So, as I said in the intro to the new Scientopical Good Math/Bad Math, I'm

going to writing some articles about fuzzy logic.

Before diving into the fuzzy stuff, let's start at the beginning. What

is fuzzy logic? More fundamentally, what is *logic*? And what problem does fuzziness solve?

So, first, what's logic?

Logic isn't really a single thing. There are many different logics, for different purposes. But at the core, all logics have the same basic underlying concept: a logic is a symbolic reasoning system. A logic is a system for expressing statements in a form where you reason about them in a completely mechanical way, without even knowing what they mean. If you know that some collection of statements is true, using the logic, you can *infer* - meaning derive from a sequence of purely mechanical steps - whether or not other statements are true. If the logic is formulated properly, and you started with an initial set of true statements, then any conclusions drawn using the inference process from those statements will also be true.

When we say logic, we tend to automatically think of a particular logic: the first order predicate logic, which is the most common, fundamental logic used in both mathematics, and in rhetoric and debate. But there are an astonishing number of different logics for different purposes. Fuzzy logic is one particular

variation, which tries to provide a way of reasoning about vagueness.

The reason for fuzzy logic is that there are some properties that we can observe, and which we'd like to be able to reason about, but which just don't make sense when they're expressed in FOPL. The classic example of this is

"tall".

My older brother is about 6 feet and 2 inches tall. I think everyone would agree that by any reasonable definition, he's tall. I'm 5 feet and 11 inches tall. Most people, but not everyone would agree that I'm tall. When I was a freshman in high school, I was 4 feet and eleven inches tall. I think no one would claim that I was tall then. Exactly when did I *become* tall? At various times, I was 4 foot 11, then 4 foot 11 1/2, then 5 feet, then 5'1, then 5'1 1/2... At some point in time, I clearly was not tall. At some later point in time, I clearly *was* tall. But what was I in between? Was there some specific day, some specific moment, where I crossed a magic boundary, and became tall?

Of course there isn't.

Similarly, you can pick out a bunch of people of different heights, and

ask whether they're tall. Someone who's 6 feet 6 inches is clearly tall. What if I take off one eighth of an inch of height from a tall person. Are they still tall? If

so, then I can take another eighth of an inch. Are they still tall? If I keep doing that, eventually I'll get to four feet, which is obviously not tall. And yet, where is the boundary?

Tall just doesn't work very well as a predicate in FOPL. It doesn't have a crisp boundary. It's intrinsically *fuzzy*. It's actually something which, in some sense, has a kind of numeric value. You're tall to a certain

degree - and that degree is intrinsic to how true the statement "You are tall" is.

The real world is full of things like that - properties that instead of

having a crisp boundary between true and false, have a vague boundary, but

also have some sort of valuation of *how true* they are. And the way that you reason about those statements needs to take the valuation into account. If you take a 6'6 person, and remove 1/8th of an inch, you get a person who is *slightly less* tall. And the valuation continues

to change through a reasoning process, so that by the time you get to someone whose height is four feet, the statement that they're tall has no truth left.

So the entire system of the logic needs to change, to reflect the idea that nothing is really strictly true or false; things have *degrees* of truth and falsehood. But going directly from crisp to fuzzy is confusing - it's too much change all at once. Instead, we'll try to make a sort-of fuzzy transition.

The way that we'll approach understanding that is by starting off with

something that's less crisp than FOPL, but more crisp than true fuzzy logic. What we'll look at is a three-valued logic, where each statement can be "true", "false", or "maybe". In fact, we'll look at a few of those: there's a whole family of closely related three-valued logics. So we'll look at those, and see what it means to lose the strict true/false dichotomy. From there, we'll gradually move to full blown fuzziness.

Cool! I'm looking forward to reading this series; fuzzy logic is something I ought to know about as a computer guy, but I never learned much about it. I'm curious if it will work like I'm guessing it will.

nice intro... looking forward to this as well

(and good to know "fuzzy logic" isn't just a term for Republican thinking ;-))

Stupid question - what is FOPL?

@Christina Pikas

FOPL = First Order Predicate Logic

I'm somewhat familiar with fuzzy logic, but I've never (as far as I can recall) worked with a three-value logic. I think that I'm really going to enjoy this series.

"I’m somewhat familiar with fuzzy logic, but I’ve never (as far as I can recall) worked with a three-value logic."

Probably you did. If you have ever used SQL, it uses three-valued logic (true, false, null).

Of course, longtime readers of The Daily WTF website know all about three-valued logic: http://thedailywtf.com/Articles/What_Is_Truth_0x3f_.aspx

Something I've never really understood in my (albeit limited) exploration of fuzzy logic is how it differs with probability theory. Specifically I'm thinking of E. T. Jaynes' "Probability Theory as Logic".

Stephen,

Probabilty logics are designed to discuss events where the outcome is unknown. Their values can change temporally. Thus, you might be 60% sure a person can make a free-throw in their first NBA season, but if the person works at this skill, 75% sure in their tenth season. So, you might choose to not foul them in the tenth season, when you might have fouled them in the first season. However, the results of the individual free-throw can be known after it has been taken.

Fuzzy logics typically describe events that have values, but don't have good divisions. Using Mark's example, you might describe him as 40% tall, and his brother as 65%, based upon some metric. That number will not change unless you change the metric.

So, while they use the same range from which to draw values, probability and fuzzy logics interpret those numbers differently.

FOPL is First Order Predicate Logic. "Predicate" because it reasons about predicates - properties. T(x) is "x is tall". B(x) is "x is blue". C(x) is "x is a car". Unlike fuzzy logic, every predicate is either true or false about a certain x -- either x is a car or x isn't, and there's no inbetween.

"First Order" means you can say things like "for every x" and "there is an x". So you could say "For every x, C(x) implies B(x)" (all cars are blue). You could also say "There is an x such that C(x) and B(x)" (there is a blue car). You can't say "for every set of objects"; that would be Second Order.

"The real world is full of things like that – properties that instead of

having a crisp boundary between true and false, have a vague boundary"

But the real world

isn'tlike that. Every proposition is either true or false - it's only when we have ambiguousdefinitionsthat you can't decide. Humans are all about fuzzy definitions, which is why we have this problem, but thinking intelligently about a matter always starts with defining what you're talking about. To the layperson, a tyrannosaurus is more of a dinosaur than Archaeopteryx, which is more of a dinosaur than a robin, but as soon as you define what you actually mean by 'dinosaur,' you have to admit that a robin either fits that definition or it doesn't - no middle ground.I don't doubt that fuzzy logic is useful, but I don't see how you can say it reflects reality. It seems to me to reflect humans' imperfect way of conceiving of reality.

@ Tim Martin:

"You have to admit that a robin either fits that definition [dinosaur] or it doesn't – no middle ground."

I disagree. For your statement to be true, at some point in evolution, what we would call a 100% dinosaur lay an egg from which was hatched a 100% robin. That is preposterous of course.

Now a robin may very well be 0.000001% "dinosaur" at this point in evolution, and nobody could ever look at a robin and call it anything but a robin – but there absolutely is a middle ground.

I think you're misunderstanding the fact that birds

aredinosaurs, according to modern classification systems. Dinosaurs (like all clades) are defined by a large set of morphological and physiological features. "To be a dinosaur, you must have this and this and this and this," is basically how the definition works. If you list all the features that make dinosaurs dinosaurs, you can see that modern birds possess all of them as well (which is unsurprising, since birds inherited those features.) Thus, birds fit the (unambiguous) definition of a dinosaur. Or, if you wanted to define a dinosaur as an organism that has X list of features and went extinct however many millions of years ago, you could do that as well. Under that definition, modern birds would unambiguouslynotbe dinosaurs. But the only way there would be a middle ground is if you haven't adequately defined what you're talking about."For your statement to be true, at some point in evolution, what we would call a 100% dinosaur lay an egg from which was hatched a 100% robin. That is preposterous of course."

Consider why you think this is preposterous. In your head, what definitions of robin and dinosaur are you using? If a robin is defined by a certain list of features, then an organism that had all of them but ONE, would still be not a robin. But what if that bird gave birth to progeny that had that final feature, due to some mutation? Then we can say a not-robin gave birth to a robin. This is exactly how evolution works.

