Qualification as Proportionality

There are different views on how to formalize qualifications, which can be made as, “A is B qua C”, but are also commonly made with English glosses like “A is B insofar as A is C”, or “A is B, when A is considered in its respect of being 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 wasn’t in the literature. My proposal relies on “degree-theoretic semantics”, which is the theory that sentences have “degrees of truth” instead of just being either 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, but rather that 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, then, 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, $\mathcal{I}(⌜Ba⌝)\propto \mathcal{I}(⌜Ca⌝)$.

That’s the entire proposal, and it seems easy enough to apply to the examples:

• Descartes claims that the degree to which he is a thinking being is proportional to the degree to which he is not made of matter. With $T$ for being a thinking being, $d$ for Descartes, and $M$ for being made of matter, we have $\mathcal{I}(⌜Td⌝)\propto \mathcal{I}(⌜\mathrm{\neg }Md⌝)$.
• Kant claims that the degree to which something is knowable is not proportional to the degree to which it is itself, but it is proportional 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 $\mathrm{\forall }t[\mathcal{I}(⌜Kt⌝)\propto ̸\mathcal{I}(⌜It⌝)]$, and $\mathrm{\forall }t[\mathcal{I}(⌜Kt⌝)\propto \mathcal{I}(⌜At⌝)]$.
• 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 $\mathrm{\forall }t[\mathcal{I}(⌜At⌝)\propto \mathcal{I}(⌜Tt⌝)]$, and $\mathrm{\forall }t[\mathcal{I}(⌜At⌝)\propto ̸\mathcal{I}(⌜It⌝)]$.
• 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’s use $H$ for the property of being honest, and let’s assume being corrupt just means not being honest, i.e., $\mathrm{\neg }H$. With $j$ for Jane, we have $\mathcal{I}(⌜Hj⌝)\propto \mathcal{I}(⌜Mj⌝)$, and $\mathcal{I}(⌜\mathrm{\neg }Hj⌝)\propto \mathcal{I}(⌜Jj⌝)$.

I don’t know any objections to this, as of yet. 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 due to 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 insofar as x is honest”, which in turn means that “for any x, the degree to which x is desirable is proportional to the degree to which x is honest”, i.e., $\mathrm{\forall }x[\mathcal{I}(⌜Dx⌝)\propto \mathcal{I}(⌜Hx⌝)]$. 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 don’t need further ‘metaproperties’ to be understood.