## March 15, 2014

### Fuzzy Logic and Enriching Over the Category [0,1]

#### Posted by Simon Willerton

Standard logic involving the truth values ‘true’ and ‘false’ can make it difficult to model some of the fuzziness we use in everyday speech. If you’d bought a bike yesterday then today it would be truthful to say “This bike is new”, but it wouldn’t be truthful so say it in 20 years’ time. However, between now and then there won’t be a specific day on which the statement “This bike is new” suddenly switches from being true to being false. How can you model this situation?

One approach to modelling this situation is with fuzzy logic where you allow your truth values to be things other than just true and false. For instance, you can take the interval $[0,1]$ as the set of truth values with $0$ representing false and $1$ representing true. So the truth degree of the statement “This bike is new” would vary, being $1$ today and decreasing to something very close to $0$ in 20 years’ time.

This post is an attempt by me to understand this fuzzy logic in the context of enriched category theory, in particular, using $[0,1]$ as a monoidal category to enrich over. We will see that categories enriched over $[0,1]$ can be interpreted as fuzzy posets or fuzzy preorders.

This was going to be a comment on Tom Avery’s Kan Extension Seminar post on Metric Spaces, Generalized Logic, and Closed Categories but grew too big!

I have been trying to understand a bit of fuzzy logic because it feeds into fuzzy concept analysis which I’m also trying to understand. As ever, I understand the category theory better than the logic. In terms of references, I’ve picked up bits and pieces from Belohlavek’s Concept lattices and order in fuzzy logic and Hájek’s Metamathematics of fuzzy logic.

### Truth degrees in $[0,1]$: fuzzy logic

As mentioned above you can replace your set of truth values $\{ \mathrm{false},\mathrm{true}\}$ by the interval $[0,1]$, with $0$ representing completely false and $1$ representing completely true. Thus an element of $[0,1]$ represents a ‘degree’ of truth. The idea is that this represents the degree of truthfulness of statements like “George is old” or “The film is good”: it is not supposed to represent than the probabitlity of the truth of a statement like “I will receive an email from a student tomorrow”. We are modelling vagueness and not probabilty. Also, we are supposed to be modeling ojectivity so the truth degree of the statement “George is old” will depend on the age of George and not on anyone’s opinion of how old he seems to be.

Being category theorists, we might try to fit this into Lawvere’s framework of generalized logic, where we think of our truth values as forming a closed monoidal category. We can make $[0,1]$ it into a poset — and hence a category — by using the order $\le$, this will be our notion of entailment. So if $P$ and $Q$ are statements then $P\vdash Q$ precisely when the truth degree of $P$ is less than or equal to the truth degree of $Q$. However, I’m not entirely sure how to interpret that.

The observant amongst you will have noticed that this poset $[0,1]$ is isomorphic to the poset we use for metric spaces $([0,\infty ],\ge )$, via the maps $x\mapsto \exp (-x)$ and $a\mapsto -\ln (a)$. So an alternative interpretation of these truth values will be as ‘proximities’ — proximity approximately $0$ meaning not at all close, and proximity approximately $1$ meaning really, really close. [Switching from distances in $[0,\infty ]$ to proximities in $[0,1]$ is an important step in calculating magnitudes of metric spaces, but I won’t say any more about that.]

Next we want some notion of ‘conjunction’ $\otimes$ and ‘implication’ $\Rightarrow$. In category theory terms we want a closed monoidal structure on the category $[0,1]$. There are at least three such structures that are well studied.

The product structure: Here $a\otimes b \coloneqq a\cdot b$ and $a\Rightarrow b \coloneqq 1$ if $a\le b$ and $b/a$ otherwise. Via the exponential and logarithm maps this corresponds to the usual closed monoidal structure on $[0,\infty ]$ of addition and truncated subtraction.

The Gödel structure: Here $a\otimes b \coloneqq \min (a,b)$ and $a\Rightarrow b \coloneqq 1$ if $a\le b$ and $b$ otherwise. Via the exponential and logarithm maps this corresponds to the closed monoidal structure on $[0,\infty ]$ which gives rise to ultrametric spaces.

