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## 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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text:
### Home > APCALC > Chapter 2 > Lesson 2.1.3 > Problem2-40

2-40.

For the given function, write an expression using summation notation that will approximate the area under the curve for $3 \le x \le 7$ using eight rectangles. Specify if you use left, right, or midpoint rectangles. Then enter this summation expression into your graphing calculator and evaluate the approximate area.

$f(x) = \frac { 2 ( x + 4 ) } { x + 6 }$

Recall problem 2-17:

i. For left-endpoint and midpoint rectangles, let the index start at $0$ and end at $n - 1$.
For right-endpoint rectangles, let the index start at $1$ and end at $n$.
$n = \text{number of rectangles}$.

$\text{ii. The }\frac{1}{2}\text{ represents the WIDTH/BASE of all rectangles.}$

To determine the width of each rectangle, divide the domain by the number of rectangles:

$\frac{7-3}{1}=\frac{1}{2}.$

$\text{Each rectangle has a width, }\Delta x \text{ is } \frac{1}{2}.$

iii. The transformed function represents the HEIGHT of each rectangle.
For left- and right-endpoint rectangles, the heights are represented by $f(a + \Delta xi)$.
For midpoint rectangles, the heights are represented by

$f \left(a+\frac{\Delta x}{2}i\right).$

$a = \text{starting value}$
$\Delta x = \text{width of each rectangle}$.

format: plain


text:
# Multiplying functions

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##### Examples
###### Lessons
1. Determine the Product of Two Functions Algebraically
If $f(x)= 2-\sqrt{x}$
and
$g(x)=7\sqrt{x}-3$
Determine:
1. $(fg)(x)$
2. $(fg)(25)$
evaluate in two different ways
2. Determine the Product of Two Functions Graphically
Given the graph of functions $f(x)$
and $g(x)$,
sketch the product of these two functions$(fg)(x)$

###### Topic Notes
multiplying functions: $(fg)(x) = f(x)\cdot g(x)$


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## Full Text for History A Proof for the Christian Faith (Text)

