### 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 $A$, $a$, has the property $P$

as shorthand for the still sugared

The $A$, $a$, has the property($A$), $P$.

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

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

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

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 $X: Type$, we have

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

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 $A: Type, P: [A, Prop], a: A$. Then

$a$ has the property (of $A$) I just mentioned (that is, $P$)

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

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

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

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

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

$X: Type, x: X, (X, P): Property \vdash$ ‘$x$ has $(X, P)$’ $\equiv P(x): Prop$.

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.

## 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.