The Łukasiewicz structure: Here $a\otimes b \coloneqq \max (a+b-1,0)$ and $a\Rightarrow b\coloneqq \min (1-a+b,1)$. Via the exponential and logarithm maps this corresponds to the monoidal structure $x\otimes y\coloneqq -\ln (\max (e^{-x}+e^{-y}-1,0))$ and that doesn’t look at all familiar!

If the truth degrees of two statements are $P$ and $Q$ then the truth degree of both statements together should be given by the conjunction $P\otimes Q$, but I don’t know examples of real world models with these three different logics. Anybody? I would be very interested in hearing thoughts and ideas on this.

In all of the above three cases the closed monoidal category $\{ \mathrm{false},\mathrm{true}\}$ of classical truth values embeds closed monoidally.

### Enriching over $[0,1]$: fuzzy preorders

We can now think about what categories enriched in $([0,1],\le )$ are, as generalizations of preorders. Such an enriched category consists of a set $C$ and to each pair of elements $c,c'\in C$ we associate a truth degree $C(c,c')\in [0,1]$, we want to think of this as the degree to which $c$ is ordered before $c'$. This is what we will call a fuzzy preorder. It will satisfy fuzzy transitivity: $C(c',c'')\otimes C(c,c')\le C(c,c'')$. How this is actually interpreted will depend on which product $\otimes$ we took.

Here’s an example of a fuzzy preorder. Suppose that Bart is aged 9, Marge is aged 39 and Homer is aged 40. Is it true that Homer is (at least) as old as Marge? Yes. Is it true that Marge is (at least) as old as Homer? Well, nearly. Is it true that Bart is (at least) as old as Marge? Well, not really at all. We can formalise this into a fuzzy preorder $\succeq$ “is at least as old as” where \begin{aligned} (\text {Homer}\succeq \text {Marge})&=1 \\ (\text {Marge}\succeq \text {Homer})&=\text {a tiny bit less than}\, \, 1\\ (\text {Bart}\succeq \text {Marge})&=\text {much less than}\, \, 1. \end{aligned} Algebraically we could define $A\succeq B\coloneqq \begin{cases} 1 &\text {if}\, \, \mathrm{age}(A)\ge \mathrm{age}(B)\\ \mathrm{age}(A)/\mathrm{age}(B)&\text {if}\, \, \mathrm{age}(A)\lt \mathrm{age}(B) . \end{cases}$ This gives a fuzzy preorder on the set $(\text {Bart}, \text {Marge}, \text {Homer})$ with respect the product monoidal structure $\cdot$ on $[0,1]$, or in other words, it gives a category enriched over the monoidal category $([0,1],\cdot ,1)$.

There should be some more convincing examples, but I haven’t found any yet.

An exercise at this point is to look at the dictionary in my post on Galois correspondences and enriched categories and extend it by another column for categories enriched over $[0,1]$.

Posted at March 15, 2014 5:51 PM UTC

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### Re: Fuzzy logic and enriching over the category [0,1]

I don’t claim to understand fuzzy logic, but there is a renewed interest in such issues in effect algebra of predicates.

Steve Vickers has work on fuzzy sets and geometric logic.

Both are at least in a categorical framework.

Posted by: Bas Spitters on March 15, 2014 9:13 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Thanks Bas. Unfortunately, I know very little topos theory, so can’t really try to tie any of that in.

Posted by: Simon Willerton on March 17, 2014 10:27 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

I suspect that as some sort of offshoot of the microcosm principle, certain forms of logic can only be understood using metamathematics based on similar forms of logic.

For example, people who are serious about understanding topos theory often like to use intuitionistic logic as part of their metamathematics. And once on this blog we saw a fistfight that started when someone claimed ultrafinitism is ‘incoherent’—because they were using a metamathematical framework in which one could prove there were infinitely many well-formed formulas!

It seems fine to study system A using metasystem B if we can, just to see what happens—but it seems unfair to claim system A is ‘bad’ because metasystem B adopts radically different principles. So perhaps we should instead develop an ultrafinitist metamathematics to study ultrafinitist mathematics, and so on. In intuitionistic logic we can already start to imagine a topos object in a 2-topos object in a 3-topos, etcetera—though I haven’t seen people carry this very far in a systematic way.