History A Proof for the Christian Faith Vice-Pri~zcipnl, %1ttJ7er(t7l Seminary, Adetclidc, Az~sfr.r7Zia ad the Editor of tlzc TA7~t1~~r(~r1 Thcologic~Z Jozi~.~?(ll, zvhcrc this article first tzppcal-cd. I N 111s FORWOIID to Jesus of Nnznveth G~lcntllci: Bornkamnl 111akes 3 number of statements which are thc stiixfi~~g-l)oint for this essay and in a sense the reason for it. Bornkan~m writes: Allany are of the opi~lioi~ that the way of historical critical re- search has provecl a false path for this subject matter 2nd should be given up for good. I do not hold this view and carlnot see at all that it is necessarily a way of unbelief, and that faith should fors.iirc:ir it and is bound to cio so. How could faith of a11 things be content 'ivitll mere tradition, even though it be that contained in the Gospels? It nlust break thtougli it and seek behind it to see the thing itself, and perhaps in this way to understand the tradition afresh and to regain it. In this attempt faith is on common ground n~itll all who are gcn- tlinclrr concerned with historical knowledge. CERTAINLY FAITI-I CANAOT AND SHOULD NOT BE DEPEWDEXT ON 'THE CHANGE AWJI UNCERTriIXTY OF HISTORI- CAL RKSEAIICH. 'ro expect this of it would be presump- tuous and foolish. But no 'one sl~ould despise thc help of his- torical research to illulnintt the truth with which each of us should hc concerned. It is thc en~phasizccl jvords ~iihich l~articul;~rI\' are germane to this in\lcstigation. There arc some very goocl reask ~r~hy faith is or may be affected by tlic changes and uncertaintics of historical research. It is simply the case that the Christian rcligion is inti- mately related, inextricably bound up 'ctritll certain events of the past, so i1.i t11 Ilistory , ivith historical events or occurrences. This state of affairs is given graphic expression in the ApostIcs' and Ni- cene Creeds: ". . . who for us men and for our salvatiorl came down from heaven, and was incarnate by the Holy Ghost of the Virgin Mary ancl was nlacle man, and was CI-ucified for us under Pontius Pilate. He suffered and was buried, and the tllii-d day he rose again . . ." The Christian church from the beginning has clcpended fur- ther, on a reporting of these events, an historical witness to these events, which in its turn has a history liltcwise. To ask for n faith unaffected by history or historical investigation is to ask for ~21at is impossible in the very nature of the case. Since, according to the Christian Gospel, God has chosen to act for the salvation of men in and through history, His action comes under the same laws as history generally; it is subject to the same l7ossibilities of investiga- tion, scrutiny, and criticism; one cannot protect this action from the necessity to meet the doubts which scholars may cast on it or with whicll they may surround it. f9istol.j .A I'~.oof Fol- 1'11~ Christian Faith ... - 201 T11c question to which these obseri atioas on the con1n1ent of Bornkainn~ Icad us nla) bc formulated as follons: Docs faith dc- pfnd upon the truth, the actuality, the incontro.i7crtible accuracy of certain historical f:jcts: ilnd the an\n.cl-, 01. ansn ers, to be clcfcndeti in this cssay, niay be stated thus : 1. I:,-om ~vhat 1 shall call the negative point of view-al- I.~,OLI~~I this phraseology is far from accurate-we nlrist an- s\\ cl- the cpcstion with YES. 2. I.:j-ci~l~ the opposite point of view the answer is XO. ?he two points of view more explicitly stand, with the an- swer inbuilt in ther~, may be for~~lul;lted as follo~vs: 1. 'I'herc can be no Christian faith, if the facts on ~vllich it rests arc no historical facts, or if what is regardecl as ccn- trnl to the faith is actually not a fact at all. 2. Faith cannot be proved, or demonstrated or macle into a logical certainty by the establishing as certain of the facts ccntl-a1 to the Christian faith. \Ire may present the first of these contentions in the words of the Paulinc test: "If Christ has not 11cc11 raised, then our preaclling is in vain and Sour f'tith is in vain." (1 Cor. 15 : 14 RSV). The sec- ond also we can put in words of Paul: "So faith coines from what is heard, and what is heard comes by the yreaching of Christ." (Ram. 10: 17 RSV: cf. the whole section, vv. 5-1 7) In developing the first of the ansi\.ers just supplied to the ques- tian that is our problei?~, .tve nlay con~~eniently begin lvith the re- d~ictio nd rrbsz~rdz~.r-~z advanced by James Peter in his ~r~ork Finding the Historic,al Jeslls. 1-Ie has argued that "some nlinimulll ki~o~vledge of the object is required. The interrelatio~lships of kno~vledge, as- sent and trust, of fides qua and fides qzlne, of event and fact, make that conclusion i~~cscapable." lie goes on to put the vedzsc~.io ad (177- surdu~~z: Lct us suppose that It\. tomb. The importance of the second of these lines of cvidcnoe is not ;\Z\Y~\YS appreciated, but it bcconles clear .ivhcll .tvc put the clucstion: If the tomb n.erc not or had not bccrl cmpt.i7, what then? Ob\.io~~sly, that there would have been no reason \voilth serious consideration for the apostles to proclaim the resurrection of Jesus of Nazarctlh. 