PS: See, Ryan North gets it!

Actually, in modern cladistics, a robin

isa dinosaur. Fact remains that boundaries between species are fuzzy.Only because our definitions are inadequate. Let's say one species of fruit fly will only mate with a mutated version of the same species 40% of the time. Is this therefore a new species? I'll be happy to answer the question as soon as you tell me how you define "new species." Do you define it as a species with ZERO gene flow from the old? Then it's not a new species because the two versions of fruit fly still mate 40% of the time. Or do you define new species as one that would rather mate with members of a certain subset of the overall population (which they do 60% of the time)? Then it is a new species. As always, there's no fuzziness here once I know what you're talking about. If you want to argue otherwise, you're going to need an example of something that is necessarily and incontrovertibly fuzzy.

If you want a *really* fuzzy group, take Oaks. Then you've got no choice but either fuzzy logic or a big disjoint, since they aren't related cladistically.

If they aren't related then they're just an example of two different things with the same name - like cranes for construction and cranes that are birds. This isn't fuzzy, it's polysemy.

That your dinosaur example works doesn't prove that fuzzy logic is unnecessary. In first-order logic terms, you've gone from "There is an X such that many people think cannot be treated by first-order logic, but that can be treated by first-order logic" to "There are no X such that X cannot be treated by first-order logic."

Let's go back to that dinosaur example. Take the predicate "X is a bird." Yes, the robin is a dinosaur, cladistically. But at what point do you say "OK, from here on these specific dinosaurs are also members of the clade Aves"? Or, work backwards from Dinosaurs. Dinosauria is a subclade of amniotes, which in turn is a subclade of vertebrata, which is a subclade of Animalia.

Someone has to decide where the lines are drawn. In actual paleontology, they talk about chronospecies, which seems to be a way of dealing with the facts that populations change gradually, but the fossil record is limited, so it makes sense to group organisms in part by whether they lived close together in time.

Also, the physical world contains a whole lot of organisms that we have defined as "robins" and "seagulls" and "horses" and "oranges," but it doesn't contain the concepts "robin," "seagull," "horse," and "orange." Yes, you can define" robin precisely enough that you can say, for any given organism O, either "O is a robin" or "O is not a robin." (If necessary, define Robin as {R: the set of items listed here, R1, R2, R3,…,R9,332,290,412}. Any finite set can be defined by enumeration.) If you look at it that way, you can define tall as "more than 1.7 meters tall" and say that someone who is 171 cm is tall and someone who is 169 cm is not, even if the 169 cm person is an 11-year-old girl and stands out among her classmates. The question is whether that's the most useful way of looking at it.

You seem to be arguing that fuzzy logic is useful, which is something I stated in my first comment. I'm arguing that fuzzy logic does not reflect reality; FO logic does. That's why we're using FO logic to have this conversation, that's why you use FO logic to plan your grocery list, and that's why scientific papers are written adhering to the laws of FO logic.

“There are no X such that X cannot be treated by first-order logic.”

That's right. If you disagree, it should be easy enough to show me one.

Ordinary language indicative conditionals. Propositional attitudes. Possibility and necessity. Generics. As Mark pointed out, vagueness. Liar sentences.

Of course, there are lots of attempts to treat these issues in first order predicate logic, but they're not clearly successful.

Can you give/explain a specific example? I don't know what most of those terms mean.

Indicative conditionals: If the butler didn't do it, the gardener did.

(Indicative conditionals don't behave the way the material conditional--the conditional of FOL--does.)

Propositional attitudes: Bill believes Ortcutt is a spy.

Possibility and necessity: Bill could have been a philosopher. Bill couldn't have been a roller skate.

Generics: Birds lay eggs.

Mark has already covered vagueness.

Liar sentences: This very sentence is false.

Here are a few more additions to the list...

Talk about non-existent objects: Sherlock Holmes is fictional.

Various quantifiers: Most dogs have tails. Few cats have wings. More mice than rats like cheese.

1. I don't know much about indicative conditionals, but a few sources I checked online say that there's no agreed upon defnition for them...? So how are we supposed to conclude anything about them? Specifically, how does FOPL not work here?

2. "Propositional attitudes: Bill believes Ortcutt is a spy." Nice name. But this statement is either true or it isn't. Again, how does FOPL not work here?

Ok, I'm going through the list here, and I don't see how any of these are a problem for FOPL. Could you take at least one of them and

explainwhat you mean?"Liar sentences: This very sentence is false."

It seems to me that this just shows that it's possible to state a contradiction. How does this break FOPL any more than "A & ~A" does?

That should be "[the material conditional] is false only when the antecedent is true and the consequent is false."

Indicative conditionals. Consider the following exchange.

A: If the butler didn't do it, then the gardener did.

B: That's not true. The cook might have done it.

The material conditional--the conditional of FOL--is true whenever the antecedent is false or the consequent is true; so it's false only when. Thus, by denying the conditional B has asserted both that the butler and the gardener are innocent. But surely this isn't right--B's utterance doesn't in fact commit him to anybody's innocence. There's another way to see a problem here, too: suppose that the butler *did* do it--is A's utterance true or false? Most speakers judge that the conditional is without a truth-value in this case.

Propositional attitudes. The following sentences can have different truth values:

(1) Bill believes Twain wrote Tom Sawyer.

(2) Bill believes Clemens wrote Tom Sawyer.

(Suppose, for example, that Bill has never heard the name 'Sam Clemens'.) But, since Twain *is* Clemens, (1) and (2) are logically equivalent in FOL--they must have the same truth value.

The Liar.

(3) This very sentence is false.

Suppose (3) is true--then it is false. But no sentence can be both true and false, so (3) cannot be true. Suppose instead that (3) is false--then it is true! So (3) can't be false, either. (The case differs from ordinary contradictions--'(p & ~p)'--since these are straightforwardly false.)

My favorite example of your second class is the story that Ronald Reagan believed that Copenhagen was in Norway. Assuming that that's true, and given that Copenhagen is the capital of Denmark, it's still not true to say that Ronald Reagan believed that the capital of Denmark was in Norway.

"Thus, by denying the conditional B has asserted both that the butler and the gardener are innocent."

How is this true? All B did was state that the causal connection sketched by A is inaccurate. B is saying that IF the butler is innocent, it does not follow from that that the gardener is guilty. Where's the problem here?

Liar sentences:

I already dealt with this. You didn't answer the question "How does this break FOPL any more than “A & ~A” does?" You said that A&~A is "straightforwardly false," but what does that mean, exactly, and how does it make two different types of contradictions different? You can't just give one of them a different name and expect that to make your argument for you. Yes, liar sentences accomplish contradiction via recursivity. So what? How is that any more of a problem for FOPL than contradictions that are accomplished via any other means?

"Propositional attitudes. The following sentences can have different truth values"

But do they, actually? Let's investigate the details of Bill's belief that "Twain wrote Tom Sawyer." What information in his brain is this actually based on? Most likely, Bill believes that there was a man, a writer, who lived some time ago and who wrote the book Tom Sawyer, and that this man's name was Twain. Now what if Bill believed that Clemens wrote Tom Sawyer? We would day that Bill believes there was a man, a writer, who lived some time ago and who wrote the book Tom Sawyer, and that his name was Clemens. So really, the two beliefs differ in one detail - what Bill thinks the man's name was. The contradiction comes in when you add in

yourbelief that Twain and Clemens are the same person - but that information was not part of Bill's belief, so there is no problem.Same thing with John's example:

When John says "Ronald Reagan believed that Copenhagen was in Norway," what he's saying is that "Ronald Reagan believed that Copenhagen, which is a city in Denmark and, in fact, its capital, is in Norway." This is a mental sleight of hand. Reagan did not believe this, obviously. Reagan's beliefs included only that there was a random city in Norway by the name of Copenhagen. John

thinksthere is a logical puzzle here becausehisbeliefs about Copenhagen, when combined with Reagan's, lead to a contradiction. But really there isn't one, because Reagan's beliefs didn't contain the information that causes the contradiction to exist.-------------------

Sorry about the 3rd-person, John; it was the easiest way.