Anyway, all this is leading up to a kind of joke theory, which may actually be serious: to understand fuzzy logic, we need fuzzy thinking!

Or in other words: there is no way to reason precisely about vagueness, so we should reason about it vaguely!

Unfortunately, I’m not sure how to make this idea precise…

Posted by: John Baez on March 17, 2014 6:23 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

The best description of mathematics that I can think of is “the precise study of precise ideas”. The apparent necessity of saying “precise” twice backs up your point, John.

It’s no coincidence that mathematics exams are more amenable to numerical marking than exams in the humanities.

Posted by: Tom Leinster on March 17, 2014 7:19 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Nice joke John, but I don’t think that’s the case here. Fuzzy logic doesn’t say you can’t have things being definitely true or definitely false, just that sometimes things aren’t so black and white.

I think it is standard to use classical logic in the metamathematics, indeed on page 6 of Gottwald’s A Treatise on Many-Valued Logics (see this 600 page download) he says the following.

Our metalanguage which we use to discuss systems of many-valued logic together with their formalized languages is English combined with a portion of formalized [classical, two-valued, first order logic] which usually is used in a semiformalized manner.

Posted by: Simon Willerton on March 17, 2014 10:38 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Well, intuitionistic logic doesn’t say you can’t have decidable propositions either, just that sometimes things aren’t decidable. And just because an approach is standard doesn’t mean it’s necessarily best… (-:

Posted by: Mike Shulman on March 18, 2014 1:03 AM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Among philosophers, the range of opinions on how to interpret vagueness is “desperately wide” (for the quote see the last sentence of section 3). In fact one of the important issues is “higher order vagueness”: if $P$ is a vague predicate, then it might be clear for some $x$’s whether $P(x)$ holds, but there may be other $x$’s for which it’s not clear whether $P(x)$, and still other $x$’s for which it’s not clear whether it’s clear… Perhaps this could be interpreted as vagueness invading the metalogic.

Simon, what sort of thing are you asking for when you ask for real-world models? Are you asking for a set of propositions and some sensibly-defined $[0,1]$-valued function on them which sends conjunction to $\otimes$?

Many-valued logic such as we’re discussing is the simplest of the philosophical approaches to vagueness discussed in the SEP article linked to above. The article seems kind of dismissive of it as an approach to vagueness. But on the other hand, I have the sense that formal fuzzy logic systems are actually used by engineers for system design and machine learning, so they must have some kind of semantics worked out and examples aplenty – right?

Posted by: Tim Campion on March 18, 2014 5:44 AM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

what sort of thing are you asking for when you ask for real-world models? Are you asking for a set of propositions and some sensibly-defined $[0,1]$-valued function on them which sends conjunction to $\otimes$?

Yes, exactly.

We can check that classical logic agrees with our everyday experience by giving some propositions “It is raining”, “I had eggs for my breakfast”, and comparing the truth value of the conjunction of these statements “It is raining and I had eggs for my breakfast” with the truth values of the individual statements to see that

• the truth value of “It is raining and I had eggs for my breakfast” depends only on the individual truth values of “It is raining” and “I had eggs for my breakfast”

• as a function of the two truth values the truth value of the conjunction is given by the logical operation AND (which we can call $max$ if we don’t want to overuse the word ‘and’).

In the fuzzy case, I would like similar set of propositions and a truth table, with any of the candidate operations for $\otimes$, to show how this logic is modeling some reasoning from the ‘real world’ (!)

Posted by: Simon Willerton on March 19, 2014 11:05 AM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

This “truth-functional” property – that the truth value of “A and B” depends only on the truth value of “A” and the truth value of “B” – sounds like an important restriction. For example, we can’t take the truth value of A to be the “probability that A is true” – because the probability of “A and B” does not follow just from the probability of A and the probability of B when A and B are correlated. This ties in nicely with your comment below that fuzzy logic is not about uncertainty.