'Tile first thing a mociern would do, if 71c were the high-pricst of thc time and the ~nessagc of resurrec- tion came to lli171, YT?OUICI be to go to the Otrrial placc, ant1 if mould- wing corllse or even a pilc of bones were there, he nrould hai~ all thc cvidcnce 11tccl.et1 to dismiss at once as ~vitho~it any foundation \i.-l~;~tc\~cl: thc report that Jcsus had risen. Caiaplins r~s it good Sad- ducee \t.oulcl unclo~~btedly ha~c donc exactly the same himsdlf. 'I'he argunient so far hiis bccn pretty evident and straight- for~vartl. Complications, hon.e\.er, arise when wct t~irn to the whole of thc Gospel tradition ant1 thc considerable body of tnaterial that is set forth there as having historical character. Does the argunlent still hold true: that the doubts as to the historical character of this ~naterial affect faith, call faith into clucstion, antl may even destroy it? If ;i hypotl~ctical eliinination of 'the resurrection as an histori- cal fact can destroy faith, \\.hat of a hypothetical elimination of facts of Jcs~~s' lifc leading up to tllc resurrcction antl affecting it. Does no dan1i9gc to faith result froni scepticisill concernii~g thc witness to Jesus' lifc as a n.holc and in its various reported aspccts, as long as the ~vitncss collcerning the end, His death and resurrcction, is I~istorically sorrl~d? Undo~~btedl~~ some facts reported about Jesus are illore iml>ortant than others in any survey of Ilis life. The e-cran- gclists themselves selected what the!' thouqht im13ortant from the mass of reports a17ailable for inclusion in their Gospels. On the other hand thc ~vitncss concerning the rest of Jesus' life is founcl in the same writings as the witness concerning the cleat11 ancl the resurrec- tion. So ~vhy accept one portion of the lnaterial while discarding the rest? h4ust not the same principle be invoked throughout: an at- tack 011 the historical reliability of the material affects the faith? Some people would probably argue at this point that the apostlc Paul actually does what is suggested in the lost paragraph, that he concentrates his interest on the death and resurrection of Jesus, while remainin4 profoundly indifferent to the rest, and that he actualIy elevates thls procedure into a principle. He is supposed, Histol-11 /2 l'l-oof TJor 'l'hc CI7~istiun Faith ? according to 2 Cor. 5 : 16, to have 110 interest in Christ accorcling to the flesh. Homc.cier, we may dismiss this point of view with the observations, that it is not very lilared rv-ith others within thc same writings.) The cluestion that 11:is bccn aslced is a ccry difficult one to answer, incleed. There is the danger on the one hand of an cscess.i\le sccpti- cisnl, :I danger already touched on. 011 the otllcl- hand, [vhere is one to stop, once one llns set out on the ~:oad of historical sceptic*ism? \&'hv this incident ~1nc1 not t't~;~t? f 11ai.e no answer 3t this point be- yonJ the crencral colnnlent that-, if \?LC lniist CIY, it is better to err >. by an uncritical acccptclnct. of everything than by a critical rejection of! crerything. 'This ~.i:hole ~~roblc'l~l impinges here, obviously, on the mat- te~: of inspiration of tllc Bible and its authority and inerrancy. This is a question for itself and is not within the scope of this article, except in onc particular. Some theologians try n despcr;~te inanoeuvre to avoid the difficulty n-c have encountered. They try to occupy t~vo l~ositions at once: acceptance of all the results of historical criticism inclucling nl~~oh of its radical scepticism, and of the impl.ications of thc inspiration doctrine. "T'his is to acccpt t\.i-o contradictory posi- tions nt one and the sallie timc, one tvith one compartment of the ~nincl, the second with another. As Bcrnliamm puts it: But ~vhen anyone, out of a concern for the understancling of history, has embarked upon these questions, he ~vill hardly kcel, a good conscience, if thereafter he is driven in despera- tion to take refuge froin the problem of illvestigation and its frequently contro~~ersial results in what is considered the safe fold of Church tradition.' 