Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

February 14, 2020

Magidor on Category Mistakes and Context

Posted by David Corfield

The previous discussion on category mistakes got me reading Ofra Magidor’s SEP article on the subject. Magidor was the right choice to produce this article as the author in 2013 of an OUP book Category Mistakes. She is the Waynflete Professor of Metaphysical Philosophy at the University of Oxford (website), a chair once held by one of my favourite British philosophers, R. G. Collingwood.

Now Collingwood came up before in a post of mine as someone who thought that the representation of propositions in Russell’s logic was totally misguided. Rather than freestanding statements, for Collingwood, propositions only make sense in the context of a series of questions and answers. In part, it was thinking through his insights in terms of type theory that got me started on the idea of proposing the latter as a new logic for philosophy of language and metaphysics.

Returning to Magidor’s SEP article mentioned above, we read

Another potential problem for the syntactic approach has to do with the context sensitivity of category mistakes. Consider the following examples:

(11) The thing I am thinking about is green.

(12) That is green.

(13) The number two has the property I’ve just mentioned.

Whether or not these sentences exhibit the kind of infelicity associated with category mistakes depends on context: (11) and (12) are perfectly felicitous in a context where it is clear that the thing I am thinking about/referring to is a pen, but not so in a context where it is the number two. Similarly, (13) is felicitous in a context where it is clear that the property I’ve just mentioned the property of being prime, but not so if it was the property of being green. This is a problem for the syntactic approach, because whether a sentence is syntactically well-formed or not is a context-invariant property: it cannot vary according to context in this manner.

From the type-theoretic perspective, this sounds very strange. There we have the term ‘context’ used in a technical sense to express the conditions that are required in order to form the type that is the proposition, as represented by the sentence. I don’t know the origins of the term in the type-theoretic tradition, but it surely arises from something close to the informal sense of the situation in which judgements are expressed.

Let’s consider proposition (13). Natural language uses sugared turns of phrase for what is represented in type theory, often leaving out parameters. It seems to me reasonable to take

The AA, aa, has the property PP

as shorthand for the still sugared

The AA, aa, has the property(AA), PP.

In a situation in which we have a:Aa:A and P:Property(A)P: Property(A), we can form the proposition ‘The AA, aa, has the property PP’, as a longhand version of P(a)P(a). We may then judge this proposition to be true.

When I say ‘The BB I just mentioned’, for a type BB, this means that we have a recent judgement b:Bb:B, whence

‘The BB I just mentioned’ :b:B:\equiv b: B.

In the current case, I must have recently mentioned some property. So is there a type of properties? Well, a reasonable first step is to say that in the context X:TypeX: Type, we have

Property(X):[X,Prop]:TypeProperty(X) :\equiv [X, Prop] :Type,

where Prop is the type of propositions. (Think subobject classifier, or just the two-element set if you like.)

Let us imagine then that we have judged A:Type,P:[A,Prop],a:AA: Type, P: [A, Prop], a: A. Then

aa has the property (of AA) I just mentioned (that is, PP)

is well-formed. Indeed, we can see that P(a)P(a) is a proposition, and now the claim is being made that P(a)P(a) is true.

Is there a more general type of properties? I suppose one might form the type X:TypeProperty(X)\sum_{X: Type} Property(X), of pairs (A,P)(A, P) such that PP is a property of AA. One might even call this dependent sum, PropertyProperty.

Then if I’ve been speaking about a property (B,Q)(B, Q),

‘The property I just mentioned’ :(B,Q):\equiv (B, Q): Property.

But then I can’t form the proposition ‘aa has (B,Q)(B, Q)’ if I’ve been introduced to aa, as a:Aa:A, since it does not arise from the rule:

X:Type,x:X,(X,P):PropertyX: Type, x: X, (X, P): Property \vdashxx has (X,P)(X, P)P(x):Prop\equiv P(x): Prop.

I find explaining such things in type theory an odd affair. It feels like you’re making a huge fuss about very little, here that we really have no right to be saying ‘The number two has the property I’ve just mentioned’, when the property I’ve just mentioned is ‘being green’. And yet if we need to say these things, then we had better do so.

I suppose one might question whether these considerations were all ‘syntactical’, but I don’t see why we shouldn’t see them as such. For type theorists, sentence formation is utterly context-dependent.