In the proto-example you mention below, we have two atomic propositions, “The room is too hot” and “The engine is not too hot”. To turn this into a fully-realized model, we need to choose two functions $f: \{\text{room temps}\} \to [0,1]$ and $g: \{\text{engine temps}\} \to [0,1]$. Then once we choose a function $\otimes: [0,1] \times [0,1] \to [0,1]$ modeling “and”, we get a definition of a function $f \otimes g: \{\text{room temps}\} \times \{\text{engine temps}\} \to [0,1]$ which defines a truth value for “the room is hot and the engine is not too hot”.

If the goal is to use this proposition to trigger turning on the AC (you only want AC when it’s hot and the AC puts extra strain on the engine), then maybe in this case you want to use the Gödelian $\otimes$? You have to be careful that the functions $f$ and $g$ are properly “calibrated” to one another…

There are a lot of degrees of freedom inherent in the choice of functions $f$ and $g$. And offhand, it sounds like the right approach for engineering applications would be to select an $\otimes$ function on a case-by-case basis depending on the context the proposition is being used in – which leaves even more degrees of freedom. It’s not even clear to me, for example, that “A and B and C” should be modeled by $[ A ] \otimes ([ B ] \otimes [ C ])$ rather than by using some ternary $\otimes$ created de novo. I suppose there are probably certain formal properties enjoyed by the standard choices for $\otimes$ which can help structure these choices. Maybe there is a more principled way to do things?

Posted by: Tim Campion on March 19, 2014 4:50 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Just a wee comment on the Łukasiewicz structure: as you may have observed, it simplifies if we conjugate by $u \mapsto 1 - u$.

Specifically, for $u, v \in [0, 1]$, define

$u \square v = \min\{u + v, 1\}, \qquad u \to v = \max\{v - u, 0\}.$

This is a monoidal closed structure on $([0, 1], \geq)$. Transporting it across the isomorphism $u \mapsto 1- u$ gives the Łukasiewicz monoidal closed structure on $([0, 1], \leq)$:

$a \otimes b = 1 - ((1 - a) \square (1 - b)), \qquad a \Rightarrow b = 1 - ((1 - a) \to (1 - b)).$

But I don’t have a good explanation for the origins of $\square$ and $\to$.

Posted by: Tom Leinster on March 17, 2014 8:58 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Tom said

as you may have observed, it simplifies if we conjugate by $u\mapsto 1−u$

Well maybe I had observed it subconciously as it is shouting out for this. Anyway, it seems (according to Section 9.1 in Gottwald’s book) that the connectives in the Łukasiewicz system are naturally derived from the definition of implication. So start, for some reason, with the function corresponding to implication:

$a\rightarrow_{\L} b \coloneqq \min (1-a+b,1).$

We can define negation using this: $\not a \coloneqq a\rightarrow_{\L} 0.$

So that gives $\not a = 1-a.$

Now ‘weak’ disjunction and ‘weak’ conjunction are defined to satisfy the following relations taken from classical logic.

$a\vee b\coloneqq (a\rightarrow_{L} b)\rightarrow_{\L} b \qquad \text{and}\qquad a\wedge b\coloneqq \not(\not a \vee \not b).$

That gives

$a\vee b = \max(a,b)\qquad \text{and}\qquad a\wedge b = \min(a,b).$

However, classical disjunction and conjunction also satisfy other relations and so ‘strong’ disjunction and ‘strong’ conjunction are defined to satisfy the following relations:

$a\veebar b\coloneqq \not a\rightarrow_{\L} b \qquad \text{and}\qquad a\& b\coloneqq \not( a \rightarrow_{\L} \not b).$

That gives

$a\veebar b = \min(1,a+b)\qquad \text{and}\qquad a\& b = \max(a+b-1,0).$

It then turns out that $\&$ and $\rightarrow_{\L}$ give us our closed monoidal structure.

This still doesn’t tell me why this gives us a useful notion of conjunction, though!

Posted by: Simon Willerton on March 23, 2014 12:08 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

What is the motivation for assuming that the poset of truth degrees is totally ordered?

Posted by: Mike Shulman on March 18, 2014 1:09 AM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

I have trouble believing there’s some deep justification for requiring the truth degrees be totally ordered. There could be incomparable kinds of truthiness, like “nine out of ten doctors say” versus “I read it on Wikipedia”.