'To hold to the view of form-criticism which makes no st of the Gospel tradition church creation and then to say: "Hon7ever, this Gospel tradition is still God's ~irord and true beca~~sc of its inspiration, and me can rely upon it," is to take a position which is mentally and orally stultifying. One and the same worcl or incident cannot be a creation of the church-the form-critical result-and an au- thcntic tradition going back to Jesus-what the Gospel material in the obvious meaning of its words declares-both at the same time. Nor can both views of the word or incident be true at the same timc. And if the form-critical conclusion is accepted as valid, no truth, in any non-prevaricating use of that term, can be seen in the opposing view. The conscientious line to talte in such a conflict is tj~at of Ethelbert Stauffer and Joachim Jeremias, to work back through the tradition so coloured by concerns and confession of the early church to the original and authentic words and works of Jesus. The position is actually even worse than this, if such n thing is possible. The: l~osition described is to mal>ptic cspcctation. On 111e other hand Jesus is the final revelation of God to the CS~(:II~ that His i~linistrv ancl His history have eschatological c-11 o n~cter. His mjllistry' 11i1d eschatolog$c;il cllal-:icter because I-Ir :lnno~~nccd the coll~ing of Gotl's rc>lgn as the beginning of tl~c: Fnd, ant1 thcrcforc FIe understood 111c'n's attitilde to\v:u(l llis niessage to be a prcdeciding of the final ju(lgrne~~t." 'I'hc. I)istot-ical event of the resurrection is absolutely ~~ecessary for this vicn: of the revelation of God in history and especially in the history of Jcsus. LVitliout His resurrection the only historical judg- ment concerning Jesus that coulti be nlade js that He was a tlrealncr and \iisionnry. Not only that: the resurrection of Jesus itself is an nntici1)ation of the End 2nd of the iniminent rule of God. \\lithout thc resurrection of Jesus His inessage \-lloulcl hale tun~ccl out to be a fanatical audacity, But in a certain sense . . . the resrirrectioli did justify Jesus' expectation of the near End. It was in Him that it was fulfilled. ildn~ittedly, this happened otherwise than Jesus and His disciples probably had illiagincct the a~liloullced future . . . Nevertheless, in view of the resur- rection of Jesus and the eschatological quality of that event, \rc cannot be satisfied with the simple judgnlcnt that Jesus' esivc:;ttion of the umi- Entl 1-eillaiiit.tl unfulfilled. The csclla- tologicrll resurrection of the clcad becanle an event in Tcjlls' cnsc, and thus thc Gorl ~vhose ncarlless Jesus hati pro- clail,-ic:tl ticc.la~.cd itself' for Him.!' 11. is to LC: cspressly noted that T.)anncnberg will ha\.e nothing of a [\istinction brtn-een a sacrctl history !HeiZsgeschicrhtc) as apart prof;ric history. He rv:~nts nothing of "some sheltered area wllcrc fjle (;hristian faith \vould Ilc in~mune from historical criti- cis~~l."'O Further, it is Ileld also bv him and just as st-rongly that tj[le mcaning of I~istory (and the revelation of God in it) is contained \vitllin Ilistory an1c-i is not imported illto it from the outside, say, from all olltside il?terpretation or Tl'orci of God. Thus he rejects tlli> distil>ctiol: 1~;flicJa Richarc1 Rothc nlade "het~vecn the jpznl~i- f o i the external facts of ]listor); :ind the illspi~.citio~~ of tllCs I',iblic;lJ II itness, \\rho teach us thc meaning of the facts and ll,Ilosc tc.;lclji,rfi is ahsol~it~lp necessi~s); if we are to recogilizc the facts as lll;iliifcstatian of God.!"' IS~~il('ling on 13. 6, Colling\vood, ?'he lilcll ut [-{i.sio~?,, 'I)~IIL~C'II~)CI:~ 110lils t(l thc unity of facts ;111d their 111eanillg. ..One may ,lot arbitrarily attach whatever illcaning one -rill to a sir c11 fil~t."'" Now let .us ;lssume that the historical cluestion concerning the res~lrrectioll of Jesus has bcen clecidecl positively, then the nlenning of this event as God's final revelation is not some- thing iliiii lnust bc ;~cldcd to it; rather, this is the original 111e;1nillg inherent in that event nlithin its own contest of- his- tor!. ;~iicl trntl ition. ':' It \\;ill 11t: ~eii(lily seen that this \.ie\.i: of the re\ielation of God in history in\.ol\.c.s immediately the l~~cjaning of faith and the rela- tion of fnitli ;11i(1 1;noivledge. And Pannenbcrg's view here is neccs- sary to 1.o~rncl out this short sketch of his theology niitllin the con- text of our prob1c.m. It is in brief that faitlt is dependent ul~on .tr.ell- basecl l c~ccisions ant1 ultin~atc conln~itnic.nts cannot be tlctel-~niilcd on thc Ixlsis of objective data, and they . . . a.i~oid such statcmen ts as P:~nncnbcrg's that "tl~c.: Jews knew their God, and yet they did not: lmow Hi111 i11.iglit; otherivisc they ~vo~~ld not halie rejectcd Jesus."" Another criticism conics fro111 \Villiam Han~ilton, an attack on the ~nethodology itself of I'annenbcrg : Goc-I, in son~c sense, IS 1listol:y; or at least 1le is by definitiou in history, and Iiistorical method can find Him therc. This rnethod can ncvcr conclude that God is not there; a ~ilethod that conclutiecl thit t nfould be tlismissed as inadequate. " Nolr;, it is true that Pannenberg himself rejects this criticisnl of Hamiltoll in ;I concluding "Response to the Disc~~ssion."'