## February 23, 2015

### Concepts of Sameness (Part 2)

#### Posted by John Baez

I’m writing about ‘concepts of sameness’ for Elaine Landry’s book Category Theory for the Working Philosopher. After an initial section on a passage by Heraclitus, I had planned to write a bit about Gongsun Long’s white horse paradox — or more precisely, his dialog When a White Horse is Not a Horse.

However, this is turning out to be harder than I thought, and more of a digression than I want. So I’ll probably drop this plan. But I have a few preliminary notes, and I might as well share them.

### Gongsun Long

Gongsun Long was a Chinese philosopher who lived from around 325 to 250 BC. Besides the better-known Confucian and Taoist schools of Chinese philosophy, another important school at this time was the Mohists, who were more interested in science and logic. Gongsun Long is considered a member of the Mohist-influenced ‘School of Names’: a loose group of logicians, not really a school in any real sense. They are remembered largely for their paradoxes: for example, they independently invented a version of Zeno’s paradox.

As with Heraclitus, most of Gongsun Long’s writings are lost. Joseph Needham [N] has written that this is one of the worst losses of ancient Chinese texts, which in general have survived much better than the Greek ones. The Gongsun Longzi is a text that originally contained 14 of his essays. Now only six survive. The second essay discusses the question “when is a white horse not a horse?”

When I first heard this ‘paradox’ I didn’t get it: it just seemed strange and silly, not a real paradox. I’m still not sure I get it. But I’ve decided that’s what makes it interesting: it seems to rely on modes of thought, or speech, that are quite alien to me. What counts as a ‘paradox’ is more culturally specific than you might realize.

If a few weeks ago you’d asked me how the paradox goes, I might have said something like this:

A white horse is not a horse, because where there is whiteness, there cannot be horseness, and where there is horseness there cannot be whiteness.

However this is inaccurate because there was no word like ‘whiteness’ (let alone ‘horseness’) in classical Chinese.

Realizing that classical Chinese does not have nouns and adjectives as separate parts of speech may help explain what’s going on here. To get into the mood for this paradox, we shouldn’t think of a horse as a thing to which the predicate ‘whiteness’ applies. We shouldn’t think of the world as consisting of things $x$ and, separately, predicates $P$, which combine to form assertions $P(x)$. Instead, both ‘white’ and ‘horse’ are on more of an equal footing.

I like this idea because it suggests that predicate logic arose in the West thanks to peculiarities of Indo-European grammar that aren’t shared by all languages. This could free us up to have some new ideas.

Here’s how the dialog actually goes. I’ll use Angus Graham’s translation because it tries hard not to wash away the peculiar qualities of classical Chinese:

Is it admissible that white horse is not-horse?

O. Why?

S. ‘Horse’ is used to name the shape; ‘white’ is used to name the color. What names the color is not what names the shape. Therefore I say white horse is not horse.

O. If we take horses having color as nonhorse, since there is no colorless horse in the world, can we say there is no horse in the world?

S. Horse obviously has color, which is why there is white horse. Suppose horse had no color, then there would just be horse, and where would you find white horse. The white is not horse. White horse is white and horse combined. Horse and white is horse, therefore I say white horse is non-horse.

(Chad Hansen writes: “Most commentaries have trouble with the sentence before the conclusion in F-8, “horse and white is horse,” since it appears to contradict the sophist’s intended conclusion. But recall the Mohists asserted that ox-horse both is and is not ox.” I’m not sure if that helps me, but anyway….)

O. If it is horse not yet combined with white which you deem horse, and white not yet combined with horse which you deem white, to compound the name ‘white horse’ for horse and white combined together is to give them when combined their names when uncombined, which is inadmissible. Therefore, I say, it is inadmissible that white horse is not horse.

S. ‘White’ does not fix anything as white; that may be left out of account. ‘White horse’ has ‘white’ fixing something as white; what fixes something as white is not ‘white’. ‘Horse’ neither selects nor excludes any colors, and therefore it can be answered with either yellow or black. ‘White horse’ selects some color and excludes others, and the yellow and the black are both excluded on grounds of color; therefore one may answer it only with white horse. What excludes none is not what excludes some. Therefore I say: white horse is not horse.

One possible anachronistic interpretation of the last passage is

The set of white horses is not equal to the set of horses, so “white horse” is not “horse”.

