David Corfield univocal-equivocal

Metaphysics

Is Being one thing or many? Are there ways of Being?

Type theory

Intrisic/extrinsic, refinement types.

Univocality of ‘:’. All elements are in $\widehat{Type} = \sum_{X: Type} X$ as $(A, a)$, with $a:A$. $\widehat{Type}$ as pointed types.

$Entity = \widehat{Type}$ is the type of pairs $(A,a)$, for $A$ a type and $a$ an element of $A$. To belong to $Entity$ is to be generically, and yet at the same time, elements $a$ only exist as belonging to a type, kinds of being.

Bundled together are object, facts, properties, etc. Books as objects, books as information sources.

To belong to $Entity$ is to be generically, and yet at the same time, elements $a$ only exist as belonging to a type.

So first comes a type of types, $Type$, and then the dependent sum $Entity = \sum_{X: Type} X$. $Entity$ varies as new types emerge, but qua dependent sum would remain the same.

Maps from $Type$ to a type of grades, etc. h-level is one.

Category theory

Category as collection of morphisms (terms), or as family of collections indexed over (pairs of) objects (types).

Image of arrow onto terminal category is $id_{\ast}: \ast \to \ast$.

Theories

Carnap/Quine. Structured or flat network. Presuppositions, contexts.

On Carnap’s Views on Ontology Willard van Orman Quine Philosophical Studies 2 (5):65–72 (1951)

This is exactly what’s at stake. But Quine is completely wrong as to the evaluation.

“Whether the statement that there are physical objects and the statement that there are black swans should be put on the same side of the dichotomy, or on opposite sides, comes to depend on the rather trivial consideration of whether we use one style of variables or two for physical objects and classes.”

Trivial!

This is supposedly backed up by the observation that mathematicians can go either way between

‘If x is a real number between 0 and I, then,’

and variably, p, q, constrained to be real numbers between 0 and 1.

Is he really having us believe that in the first case this is

For all x( x is a real number between 0 and I -> …)

Nonsense. In fact, which mathematician would even start that way? Far more common is

Let x be a real number between 0 and I. Then,…

Which of course is most naturally represented as a type declaration.

“Carnap’s distinction between internal and external, based as it is upon a distinction between category questions and subclass questions, is of little concern to us apart from the adoption of something like the theory of types. I am one of those who have tended for many years not to adopt the theory of types.”

(It suggests there was a reply from Carnap in this journal.)

“I argued before that the distinction between category questions and sub-class questions is of little concern apart from the adoption of something like the theory of types. But what I now think to have shown is that it is of little concern even under the theory of types. It is a distinction which is not invariant under logically irrelevant changes of typography.”

I think we’re seeing hints there of the temporal dimension to our endorsement of the use of a language, surely a necessary aspect to those of us who approach these matters via the notion of enquiry conducted through time.

If you use an untyped framework, at any point you’ll have FOL + some sentences specifying what you think you know. Some of these will include things more akin to definitions (All x. Bachelor(x) <–> Unmarried(x) & Male(x)); some will be factual statements (exists x. x is a blackbird singing in a tree here and now).

As time passes, (let’s assume we don’t do anything as radical as change our logic) I may modify my definitions and empirical sentences (and I take it the fact that we do modify both is behind the challenge to the analytic/synthetic distinction). But in some sense I’m not changing what I take there to be. It’s always ‘everything’.

If you use a typed framework, at any point you’ll have your version of TT + declaration of some types + sentences specifying what you know.

As time passes, you may modify these sentences, but more radically you may decide to change the types you declare. It’s the latter kind of consideration that’s close to Carnap’s external questions, and the former to his internal questions. Once we have decided on our language (our types) then internal questions arise concerning elements of them. Our choice of language concerns external questions, should we endorse the inclusion of these types in our language.

As Michael Friedman points out, it’s more layered in the Carnapian approach – the Duhem-Quine thing is wrong. Our theories don’t confront the world as a large conjunction.

Time for an extract of my book

Another philosopher arguing along similar lines to Collingwood in the context of the natural sciences is Michael Friedman. Friedman (2001) contrasts his own views with those of Quine, who famously proposed that we operate with a connected web of beliefs (Quine and Ullian 1970). For the latter, when observations are made which run against expectations derived from these beliefs, we typically modify peripheral ones, allowing us to make minimal modification to the web. However, we may become inclined to make more radical changes to entrenched beliefs located at the heart of the web, perhaps even to the laws of arithmetic or the logic we employ. Friedman insists, by contrast, that there is a hierarchical structure to our (scientific) beliefs. This entails that, without the availability of fundamental modes of expression, including the resources of mathematical languages, and constitutive principles which deploy these resources, statements concerning empiricial observations and predictions cannot even be meaningfully formed. In his (2002), while discussing the formulation of Newtonian physics, he says

It follows that without the Newtonian laws of motion Newton’s theory of gravitation would not even make empirical sense, let alone give a correct account of the empirical phenomena: in the absence of these laws we would simply have no idea what the relevant frame of reference might be in relation to which the universal accelerations due to gravity are defined. Once again, Newton’s mechanics and gravitational physics are not happily viewed as symmetrically functioning elements of a larger conjunction: the former is rather a necessary part of the language or conceptual framework within which alone the latter makes empirical sense. (Friedman 2002, pp. 178–9)

To see how similar this case is to that of the beaten spouse, we might imitate Friedman thus:

It follows that without the concepts of personhood, of marriage, of action, of intention, etc., the idea of someone leaving off beating their wife would not make any sense, let alone give a correct account of the empirical phenomena: in the absence of these concepts, we would simply have no idea what the relevant moral–legal–ontological frame of reference might be in relation to which the claimed cessation of violence has taken place. Once again, facts about personhood, intention and action are not happily viewed as symmetrically functioning elements of a larger conjunction: the former is rather a necessary part of the language or conceptual framework within which alone the latter makes empirical sense.

I just finished Thomasson’s piece. Very interesting! I could sign up to ‘easy ontology’. It fits very nicely with the outlook I was suggesting before. There are external questions about the types we introduce. But once they’re in play, then internal questions concerning them are readily answerable.

‘easy ontology’ looks good.

Type theory also gives you what’s right about quantifier variance and about quantifier invariance.

All As are B

is formed by a single rule, but depends on the A and B.

For a type A, and a predicate B that depends upon it, there is a type ‘All As are B’.

Questions about whether it’s true that All As are B are internal. Questions about whether we ought to have type A or predicate B in our language are external.