Posted at February 14, 2020 10:14 AM UTC

TrackBack URL for this Entry:

5 Comments & 0 Trackbacks

Re: Magidor on Category Mistakes and Context

One could have as context a list of numbers written with different coloured pens (with no repetition). Then, one could refer to “the number 2” as being one of the items on that list, and the proposition whether it was green or not would then be well-formed. But if the context was \mathbb{N}, as you have implicitly above, then it doesn’t make sense, as you say.

Posted by: David Roberts on February 14, 2020 9:12 PM | Permalink | Reply to this

Re: Magidor on Category Mistakes and Context

For people with grapheme-color synesthesia, the digit 2 can already be blue/green/etc. regardless of its physical color. In other words, “the number two is green” is not automatically a category mistake.

It’s clear to me that we should consider a type of contexts, built from the type of properties, in order to discuss situations like this in which you’d want to agree with someone on most category errors but only disagree on whether numbers can have colors.

Posted by: unekdoud on February 15, 2020 6:14 AM | Permalink | Reply to this

Re: Magidor on Category Mistakes and Context

One would certainly need to distinguish a type of numbers from a type of instances of numerals.

Posted by: David Corfield on February 15, 2020 8:12 AM | Permalink | Reply to this

Re: Magidor on Category Mistakes and Context

One argument that the issue of the well-formedness of (13) is syntactical would reinterpret things category-theoretically.

The case is, in more general terms, as though I have arrows f:ABf: A \to B and g:ACg: A \to C, and I wonder whether it’s well-formed to say

The composite of ff and another arrow that I’ve just mentioned is equal to gg.

Posted by: David Corfield on February 15, 2020 3:15 PM | Permalink | Reply to this

Re: Magidor on Category Mistakes and Context

Later on in the article we are closer to home:

A second question which separates different versions of the presuppositional account is what kind of presuppositions are involved in the case of category mistakes. A natural hypothesis (endorsed by Asher (2011)) is that these are type presuppositions. For example, we might suggest that ‘is green’ triggers the presupposition that its subject is a concrete object; that ‘is prime’ triggers the presupposition that its subject is a number; and that ‘is pregnant’ triggers the presupposition that its subject is a female.

Presuppositions are something I treat in my book (p. 59) and appear in the (type-theoretic) context that needs to be in place for a type to be formed. Ranta had also reached this conclusion.

Now, apparently, the following causes a problem for Asher:

(36) Mathematician: you know, not only numbers, but also polynomials are prime or composite. This polynomial, for example, is prime.

Of course this is a reasonable thing to say. We can all imagine trying to expand the mind of a teenager by informing them of the extension of the concept of prime elements to kinds of collection, known as ‘rings’. We might also tell them about the idea of prime knots.

Staying with rings, a mathematician isn’t going to speak of something being prime without having specified the ring, since, e.g., 22 is prime in \mathbb{Z} but not in [i]\mathbb{Z}[i]. So there is a tacit parameter when speaking of prime elements:

R:Ring,r:Rprime R(r):Prop. R: Ring, r:R \vdash prime_R(r): Prop.

Returning to the article

These discourses [such as (36) - DC] seem entirely felicitous even though the final sentence in each discourse violates the proposed type presupposition. In response a proponent of the type-presuppositional hypothesis might suggest reverting to a different type-based hypothesis. For example, in response to (36) they might suggest that the presupposition generated is that the subject-term is a mathematical object. However, this new proposal might be too liberal and because it fails to account for why ‘2.145 is prime’ is a category mistake and it might also be too restrictive because it fails to account for other versions of the argument, where an expert suggests that a non-mathematical object can be prime as well.

That would be misguided to suggest that the presupposed type is any mathematical object. Primeness is relative to a ring. In fact, ‘2.145 is prime’ need not be a category mistake in that sense. Qua real number, the sentence is well-formed. It’s just false – fields, while they are rings, do not have any primes.

Outside of mathematics, I guess we say things like ‘prime real estate’, but that’s a homonym. We really shouldn’t expect each lexical item to have a single type reading.

Posted by: David Corfield on February 16, 2020 3:32 PM | Permalink | Reply to this

Post a New Comment