This makes sense, but it seems like a way of saying we can have $S \subseteq T$ while also $S \ne T$. That would be a worthwhile observation around 300 BC — and it would even be worth trying to get people upset about this, back then! But it doesn’t seem very interesting today.

A more interesting interpretation of the overall dialog is given by Chad Hansen [H]. He argues that to understand it, we should think of both ‘white’ and ‘horse’ as mass nouns or ‘kinds of stuff’.

The issue of how two kinds of stuff can be present in the same place at the same time is a bit challenging — we see Plato battling with it in the Parmenides — and in some sense western mathematics deals with it by switching to a different setup, where we have a universe of entities $x$ of which predicates $P$ can be asserted. If $x$ is a horse and $P$ is ‘being white’, then $P(x)$ says the horse is white.

However, then we get Leibniz’s principle of the ‘indistinguishability of indiscernibles’, which is a way of defining equality. This says that $x = y$ if and only if $P(x) \iff P(y)$ for all predicates $P$. By this account, an entity really amounts to nothing more than the predicates it satisfies!

This is where equality comes in — but as I said, all of this is seeming like too much of a distraction from my overall goals for this essay right now.

### Notes

• [N] Joseph Needham, Science and Civilisation in China vol. 2: History of Scientific Thought, Cambridge U. Press, Cambridge, 1956, p. 185.

• [H] Chad Hansen, Mass nouns and “A white horse is not a horse”, Philosophy East and West 26 (1976), 189–209.

Posted at February 23, 2015 3:31 AM UTC

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### Re: Concepts of Sameness (Part 2)

One could view this as an early instance of structural set theory: a white horse is not an element of $Horses$, but an element of $WhiteHorses$. We have a canonical injection, $WhiteHorse \to Horse$, but not a literal subset in the material sense.

Posted by: David Roberts on February 25, 2015 8:36 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

That’s a nice modern interpretation!

Posted by: John Baez on February 25, 2015 3:58 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

So one could say that the type $WhiteHorse$ is the dependent sum $\sum_{x: Horse} White(x)$, hence strictly a white horse is not a horse, as really it’s a pair of a horse and warrant for its whiteness.

But applying the coercion from the canonical injection (or first projection), we can count it as a horse, and apply horse predicates to it.

Posted by: David Corfield on March 2, 2015 10:11 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

I’m not an expert, but I would say that “classical Chinese does not have nouns and adjectives as separate parts of speech” is not correct. (If anything, I would say such a statement betrays an Indo-European mode of thinking!) I would rather argue that adjectives are a special kind of verb (which is certainly something which is said of Mandarin). For instance, compare the syntax of negation:

• To say “is not [noun]”, one uses 非, e.g. 道可道，非常道 “The way can be expressed, but it is not the eternal way”.
• To say “does not [verb]” or “is not [adjective]”, one uses 不, e.g. 名不正，則言不順；言不順，則事不成 “If names are not correct, then speech will not be understood; if speech is not understood, then things will not be accomplished”.

Regardless, it seems clear to me that O(bjector) is assuming that “A is B” implies “A and B are inter-substitutable”, or at least “B can substitute for A”, when he (later) argues that a white horse is not a horse by pointing out that, when asking for a horse, one could be sent a brown or black horse, but when asking for a white horse, one would not be sent a brown or black horse. What this really reveals is that the subtyping relation does not play well with complex predicates.

Posted by: Zhen Lin on February 25, 2015 12:18 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Thanks! I don’t know Chinese, which is one reason I should probably stay away from this subject in my paper. (I don’t know Greek either, but I can’t resist talking about Heraclitus.) My wife knows classical Chinese and Greek, and she’s mentioned that in classical Chinese there’s a lot more lexical flexibility than modern Mandarin: for example, there’s a passage where ‘ten thousand’ seems to be used as a verb: “the army ten thousands”.

Anyway, I agree with your interpretation of the passage, which makes it less relevant to my goals than Chad Hansen’s interpretation in terms of ‘mass nouns’.

Posted by: John Baez on February 25, 2015 4:08 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

What this really reveals is that the subtyping relation does not play well with complex predicates.

In more categorical language, that sounds like one needs to account for contravariance. (-: So it’s not entirely unrelated to a book about category theory for philosophers, although I guess it may not have much to do with John’s goal to talk about equality.

