The n-Category Café

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

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.

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.

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… (-:

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?

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!

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.

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.

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.

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.

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?”