~ But Ile does this, so it sci:111s to II~C, at the price of his whole position. He is ;111s\\.cring the (lllestioll whether historical ltnowledge at best attains probability, so that one has to find solllc other basis for the certainty of faith. In his answer Pannenberg distinguishes between histo~icnl ccrtninty and the certai~zty of faith and between "the eschatological meaning of the history of Jesus" and "historical knowl- edge of the history of Jesus." The certainty of faith and the coin- 117eteness of trust is based 011 the former; our ltnowledge of the lat- ter is only probable. He then declares: In 1-11.incjpl~ ti~c l'osil~ii it: cannot bc excludecl that the 1listol:ical probal>ilit);. of those traits of the history of Jesus in whicl~ its esci~ntologic.;ll nlcan ing is grouiiciecll will sornc (la!., fro113 so~ne points of l7ic\.i., I~ecomc doubtful; to tlie c1cgl:cc that a concep- tion of' tl~c llisl-orical fol-HI of Jesus conlcl or ~\.ou.ld 11a1:c to seclll prol~al>le, ~i-hich \voulti makc the cal-ly Christian faith seem .(,\-ithotlt SLT pport, \vitl~o~~ t b;lsis in the historical form of Jcs~ls. ! see no occ;~sion for al?lx-chcnsiol~ that sucl~ a posi- tion of !-cscarcll sho~lIcl cmcrgc. in t11c forc.sce;~ble futurc. But: in l~~.ill('ipl~ it cnni?ot hc excluclecl. 111 sucll a ci~sc. the fountla- tion for t-l~e certainty of faith, trust in the cschntological power and nieanjr1g of tl~c 11istol.y of Icsus, 1\-oulc1 bc rcmo~~ccl.~~ To lca\rc Panncnberg and to come to the fu~~damcntal Chris- tian facts, the clcittlr of Jesus and His resurrection--it sccms to nle to be aliilost self-evidcnt that conlmon incthods of historical iii\t:sti- gation canl~ot colni: to tl~c coricl~~sjon that "Gotf \\:as in Clilirist ~.c..con- cilillg thc ~vorl<'f unto Flilnself, not imputing their trcspassvs unto them." (:an thc f;?cf-s of the resurrection (granted that the): arc facts, the appei'rmces mci thc empty tomb, force tlic conclusion that thc uevv worlcl of God has been inauguratccl in thc rvsurrection of Jesus? Hardly. Historical facts cortld cicn~onst~ritc, if the c\idcnce were con\!inci~~g, enough, that St;-;)nil-So, I .nz:irus for instance, hati 1)ccn res~iscit;~tct-I, restowd to tki~ lift:. Ii' [his IC.C'~C p~:o\.c(t, jt 11.0~ild 1~ c? fact fol: 1)iologist.s to talic note oi' an- dcmon- strable. 'She esistci~cc of the Christian ch~lscli do~i;i~ the ages to our tiinc is proof cnough. f~ I)~:.ic!flv, the word of the Gosl~el itself, tl~e \\.itncss of tllc apostles, .ivhich is identical ~vitk the n-itncss of the Holy Spirit, js a po.cccl:ili.rl ~vortl, ptvr4-cl-ful to atousc n~id cxcite faith in those \I-ho he:rr i~. So I\ e ha\-e tlnc asscl-tions of tllc X\;e\'c: Tcstn~llcnt: "Faith comcth by hearing, and heal.ing by the wol-ct oi.! God" (lio111. IO: 17); "Yo one call say 'Jesus is l.ord!' except by tllc x-Ioljr Spi~.it" ( 1 Coy. 12 : 3 17SV); "But vou sli;~'tl r:cccive p6n.el- n:I.)cn the Holy Spirit Elas come upon you; jilt1 J'OL~ ~11~11 bc 11))' witllcsses . . .') (jlcts i : 8 J:SIi); "l3ut \!.hen the Counsclo:: cones, ~,vliom .I shall scntl to pou fro111 the Father, e\-en thc Spirit of tl-uth, .tr:ho proceecls from tJ~e F'ittJlcr, 1.1c will bear wit- ness to lne; and you illso are ~ti.ittlesscs . . .)) (John 15 : 26, 27 HSV); "So we :Ire a~~it)assiido~-s for Chris(-, God nlali(l., 127. 13. [I)J~., 128. 11. Ibicl., pp. 130-131.. 15. "Thc hlcnnjng :)f Fiistor!" in TEIEOL.OC;Y :IS FflSTIOIZY, pp. 152-153. 16. '"l'he Character of I'anncnberg's Theolog!." in THEOLOGY AS 1-11s- ?'OllY, 1'1'. 385-186. 17. THEOLOGY AS I-IISI'OI:Y, pp. 221-276. 18. This quotation ;tnci :he prccr:djng material from Pannenberg, "Response to the l)isc~~ssion," p~. 272-271. 19. Sir Edwyn Hosk!:~is and Xoei Ilave!., THI; RIDDLE OF THE iXI'i\V TES'TRMEhT (1,ontlon : 1;nber paper-coverecl Editions, 1958), 13. 18 1. 20. Hosliyns and I)avey, p. I8 1. 71. l'!~ rest of t1.1~ ~j~10t;lti0iis ;~t this point, inc1udj11g thc cztcndccl citation, all ft-om Hoskyns and Dave!., 1713. 18 1-1 82. 22. Scc, for jnst:~ncc, I!ornkr~mi~l, p. 14 : .-\ltho~rgh their ].ciation to I~istol-y is ;i tliffct-cnt onc from that of John, they none thc lcss unitc to a rcmarltablc degree both rccord of Jesus Christ 2nd ~vitncss to Him, testimony of the church's faith in fliln :tnd narration of His his to^:!.. Both shor~ltl be continually distinguished in the understandjng of - tl~e Gospcls . . . ; on the ot-hcr li;tnct, 1)oth arc so closely intcrn-oven that it is oftcn cxccrtljngl!~ hard to say whcrc o11c ends anc1 thc otllcr 1)egins . . . Wc possess no singlc wort1 of Jcsus and no single story of Tcsus, no mat- tc.1. !io\~ incontcstal>ly gcnlline they ma): be, which clo not contain at thc siilllc tilnc the confession of thc bclic.iring congregation or which arc not i~t ltrast cml~cddecl therein. 'This maltcs thc search after the barc facts of Iiistorjl difficult 2nd to a large cxtcnt futile. -- .- . - -. - . -. - - - - - . - -. - -. - . . - . . - - - - BOOKS I3EClIIIVI:D 'l'irc (;li~.i.~litrt~ El-rcolrrrtcrs Org,(l~iized C~~~ILC. By Richard D Klludteli. Coiicordia l'ttblish- ills Ilolisc, St. Louis, 1372. 9.5 pngcs. l'apcr.$1.75. .\l~lo\ I'J.oIJ~J~'~ of Socinl Justice. By Pngc 11. I