Posted by: Mike Shulman on February 25, 2015 6:55 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

That did come to mind – after all, it comes up in programming – but I think it’s even more complicated than that. The action “ask for an $X$” is covariant in $X$, but “$X$ is acceptable” is contravariant in $X$, so “if I ask for an $X$, then $X$ is acceptable” is neither covariant nor contravariant in $X$.

Posted by: Zhen Lin on February 25, 2015 9:43 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

I would classify problems of that sort as part of “accounting for contravariance”. (-:

Posted by: Mike Shulman on February 25, 2015 11:32 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Lately I’ve frequently heard my three-year old say something like, “No, I don’t want juice, I want apple juice!” So this paradox had a very familiar feel to me.

Posted by: Mark Meckes on February 25, 2015 6:24 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Wow, that’s great! Have you ever tried saying “but apple juice is juice”? I suppose the mere fact that I’m asking this is a sign that I’m not wise enough to have children.

Posted by: John Baez on February 25, 2015 7:24 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Some times “apple juice” is not juice.

As to the relation between wisdom and having children, that’s a fraught topic :).

Posted by: Eugene on February 25, 2015 7:42 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Have you ever tried saying “but apple juice is juice”?

Often, with mixed results. Whether or not this is a good idea at a given moment depends on just how upset my son is about the type of juice being offered. But judging that is a matter of experience, not wisdom.

The linguistic issues here are more subtle than they may appear at first (even leaving aside the point that Eugene brings up — the “apple juice” in our house really is juice). We’re used to thinking that an English phrase or compound of the form modifier noun denotes a type of noun, when modifier isn’t explicitly negative. But there are many phrases where that’s not the case; for instance, horseshoe crabs aren’t crabs, and it’s questionable whether horseshoes are shoes. This shows up in mathematical English as well: in a vector space, subspaces are affine subspaces, but most affine subspaces aren’t subspaces. I can’t decide whether I think this has any bearing on the white horse paradox.

Sorting out the general rules from all the exceptions in language is a phenomenally complex task. In my son’s case, I think that he currently believes that “juice” by itself means “orange juice”, because until fairly recently that was the only type of juice he had had at home.

Posted by: Mark Meckes on February 25, 2015 8:30 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Cf. the red herring principle.

I can’t decide whether I think this has any bearing on the white horse paradox.

I don’t think it does, because white horses are horses.

Posted by: Mike Shulman on February 25, 2015 9:13 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

I don’t think it does, because white horses are horses.

They are, but a priori they don’t have to be. To say whether this has anything to do with Gongsun Long’s point would probably require knowing more than I do (i.e., anything at all) about classical Chinese.

Posted by: Mark Meckes on February 26, 2015 1:14 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

It’s a straightforward translation. “White horse” is 白馬, “horse” is 馬. But I think this is a non-issue. Stallions are horses, but there’s no etymological relation between “stallion” and “horse”!

Posted by: Zhen Lin on February 26, 2015 8:11 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Whether some horses can also be referred to by a completely different word is a separate issue. The (first) relevant question here is whether it is possible, or even common, for a phrase of the form adjective noun in classical Chinese to mean something which is not adjective, not a noun, or both. In particular, whether this is possible, or even common, for 白 in combination with other nouns or for 馬 in combination with other adjectives.

This may or may not have anything to do with the point Gongsun Long was making. But it would definitely be interesting information about the linguistic context in which he conceived the paradox.

Posted by: Mark Meckes on February 26, 2015 3:01 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Of course it’s possible. For instance, the passage speaks of 黑馬 “black horse” and 黃馬 “brown horse”. It goes without saying that 白, 黑, and 黃 are all ordinary colour words as well. I reiterate that the “paradox” does not depend on any special features of classical Chinese. (Proof: it can be translated into English!)

Posted by: Zhen Lin on February 26, 2015 4:44 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Of course it’s possible. For instance, the passage speaks of 黑馬 “black horse” and 黃馬 “brown horse”.

Unless those phrases ordinarily mean something other than “a horse which is black/brown”, they aren’t examples of what I’m asking about.

I reiterate that the “paradox” does not depend on any special features of classical Chinese.

I understand that. I’m merely curious about the extent to which features of the language may have helped to suggest the paradox in the first place. (In particular, I’m not suggesting or assuming that classical Chinese would do so any more than modern English.)

