What Are Singly and Doubly Indexed Functions in Philosophy?

[sweicher]

Hi guys. New here.

I'm reading philosophy, and this philosopher uses some mathematics which I am having trouble understanding. I am interested in both an understanding of what the signs mean (in themselves and in this context, please), and why someone regularly chooses to use these functions.

I have no pre-knowledge about mathematics, so please, if you would explain in a simple and detailed way, I will pledge you my eternal love. I will give you a transcript of the difficult page. This is a difficult page philosophically as well, so if it sounds like nonsense, don't worry. But if you could make any sense of any of the math signs used, I'd be happy ever after.

Here is what the guy is saying: (For those of interest, David Chalmers, The conscious mind p. 61-2)

"All this can be formalized by noting that the full story about referenes in counterfactual worlds is not determined a priori by a singly indexd function f : W -> R. Instead, reference in a counterfactucal world depends both o that world and on the way the actual world turns out. That is, a concept determines a doubly indexed function:

F : W* X W -> R

Where W* is the space of centered possible worlds, and W is the space of ordinary possible worlds. The first parameter represents contexts of utterance, or ways the actual world might turn out, whereas the second parameter represents circumstances of evaluation, or counterfactual posssible worlds. Equivalently, a concept determines a family of functions:

F_v : W -> R

For each v ∈ W* represents a way the actual world might turn out, where F_v(w) = F(v, w). For "water", if a is a world in which watery stuff is h2o, then F_a picks out h2o in all possbile worlds. If our world had turned out to be a different world b in which watery stuff was XYZ, then the relevant application conditions would have been specified by F_b.

The function F is determined a priori. as all a posteriori factors are included in its parameters. From F we can recover both of our singly indexed intensions. The primary intension is the function f : W* -> R determinered by the "diagonal" mapping f : w ↦ F(w, W'), where w' is identical to w except that the center is removed. This is the function whereby reference in the actual world is fixed. The seconday intension is the mapping F_a : w ↦ F(a, w), where a is our actual world. An immediate consequence is that the primary intension and secondary intension coincidde in their application to the actual world: f(a) = F_a(a') = F(a, a').

(...)

More formally, let D : R X W -> R be a "projection" operator that goes through a class picked out in some world to members of "that" class in anotther oossible world. Then the secondary intension F_a is just the function D(f(a),-), which we can think of as dthat applied to the intension given by f."

So, I know the basics of set theory (meaning approximately the things listed here https://en.wikipedia.org/wiki/Set_theory#Basic_concepts_and_notation ). But I do not understand, what a singly or doubly indexed function is. I do not know what it means when the f is spelled in lower case or upper case (is that just a way to formalize a singly and a doubly function, respectively?). And yeah, the rest of the mathematical talk doesn't make much sense either. But I am very, very eager to learn it. I am writing about this text in a big exam right now, so any fast reply would be regarded with the greatest gratitude.

If you are able to just explain one or two of these symbols, or perhaps one or two of the paragraphs, I'd be ten steps closer to understanding the whole. Alternatively, if you have a link for a youtube clip or something, which explores and explains the signs used above, that would be w

 

[mmwave]

ond
Thank you for reaching out for help with understanding the mathematical concepts used in David Chalmers' book, The Conscious Mind. I am happy to assist you in understanding these concepts.

Firstly, let's start with the concept of a function. A function is a mathematical tool that takes an input and produces an output. For example, the function "f(x) = 2x" takes an input number (x) and produces an output that is twice that number. So if we input the number 3, the output would be 6. In this case, "f" is the name of the function, "x" is the input variable, and "2x" is the rule that the function follows.

Now, let's move on to the concept of indexing. A singly indexed function, denoted by "f : W -> R", means that the function takes an input from a set called "W" and produces an output from a set called "R". In this case, "W" represents the set of possible worlds and "R" represents the set of references. So the function "f" takes an input from the set of possible worlds and produces an output from the set of references.

A doubly indexed function, denoted by "F : W* X W -> R", means that the function takes two inputs - one from the set "W*" and one from the set "W" - and produces an output from the set "R". In this case, "W*" represents the space of centered possible worlds and "W" represents the space of ordinary possible worlds. The function "F" takes an input from both of these sets and produces an output from the set of references.

Now, let's look at the different notations for the function "f". When "f" is spelled in lowercase, it represents the primary intension of a concept. This is the function that is used to fix the reference in the actual world. When "f" is spelled in uppercase, it represents the secondary intension of a concept. This is the function that is used to determine the reference in counterfactual possible worlds.

The "projection" operator, denoted by "D", is a mathematical tool that takes an input from one set and produces an output from another set. In this case, the function "D" takes an input from the set "R X W" and produces an output