There are different views on how to formalize qualifications, which can be made as “A is B qua C”, or more idiomatically in English, “A is B, when A is considered in its respect of being C”. A related, but distinct phrase is “A is B insofar as A is C”. They are reviewed in Hennig (2024). The following are some examples of qualification adapted from the paper:
- Descartes claims that, insofar as he is a thinking being, he is not made of matter.
- Kant claims that we can know things as appearances, but not as they are in themselves.
- Aristotle claims that an isosceles triangle has internal angles that add up to two right angles, but it has this property qua triangle, not qua isosceles.
- Suppose: “Jane is a corrupt judge, but an honest merchant”, which is to say, “Jane is corrupt qua judge, but honest qua merchant”.
The paper reviews many proposals for translating such qualifications into formal languages. It does not review the one I propose here, because it was not in the literature. My proposal relies on degree-theoretic semantics, which is the theory that sentences have degrees of truth instead of merely being true or false. This semantics arises in the philosophical theory of vagueness, which is concerned largely with solving the well-known sorites paradox:
- how many grains of sand does it take for you to have a heap of sand?
- at what precise instant does a child become an adult?
- how many hairs on his head does it take for a man to no longer be bald?
Degree-theoretic semantics approaches this problem by saying that there is no such number. Rather, the truth-value of “x is a heap of sand”, “x is not bald”, or “x is an adult” moves continuously from less true to more true as more grains of sand, hairs on the head, or years of age are added to x. Its oldest version seems to come from this 1976 paper by Kenton F. Machina, but probably its most famous version came from Dorothy Edgington in 1996.
My proposal is to translate “A is B qua C” into: the degree of truth of “A is B” is proportional to the degree of truth of “A is C”. If we take the properties B and C as predicates \( B \) and \( C \), the object A as a constant \( a \), and \( \mathcal{I} \) as the interpretation function that assigns degrees of truth to statements, then formally:
\[
\mathcal{I}(\ulcorner Ba \urcorner) \propto \mathcal{I}(\ulcorner Ca \urcorner).
\]
I propose this formalization only for “qua”, or “considered in its respect as”. For the related “A is B insofar as A is C”, although at first I thought it could be treated uniformly, my proposal is instead to express it by equality rather than proportionality:
\[
\mathcal{I}(\ulcorner Ba \urcorner) = \mathcal{I}(\ulcorner Ca \urcorner).
\]
The proportionality claim is implied by the equality claim, but not conversely, so “insofar as” is stronger than “qua”.
This seems easy enough to apply to the examples.
-
Descartes claims that the degree to which he is not made of matter is equal to the degree to which he is a thinking being. With \( T \) for being a thinking being, \( d \) for Descartes, and \( M \) for being made of matter, we have:
\[
\mathcal{I}(\ulcorner \neg Md \urcorner) = \mathcal{I}(\ulcorner Td \urcorner).
\] -
Kant claims that the degree to which something is knowable is not equal to the degree to which it is itself, but it is equal to the degree to which it is an appearance. With \( I \) for a thing’s being itself, \( K \) for a thing’s being knowable, and \( A \) for a thing’s being an appearance, we have:
\[
\forall t \big[\mathcal{I}(\ulcorner Kt \urcorner) \neq \mathcal{I}(\ulcorner It \urcorner)\big],
\]
and
\[
\forall t \big[\mathcal{I}(\ulcorner Kt \urcorner) = \mathcal{I}(\ulcorner At \urcorner)\big].
\] -
Aristotle claims that the truth of something’s having angles summing to two right angles is proportional to the truth of its being a triangle, but not proportional to the truth of its being isosceles. With \( A \) for the property of having angles that add up to two right angles, \( I \) for the property of being isosceles, and \( T \) for the property of being a triangle, we have:
\[
\forall t \big[\mathcal{I}(\ulcorner At \urcorner) \propto \mathcal{I}(\ulcorner Tt \urcorner)\big],
\]
and
\[
\forall t \big[\mathcal{I}(\ulcorner At \urcorner) \not\propto \mathcal{I}(\ulcorner It \urcorner)\big].
\] -
The degree to which Jane is honest is proportional to the degree to which she is a merchant, while the degree to which she is corrupt is proportional to the degree to which she is a judge. Let \( H \) be the property of being honest, and assume being corrupt just means not being honest, i.e. \( \neg H \). With \( j \) for Jane, \( M \) for merchant, and \( J \) for judge, we have:
\[
\mathcal{I}(\ulcorner Hj \urcorner) \propto \mathcal{I}(\ulcorner Mj \urcorner),
\]
and
\[
\mathcal{I}(\ulcorner \neg Hj \urcorner) \propto \mathcal{I}(\ulcorner Jj \urcorner).
\]
I also now propose to render the reduplicative “A qua A is B” (which is the same as “A is B qua A”) as:
\[
\mathcal{I}(\ulcorner Ba \urcorner) \propto \mathcal{I}(\ulcorner a=a \urcorner).
\]
And “A is B insofar as A is A” accordingly becomes:
\[
\mathcal{I}(\ulcorner Ba \urcorner) = \mathcal{I}(\ulcorner a=a \urcorner).
\]
So if A, considered as itself, is B, it need not be fully B. But if A is B precisely insofar as A is itself, then A must be fully B, since, I think, everything is always fully itself: \( \mathcal{I}(\ulcorner a=a \urcorner) = 1 \). When we say something “is not itself” to some extent, I take that to be a figure of speech; properly speaking, everything is always fully itself.
From the theory as I now have it, together with the mathematical properties of the proportionality relation, I derive the following properties of qualification:
- Always, A qua A is A.
- If A is B, then A qua B is B.
- A qua B is C if and only if A qua C is B.
- If A qua B is C, and A is not C at all, then A is not B at all.
- If A qua B is C, and A is B to some extent, then A is C to some extent.
- If A qua B is C and A qua C is D, then A qua B is D.
- If A qua B is C and A qua B is D, then A qua D is C.
- If A qua B is C, then A qua D is B if and only if A qua D is C.
Assuming additionally, as a property of negation, that
\[
\mathcal{I}(\ulcorner \lnot Ba \urcorner) = 1 - \mathcal{I}(\ulcorner Ba \urcorner),
\]
we also have:
- If A qua B is C, and A qua non-B is non-C, then A is B precisely insofar as A is C.
I do not yet know any objections to this. If I knew of any, I would try to answer them, which would give me enough material to try to publish this as a paper instead of a blog post.
This came up in conversation earlier because of my further view that statements involving higher-order properties should be interpreted as qualified general statements about properties. For instance, “honesty is desirable” should be interpreted as: for any \( x \), \( x \) is desirable qua honest. This in turn means that, for any \( x \), the degree to which \( x \) is desirable is proportional to the degree to which \( x \) is honest:
\[
\forall x \big[\mathcal{I}(\ulcorner Dx \urcorner) \propto \mathcal{I}(\ulcorner Hx \urcorner)\big].
\]
In my view, this makes higher-order properties unnecessary. This is motivated by my view that the reason we can use properties to understand things is because properties are intrinsically understandable in themselves. If this is so, then properties do not need further metaproperties to be understood.
Finally, if one wants to distinguish “A qua A is B” from “A is B, but not qua A”, it may be clearer to avoid “qua” entirely and instead speak of “A is B de dicto” and “A is B de re” respectively. I am not sure these formulations are truly equivalent, and that remains room for further research.
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