Oh, don't worry about the third-person, Tim. You only need to apologize for putting words into my mouth.

Ah, apologies for that too then.

"How is this true? All B did was state that the causal connection sketched by A is inaccurate. B is saying that IF the butler is innocent, it does not follow from that that the gardener is guilty. Where’s the problem here?"

Not if "if... then..." is a material conditional. Here's how conditionals work in first order logic: a sentence of the form [If P then Q] is true just in case P is false or Q is true, and so is false just in case Q is true and P is false.

Now, suppose that we take A's utterance to contain a material conditional (the only conditional that FOL has!): When B deny's A's claim, he's asserting that the conditional is false; but there's only one way for the conditional to be false--the antecedent (the butler didn't do it) is true, and the consequent (the gardner did it) is false.

But, you say, that's not what A's utterance really means! And that is precisely the point: ordinary indicative conditionals aren't handled by first order logic.

On liar sentences: Supposing that '(p & ~p)' is true leads to a contradiction; but supposing that it is false does not lead to a contradiction--we can coherently assign a truth value to the sentence. Supposing that the liar sentence is true leads to a contradiction--and so does supposing that it is false! We can't coherently assign any truth value to the liar.

On belief: Here's a feature of first order logic: if two names co-refer--that is, if they name the same thing--then substituting one for the other will never transform a true sentence into a false one, or vice versa. This is exactly what we want in most contexts: 'Twain had white hair' is true, and since Twain is Clemens, 'Clemens had white hair' better be true as well. But there are some contexts--called opaque contexts--where substitutivity fails. 'believes' creates such a context: the truth of 'Bill believes Twain wrote Huck Finn' doesn't guarantee the truth of 'Bill believes Clemens wrote Huck Finn', even though Clemens just is Twain. (Another example just to drive it home: Lois Lane thinks Superman can leap tall buildings in single bound, but she doesn't think that Clark Kent can do so).

I assure you, these are all real problems that logicians, linguists, and philosophers have been working on for about a century.

Tim,

Here's just a few possibilities:

The moon is a very big object in the way I perceive it right now. Usually Christians consider Sunday as their day of worship. Often you will not understand that some (not "there exists" I mean some) of these statements reflect statements which describe reality. Almost surely, you will eat a piece of bread tomorrow. The time is *about* 8:30 P. M. EST. In a few astronomical texts, 3 is much smaller than 3*10^60. Crows often chirp loudly. In comparison to objects seen with normal eyesight, atoms are very, very, very, very tiny.

Suppose Christina likes only red and green shoes. She has a large number of red shoes and a medium number of green shoes in one box A. In another box B she has a small number of green shoes and a large number of red shoes. Instead of posing a question about the probability of finding a red shoe in box B, I'll say that the probability of finding a red shoe in box B is large. It is not extremely large or almost equal to 1, but it is also not almost equal to .5. My apologies to anyone named, or having a good friend named Christina.

6 encyclopedia volumes carried at once is a very heavy load. Pine trees often got cut several years before they grow to maximum height. Your pink finger is a rather small finger. Your thumb is a somewhat wide finger. There exists a person you know, or knew, who is rather hairy. European people often have very pale skin. A deck of cards feels rather light in your hands. A currency bill feels very, very light in your hands, is somewhat dirty, but is not quite so dirty as a sewage tank. An ant is extremely strong for its body size. A car usually travels much faster than a person on foot does. A rocket travels much, much faster than a car does.

You can tap on things much, much harder than how hard you tap the keys when you type. Trees usually look tall to us (here the reality lies in the perception). Almost always, countries have had an established religion. Yellow is a much, much brighter color than black. Forms of a little bit of water can often make your skin wet. There exists little space between symbols in this reply (notice, the space is not uniform). Apartments are often not very big. Massive gravitational objects larger than Jupiter or other gas giants often emit massive amounts of radiation. Cars do not usually travel very close to their top speeds.

I hope that you and anyone else can see that *at least* one of these statements reflects reality. The quantifiers here come as words like "very, often, almost always, "much, much", usually, little, etc." I don't have a clue how these can or could possibly exist in first order logic (note, I'll give you the quantifier "most"). You might disagree with my assessments here, but do you really still maintain that not one of these holds in reality, or that any of them that do, that first-order logic comes as sufficient here? Perhaps some of them can get quantified in a way that I would agree with in first-order logic, but what other word than "miracle" would describe a situation where *ALL* of these statements come as *properly* describe-able in first-order logic?

Last I checked FOPL has quantifiers "for all" and "there exists" at its base, and then relies on creative ways of using them to describe other quantifies. I'll give you "most" and maybe some other quantifiers can get described, but seriously... all other natural language quantifiers, in *absolutely all* contexts in which people use them *rather* well?

Tim,

I'm going to expand on what the The Science Pundit said below.

Logics, whether first-order predicate calculus, intuitionist, fuzzy, or whatever else, are created by us as models that we use to simply reality into a form we can discuss. We choose what propostions look like and what sorts of truth values we will attach to them. You can certainly create a model where no propostion is ambiguous and each has exactly one of true truth values. Then, when you examine reality within that model, it will seem that no proposition is ambiguous and each has exactly one truth value. However, that's a feature of the model, and not necessarily the reality underlying it. Models are chosen by how useful they are. If we could create a useful model that perfectly mirrored every aspect of reality, we wouldn't need a model at all.

As another example, while it looks like Mark will be staying with one-dimensional logical models in this series, you can even create multi-dimensional logics. For example, you might use a logical system to evaluate "It is raining" based on three variables (longitude, latitude, and how hard it is raining), which would then allow you to make the statement accurate over every point in the world at the same time.

Then, when you examine reality within that model, it will seem that no proposition is ambiguous and each has exactly one truth value. However, that’s a feature of the model, and not necessarily the reality underlying itEven if what you're saying is true, what if it is also false?

"Something can't be both true and false at the same time!"

Not in FO logic, it can't. But you said yourself that that is a feature of the model - in reality, maybe something

canbe both true and false."Ok, but what would follow from that?"

Nothing would. That's why FO logic is the logic of reality, or reality as far as humans can conceive of it - because it doesn't mean anything at all to say that A is true and it is false.

-----------------------

Bottom line: You can't use any form of logic to defeat itself.

Tim,

Sometimes, separating the model from reality can be tricky.

Not in FO logic, it can’t. But you said yourself that that is a feature of the model – in reality, maybe something can be both true and false.Reality has no "true" and "false", nor even any propostions. The very notion of a proposition can only be in the model we create. Can you create a model where a propostion can have nmore than one truth value (and in particular, can be true and false)? Yes.

“Ok, but what would follow from that?”Nothing would.Why not? I suppose you could set up a model where you could both say propostions can have two truth values, and also say that nothing can be derived from having two truth values, but whjy would you bother?

That’s why FO logic is the logic of reality, or reality as far as humans can conceive of it – because it doesn’t mean anything at all to say that A is true and it is false.It doesn't mean anything in first-order logic, you mean.

"Reality has no “true” and “false”, nor even any propostions. "

Here we go again. "The Earth has one moon" is a statement that describes the actual physical world. Tell me how this is not the "true" reality.

"Can you create a model where a propostion can have nmore than one truth value (and in particular, can be true and false)?"

Yes. Except multi-valued logic doesn't preserve truth; it preserves something else (whatever other thing you set it up to preserve). From Wikipedia: "For example, the preserved property could be justification, the foundational concept of intuitionistic logic. Thus, a proposition is not true or false; instead, it is justified or flawed."

Since we're talking about reality, we're interested in preserving truth. There's a logic for that. We're using it to have this conversation. If other logics describe reality just as well, why don't you use them in your everyday life? My contention is that you can't. (More on that below)

"Models are chosen by how useful they are. "

You didn't choose your model. Here's a test - try to convince yourself that you are sitting there reading this comment, and you

aren'tsitting there reading this comment, at the same time. Can you do it? Or how about this. Imagine that "the Earth has one moon" is neither truenorfalse....Can you do that? If you can't, is it because you

choseto think this way? You chose to put this limitation on yourself? No,evolutionput this limitation on your thinkingbecause it meshes with reality. Getting eaten by a tiger is NOT the same as not getting eaten. That truth is not a choice.FO logic isn't necessarily true, though - WVO Quine put forward probably the most obvious example of FO logic not holding true in a scientific statement.