Posted by: Mark Meckes on February 26, 2015 7:33 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Ah. So I suppose you are asking about bahuvrihi compounds and the like. I don’t know whether they were common in classical Chinese. Modern Chinese probably has more, e.g. 赤字 “deficit” (lit. “red characters”), 青年 “youth, young person” (lit. “green year”), 黃色 “obscene, pornographic” (but more commonly “yellow colour”). Of course, English has them too: “redcoat”, “greenback”, “blackhead”, etc.

Posted by: Zhen Lin on February 27, 2015 2:26 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Mark wrote:

I’m merely curious about the extent to which features of the language may have helped to suggest the paradox in the first place.

As I mentioned, this paper suggests that special features of classical Chinese did help suggest the paradox:

• Chad Hansen, Mass nouns and “A white horse is not a horse”, Philosophy East and West 26 (1976), 189–209.

But right now, I’m not convinced.

(If you don’t have access to this journal on JSTOR, let me know.)

By the way, old-timers here often reply to the next to last comment, to avoid a huge buildup of vertical bars at the left.

Posted by: John Baez on February 26, 2015 8:04 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

By the way, old-timers here often reply to the next to last comment, to avoid a huge buildup of vertical bars at the left.

Yes, I know that, and I sometimes do the same. But I’ve also been around long enough to recall a discussion about the practice, in which someone (Mike, maybe?) complained that doing so messes up the threading, which is more inconvenient than the vertical bars for people reading the blog using some other program.

Posted by: Mark Meckes on February 26, 2015 8:30 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Replying to the next-to-last comment is appropriate when a discussion is mostly linear. When it starts to get all fractally, then I get confused no matter what, but I guess threading helps a bit. My rule is that if I’m replying to the last comment in a particular thread, then I actually reply to its parent, whereas if I’m replying to a comment in the middle of a thread, then I reply to it directly, thereby creating a new branch.

Posted by: Mike Shulman on February 26, 2015 10:47 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

I’ve started to suspect that in many situations, it makes sense to use notions of “sameness” that are not necessarily symmetric. If you find this phrase too provocative, then take it as saying that ordering relations are sometimes more natural and fundamental than equality or equivalence relations.

For example, “white horse” is more specific than “horse”, and hence any predicate which has a definite value on “horse” will also have a definite value on “white horse”. In analogy to Leibniz equality, we could introduce a Leibniz ordering $x\leq y$ as shorthand for $P(x)\Rightarrow P(y)$ for all $P$. This captures the horse vs white horse example if you assume that predicates with unspecific value are taken to be false. The ordering here goes from the less specific to the more specific, but of course this is purely conventional.

We could also say that a white horse can substitute for a horse, but not conversely. So in general, substitutability is not a symmetric concept. Have logicians studied substitutability under this aspect? I think that this is related to the problem of coming up with directed type theories: the usual identity types are what formalizes substitution in many interesting type theories, but the induction principle of the identity type makes this notion of substitutability automatically symmetric. I don’t know how the existing proposals for directed type theory get around this.

As another intuitive example, take a firm with employees Alice and Bob. Imagine that Alice can substitute for Bob when he is away, but Bob does not have the skill necessary to stand in for Alice when she is off. I think that this can be substituted for the horse-vs-white-horse example ;)

Well, most of this will be familiar to any category theorist who’s not just a groupoid theorist. Which seems to mean that we now have a meta-example of a non-symmetric substitutability relation…

Posted by: Tobias Fritz on February 25, 2015 7:03 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Tobias wrote:

In analogy to Leibniz equality, we could introduce a Leibniz ordering $x\leq y$ as shorthand for $P(x)\Rightarrow P(y)$ for all $P$.

That’s interesting.

Of course in classical logic this is rather dull, since if we assume $P(x)\Rightarrow P(y)$ for all $P$ then we either also have $P(y)\Rightarrow P(x)$ for all $P$, in which case we say $x = y$… or we don’t, in which case we have $Q(y)$ but not $Q(x)$ for some $Q$, and thus $\not Q(x) \nRightarrow \not Q(y)$, contradicting our assumption.

But maybe in intuitionistic logic, or the internal logic of some topos of sheaves, we can have one individual who has all the properties of another, and more.

Or, you might try to distinguish ‘positive’ properties from ‘negative’ ones — for example, abilities versus disabilities — and write $x\leq y$ as shorthand for $P(x)\Rightarrow P(y)$ for all ‘positive’ $P$.

Imagine that Alice can substitute for Bob when he is away, but Bob does not have the skill necessary to stand in for Alice when she is off.