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# Dark matter

1. Jun 10, 2006

### wolram

Can any, qg or string theory reproduce dark matter ?

2. Jun 10, 2006

### marcus

I can only say as regards what i am familiar with in non-string QG, and
AFAIK the answer there is NOT SO FAR. There is not any non-string QG so far, that I know of, that reproduces dark matter.

There is quite a lot of development going on in QG these days. One might do that in future.

A footnote to that concerns various MOND theories like Bekenstein and also like that of Moffat. These are not quantum gravity theories but are modifications of the classical theory of gravity. I think of them as slight "corrections" to General Relativity.

These MOND theories avoid the need for Dark Matter and they are TESTABLE within the solar system. (Bekenstein Magueijo paper on MOND habitats in the solar system).

It would help if some MOND theories could be tested and would either pass or fail. If some version of MOND gained credibility then one could try quantizing THAT instead of the plain vanilla Gen Rel. In that case the goal would be a quantum gravity that takes care of dark matter, simply by reproducing MOND at large scale.

3. Jun 11, 2006

### francesca

I'm going to attend this seminar:
Luca Lusanna
Canonical gravity, York map and non-inertial frames: is dark matter a relativistic inertial effect?
Abstract: After a review of the simultaneity problem from Newton to special relativity, I describe the status of the Hamiltonian description of ADM metric and tetrad gravity emphasizing the role of non-inertial frames, the only one allowed by the equivalence principle, which replaces the relativity principle in general relativity (GR). After having shown the dynamical nature of the chrnogeometrical structure of spacetime in GR, I show that the natural Hamiltonian for the spacetimes allowing the incorporation of particle physics is the weak ADM energy (the energy problem in GR is connected with the dependence upon the inertial potentials of the ADM energy density, whose role in the dark energy problem is unknown). Having built the York map, there is a clear separation between the physical tidal effects and the gauge variables inside the gravitational field. Which types of relativistic inertial effects are described by the gauge variables is explicitly shown. One of them (the trace of the extrinsic curvature) describes the freedom in the choice of the convention for the synchronization of distant clocks, namely in the definition of the instantaneous 3-space. This freedom is absent in Newton theory, where the Euclidean 3-space is absolute. This gauge variable, appearing as a relativistic inertial force in Hamilton equations, may change from attractive to repulsive in different regions of the simultaneity surface. It could be the origin of dark matter in the rotation curves of galaxies in a framework more general of the non-relativistic MOND model and less naive of the Cooperstock-Tieu proposal.

