Deleting paradoxes with fuzzy logic

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, clem2, 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.