For example if Alice is a category theorist, but Bob is a mere groupoid theorist.

Posted by: John Baez on February 25, 2015 7:21 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

John wrote:

Of course in classical logic this is rather dull, since if we assume $P(x)\Rightarrow P(y)$ for all $P$ then we either also have $P(y)\Rightarrow P(x)$ for all $P$

Good point. Another way to phrase that is to say that the Leibniz ordering is “accidentally” symmetric in classical logic. This is similar to how any given category can accidentally turn out to be a groupoid.

To capture the horse example, I was secretly thinking of a logic with an additional truth value of “unspecified” or “don’t know”, so that we have, for example, $IsBlack(horse) = unspecified, \qquad IsBlack(white horse) = false.$ In this setup, we would have $horse \leq white\, horse, \qquad white\, horse \nleq horse.$

Posted by: Tobias Fritz on February 25, 2015 8:41 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

I’ve been thinking along similar lines. Rescher and Brandom have an interesting book, The Logic of Inconsistency in which they develop a logic roughly based on modal logic but with the addition of impossible worlds. Some impossible worlds are ‘gappy’ (both a proposition and its negation fail at the world) and others are ‘glutty’ (both a proposition and its negation hold at the world).

In this framework you can have ‘gappy’ and ‘glutty’ individuals. (Just start with the Russellian idea that an object is the bundle of properties it instantiates, a gappy object then is one that lacks both a property and its negation and a glutty object one that includes both.) Rescher and Brandom are interested in ‘glutty’ objects because they want to reconstruct Meinong. However, we can allow in your terminology $horse$ and $whitehorse$ to be ‘glutty’ objects, the first of these the unioning of the class of all possible horses and the second the unioning of the class of all possible horses. Then we will have

(1)$P(horse) \Rightarrow P(whitehorse)$

for all $P$ but not vice versa. (We needn’t even introduce an ‘unspecified’ value. We would just have $IsBlack(horse) = false$ and $not-IsBlack(horse) = false$ while $not-IsBlack(whitehorse) = true$.) This logic would do nice things like assign unique states of affairs to both incomplete and inconsistent worlds and give us abstract and Meinongian objects. (Of course whether we actually want these is a separate issue.)

Posted by: Damon Stanley on March 3, 2015 5:01 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

In this setup, we would have $horse \lt whitehorse$, $whitehorse \nless horse$.

Something like that actually goes on in conversational implicature involving Grician maxims.

Let’s say Charles has a beard and a mustache. It would normally be infectiousness to introduce a reference to him as “the man with the mustache” (even though he may be the only man in scope with a mustache) because a beard is more salient than a mustache and it is often safe to assume that a man with a beard does have a mustache. Not having a mustache to go with a beard is so uncommon that its absence may need to be noted in certain situations.

Posted by: RodMcGuire on March 3, 2015 4:12 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Thanks very much, David, Damon and Rod, for that valuable information!

I’m currently writing a paper in which a non-symmetric notion of substitutivity is the main theme (although it doesn’t carry that name, for good reason). I’d like to mention that non-symmetric notions of substitutivity have been investigated by Brandom, and so I’m currently reading parts of Brandom’s Articulating Reasons. Is there other philosophical literature in which the failure of symmetry of substitutivity is explicitly discussed?

The nForum seems to be down again, or I’d be looking for a place where you talk about this ;)

Posted by: Tobias Fritz on March 5, 2015 5:35 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

The beard vs. mustache example comes from Jim McCawley. I’m sure he explained it better along with a theoretical framework in which it is an issue. I probably have the book or paper it appears in but it is packed up somewhere I can’t lay a hand on.

Posted by: RodMcGuire on March 5, 2015 11:20 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

This is just the argument the philosopher Robert Brandom gives for why there must be a symmetric kind of substitutivity (applicable to singular terms) and a possibly non-symmetric part (applicable to predicates, etc.)

nForum seems to be down or I’d point to a place where I talk about this. I was hoping it might provide grounds for an account of why in type theory there are terms and types.

Posted by: David Corfield on March 2, 2015 10:03 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

I don’t think “Leibniz ordering” is any more interesting intuitionistically: try $P(z) = (z\le x)$.

Distinguishing properties by variance, on the other hand, leads to directed type theory. (-:

Posted by: Mike Shulman on February 25, 2015 9:15 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Mike wrote:

I don’t think “Leibniz ordering” is any more interesting intuitionistically: try $P(z) = (z\le x)$.