4. Jun 11, 2006

### f-h

Marcus, MOND is a modified Newtonian mechanics. It's Newtonian gravity with an extremely strange modification in the low accelaration regime. It's not even specially relativistic. It's a fascinating phenomenological tool but it's certainly not something that makes sense to quantize IMO (plus it would be difficult to do so)

5. Jun 11, 2006

### marcus

You might be interested in having a look at this
http://arxiv.org/abs/astro-ph/0509519
A Primer to Relativistic MOND Theory
Jacob D. Bekenstein (Jerusalem), Robert H. Sanders (Groningen)
6 pages, 1 figure, to appear in proceedings of IAP05 in Paris: Mass Profiles and Shapes of Cosmological Structures, G. Mamon, F. Combes, C. Deffayet and B. Fort (eds), (EDP-Sciences 2005)

"We first review the nonrelativistic lagrangian theory as a framework for the MOND equation. Obstructions to a relativistic version of it are discussed leading up to TeVeS, a relativistic tensor-vector-scalar field theory which displays both MOND and Newtonian limits. The whys for its particular structure are discussed and its achievements so far are summarized."

Wikipedia (not as authoritative as Bekenstein but indicative of common language usage) classifies TeVeS as a type of relativistic MOND.
http://en.wikipedia.org/wiki/Tensor-vector-scalar_gravity
Wiki says: "Tensor-Vector-Scalar gravity (TeVeS) is a proposed relativistic theory of Modified Newtonian dynamics (MOND),..."

MOND (like "Kleenex") has become a generic term that includes relativistic theories very different from the original MOND of Moti Milgrom in 1981. There is no other term for these theories in wide use, so for the present people are pretty much forced to say MOND.

Maybe you or someone will eventually get the terminology straightened out and make up a better word and get the experts to accept it. Language reform is difficult

Last edited: Jun 11, 2006
6. Jun 11, 2006

### marcus

I am worried about Lusanna because he sounds out of touch with current MOND developments. He mentions only "non-relativistic MOND" and does not mention relativistic MOND in his abstract!

There have been some interesting talks and papers at Perimeter Institute about this recently. The authors to check, if you do a search, are John Moffat, Jacob Bekenstein, Joao Magueijo.

Bekenstein is famous because of the Bekenstein-Hawking BH entropy formula (his Princeton thesis introduced the ideas of BH thermodynamics). He has developed a relativistic MOND he calls TeVeS (tensor-vector-scalar).

Magueijo (Perimeter and London Imperial) is also prominent. He recently co-authored this
http://arxiv.org/abs/astro-ph/0602266
MOND habitats within the solar system
Jacob Bekenstein, Joao Magueijo
Phys.Rev. D73 (2006) 103513

"MOdified Newtonian Dynamics (MOND) is an interesting alternative to dark matter in extragalactic systems. We here examine the possibility that mild or even strong MOND behavior may become evident well inside the solar system, in particular near saddle points of the total gravitational potential. Whereas in Newtonian theory tidal stresses are finite at saddle points, they are expected to diverge in MOND, and to remain distinctly large inside a sizeable oblate ellipsoid around the saddle point. We work out the MOND effects using the nonrelativistic limit of the TeVeS theory, both in the perturbative nearly Newtonian regime and in the deep MOND regime. While strong MOND behavior would be a spectacular 'backyard' vindication of the theory, pinpointing the MOND-bubbles in the setting of the realistic solar system may be difficult. Space missions, such as the LISA Pathfinder, equipped with sensitive accelerometers, may be able to explore the larger perturbative region."

the point is that the general category MOND includes relativistic MOND, such as TeVeS, and a relativistic MOND has a NONRELATIVISTIC LIMIT which is similar to but not exactly the same as Newton gravity, and this nonrelativistic limit can be TESTED by spaceprobe observations within the solar system. In fact, according to him, by the LISA probe in the inner solar system.