On the other hand, I think it’s valuable to carefully explore what we can do using the toset $[0,1]$ as your set of truth degrees. It’s just such a great example, after all.

(Here I’m using the word toset to mean ‘totally ordered set’, just as poset means ‘partially ordered set’. Why don’t people do that more?)

I suspect that people in fuzzy logic are drawn to using $[0,1]$ mainly because everyone knows this set and a bunch of operations on it.

However, in Oxford I talked to two linguists named Ash Asudeh and Gianluca Giorgolo, who wrote a paper called ‘One semiring to rule them all’. This studies a toset

{false, unlikely, could go either way, likely, true}

and defines operations ‘and’, ‘or’ and ‘not’ on it, in the fairly obvious way. They claim people often use this form of logic.

Posted by: John Baez on March 18, 2014 10:10 AM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Here I’m using the word toset to mean ‘totally ordered set’, just as poset means ‘partially ordered set’. Why don’t people do that more?

You’ll be glad to know then that the nLab is part of the vanguard: see total order. Do you also like ‘woset’? That’s also in the nLab.

We mustn’t forget the related concept of loset. You have to be a somewhat sophisticated to know the difference between a toset and a loset.

Posted by: Todd Trimble on March 18, 2014 12:51 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

There could be incomparable kinds of truthiness

Yes, that was exactly my thought. I would be more inclined to study categories enriched over posets in general, and ask questions like “what properties does the enriching poset have to have in order that the resulting logic has desirable property X?” At the extreme, one might hope for a characterization theorem of “the best poset for fuzzy logic”, like Cox’s theorem for probability.

Posted by: Mike Shulman on March 18, 2014 4:03 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Keynes believed that probabilities as degrees of partial entailment were not necessarily totally ordered. Two degrees might not be comparable.

Posted by: David Corfield on March 18, 2014 6:34 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Yes, it’s true that even Cox’s theorem already explicitly assumes that probabilities are totally ordered (indeed, that they are real numbers). Has anyone proposed a formal system deserving the name “probability” in which “probabilities” are not totally ordered?

Posted by: Mike Shulman on March 18, 2014 7:59 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

There is also the formalism of plausbility measures, which replaces the unit interval by an abitrary poset and seems very similar to what Mike has in mind. I can’t say any more about this, since I just learnt about it from Matt Leifer.

Posted by: Tobias Fritz on March 18, 2014 9:58 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

I guess that depends what you mean by “formal” and by “probabilities”.

I have to admit I’ve never looked closely enough at A Treatise of Probability to see if any of the formalism, which comes in later in the book, is dealing with incomparable probabilities.

On the other hand, there are interval-based theories, such as Dempster-Shafer, but maybe this is not about “probabilities”.

Posted by: David Corfield on March 18, 2014 9:38 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Thanks for the references! Plausibility measures do seem the most similar to what I had in mind.

Posted by: Mike Shulman on March 19, 2014 4:02 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

I have trouble believing there’s some deep justification for requiring the truth degrees be totally ordered. There could be incomparable kinds of truthiness, like “nine out of ten doctors say” versus “I read it on Wikipedia”.

Firstly I don’t think anyone said there was any deep justification for, in general, requiring truth degrees to be totally ordered.

Secondly, I would like to stress that this specific example of multi-valued logic is supposed to be modeling vagueness rather than uncertainty. So, in this context, the truth degree of “My bike is new” is not supposed to be based on my lack of knowledge (eg having asked a load of people on the street), but on a certain vagueness – “It’s a bit new”.

This is the kind of vagueness they build into fuzzy control systems, “if it’s hot in here and the motor is not overheating then turn on the air-conditioning”.

Historically, this grew out three and four valued logics as another family of examples, it would seem.

I should mention why I am interested in the truth values $[0,1]$. I’m interested because it fits into a rather nice mathematical story involving metric spaces, category theory, tropical geometry, formal concept analysis, duality theorems for convex analysis, etc, etc. And it possibly has novel applications to data analysis.