I can see that that is feasible in type theory, and emulates the construction of path inversion $(x=y)\to (y=x)$ in HoTT. But in other setups, such as first-order logic, referring to the ordering itself may not be a valid predicate and then that argument doesn’t work.

But admittedly, other than horsing around with horses and groupoid theorists, I don’t know of any situations in which the Leibniz ordering would not be symmetric.

Distinguishing properties by variance, on the other hand, leads to directed type theory. (-:

Aha, interesting! Thanks.

Posted by: Tobias Fritz on February 25, 2015 9:33 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

in other setups, such as first-order logic, referring to the ordering itself may not be a valid predicate and then that argument doesn’t work.

Well, John’s argument in classical logic involved a quantifier over all predicates, so it’s not first-order either. But anyway I don’t think it’s very interesting if the definition of the ordering only happens at a meta-level.

Posted by: Mike Shulman on February 25, 2015 11:35 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Mike wrote:

But anyway I don’t think it’s very interesting if the definition of the ordering only happens at a meta-level.

Doesn’t conventional Leibniz equality live at the meta-level as well?

A web search suggests that whether Leibniz equality is first-order definable or a second-order property depends on the particular theory under consideration. There’s a paper studying this kind of question, and it seems to be quite non-trivial.

In any case, I don’t see how my “Leibniz ordering” would be different from Leibniz equality in that aspect.

To make the discussion a bit more concrete, let me give a mathematical example of a situation in which I believe that using a non-symmetric notion of substitutability may be enlightening and useful. In a topological space, two points $x$ and $y$ are Leibniz-equal if and only if they have exactly the same open neighbourhoods. Equivalently, the Leibniz equality $x=y$ holds if and only if $y$ lies in the closure of $x$ and $x$ lies in the closure of $y$.

However, we retain more information about the topology if we take this conjunction apart. So we declare $x\leq y$ whenever $y$ lies in the closure of $x$. This is the specialization preorder. By construction, this results in $x=y$ if and only if $x\leq y$ and $y\leq x$.

If $x\leq y$ holds, then every open neighbourhood of $y$ is also an open neighbourhood of $x$. Hence every sequence/net/filter which converges to $x$ also converges to $y$. So as far as being a limit is concerned, $y$ can be substituted for $x$.

I suspect that the specialization preorder becomes an instance of a Leibniz ordering if one implements John’s idea of distinguishing ‘positive’ predicates from ‘negative’ ones, but so far this is just speculation.

Posted by: Tobias Fritz on February 26, 2015 1:07 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

John’s idea of distinguishing ‘positive’ predicates from ‘negative’ ones

On second thought, this may be closely related to Mike’s comment on variance and directed type theory.

Posted by: Tobias Fritz on February 26, 2015 1:41 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Tobias wrote:

A web search suggests that whether Leibniz equality is first-order definable or a second-order property depends on the particular theory under consideration. There’s a paper studying this kind of question, and it seems to be quite non-trivial.

Thanks! Later in this paper I plan to get into Leibniz’s definition of equality, so I want to gain expertise on this subject.

Once upon a time on G+ I claimed that Leibniz’s idea only lets you define equality in second-order logic, via:

$x = y \; \iff \; \forall P (P(x) \iff P(y))$

But Toby Bartels pointed out that we can do something similar in first-order logic if we allow ourselves an axiom schema that has the effect of quantifying over definable predicates.

(I could make this more precise, at some cost in pain to both me and my readers, but I’m hoping it’s sorta clear enough for now.)

Posted by: John Baez on February 26, 2015 2:18 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Tobias wrote:

I suspect that the specialization preorder becomes an instance of a Leibniz ordering if one implements John’s idea of distinguishing ‘positive’ predicates from ‘negative’ ones, but so far this is just speculation.

Here’s one famous instance of distinguishing ‘positive’ predicates from ‘negative’ ones. Popper said that a theory should be falsifiable. Presumably he meant that it can contain statements like

for all $x$, if $x$ is a raven then $x$ is black

since discovering one green raven would falsify it. Let’s call a falsifiable claim a ‘positive’ statement. Then its negation is typically not falsifiable:

there exists a raven that is not black

and apparently Popper didn’t want these ‘negative’ statements in a scientific theory.

