Can set theory formulate causal proximity?

Question

We often speak of causes being direct and indirect, remote and proximate. The Big Bang is (presumably) causally related to my ability to type this question. But it isn't related as immediately as my use of a computer. Is there a way we might use set theory (or even FOL) to more precisely represent these sorts of relation, these relations of causal remoteness and proximity? Might causal proximity be something like a set ordered temporally (and perhaps some other way as well)? Might we, or must we, pattern this idea after or upon the integers (especially in speaking of the nearness or distance of a cause to or from some effect)? How does one formally specify a cause as this near or that far from an effect?

Answer 1

You ask:

Can set theory formulate causal proximity?

Yes. Kristian Berry provides in his answer an exotic system which sits atop set theory; generally, since causality is often taken in conjunction with time, it is easier to think of causes and effects as sequences whose order is determined by time of occurrence. Of course, a set of ordered pairs of events with times is functionality equivalent. And generally, such discussions of cause and effect are represented in graph-theoretic diagrams where order of nodes and edges across a plane serve the same purpose. Such diagrams are used in Pearl's Causality (GB), for instance. Pearl, who is a computer scientist by trade also uses data structures that are consistent with mathematical intensions and extensions.

Remoteness and proximity in the general scheme is simply held in accordance with what is the system we use to represent time, our calendar and clocks. According to many philosophers, causality is held to require time and space as metaphysically prior (SEP). Consider that in a legal case, the motion filed on Monday may cause a response by Wednesday by opposing counsel triggering a hearing by an adjudicator on Friday resulting in a decision and a legal action. In science, we might measure chemical reactions caused by adding reagents at 10 am and 11:30 am and note the final outcome after stirring sufficiently by 12 pm.

Can we use integers? Certainly, if the subject matter calls for it. For instance, we could use the integers 1-24 and then decimal representations of a fraction of an hour, minute, or second, but such time keeping would be a bit unconventional for anything more than a fraction of a second. But the principle is the same, where our temporal continuum is held to be an ordered sequence of numbers representing some division of time. In relativistic physics, its even necessary to maintain multiple and distinct timelines given time dilation.

We can also construct theories to reason about time formally. You might be interested in the notions of temporal parts (SEP). It holds that the same object can be seen as having different parts at different times. And we can use temporal logic (SEP) for more formal treatments of the language of the passage of time. From the SEP:

Broadly construed, Temporal Logic covers all formal approaches to representing and reasoning about time and temporal information. More narrowly, it usually refers to the modal-logic style approach introduced by Arthur Prior in the 1950s under the name Tense Logic and subsequently developed further by many logicians and computer scientists. Temporal Logic has been widely used as a formalism for clarifying philosophical issues about time, as a framework for defining the semantics of temporal expressions in natural language, as a language for encoding temporal knowledge in artificial intelligence, and as a tool for specification and verification of computer programs and systems.

Answer 2

Yes.

You'd have to define a distance metric D(x,y) and a boolean function NEARBY that acts on that metric.

Even better, define NEARBY as a mapping to [0,1]. 0 Is infinitely far away and 1 is the same location.

That mapping would be hyperbolic, though.

Answer 3

This is nothing less than causal set theory:

The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete (a collection of discrete spacetime points, called the elements of the causal set) and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events.

Here is the basic theory of causal-set geodesics:

Answer 4

Can set theory formulate causal proximity?

Formal systems can be used to model the structure of phenomena, in general. The answer to most questions of the form "Can [formal system X] model [phenomenon Y]?" is "yes". Formal systems like first-order logic, intuitionistic type theory, and lambda calculus are designed to be languages you can describe all of mathematics in. As long as you can abstract a phenomenon into the kinds of things mathematics can express, like functional relationships, then yes, a formal system can model it.

Is there a way we might use set theory (or even FOL) to more precisely represent these sorts of relations, these relations of causal remoteness and proximity?

First-order logic is a language, for talking about the properties of some system. Set theory is a theory about a certain kind of object - called a 'set' - which is being described by statements in first-order logic. When you make a statement about sets, you do so in the language of first-order logic. They go hand-in-hand.

How does one formally specify a cause as this near or that far from an effect?

In general, a theory in first-order logic begins with a signature. This is where you declare the essential concepts in your theory. You state what constants, functions, and relations there are, in your theory.

Constants are like things. In this theory, you might consider an "event" to be a class of things of interest to your theory, so you could include a set of all events, E, in your signature.

Functions are like relations where there is a unique referent. For example, if we assume that a human can only have one biological father, we could declare "father" to be a function from the set of all people to the set of all people. That way, for any person p, I can apply the function father(), and this term, father(p), will be equivalent to the biological father of p. Functions allow you to relate different classes of things to each other.

Relations are statements. I could have a relation "likes(x, y)", which requires 2 arguments, both of type person. The inclusion of the statement likes(Peter, Pat) becomes a fact, in the theory. In your case, you could begin by conceiving of causation as a relation, cause(x,y), where x and y are events.

Relations can have various properties which are expressible through logical statements. For example, some relations are reflexive: for any element x, it is the case that R(x,x). This could be the case in a world where we imagine every person likes themselves. There are many other properties a relation can have, such as being symmetric, transitive, total, etc.

You build the structure of your model of some phenomenon in this way. You use logical statements like "if X and Y, then Z" to establish relationships between all the functions, relations, and constants of your theory.

In order to build the theory you have in mind, you should begin studying logic. Two classic textbooks are by Mendelssohn and Enderton.

Answer 5

You could try a tree graph.

Each node or vertex represents a cause or effect. The green edges represent the link from cause to effect and these can be given weights or lengths. You could use arrows. The proximity would be a function of the connected path length.

Answer 6

My first thought is that if I were to take away the Big Bang it would stop you from writing your question, but it would also change a whole load of other things (everything, basically). If I took your computer away it would also stop you from writing this and it would also change a lot of other things, but far fewer of them, and most of them only in your future light cone. This makes me think of defining a partial order on "causes", according to how many other things they change if you intervene in them.

But to cash that out formally we'd have to already have a theory of what it means for one thing to "cause" another, in terms of intervention. Such a formalism already exists, in the form of Pearl's causal Bayesian networks, but it's quite pragmatically motivated and because of that it's somewhat restrictive in what it can express and might not be completely suitable for this kind of thing. You probably could cash out this idea in terms of Pearl's formalism if you wanted to, though.

I'm not sure whether that would count as using set theory or FOL - in one sense it does because it's maths and all maths can be expressed in terms of set theory, but in another sense it's not, because it wouldn't use set theory very directly at all. That might change if you used a different model of causality though, one that's "closer to the metal" than Pearl's. (I don't know if a suitable more 'logical' approach to causality exists.) This is quite an interesting thing to think about, and I can imagine trying to work on it.