. Instead, all we can do is get arbitrarily close to saying that, by having continuous functions that are 1 at X and fall off sharply around X... this, I admit, is rather nifty, but it is still a reference gap. Warrigal refers to actual values when describing the logic, but the logic itself is incapable of doing that without running into paradox.

I think the original post is not specific enough to be useful.

I see two essential moot points:

1) Why should be there a system of continuous correspondences between the truth values of sentences that have to do anything with some intuitive notion of truth values?

2) Are the truth values (of the sentences) after taking the fix point actually useful? E.g. can't it be that we end up truth values of 1/2 for almost every sentence we can come up?

Before these points are cleared, the original post is merely an extremely vague speculation.

A closely related analogue t... (read more)

Nitpick: Since this sentence doesn't refer to

the truth value ofany English sentence, you'd still be able to write it even if you were using type hierarchies or the like. I think.I might be missing something, but it seems as if you're needlessly complicating the situation.

First of all, I'm not convinced that sentences

oughtto be able to self reference. The example you give, "All complete sentences written in English contain at least one vowel" isn't necessarily self-referencing. It's stating a rule whic is inevitably true, and which it happens to conform to. I could equally well say "All good sentences must at least one verb." This is not a good sentence, but it does communicate a grammatical rule.But none o... (read more)

On the topic of fuzzy logic: Is there a semantics for fuzzy logic in which the fuzzy truth value of the statement "predicate

Pis true of objectx" is the expected value ofP(x) after marginalizing out a prior belief distribution over possible hidden crisp definitions ofP?There should be a way to break this system. Let's see...

"This sentence doesn't have a consistent truth value."

Did I win?

This makes it sound like you have indeed just reintroduced types under another name, patching "this statement is false" by forbidding "this statement has truth value 0.0".

It occurs to me as a "Notion" that . . .

To formulate fuzzy logic in a boolean top domain environment, I think you would need to use a probabilistic wave form type explanation. Or else just treat fuzzy logic as a conditional multiplier on any boolean truth value. To encapsulate a boolean or strict logic system into fuzzy logic is trivial and evolving. You could start with just adding a percentage based on some complex criteria to any logical tautology or contradiction. By default the truth axis of a fuzzy logic decision or logic tree is goin... (read more)

If all true statements are defined as non-contradictory, then you can ask more meaningful fuzzy logic questions about the relevance of several tautologies for applying to a specific real world phenomena. To do this you need a survey or poll of the environment and a survey or poll for determining how much the teutologies matter. For example.

Consider the following boolean true false claims we hold to be true and consider their relevance for use in locating humans statistically: our first rule or fuzzy logic heuristic is to take the first tautology that seem... (read more)

Why?

A conjecture (seems easy to prove):

"If, in a fuzzy logic where truth values range from [0,1], we allow logical operators (which are maps from [0,1] to [0,1]) or predicates that does

notintersect the slope=1 line, then we can always construct a Liar's Paradox."An example is the binary predicate "less-than", which has a discontinuity at 0.5 and hence does not intersect the y=x line.

Nitpick: Since this sentence doesn't refer to

the truth value ofany English sentence, you'd still be able to write it even if you were using type hierarchies or the like.A conjecture (seems easy to prove):

"If, in a fuzzy logic where truth values range from [0,1], we allow logical operators (which are maps from [0,1] to [0,1]) or predicates that does

notintersect the slope=1 line, then we can always construct a Liar's Paradox."An example is the binary predicate "less-than", which has a discontinuity at 0.5 and hence does not intersect the y=x line.

O...kay. It looks like you just decided to post the first thing on your head without concern for saying anything useful.

You come up with fractional values for truth, but don't think it's necessary to say what a fractional truth value means, let alone formalize it.

You propose the neato idea to use fractional truth values to deal with statements like "this is tall", and boost it with a way to adjust such truth values as height varies. Somehow you missed that we already have a way to handle such gradations; it's called "units of measurement&q... (read more)

Silas, a suggestion which you can take or leave, as your prefer.

This comment makes some sound points, but IMHO, in an unnecessarily personal way. Note the consistent use of the critical "you"-based formulations ("you just decided", "you come up with", "you propose", "you missed that"). Contrast this with Christian's comment, which is also critical, but consistently focuses on the

ideas, rather than thepersonpresenting them.I have no idea why you feel the need to throw about thinly-veiled accusations that Warrigal is basically an idiot. (How else could he or she

possiblyhave missed all these really obvious problems you so insightfully spotted?). Maybe you don't even intend them as such (though I'm baffled as to how could you possibly miss the overtones of your statements when they're so freakin' OBVIOUS). But the tendency to belittle others' intellectual capacities (rather than just their views) is one that you've exhibited on a number of prior occasions as well, and one that I think you would do well to try to overcome - if only so that others will be more receptive to your ideas.PS. For the avoidance of doubt, that final para was intended in part as an ironic illustration of the problem. I'm not

thatun-self-aware.PPS. Also, I didn't vote you down.

Fuzzy logic

isjust sloppy probability, although Lofti Zadeh doesn't realize it. (I heard him give a talk on it at NIH, and my su... (read more)