I believe that if the space of states of the world is imagined as a topological space $X$, a ‘positive’ statement should be envisioned as asserting that the state of the world lies in some closed subset $C \subseteq X$. Its negation then corresponds to an open subset, so it’s possible to check that $x \in X - C$ with a finite amount of work (e.g., measurements with finite accuracy).

This is supposed to connect to your ideas on topology. It’s also related to intuitionistic logic, but there we associate propositions to open sets and define the negation using the interior of the complement, so every statement is ‘robust’, maintaining its truth even if we perturb the state of the world slightly.

(For some reason Popper likes statements that are falsifiable, while intuitionists like statements that are verifiable!)

Posted by: John Baez on February 26, 2015 2:34 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Very cool!

I’m not sure what to make of the topological stuff at the moment. But I guess that there also is a connection to variance: your ‘positive’ statement is contravariant in raven, while the ‘negative’ statement is covariant, as follows.

The type-theoretic version of ‘for all $x$, if $x$ is a raven then $x$ is black’ is ‘for all $x$ of type Raven, $x$ is black’, $\prod_{x:Raven} isBlack(x).$ This is a type of dependent functions, and as such it is contravariant in its domain Raven; if we take e.g. the obvious function $Raven\to Bird$, then we obtain $\bigg( \prod_{x:Bird} isBlack(x) \bigg) \longrightarrow \bigg( \prod_{x:Raven} isBlack(x) \bigg).$ This means that the general hypothesis ‘all birds are black’ specializes to ‘all ravens are black’. Since there is ‘Bird’ on the left and ‘Raven’ on the right, this is contravariant.

Now let’s do the same with ‘there exists a raven that is not black’. Type-theoretically, this is described by the dependent pair type, $\sum_{x:Raven} isNotBlack(x).$ And this is indeed covariant in Raven, so that our function $Raven\to Bird$ gives rise to a function $\bigg( \sum_{x:Raven} isNotBlack(x) \bigg) \longrightarrow \bigg( \sum_{x:Bird} isNotBlack(x) \bigg).$ This means that in order to find a bird that is not black, it is sufficient to find a raven that is not black. Since there is ‘Raven’ on the left and ‘Bird’ on the right, this is covariant.

Does this capture anything of Popper? One reason for why it might not is that I have only considered the variance in “Raven”, but not e.g. the variance in “isBlack”, with respect to which both statements are covariant.

Posted by: Tobias Fritz on February 26, 2015 5:45 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

There is at least a connection between variance and topology: open sets (verifiable statements) are closed under arbitrary unions (i.e. $\Sigma$/$\exists$) while closed sets (falsifiable statements) are closed under arbitrary intersections (i.e. $\Pi$/$\forall$).

Posted by: Mike Shulman on February 26, 2015 10:54 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

I think that does capture something of Popper’s philosophy. I think he likes scientists to “stick their neck out”: to make bold claims that can be shown wrong by a single example. Since we have a map

$\{example\} \hookrightarrow \{many \; cases\}$

it seems he likes contravariant statements, where a statement about many cases pulls back to a statement about each example.

Mike wrote, below:

Perhaps the reason intuitionists like verifiable statements is that they want to do mathematics, i.e. they want to prove things, and proving is a kind of verifying. Maybe this has something to do with the opposition between inductive and deductive reasoning?

Maybe. Perhaps mathematicians prefer verifiability to falsifiability. They don’t have quite the same urge to “stick their neck out”: they like to prove things and be certain they’re true.

However, it’s a bit tricky! They do prefer bold general conjectures like:

“Every even number > 2 is the sum of two primes.”

to conjectures about unseen special cases, like:

“Some even number > 2 is not the sum of two primes.”

In this way they side with Popper, who prefers scientific theories like

“Every raven is black.”

to

“There exists a white raven.”

Posted by: John Baez on February 27, 2015 5:56 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Popper usually phrased the difference in terms of general/universal and particular/existential statements. His classic non-falsifiable statement was

There exists a finite sequence of Latin elegiac couplets such that, if it is pronounced in an appropriate manner at a certain time and place, this is immediately followed by the appearance of the Devil - that is to say, of a man-like creature with two small horns and one cloven hoof.

Naturally many people have pointed out that science is full of particular statements, e.g.,

65 million years ago an asteroid struck the Earth near what is today the Yucatan and killed off many species, including most dinosaurs.