Take the example:

R = The particle is moving to the right

P = The particle is in interval [-1,1]

So we can construct:

R&(Pv~P)

(~P = The particle is not in the interval [-1,1])

Which by distribution is equivalent to:

(R&P)v(R&~P)

There's a problem here. If we're to take uncertainty at face value, and the particle in question is in a domain governed by quantum effects, then the first statement is true, but the second false(!).

In other words, it's not hard to construct an example that *seems* to hold truth-values, but given our current scientific understanding of the world, will give false outputs from true inputs.

Actually, it wasn't WVO Quine. Although I can't remember whose example it originally was. In any case, the point stands that examples known to science don't fit FOL.

(Quantum models in general pose a whole host of problems for presuming truth-preservation of FOL and its application to "reality", I might add - the example given is one of many.)

Granted, my understanding of physics at that level isn't particularly thorough, but I've encountered the examples regarding epistemological naturalism and questioning the status of logic as non-amenable to experiential affections.

Wait, what was that Tybo?

First of all, "moving to the right" and "the interval [-1,1]" are both sufficiently vast statements that they can both be known about a particle at the same time, even considering uncertainty, so the second disjunction is perfectly possible.

Secondly, uncertainty doesn't say anything about certain statements being false, it says that certain statements are

meaningless.Here we go again. “The Earth has one moon” is a statement that describes the actual physical world. Tell me how this is not the “true” reality.It's a model of reality, under specific defintions that require a moon to be of a certain size, among other things.

Yes. Except multi-valued logic doesn’t preserve truth; it preserves something else (whatever other thing you set it up to preserve). From Wikipedia: “For example, the preserved property could be justification, the foundational concept of intuitionistic logic. Thus, a proposition is not true or false; instead, it is justified or flawed.”Those are different interpretations of the smae calculus underlying the intuitionist's model, thinking of propsitions as justified, refuted, or unknown. The attempt to reduce those three distinct values to two is one possible imposition of a two-valued model over a three-valued model. It doesn't work, in part because contraditory statements will both be "flawed" in the attempted revision.

Since we’re talking about reality, we’re interested in preserving truth. There’s a logic for that.No, there isn't. Logics are formal systems, not truth-preserving creations.

We’re using it to have this conversation. If other logics describe reality just as well, why don’t you use them in your everyday life? My contention is that you can’t.You are just wrong.

You didn’t choose your model. Here’s a test – try to convince yourself that you are sitting there reading this comment, and you aren’t sitting there reading this comment, at the same time. Can you do it?If I am reading the comment while listening to music and engaging in conversation, I could easily be 70% reading, or 40%, or 10%.

Or how about this. Imagine that “the Earth has one moon” is neither true nor false.…Can you do that?Depends on what you mean by a moon.

More seriously, you can't use the notion that one, or a few, ideas always fit well into a true/false paradigm to then say that the true/false paradigm is always an adequate model.

That truth is not a choice.How we choose to talk about truth is a choice.

Mark,

Would you do a post on Bochvar's logic where all truth values outside of classical logic get taken as nonsense?

Tim,

You said

"Nothing would. That’s why FO logic is the logic of reality, or reality as far as humans can conceive of it – because it doesn’t mean anything at all to say that A is true and it is false."

I agree with that contradictions do not come as permissible in logic or mathematics, and that contradictions do not exist. However, if you claim that you've made a claim in first-order logic, then you've engaged in a contradiction. If you say that "FO logic is the logic of reality", then either any statement form, that is proposition, is either true or false *OR* nonsense. Modern logic involves propositions, which are the *forms* of statements. That nonsense forms can and *do* exist in reality comes as demonstrable given the following rules.

1. Let the capital letter N denote a unary operation.

2. Let all other capital letters in the Latin alphabet denote binary operations (for anything not recognizable as in English, of course).

3. Let lower case letters denote logical variables.

Now, the statement *form* Ka *is* nonsense, or meaningless if you prefer to say that. It is *neither* true *nor* false. As a *form* it does exist in reality. Likewise abJK is nonsense, as is KAABjkjlkjiop also nonsense. All of these are meaningless both in prefix (or "Polish" that is Lukasiewicz) notation, infix notation, *and* suffix (or "postfix") notation. *None* of them are either true or false, and consequently lie completely outside the domain of first-order logic as *forms*. I do not understand how you can seriously disagree that *forms as forms* exist in reality unless you believe that *every single* perception you have ever had is an illusion. So, all of the above nonsense, as forms, do exist in reality.

Three-valued logic can get studied from the perspective of statement forms as either true, false, or nonsense. So far as I can tell, Tim has implicitly presupposed this throughout. Merrie Bergmann's in _Introduction to Many-Valued and Fuzzy Logic_ mentions that Bochvar in 1937 developed a three-valued logic where propositions are either true, false, or meaningless.

Mark,

In light of this comment and since so many of us discussed Tim's perspective and tried to refute it, would you perhaps do a post on Bochvar's three-valued logic where any value outside of classical logic gets interpreted as nonsense?

I find this simple to answer. Humans, including, me and you are part of the real world. Our concepts also exist in reality. We always think in our bodies, which exist in reality. So, even if only humans have "fuzzy" definitions and concepts and nothing else in reality does, to some non-zero degree, it reflects part of reality, since we are part of reality. It seems clear enough that birds and other sentient beings also have such "fuzzy" concepts, since they sometimes do things like fly into windows. So, it can get said to reflect at least part of the structure of reality in a somewhat broader sense than just that of ourselves.

Also, every proposition is true or false, then statements like "Few people like walking." "Many people read webblogs." "3 is much, much smaller than 10^6." "when running a race, 1400 meters is about the same as 1600 meters." "Usually swans are white." "Often adults eat fruit." all come as true or false, or do not qualify as propositions. If you seriously rejected all of those sorts of statements as propositions, then there exist a whole host of problems with communicating things in natural language, including everyday statements like "it's 3'oclock", which is almost always invariably false, if it's a proposition, and it's either true or false.

He didn't say "propositions" which, by definition, are crisply true/false. He said "properties", in the context of "real-world".

Interesting topic, I'm looking forward to follow up posts.

One thing I'd like you to expand upon is why we should (or should not) prefer fuzzy logic over probability theory, which it seems to me does the same job although in what seems superficially a different way.

Well, not exactly: probability theory has a way of quantifying being tall. In the 1.7 m criterion as a criterion for height, a very coarse probabilistic model would state for instance that in a certain population, a certain percentage has at least a height of 1.7 m, say 55%.

The interesting thing with probability is that if we want to consider two criteria, say height and wealth, we can state that someone is wealthy if her net worth is say $500,000. For instance in the same population, 20% have at least this net worth. But we can also quantify the proportion of people who are tall and rich, and there are rules, which tell us that we can't have really arbitrary numbers. For instance, the proportion of rich and tall people cannot exceed 20%, and the sum of the proportion of rich and tall people and the proportion of rich and short people has to be equal to 20% – otherwise our data are incoherent.

We can also model the possible correlation of wealth and height. If they were independent, then 11% among the tall people would be rich and 9% among the short people would also be rich. In other words, the proportion of rich peole among short and tall is the same.

I do not think that fuzzy logic allows for this kind of considerations.

Consider this for tallness, define the truth-value to be the cumulative normal distribution function. That makes any statement of fuzzy logic about tallness equivalent to probabilistic statements about drawing a member from the population at random. I only see a difference in philosophy, not in the maths (meaning formulas).