John Moffat, also at Perimeter, has a couple of different Scalar-Tensor-Vector versions of MOND which he has been working out ways to test in the solar system (by some outer planet mission)
He manages to fit his MOND both to the Pioneer anomaly data and to galaxy rotation curve data

http://arxiv.org/abs/gr-qc/0605141
Time Delay Predictions in a Modified Gravity Theory
J. W. Moffat
5 pages

"The time delay effect for planets and spacecraft is obtained from a fully relativistic modified gravity theory including a fifth force skew symmetric field by fitting to the Pioneer 10/11 anomalous acceleration data. A possible detection of the predicted time delay corrections to general relativity for the outer planets and future spacecraft missions is considered. The time delay correction to GR predicted by the modified gravity is consistent with the observational limit of the Doppler tracking measurement reported by the Cassini spacecraft on its way to Saturn, and the correction increases to a value that could be measured for a spacecraft approaching Neptune and Pluto."

I see that Moffat makes a helpful distinction! In effect, he says MOG instead of MOND. He calls his "Scalar-Tensor-Vector" theory a kind of "Modified Gravity". But I fear other people would refer to it as a kind of MOND because that is the usual term---as witness Bekenstein.

http://arxiv.org/abs/gr-qc/0506021
Scalar-Tensor-Vector Gravity Theory
J. W. Moffat
JCAP 0603 (2006) 004
"A covariant scalar-tensor-vector gravity theory is developed which allows the gravitational constant G, a vector field coupling $\omega$ and the vector field mass mu to vary with space and time. The equations of motion for a test particle lead to a modified gravitational acceleration law that can fit galaxy rotation curves and cluster data without non-baryonic dark matter. The theory is consistent with solar system observational tests. The linear evolutions of the metric, vector field and scalar field perturbations and their consequences for the observations of the cosmic microwave background are investigated."

If the other professionals in the field could be persuaded to say MOG instead of MOND when talking about relativistic versions of MOND, then the confusion would be straightened out.

Last edited: Jun 11, 2006
7. Jun 18, 2006

### OogieWaWa

Semi-related humorous remark:

Dark matter is really light. But so is dark energy. Its all there, but only when you look. No strings attached.

8. Jun 21, 2006

### francesca

Yesterday I've attended the Lusanna's seminar...
In the past days I've taken a look to papers about MOND that you suggested.
I have no deep knowledge, only feelings... but I prefer Lusanna approch than Bekenstein one, because the former sounds more natural: it doesn't invoke nothing more than seriusly considering the relativistic non-inertial framework. By now he hasn't given results for the DE problem, only hints, but he is firmly persuaded that this is the right way...
Maybe I can share a convinction of this kind...

BTW he has suggested me a paper also with a philosophycal point of view
"[URL [Broken] Leibniz-equivalence as difference of non-inertial appearances:
dis-solution of the Hole Argument and physical individuation of point-events[/URL]
Authors: Luca Lusanna (INFN, Firenze), Massimo Pauri (Parma Univ.)

Last edited by a moderator: May 2, 2017
9. Jul 28, 2006

### mojocujo

Is the einstein tensor unfinished? It addresses spacetime with matter in it. Did Einstein intentionally not address spacetime in the absence or near absence of matter? Assume you write the tensor while facing a black hole and then turn around and extrapolate the tensor into cosmologically empty spacetime. Do you get MOND? I would think you would at some point in the extrapolation and what do you get when the gravitational energy density nears the cosmic background gravitational potential?


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# Determining Fluorescence Quantum Yield of C60 in Toluene

For my experiment, I am trying to determine the fluorescence quantum yield of C60-fullerene in toluene (at room temperature). In order to do that, I need to have a standard sample with a valid and known fluorescence quantum yield. I have been looking for a standard sample with a known quantum yield in toluene, however, I am not sure of what to use.

Would it be better if I used a different solvent in determining the quantum yield and if so what solvent and standard should I use.

• A fairly popular standard is Rhodamine 6G, in some sort of alcohol. Be aware that if you decide to use a different solvent, you'll have to account for differences in refractive index. – chipbuster Mar 13 '14 at 2:04

## 1 Answer

Toluene instead of benzene can be a poor choice when H-atom abstraction by the excited state is possible. Here, you're safe.

The authors apparently used 9-cyanoanthracene in toluene as a standard to determine $\Phi_f$ of C60.

You might also want to have a look for review articles by Dirk Guldi from the time around 2000; I remember him giving a number of talks on C60 around that time.

• In addition you should use a standard with emission in the same region as you sample to avoid effects of detector etc effects with wavelength. Porphyrins are probably good here as there are standard corrected spectra published. Automatic correction, as used in some fluorimeters, is not always as good as its made out to be. – porphyrin Jul 7 '16 at 19:55