Posted by: David Corfield on March 2, 2015 9:58 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Well, I always tend to think in full dependent type theory these days, where you have a type of propositions that you can quantify over. Anything else is centipede mathematics. (-:O

Posted by: Mike Shulman on February 26, 2015 5:33 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

Perhaps the reason intuitionists like verifiable statements is that they want to do mathematics, i.e. they want to prove things, and proving is a kind of verifying. Maybe this has something to do with the opposition between inductive and deductive reasoning?

Interestingly, there is also a way to distinguish “positive and negative” statements inside constructive logic. Suppose we’ve specified a particular subset $\Sigma$ of the set $\Omega$ of propositions, which contains $\top$ and $\bot$ and has the property that if $p\in \Omega$ and $u\in \Sigma$ and $u\to (p\in \Sigma)$, then $(u\wedge p)\in \Sigma$. Such a set is called a dominance. Of course $\Omega$ itself is a dominance, but there are other interesting ones, such as the “Rosolini dominance” which consists of the propositions of the form $(\exists n\in\mathbb{N}) f(n)=0$ for some $f:\mathbb{N}\to 2$.

After fixing a dominance, we define a subset to be open if its characteristic function factors through $\Sigma$, and closed if it is the complement of an open subset. (Or better, we define a “closed subset” to be an open subset “thought of complemetarily” — that is, a falsifiable statement is defined by specifying a verifiable statement that falsifies it.) This is one road to “synthetic topology” — which, interestingly enough, is also going to appear in my upcoming post! I’m not an expert in synthetic topology, but one place to read some more about it is here. I expect that you could define the specialization preorder in the obvious way in this context, and that my argument for degenerating it to Leibniz equality won’t apply because it won’t itself be open.

Posted by: Mike Shulman on February 26, 2015 5:59 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

To get a better idea of how I arrived at the suspicion that non-symmetric notions of equality may be of interest, check out this draft. The discussion thread here has resulted in Remark 2.3. Any comments will be very welcome, such as suggestions on where to submit this!

Posted by: Tobias Fritz on March 12, 2015 4:51 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

This reminds me of another paradox I encountered. You think of “white horses” as a subset of “horses” and in general, “adjective noun” as a subset of “noun.” So if you go about listing facts that are true about all lions– they are carnivores, they are animals, etc… all of these facts should be true about any subset of lions (because they are true for all lions). So which of these facts will be true about “stone lions”?

Posted by: Doug Summers-Stay on February 26, 2015 12:53 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

It’s vacuously true since the subset of lions (that are carnivorous animals etc) that are made of stone is empty: if Leo(nie) is a stone lion, then it is a carnivore, and animal, etc.

Unless you are taking the union (necessarily disjoint) of living lions and other things one might call lions, but then the statement they are all carnivores etc is wrong.

Posted by: David Roberts on February 26, 2015 8:13 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

In case it’s not clear… there are stone lions in front of many buildings in China. Here is a particularly grand one:

Stone lions are to lions as Lie algebras are to algebras.

Posted by: John Baez on February 26, 2015 4:11 PM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

That looks like a bronze lion. Google suggests it is, and that it is at the Forbidden City. Google also tells me that there is stone lion very much like it at Tiananmen Gate.

Posted by: S. on March 1, 2015 1:51 AM | Permalink | Reply to this

### Re: Concepts of Sameness (Part 2)

I haven’t thought this out too well, but maybe we can think of ‘white’ and ‘horse’ as verbs in the sense that to say ‘horse’ is to say ‘it horses’ and to say ‘white’ is to say ‘it whites’. Then we could think of them as arrows in some category, where the fact that ‘white horse’ is sensible while ‘horse white’ is not means that the codomain of horse is the domain of white. Then ‘white horse’ is not ‘horse’, since they have different codomain, but they are the same in the sense that ‘white horse’ factors through ‘horse’.

It’s like the categorical reading of the classical syllogism “Socrates is a man. All men are mortal. Therefore, Socrates is mortal” as a diagram (apologies for the lack of tex, I’m not sure how to work it): 1 –socrates–> Men >-are-> Mortal. Socrates (the man) is mortal only in as much as he factors through Men >-are-> Mortal. So the white horse paradox could be grasping at the kind of contextualization of domain/codomain that category theory makes explicit, but also playing with our want to (judgementally) identify the man Socrates and the mortal Socrates.

Posted by: David Jaz on March 8, 2015 4:12 PM | Permalink | Reply to this

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