First off one can have discrete fuzzy sets, such as {(1, .2), (2, 1), (3, .4)}. The cumulative distribution function is defined over an interval of real numbers. Fuzzy sets need not happen on such an interval. You could have them only able to take rational values in [0, 1]. Second, the cumulative distribution function is monotone non-decreasing and its limit as x approaches negative infinity is 0, and 1 as x approaches positive infinity. Fuzzy sets need not do this, such as the one above. Defining the truth-value to be the cumulative distribution function also restricts truth values such that their summation or classical integral for continuous truth values to always equal 1. There doesn't exist such a restriction in fuzzy set theory. The analogue here consists of talking about *normal* fuzzy sets which have a *maximum* height of 1. But, the sum of the degrees of membership here, don't equal one, unless the function only maps to a single point.

One could define the truth value of tallness like this:

m(x)={0 when x belongs to [0, 60]

{(x/12)-5, when x belongs [60, 72]

{(-x/12)+7, when x belongs to [72, 84]

Defining the truth-value to be the cumulative normal distribution function here simply doesn't work, since this function decreases.

In the world of active mathematicians, fuzzy logic has a bad name, because you cannot do much with it. The idea may look attractive, and lots of people may have been attracted to it initially, but the output is meager. Let me take an example: if I consider the predicates "tall" and "wealthy", both of which can be measured, and I want to talk about "tall and wealthy", fuzzy logic can handle that: if for a population t(x) is the truth value of being tall for individual x, and w(x) is the truth value of being wealthy for the same individual, then the truth value for being tall and wealthy is the minimum of t(x) and w(x) for this individual. But there are other possible definitions for the "and" logical operator in fuzzy logic.

But what then? Obviously, this kind of definition does not help in recognizing possible relationships between wealth and height. In countries where not everyone is well-fed, presently and in the past, tall people are usually richer than short people.

The present situation is that the field of fuzzy logic, and more generally fuzzy mathematics is very insular. It has been used in control theory, with many industrial applications, particularly in Japan. It is not clear that fuzzy control works better than genetic algorithms or neural networks, but the essential advantage of fuzzy control is that its language is closer to human language.

Being a numerical analyst and partial differential equations person, when I look at the kind of rules used in fuzzy logic, I tend to think that it looks very much like approximation, which is adjusted through a fitting strategy. There is nothing wrong with that. If it makes people happy and it works in industry, very good. I'd just like to see more classical explanations, because it would help break the insularity of fuzzy maths, and I am convinced that insularity is bad for science and its development.

I would venture the following sociological analysis: fuzzy math has an ideological side, and mathematicians do not like to mix ideology with serious business.

I second that entirely.

Well written introduction. I wonder if an introduction to logic is a good pre-cursor to understanding type theory and reasoning about types.

Tim - that's a bit deep for me but I don't see how you could ever evolve a complex creature that doesn't divide the world into blurred categories (things to eat, things to shag, things to run away from, things to shag _then_ eat etc.). So in that sense the blurriness is intrinsic to reality, insofar as "the evolution of complex creatures" is intrinsic to reality.

"the blurriness is intrinsic to reality"

So are you saying this is unambiguously

true? Where's the blurriness?Bollocks, Tim -- that's a cheap dodge. Interpret it as "the blurriness is (to some high degree) intrinsic to reality", where "intrinsic to reality" is a fuzzy predicate, and there's nothing inherently self-defeating.

And, of course, there are propositions in FOPL for which the claim that "all propositions are either true or false" doesn't work. That's the point of Goedel's Incompleteness Theorem--there are some for which we _cannot_ unambiguously say whether they're true or false. Where's the crispness?

Um, no. GIT says that there are propositions in a system that are TRUE, but that are not provable within the system.

"Bollocks, Tim — that’s a cheap dodge. "

Ok, then name a fuzzy truth - something that is ambiguous not because you haven't been clear what you're talking about, but because reality itself is ambiguous.

A fuzzy truth? "I am tall".

The moon is a big satellite.

The sun is a typical star.

I'm happy.

There are some things in reality that are crisp, and are accurately described by FOPL and its inference mechanisms. But there are also some things that *aren't*. There are a range of classical paradoxes that are based on the idea of vague predicates.

I live in a house. There's no vagueness to that statement. I'm employed by google. The earth orbits the sun. Those are statements that are unambiguously true.

I'm healthy. It's warm outside. Those are intrinsically vague. There is no crisp boundary between "healthy" and "non-healthy". There is no crisp boundary between "warm" and "not warm". You can't capture the essentially qualities of those predicates using a FOPL.

> Um, no. GIT says that there are propositions in a system that are TRUE, but that are not provable within the system.

No, both the truth and falsity of a Goedel statement is consistent with the system. Their truth value is indeterminate.

@Mark: "I’m healthy. It’s warm outside. Those are intrinsically vague."

Yes, the

wordsare vague. What does "healthy" mean? What does "warm" mean? (Also see my comment below. Just because humans have not given these everyday words rigorous definitions does not mean that boundaries in the real world are vague. Rather, it means that we use these words across a range of boundaries because we can. But the boundaries are still there! Or do you disagree that the law of excluded middle applies to reality?@Scott: I stand corrected. Their truth value is indeterminate, but that does not mean that they are not true or false - just that we cannot prove them so using that system.

Or do you disagree that the law of excluded middle applies to reality?The Law of the Excluded Middle is an artifice of the logical model you are using to describe reality, not a feture of reality itself.

I'm looking forward to this series as well. When everything is black or white TRUE or FALSE, logic is easy to calculate and it is exact (given that the initial statements are correct). If everything is black, white or medium gray TRUE , FALSE, MAYBE, it would seem like pretty much every fuzzy problem would result in an answer of maybe since yes && maybe == maybe, yes || maybe == maybe, no && maybe == maybe, no || maybe == maybe, etc.

For what it's worth, George Boole was the inventor of the typical TRUE / FALSE logic used all over today. He lived in the first half of the 1800's, dying at the age of 49 after succumbing to a bad case of the false.

This is just silly. Propositions are

notthe real world, but rather they are descriptions of the real world. These descriptions must be in some language and hence adhere to the rules of that language, be it English, French, Chinese, FOPL, or fuzzy logic. If it seems to you that the real world is made up of onlytrues andfalses, that's because your perceptions are being filtered by the FOPL colored glasses that you're wearing. The real world is no more "unambigious" as it is euclidean or cartesian. And just as there are occasions when it is preferable to use a non-euclidean geometry or a polar coordinate system, there are occasions when it is preferable to use fuzzy logic. It all depends on perspective.Actually, I think reality is even more weird. While math has been concerned mostly with hard truths, many category boundaries are malleable. Take, for example, a morphing sequence between the letter shapes H and A. As the parameter runs from zero to one,

there is no point that separates the classes H and A. The intermediate shapes show perceptual oscillations and hysteresis, but never, ever, I can *see* a shape between H and A. I do not see a way to apply probability or fuzzy logic to this situation.

Take a look at some Fraktur fonts. Ask people what letter is, or . Use that to determine a numerical answer to the question "to what extent is this shape a representative of this letter?" that lies between zero and one.

If Mark can add the image links to

http://www.library.yale.edu/cataloging/music/v01.gif

and

http://www.library.yale.edu/cataloging/music/s01.gif

that would be great.

Funny you should come up with Fraktur fonts. I'm building an OCR system...

Of course, your proposed experiment works. It will give hard numerical evidence for subjective belief. The price, however, is some averaging out over an ensemble of observers. As a single observer I experience not fuzzy membership but perceptual oscillations. At each instant in time there is a distiguished percept, and the transitions are fast. ( think multivibrator and feedback ).

But back to mathematics and fonts.

The boxed grayscale images of letters are points in a large topological ( metrical, vector) space X. A classifier is a map into a small ( finite ) topological space Y. An example is a point of X with prescribed image in Y. A good example has a (small) neighborhood where the classifier does not change. A set of examples gives a section in the sheaf of continuous functions X → Y. Perceptual learning consists of a sequence of modifications to this section. The state space of this dynamical system is the set of sections in the sheaf of continuous functions X → Y.

A global section, however, does not exist.

If we replace the indicator functions X → {0,1} by fuzzy membership functions X → [0,1] , we may glue to obtain a continuous function on the whole space X. If we build a classifier by choosing the class with highest membership, we have extended the initial germs given by examples to a dense open subset of X.