Posted by: Simon Willerton on March 19, 2014 11:55 AM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Well, it’s not any more intuitive to me to require that one of the sentences “it’s hot in here” and “my bike is new” should always be more true than the other one. (-:

Posted by: Mike Shulman on March 19, 2014 4:07 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Let me try to state some reasons why “logicians” sometimes only consider total orders (some examples are for instance the books by Petr Hájek and Siegfried Gottwald).

I am just gonna focus at the propositional level (in first-order these remarks do not work so easily). Before going into “fuzzy” stuff, let me illustrate the phenomenon with Boolean algebras. It is well known that the only Boolean algebra which is a total order (i.e., chain) is the 2-element one; and that there are a lot of Boolean algebras which are partial orders, but not total. However, from the logic point of view it is enouh to worry about the 2-element Boolean algebra (i.e., the chain) since they have the same valid propositional formulas (i.e., whenever a propositional formula fails in some Boolean algebra it also fails in a Boolean algebra with a total order).

The same situation happens with fuzzy propositional logic: that is, whenever a propositional formula fails in some fuzzy structure (with an associated partial order, perhaps not total) then it also fails in a fuzzy structure with a total order.

Indeed, in the framework developed by Hájek and Gottwald it happens that all subdirectly irreducible algebras are in particular chains.

Posted by: boumol on March 28, 2014 6:30 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Interesting, thanks! Of course, that doesn’t work for intuitionistic logic. And predicate logic is of course important…

Posted by: Mike Shulman on March 28, 2014 8:07 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

You are right (this is not the case for intuitionistic logic). The smallest superintuinistic logic where this “chain completeness” holds is the well known Gödel-Dummet logic (fuzzy logicians usually call it “Gödel logic”): which can be semantically introduced using intuitionistic Kripke models where the accessibility relation is a total order.

Posted by: boumol on March 28, 2014 9:21 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

So perhaps one might say provocatively that the reason “logicians” sometimes only consider total orders is that they are too attached to LEM (or a weaker form of it like the Godel-Dummet axiom). (-:

Posted by: Mike Shulman on March 28, 2014 11:21 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Yes, at least for “fuzzy logicians”.

Posted by: boumol on March 28, 2014 11:56 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

It seems that all of the examples you give of vagueness are all the result of ill-defined terms, which means that different individuals might have different personal definitions, but there is no agreed upon definition.

Let’s say person A defines “this bike is new” to mean “this bike is less than one year old” and person B defines “this bike is new” to mean “this bike is less than five years old”. If the bike is less than one year or more than five years, then person A and person B will agree as to whether it is new. If the age of the bike is 1 < x < 5 years old, then person A and B will disagree as to whether it is new.

However, for each person A and B, the statement “this bike is new” is either definitely true or definitely false. However, since you did not ahead of time, objectively define “new”, you left it up to the individual to decide, which means the so-called “fuzziness” is only the result of the subjective definition of “new”. There would be no need for “fuzzy” anything if you simply define all the terms precisely.

You also assume that everyone would agree that a 20 year old bicycle is definitely not “new”. Well, if someone worked at bicycle museum, where every single bicycle was at least 100 years old, they might consider a 20 year old bike to be “new”.

Whether someone is “old” is not vague if everyone uses the same definition of “old”, such as “at least 60”. What you call “fuzziness” is simply the result of different people using different definitions. If you agree on the definition ahead of time, such as “at least 60 years old”, rather than letting each individual subjectively define it, then everyone would agree whether someone is old, and then there would be no “fuzziness”.

However, I get the impression that you are claiming something else, than you are claiming that a statement is somehow objectively “partially true, partially false”, and not just the result of different people using different definitions of the words. Well, give an example of that.

It’s true that the age difference between Bart and Marge is greater than the age difference between Marge and Homer. However, someone might say that Bart is “only a little younger than Marge” if you were comparing the age difference between Bart and Marge to the age difference between Bart and Mr. Burns.

Also, in conversation, when you are using terms that aren’t defined precisely, you usually try to guess what the person you are talking to probably means.

Posted by: Jeffery Winkler on March 19, 2014 8:34 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Jeffery, thanks for the interesting comments! When you say

There would be no need for “fuzzy” anything if you simply define all the terms precisely.

