Formalization of Causation
Hello!
I am concerned with how to formalize some definitions of causation. In David Lewis' groundbreaking work entitled 'Causation', a definition is given as follows:
c causes e iff if it were the case that it is not the case that c occurs, then it would be the case that it is not the case that e occurs. (tense can be ignored here)
I think that so-called Lewisian analyses can be best understood as formalization of 'Ramsey-Lewis method'. It is articulated as follows:
(?x)(?y)(x causes y iff if it were the case that it is not the case that x occurs, then it would be the case that it is not the case that y occurs.)
The whole formulation can be Carnap sentence as:
If (?x)(?y)(x causes y iff if it were the case that it is not the case that x occurs, then it would be the case that it is not the case that y occurs.), then c causes e iff it were the case that it is not the case that c occurs, then it would be the case that it is not the case that e occurs.
I ask whether I am right about it here. Thanks!
I am concerned with how to formalize some definitions of causation. In David Lewis' groundbreaking work entitled 'Causation', a definition is given as follows:
c causes e iff if it were the case that it is not the case that c occurs, then it would be the case that it is not the case that e occurs. (tense can be ignored here)
I think that so-called Lewisian analyses can be best understood as formalization of 'Ramsey-Lewis method'. It is articulated as follows:
(?x)(?y)(x causes y iff if it were the case that it is not the case that x occurs, then it would be the case that it is not the case that y occurs.)
The whole formulation can be Carnap sentence as:
If (?x)(?y)(x causes y iff if it were the case that it is not the case that x occurs, then it would be the case that it is not the case that y occurs.), then c causes e iff it were the case that it is not the case that c occurs, then it would be the case that it is not the case that e occurs.
I ask whether I am right about it here. Thanks!
Comments (13)
I am examining correct formalization of Lewis' counterfactual analysis of causation. Lewis gives a semi-formal analysis, a combination of schematic letters (e.g. c, e, ...) and English expressions (e.g. causes, ...). What I am looking for is a formal analysis of causation symbolized completely down to schema.
If c is not the case, then something else is the case, right? What could be the case then? And couldn't some alternate state of affairs result in the same effect? Suppose we see a red ball strike a yellow ball and then the yellow ball rolls into a pocket. We would normally say that the red ball striking the yellow ball (c) caused the latter to roll into a pocket (e). But what would be a counterfactual to that? Suppose that a blue ball struck the yellow ball instead of the red ball, with the same consequences. (c) is not the case, but (e) occurred anyway. Does this mean that (c) was not the cause of (e) after all? It seems that on this definition of causation either there is no multiple realizability of effects or there is no causation - either option is implausible.
A less troubled approach is to define cause in terms of agreed initial conditions and scientific theories. Try reading this for an approach that does it that way.
I am interested in logical forms of analyses. Or, I am interested in how to construct definitions in symbolic logic precisely.
xRy = x is correlates with y
xNy = x is correlated coincidentally with y
1. xRy
2. ~yCx
3. ~(Ez)(zCx AND zCy)
4. ~xNy
Then
xCy <-> 1 & 2 & 3 & 4
:P
1. Rxy
2. ~Cyx
3. ~(?z)(Czx & Czy)
4. ~Nxy
5. Show Cxy ? (Rxy & ~Cyx & ~(?z)(Czx & Czy) & ~Nxy)
6. [s]Show[/s] Cxy ? (Rxy & ~Cyx & ~(?z)(Czx & Czy) & ~Nxy)
7. Cxy A
8. [s]Show[/s] Rxy & ~Cyx & ~(?z)(Czx & Czy) & ~Nxy
9. Rxy & ~Cyx & ~(?z)(Czx & Czy) & ~Nxy 1, 2, 3, 4, Add
10. Show (Rxy & ~Cyx & ~(?z)(Czx & Czy) & ~Nxy) ? Cxy
11. Rxy & ~Cyx & ~(?z)(Czx & Czy) & ~Nxy A
12. Show Cxy
13. (?z)~(Czx & Czy) 3, QN
14. ~(Cxx & Cxy) 13, UI
15. ~Cxx ? ~Cxy 14, DM
INVALID
TheMadFool, your argument is not valid as shown above.
I'm defining causation.
Ah, you defined causation by biconditional (or necessary and sufficient condition) ... I got it.
However, your analysis includes free variables x and y, which is not allowed in first-order predicate logic. Of course, if you used them as names, no troubles here.
(Ex)(Ey)(xCy <-> xRy & ~yCx & ~(Ez)(zCx & zCy) & ~xNy)
Thanks for pointing out the mistake. My logic is rusty. How's the above formulation
aCb <-> (aRb & ~bCa & ~(Ex)(xCa & xCb) & ~aNb)
Two formulations are well-formed fomulae.