Sorry for this incoherent ramblings. I have built an OCR engine along these ideas, but I can't find a book that connects them to the mainstream. Do you know of any -- preferably elementary -- reference that connects fuzzy logic to the construction of topological spaces and maps between them?

Thanks.

"If it seems to you that the real world is made up of only trues and falses, that’s because your perceptions are being filtered by the FOPL colored glasses that you’re wearing."

Is this statement true or false?

"The real world is no more “unambigious” as it is euclidean or cartesian."

Is this unambiguously true?

“If it seems to you that the real world is made up of only trues and falses, that’s because your perceptions are being filtered by the FOPL colored glasses that you’re wearing.”Is this statement true or false?It is usefully descriptive.

“The real world is no more “unambigious” as it is euclidean or cartesian.”Is this unambiguously true?It is usefully descriptive.

Descriptive of what?

Reality?The word for "being descriptive of reality" is called "true."And if a statement isn't true, explain how it can be useful at all.

Descriptive of what? Reality? The word for “being descriptive of reality” is called “true.”It's descriptive of the out abiltiy to take different models and apply them to the same situation, to perform different evaluations upon them.

And if a statement isn’t true, explain how it can be useful at all.In this case, it is useful as as demonstrating how various models interact. It is not always true. When you are using a cartesian model, reality looks cartesian and does not look polar. When you are using a polar model, reality looks polar and not cartesian. So, it is true to say reality is cartesian when you are using a cartesian model, and it is true to say rality is not cartesian when you are using a polar model. Yet, your choice of a model does not actually change reality.

Tim,

You still don't seem to get it. When you ask if something is true or false or if it's ambiguous, you're referring to a

statementand not tothe real world. The "real world" is neither true nor false, or ambiguous nor unambiguous. Those qualities can only define statements within languages. The fact is that fuzzy logic can describe "the real world" just as well as FOPL; it just describes it from a different perspective. You are wrong (and that is both true and unambiguous)."You still don’t seem to get it. "

Certainly not! I have no idea what you're saying.

Statements describe reality? Ok, then it seems like a true sentence would describe reality

accurately. So when I say that "the Earth has one moon," that's "just" a statement, but if it's a true one I know that the white ball I can sometimes see in the sky is our moon and that there isn't a second one hiding out somewhere on the other side of the planet."The “real world” is neither true nor false"

Of course not. "The real world" isn't a proposition; it's a noun phrase. Statements

aboutthe real worldaretrue or false. Give me one that isn't.Incidentally, it seems you disagree with Mark, since he wrote: “The real world is full of things like that – properties that instead of having a crisp boundary between true and false, have a vague boundary” Sounds like he's talking about more than statements, no?

Re: "Statements about the real world are true or false. Give me one that isn't."

Tim, I can't take credit for this one as it's lifted directly from the original article... This is the sort of question that leads tech support folks to add "RTFM" next to your name in their database.

Try this statement: A person who is 5'11" is tall.

Is that true or is that false? According to you, it must be one, so which is it?

Now, assuming you have an answer to that question, please state exactly which measurement identifies someone as tall or short.

"This is the sort of question that leads tech support folks to add “RTFM” next to your name in their database."

In your case I would add "RMFFC," for "read my first f-ing comment."

"Try this statement: A person who is 5’11″ is tall.

Is that true or is that false? According to you, it must be one, so which is it? "

Define tall - I'll answer the question when I know what you're asking. Like I said, this is all about ambiguous definitions. It's as if you said "A person who is 5' 11" is snarfblat- true or false?" My first response would be to ask, "what does snarfblat mean?" Of course I can't answer if I don't know. You think you've asked a reasonable question just because you've used a real word - but it's still a word with multiple meanings, and therefore I cannot answer until I know what you're asking.

We often use "tall" to mean "taller than average" or "taller than the speaker expected." If you're using the first definition, then if 5' 11" is taller than average for a human, the statement is unambiguously true. If you're using the second definition, the same thing applies. At 5' 11", Tom either is or is not taller than I expected (I would say not, unless someone suggested to me before I met him that Tom was short).

Tim: "taller than average" is measurable and precise. "tall" is a relative term, is ambiguous, and is fuzzy. we agree. cool.

Too bad this series starts just now: On monday, I will have to take an examination on Fuzzy Sets and Fuzzy Logic....

In partial defense of Tim, it's actually pretty hard to make out how there could be vagueness "in the world" as opposed to vagueness "in our concepts." Recently, Elizabeth Barnes (http://www.philosophy.leeds.ac.uk/Staff/az/Elizabeth_Barnes.htm) and JRG Williams (http://www.personal.leeds.ac.uk/~phljrgw/) have tried give a rigorous account of what worldly vagueness would be, but--as with almost everything in philosophy--not everyone's convinced. For the contrary view--and indeed, for the best introduction to the topic of vagueness--see Timothy Williamson's book Vagueness.

I have a dumb question. What's the difference between logic and reason? The terms have always seemed interchangeable, splitting hairs. Also, is there such a thing as "fuzzy reason"?

I think better fuzzy concepts than "tall" or "short" would be "loud" or "soft". These two concepts are so contextually driven that the same person will perceive the same sound as "loud" or "soft" in differing situations.

This is fascinating. I read and learn here from a lit major's perspective. Tim seems to believe in an ultimate reality. I understand his argument, yet he seems to fall into reification, taking an abstract logic for a "real" description of the world.

But in order to use his version of logic, we must make absolute and somewhat arbitrary definitions. FOPL is only one way to describe the world and manipulate statements. Not the "best" way, just one way.

And in the vein of Vhurtig's "loud" and "soft," how would anyone define pain? Which is different for different people at different times under different circumstances and mental states.

Sure, English is ambiguous, that's one of its pleasures. And a logic to manipulate ambiguity is not a "lesser" logic.

Am I following this accurately?

"taking an abstract logic for a “real” description of the world. "

Yes, the reason we use logic is because it

doesmesh with the real world. You know, like math - which is just logic in numerical form. It says crazy "abstract" things like 1+1=2. You're saying it's a mistake to think this is accurate in the real world?"But in order to use his version of logic, we must make absolute and somewhat arbitrary definitions. "

Symbols

arearbitrary; that's why they're useful as symbols. There's no non-arbitrary reason why a horizontal line on top of a vertical line should symbolize a T sound, and no reason why the sound pattern "tree" should symbolize those green things outside - these are completely arbitrary pairings between shape and sound, or between sound and meaning. What's important is that when I say "tree," you know that there's this category of thing that I'm talking about, and we can share information about it.What some people here are trying to say, is that when somebody says "tall," you don't know exactly what is meant by that, and therefore there's some kind of vagueness inherent in reality... when

reallywhat it means is that the sound pattern "tall" happens to be paired with anumber ofmeanings, and so you can't always be sure exactly what someone meant unless you ask them. The fact that youcanalways make your definition explicit is proof that there is no limitation on the specificity of reality - just limitations on our poor monkey brains when it comes to dealing with complexity.In what context? Under what circumstances? As John Allen Paulos points out, one cup of water plus one cup of popcorn does not equal two cups of soggy popcorn.

"Under what circumstances?"

You seem to be suggesting that this is only true under certain circumstances, and therefore... what? Math doesn't tell us accurate things about the real world? Sure it does - what

everthe appropriate circumstances are, 1+1=2 is an accurate statementfor those circumstances.It's as if I said, "given the axioms of euclidian geometry, the angles in a triangle add up to 180 degrees," and you said "but that's not true for elliptic geometry!" Right, because I said this only worksgiven certain circumstances. And under those circumstances, our statements are accurate.Ah, so you agree that some mathematics model certain aspects of physical reality, but the mathematics are not the reality themselves? Isn't it the case that the models are only approximations, whose applicability varies?

Maybe a model is very good at describing one physical experiment, but only moderately good at describing another one. Newtonian gravity is a very good model of celestial mechanics, but general relativity is an excellent one. "Dropped objects fall at the same rate" is more true on the moon than it is on Earth.