I agree that if every predicate is interpreted as a function from its domain to the classical truth-value object $\{0,1\}$, then classical logic applies, there is no vagueness, and there is no need for fuzzy logic. This amounts to sidestepping the issue of vagueness altogether. But there are at least a couple of reasons you might not want to do this:

1. In natural language we don’t always stick to precise, classical predicates. We do in fact use vague language. So if one wants to understand natural language, then we have have to confront vagueness.

2. It is apparently fruitful sometimes to use non-classical logics such as $[0,1]$-valued logic, for example in some engineering contexts.

Ad (1), you seem to be offering a theory about the natural language semantics of vague terms when you say

Different individuals might have different personal definitions, but there is no agreed upon definition.

I believe many philosophers agree that rational agents can legitimately disagree on the meaning of vague language. But I’d urge some caution on interpreting the “personal definition” theory too literally for a couple of reasons:

• When I say “Simon’s bike is old”, I’m not sure I actually have a precise definition of “old” in mind. At least, I’m pretty sure I’m not thinking about a precise cutoff of, say 2.7 years, even within a particular context. I agree that I could settle on a more precise definition if conversational need calls for it, but it’s not clear to me that I had a precise definition in mind to start with.

• What is the relationship between my personal definitions and yours? If I’m using a different definition than you are, then how do you decide whether I’m using an acceptable variant of a vague term versus simply being incorrect?

• If we have precise definitions in mind, why do we continue to use vague language? If vague language suffices, why do we bother keeping track of precise definitions?

Turning to (2) – the status of $[0,1]$-valued logic – you say

I get the impression that you are claiming … that a statement is somehow objectively “partially true, partially false”, and not just the result of different people using different definitions of the words. Well, give an example of that.

I don’t think Simon is claiming that the fuzzy logic perspective is the unique interpretation of anything – in particular he’s not presenting it as a theory of natural-language vagueness. Rather he’s inviting us to try out the perspective it offers in examples like the ones he’s given.

Here’s one way to think about the relationship of $[0,1]$-valued logic to classical $\{0,1\}$-valued logic. A classical predicate on a set $X$ is a function $P: X \to \{0,1\}$ whereas a $[0,1]$-valued predicate on $X$ is a function $P: X \to [0,1]$. In particular, reasoning about $P$ entails reasoning about, for each $r \in [0,1]$, the classical predicate $P_r$ given by the characteristic function of $P^{-1}([r,1])$. If we need to obtain a classical truth value from a $[0,1]$-valued one for some reason (perhaps to make a choice between two options), then we can choose such an $r$ to use, but we don’t have to build those choices into the foundations of our thinking.

I think this is analogous to the way I use vagueness in my thinking and conversation: a vague term can be precisified, but I can put off that precisification to another time. I only precisify if and when it is necessary.

Posted by: Tim Campion on March 20, 2014 1:33 AM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Tim wrote:

I don’t think Simon is claiming that the fuzzy logic perspective is the unique interpretation of anything – in particular he’s not presenting it as a theory of natural-language vagueness. Rather he’s inviting us to try out the perspective it offers in examples like the ones he’s given.

Indeed, I really support what Simon is doing, for this reason… but also because he’s finding:

a rather nice mathematical story involving metric spaces, category theory, tropical geometry, formal concept analysis, duality theorems for convex analysis, etc, etc.

Clearly we can always take anything anyone says and try to force it into the Procrustean bed of classical logic. As Jeremy put it:

There would be no need for “fuzzy” anything if you simply define all the terms precisely.

Right! If Simon says “My bike is new”, we can demand that he demand his terms precisely. Exactly how many seconds old can a bike be and still count as “new”? Make him say. If he resists, tie him and down and waterboard him until he tells the truth! And if we do this to everyone in the world, we’ll never need fuzzy logic.

However, I find this approach to logic a bit depressing. I think that instead of trying to get all utterances to fit into the ‘one true logic’, it’s more fun to investigate in the way we investigate geometry. Once people worried a lot about whether Euclidean geometry was ‘correct’. Now we study dozens of different kinds of geometry—Euclidean, projective, conformal, symplectic, Riemannian, etc.—and their relations. When a practical chore comes up we grab a type of geometry off the shelf that’s suited to the task at hand, instead of using a ‘one size fits all’ approach. And even geometries that aren’t very useful for practical chores can still be very interesting to study; worrying about practicality prematurely can sometimes inhibit new developments.

