Title:
Dependence and arbitrariness
Abstract:
In previous work, I have proposed an analysis of dependence claims as strict conditionals whose antecedent and consequent are questions: “Q1 determines Q2” is true if, within a certain range of possibilities, any answer to Q1 implies some answer to Q2.
In this talk, I consider dependence statements in mathematics, such as (1).
(1) The number of sides of a polygon determines the sum of its internal angles.
I argue that (1) fits the general analysis provided we take an indefinite like “a polygon x” to introduce a range of possibilities, one for each permissible value of x. Within this range of possibilities, an answer to the question "how many sides does x have" yields an answer to the question "what is the sum of the internal angles of x". Thus, the semantics of dependence statements like (1) involves a kind of intensionality that has its source in arbitrary reference.
I implement this idea in a team-based extension of first-order logic that comprises (i) an operator [x] which introduces an arbitrary referent and (ii) questions concerning the value of this referent. In this logic, the above dependence claim can be regimented as follows:
(2) [x]( polygon(x) -> ( value-of(num-sides(x)) -> value-of(sum-angles(x)) )
I then study how, and to what extent, such intensional claims can be reduced to extensional claims in first-order logic, such as (3):
(3) ∀x∀y( polygon(x) & polygon(y) -> ( num-sides(x)=num-sides(y) -> sum-angles(x)=sum-angles(y))
As I will show, the relation between the two ways of expressing dependencies is far from straightforward.
If time permits, I will also consider a modal extension of the logic and discuss how it offers a unified analysis of different varieties of supervenience.