And anyway, why isn't language "real"? What's so offensive about the idea of modeling language as it is actually used? Engineers make as much use of verbal descriptions as of mathematical models. Why not try to come up with a way of translating the verbal descriptions humans use to communicate about the real world more directly into a branch of mathematics which can model and control the reality referred to by said verbal descriptions?

"so you agree that some mathematics model certain aspects of physical reality, but the mathematics are not the reality themselves?"

Of course. I'm not a Pythagorean; I don't believe that numbers

arereality, whatever that would mean. Throughout this thread I've said that FOPL is anaccuratedescription of reality, at least as far as our minds can conceive of reality. No where have I said that reality IS logic, or reality IS math."Isn’t it the case that the models are only approximations, whose applicability varies?"

I believe we've been talking about "laws" more than "models," so that's the word I'm going to use. Humans strive to discover what the actual, correct laws of nature are. Many times our theories are only partially correct, but we contnue doing science under the assumption that

somethingis correct and that we can eventually figure out what it is. The second law of thermodynamics is correct. 1+1=2 is correct for whatever conditions it is meant to work under (which is basically the same thing as simply stating "it is correct," because no theory or law is meant to apply to everything - everything has conditions, just as the 2nd law of thermodynamics doesn't apply to economics). The point is these things accurately describe reality, as far as we know, so if you're saying thatnothingis accurate, then you're both wrong and contradicting yourself, because it you cannot claim it isaccurateto say thatnothing is accurate."And anyway, why isn’t language “real”? What’s so offensive about the idea of modeling language as it is actually used?"

Who's putting words in whose mouth now? I never complained about modeling language, nor said it wasn't real. I said that properties such as "tall" only lack crisp boundaries because we haven't defined crisp boundaries for the word. This is no reflection of "reality," just the way humans tend to think about reality. And still no one has answered my challenge: If you want to know whether a person who is 5' 11" is tall, just define "tall," and I will happily give you a Yes or No answer every time. This is because reality does not limit me in answering the question;

not knowing what you're askingbecause the word has a vague definition (or too many definitions) is what limits me, just as you would be limited if I asked you whether a 5' 11" person is "snarfblat."Tim et alia,

Why do you keep asking for someone to define "tall" for you instead of doing it for yourself? Why don't you propose such a definition and then attempt to show that it has no ambiguity here?

Since you've made such a big deal out of defining "tall", I'll give _a_ definition here: A person who stands at 60 inches or less is not tall. I'll denote a person who stands at 60 inches as not tall by (60, 0). A person who stands at 72 inches or more is tall. I'll denote a person who stands at 72 inches as tall by (72, 1). I'll conclude this definition by saying that we have a linear, increasing function which maps from [60, 72] into [0, 1] with (60, 0) as the least point of the function and (72, 1) as the greatest point of the function (since we can represent such by a line on a coordinate graph, I can order all of the points exactly as if it had one dimension... so I can talk about all of its points in this way).

There exists only one such function, so I can summarize this definition by writing

m(x)={0 when x belongs to [0, 60]

{(x/12)-5, when x belongs [60, 72]

{1 otherwise.

Now, since m(60)=0 means not tall here, and m(72)=1 means tall here, what does... *unambiguously* m(66) mean in terms of tallness? According to my definition here, please answer, as you requested of definitions in general, YES OR NO to the following question: according to this definition above, is a person who stands at 66 inches tall? Please answer my question here *happily* as you promised you could do also.

They've got those charts with faces on them in the doctors' offices and everything; how much more precise do you want?

Tim,

That math is just classical logic in numerical form is an outdated philosophical thesis called 'logicism'. Principia Mathematica didn't end up proving what the authors of it wanted it to do. At least Jan Lukasiewicz and presumably some of his students had an intimation that the logicist position wasn't necessary BEFORE Godel, since Lukasiewicz published his "A Numerical Interpretation of the Theory of Propositions" in 1922, where some of the propositions end up having a truth value of 1/2.

"“taking an abstract logic for a “real” description of the world. ”

Yes, the reason we use logic is because it does mesh with the real world. You know, like math – which is just logic in numerical form. It says crazy “abstract” things like 1+1=2. You’re saying it’s a mistake to think this is accurate in the real world?"

If classical "abstract" logic does mesh with the real world, then all classical logic statements mesh with the real world and are either true or false. Consequently, ONE counterexample will disprove any conjecture. So, if 1+1=2 is a statement of classical logic, then 1+1=2 gets CATEGORICALLY disproved as matching the real world by a single counterexample, and thus ends up false. So, 1 table on top of another table yields 2 tables. But, you can't use both tables at the same time, so you really only have one table, even though you still have two pieces of wood.

Plenty of other examples can get supplied which refute 1+1=2 if it's a statement in classical logic *and* it meshes with the real world. The problem here comes as the unproven, unprovable, and problematic assumption that math is just logic in numerical form. 1+1=2 holds, DESPITE the fact that real-world counterexamples exist to it, precisely because it doesn't concern all of reality which classical logic would suggest, and only meshes with *parts* of reality.

"Symbols are arbitrary; that’s why they’re useful as symbols. There’s no non-arbitrary reason why a horizontal line on top of a vertical line should symbolize a T sound, and no reason why the sound pattern “tree” should symbolize those green things outside – these are completely arbitrary pairings between shape and sound, or between sound and meaning."

No, because all of us exist inside a social context of some sort. The social context here comes as that of using the English language, or something close enough to English that English language readers can understand what the author means by his/her writing. In the social context here that T should symbolize a T sound comes as non-arbitrary for those readers who speak to other people in English, so that they can have the possibility of communicating some of this discussion to someone else in real life. The same holds for "tree". Those sounds DO work "arbitrary" in other contexts, but not for a native English speaker here.

"What’s important is that when I say “tree,” you know that there’s this category of thing that I’m talking about, and we can share information about it."

Nope, the person you've talked to MIGHT refer to a specific object external to us, and not a category of those objects.

"What some people here are trying to say, is that when somebody says “tall,” you don’t know exactly what is meant by that, and therefore there’s some kind of vagueness inherent in reality… when really what it means is that the sound pattern “tall” happens to be paired with a number of meanings, and so you can’t always be sure exactly what someone meant unless you ask them. The fact that you can always make your definition explicit is proof that there is no limitation on the specificity of reality – just limitations on our poor monkey brains when it comes to dealing with complexity."

We don't have monkey brains, and not even a biologist thinks so exactly, because they classify as hominids, which doesn't mean the exact same thing as the common term "monkey". Also, let's at least try to get a better hand on "tallness" here by considering something more concrete.

Say you end up in a room with Bob, Betsy, Steve, and Sue. Bob's height got *measured* at 5"5', Betsy at 5"2', Steve at 5"6', and Sue at 5"1'. Now, physics textbooks that I've seen these days will tell you that there exists uncertainty in measurements like that, because the ruler or other measuring device used only has a finite amount of precision. Now, in this imagined room, is Bob tall, Betsy short, Steve tall, Sue short? I can't tell. Why? Because you're in the room, and I have no clue how tall you are, nor how you would perceive these people in terms of their tallness. I don't want to prescribe to you how to view tallness here either. Even if I ask you to describe your perception here, that doesn't mean you'll necessarily tell me your real perception. On top of this, you might not even know how you would actually perceive this sort of situation in a real-world context, because you simply don't know what sorts of other factors around this. Even given previous conscious perceptions here, you might still change such later.

So, I don't want a binary language, like that of classical logic, to talk about such a situation, because I don't know enough about you, and so that you can change your description of tall people in the room if you learn something new about yourself in the future, or you change in this respect. I don't see why we want an untenative description here, where perhaps we would write {(Bob, 1), (Sue, 0), (Betsy, 0), (Steve, 1)} where 1 corresponds to x being tall and 0 corresponds to short. Instead, it makes more sense to have a description which can easily get modified if new information comes to light which indicates that you want to or should change it. If we instead write {(Bob, .6), (Sue, .1), (Betsy, .2), (Steve, .5)}, where .5 indicates that Steve has .5 degree of tallness (or is viewed as .5 by our hypothetical observer), it comes as much more clear that we can change these numbers if needed to better describe our perception, and that such change may come as desireable, since we have little pretension to have the absolute truth of the situation, as any description suggested by classical logic *alone* would give us.