Similar, I think we should study classical logic, quantum logic, intuitionistic logic, paraconsistent logic, multi-valued logic, fuzzy logic, Lawvere’s generalized logics, algebraic theories, doctrines, cartesian closed categories, topoi, 2-topoi, homotopy type theories, and more… sometimes led purely by the desire to try interesting new things, sometimes seeking mathematical elegance, sometimes seeking conceptual clarity, sometimes trying to systematize and unify everything that seems good so far, etcetera.

Posted by: John Baez on March 20, 2014 7:40 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

It seems to me that ‘ill-defined’ could be seen as ‘locally well-defined’. So if I and my friend have a definition of a ‘cool kite’, which when we fall into a discussion about cool kites we discover are different, and neither are kind nor nice enough to accept the definition of the other; then one need not say that a cool-kite is badly defined, since I have a perfectly precise definition, and so does my friend; but that they are locally well-defined, and if we do happen to agree - that they are globally well-defined.

It has nice resonances with local & global truth in toposes.

Posted by: mozibur ullah on March 20, 2014 11:14 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

Good point! Indeed, we should expect most groups of speakers to define their terms no more precisely than needed to ‘get the job done’, whatever they’re trying to do. Only mathematicians and (especially) logicians pursue the attractive mirage of complete precision.

Posted by: John Baez on March 21, 2014 10:26 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

I can’t resist saying that precision is a fuzzy concept.

Posted by: Eugene Lerman on March 22, 2014 12:41 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

one need not say that a cool-kite is badly defined, since I have a perfectly precise definition, and so does my friend; but that they are locally well-defined, and if we do happen to agree - that they are globally well-defined.

The problem is that notions like “coolness” are scalar values, and “being cool” means that “coolness” is quantized into having exceeded some fixed (personal) threshold. Global agreement should mean having the same map to “coolness” and same threshold but all one can observe is binary classifications.

Moreover such scalar evaluations as “coolness” appear fuzzy - otherwise a person given two arbitrary objects would consistently be able to decide which is the “coolest”.

Posted by: RodMcGuire on March 22, 2014 4:13 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

You don’t have to tie someone up to find out their personal definition of a vague term. Let’s say you are trying to determine how tall someone has to be for someone else to refer to them as “tall”. You can gather up a group of people of ascending height, and have them walk past the person you are questioning. They might describe each person as “as on the tall side”, “tall-ish”, “almost tall”, “close to being tall”, “as close to tall as you can be without being tall”, but then at some point, they just say, “That person is tall”. So there you have determined that person’s personal threshold for how tall someone has to be in order to be called “tall”. Someone else might have a different threshold. Someone else might consider a shorter person to be tall. You can easily find out someone’s person’s personal definitions of vague terms just by listening to them talk. What is the tallest person that you have heard them call tall? What is the newest bike you heard them call new? What is the oldest person you have heard them call old?

Let’s say you have a statement such as “This bike is new”, “That person is old”, or “That person is tall”, and you asked a group of people whether the statement was true or false. Perhaps 75% of people would say it was true, and 25% of people would say it was false. However, not one person would say, “That statement is 75% true and 25% false”. People might not agree whether it was true or false but everyone agrees that it’s either true or false.

The interval [1, 0] might measure the percentage of people who say it was true, or the probability that someone would say it was true, but it does not represent the extent to which the statement is true. If there is a statement where people can not agree whether it is true or false, does that mean the statement is neither true nor false but is instead something in between true and false?

Posted by: Jeffery Winkler on March 25, 2014 8:36 PM | Permalink | Reply to this

### Re: Fuzzy logic and enriching over the category [0,1]

I meant to say “what is the shortest person you heard them call tall?” “What is the youngest person you heard them call old?” “What is the oldest bike you heard them call new?”

Posted by: Jeffery Winkler on April 1, 2014 7:49 PM | Permalink | Reply to this

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