Not that anyone cares about the opinion of a long-time lurker, but I mostly agree with Tim on this. "Tall" is indeed just a sorites paradox, which is just a problem of vagueness. This has been said several times already in this exchange, so I'll approach it in a different way.

Language is how we communicate our experience of this so-called reality. The better the communication, the better the person on the receiving end can simulate the communicated experience in their own mind to approximate the original experience. All communication between people (and within our selves and our own memories) is lossy.

Fortunately, we each have enough similar experiences, that this loss is often okay. If we both already know how a particular tune sounds, you can mention the name or hum a few bars from the melody, and I already know how the percussion and harmony and vocals go; you don't need to communicate the entire thing to me.

However, for every word in our shared language, there is a discrepency, however miniscule, in the symbol(s) it maps to in each of our brains. Usually it's irrelevant, but it's there, a collective Venn diagram of what a word in the population actually means in space and time. Some words, like "chair" and "sky" and "apple", have so much universal agreement, that we apply FPOL without conflict. Other words, like "tall" and "smart" and "sexy", have less agreement according to each of our individual experiences.

Every single word has a probability distribution of meaning. The "best" words have small distributions inside, say, a bulls-eye, and the "worst" words--the vague and equivocated--have distributions all over the board. Usually the more common a word, the smaller its distribution and the better the communication, but not always; "god" is the perfect example of an exception to that.

In short, I agree that the problem is not that reality is fuzzy, but that our communication is fuzzy. Necessarily, my immediate recall of an event is severely compressed, otherwise my brain would quickly run out of memory; even storing just the differential changes in my environment ("internal" and "external") as referenced to some original perfect snapshot. My communication of this event is even moreso. My communication in this post, likely more still.

In short short, I would say that fuzzy logic models how we communicate about reality, not reality itself. Even then, however, it seems a bit odd to me to assign "He is tall" a truth value when the meaning of "tall" is widely distributed. In other words, without a context. "He is tall according to 55% of the population of America", that makes more sense.

That will only work if communication doesn't happen in reality. But, if it doesn't happen in reality, then such communication happens only in imagination. Then, we aren't really having a conversation or communicating anything, we just believe we've communicated something to each other. But, undoubtedly, you read a re-production of computer symbols which I wrote on a computer to get transmitted. So, something, even if seemingly small and insignificant, got communicated in reality. Consequently, at least this communication happens in reality. The same argument form works for all other communications, though computer symbols may change to sound symbols or visual symbols. So, all communications happen in reality. Therefore, even if fuzzy logic only models how we communicate about reality, it STILL models something about many aspects of reality itself, since at least many communications happen in reality.

Though my opinion is that if the meaning of a word doesn't have 95% agreement among the people using it, then it should be tossed, and new words invented to cover the range of disagreement. 😛 If only language were invented so deliberately.

This post seems to only deal with fuzzy logic in what Zadeh has called "the narrow sense." In a more wide sense fuzzy logic refers to any use where fuzzy sets have gotten used. I certainly don't see how fuzzy logic in the broad sense comes as mathematically constraining. People have started to develop fuzzy (or graduated) geometries, fuzzy ring theory, fuzzy group theory, fuzzy topology, fuzzy systems of linear equations (which seems like basic linear algebra), fuzzy differential equations, fuzzy graphs, fuzzy probability theory, etc. at a mathematical level (Buckley, Kaufmann, Gupta, Klir, Yuan, Zadeh are some authors names to search). Do there exist extensions of these and other mathematical fields to a potentially-infinite valued domain in mathematics based on classical logic or classical set theory?

Even in fuzzy logic in the narrow sense, statements can STILL be strictly true or false. To clarify this, no serious author that I know of (and I doubt you'll find one), represents the truth values for a fuzzy logic on an open interval. They all define those truth values on a closed interval or lattice or partially ordered set, where the endpoints (or their lattice-theoretic analogues or partially ordered set analogues) behave just like F and T in classical logic. Usually they represent truth values as taking truth values in [0, 1], NOT (0, 1).

The difference lies in the expectation. In classical logic one could expect propositions to either come as true or false (at least eventually). In fuzzy logic, only in rare or special cases do propositions have truth values of 0 or 1, since there exists at least a potential infinity of truth values other than 0 or 1 for statements.

I once searched possible truth tables for a three-valued logic that satisfied certain contitions (eg: [(a implies b) and maybe a ] implies maybe b). I found two logics different in one cell of the table. Must see if I can dig it up again.

People who've studied three-valued logic thoroughly (not just one author's 3-valued logic such as Kleene, Bochvar, Heyting, Reichenbach, or Lukasiewicz, according to Klir and Yuan's: Fuzzy Sets and Fuzzy Logic: Theory and Applications) have found this phenomenon popping up all over the place. It doesn't happen in just an implication table, but in any truth table... even logical equivalence, as authors haven't agreed on how logical equivalence works when we have the third term at work, which one can denote as 1/2. Fuzzy set theory has made this simple to see without the technical arguments which some of those authors may have used to establish their tables (I know Kleene tried to give reasons, I don't know about the others, though I doubt any of them thought such arbitrary).

Basically in fuzzy set theory, one *can* (fuzzy set theory, of course, does more than this) set up arithmetical functions to model how classical truth values behave if 0 gets taken as meaning false and 1 gets taken as meaning true. All these functions can get represented in a 3x3 relational matrix (or table). So, we have 9-4=5 spots open for truth values. So, there exist 3^5=9^2*3=81*9=243 possibilities for 3-valued logic tables for... here's the catch... *any* logical connective. So if we have 5 basic logic connectives in the rules of inference of a logic, such as AND, OR, NOT, IMPLIES, EQUIVALENCE, then there exist 243^5 which gives us slightly more than 800 billion possibilities for truth tables. If we just have AND, OR, NOT, as most logic text authors seem to prefer, we have 243^3=14,348,907 possibilities for 3-valued logic tables. Even though those numbers aren't technically infinite, they end up so large to most ways of thinking, that this perhaps explains how fuzzy set theory has helped multi-valued logics to become more studied.

To get a hint of this phenomenon *without* having to dig anything up, just use 1-a for NOT a, and look at how the following four functions for a AND b differ in a three-valued truth table:

min(a, b), max(0, a+b-1), ab, {a if b=1, b if a=1, 0 otherwise},

for a OR b compare max(a, b), min(1, a+b), a+b-ab, {a if b=0, a if b=0, 1 otherwise}.

If you want to see how implication can differ here, select an AND function, an OR function, and use NOT a OR b for implication, as well as possibly NOT(a and NOT b).

For the doubters, here's a mathematical *statement* which can't hold as true in two-valued logic, but can't end up false either:

Usually infinite series are divergent. This implies "infinite series diverge" as a true statement, but not a wholly true statement. By usually here, I mean substantially more than "most" but less than all.

Argument: That some infinite series converge immediately implies that less than all infinite series diverge. The sum of an infinite series, by definition, is what number we would get were it the case that we could add all numbers of a infinite sequence. If an infinite series has a sum, its corresponding sequence converges to 0. So, all convergent infinite series can get placed into a one-one correspondence with their corresponding convergent infinite sequences (not onto!). Now take the reciprocal of each term of each of these sequences and form a second set of sequences S2. Each member of this sequence diverges. Each of these sequences corresponds to a divergent infinite series. Thus, there exist at least as many divergent infinite series as convergent infinite series. Since 1+1/2+1/3+...+1/n+... diverges, there exists at least one more divergent infinite series than convergent infinite series. So, more than most infinite series diverge. That we have *substantially* more infinite series which diverge comes as evident from the class of infinite series r-r+r-r+..., all of which diverge, and the reciprocal series (we reciprocate each term of the series and then sum them in the same order as before) of this class of infinite series also diverge. So, more than most, and substantially more than most, but less than all, that is, usually infinite series diverge.

So, if one says "infinite series diverge" one has a mathematical statement neither true, nor false in the classical sense, but still substantially more true than false. If what gets argued for here for aren't statements, then what in the world does "statement" refer to?

Of course, I mean *real* infinite series here.