Causality (a question for the logicians among us)

[memimemi]

Prove causality, without stating causality as an axiom.  As a bonus question, prove that causality is more than a perceptual flaw in any conceivable universe, and not just our own.

This one's been bugging me for a long time, and I'm still stumped.

[Bloodshadow]

You need to define causality more clearly. Just exactly what are you asking us to prove, and in what contexts?

[memimemi]

You need to define causality more clearly. Just exactly what are you asking us to prove, and in what contexts?

I am asking for a proof that 'if x, then y' is a valid inference - that causality itself is intrinsic to the universe; a proof that doesn't take causality for granted from the outset.

Without a priori assumptions, can one prove, logically, that cause -> effect? 

A simple dictionary definition of causality will suffice:

causality  (kɔːˈzælɪtɪ)
 
— n  , pl -ties
1.    a. the relationship of cause and effect
    b. the principle that nothing can happen without being caused

This may well be a poorly formed question.  I'll see if I can clarify a little more, later.

[OldTrees]

Following.

[Bloodshadow]

I am asking for a proof that 'if x, then y' is a valid inference - that causality itself is intrinsic to the universe; a proof that doesn't take causality for granted from the outset.

In propositional logic, "if p then q" does not necessarily mean "p physically causes q". It simply means that if that statement ("if p then q") is true, then q must be true whenever p is true; the events p and q do not need to have any actual relations at all. The truth of that statement is only violated when p is true and q is false. If p is false, the whole statement is true regardless of whether q is true or false, as the statement is not violated. Logically, "if p then q" is equivalent to "~p or q", where the "~" symbol means "not".

A rule of inference is composed of any number of premises and a conclusion. It is valid if and only if the truthfulness of its conclusion is guaranteed by the truthfulness of all of its premises. In other words, given the premises a and b, and the conclusion c, the statement "if a and b then c" is a tautology (i.e. a statement that is always true regardless of the truth values of the variables inside it). Even in this case, the truthfulness of the premises do not physically "cause" the conclusion to be true. The conclusion is only true because of certain rules we've defined in our system of logic; in a way we've defined that the conclusion is true whenever the premises are true. Otherwise our rules will be violated, and we can't have that. The statement "if p then q" is not a rule of inference, because it is not composed of premises and a conclusion.

There is no room for physical causality in logic. To assert that p physically causes q, we'll have to work within a model of the physical universe where causality like such is intrinsic in the rules governing how the model functions. Causality is in no way a necessary ingredient of all possible models.

If you're talking about causality in real life, you're out of luck. In special relativity, events simultaneous in one frame of reference may not be simultaneous in another frame of reference.  And it gets even more tangled up when you have uncertainty on a quantum level. We can't prove causality in real life because we don't yet fully know how real life works, and that's what many physicists are working on right now.

[AnonymousRevival]

Uhhh....god. I get confused easily sorry so I don't really understand this. But when I seem to comprehend it more, maybe I can use modal logic to contribute.

[memimemi]

I am asking for a proof that 'if x, then y' is a valid inference - that causality itself is intrinsic to the universe; a proof that doesn't take causality for granted from the outset.

In propositional logic, "if p then q" does not necessarily mean "p physically causes q". It simply means that if that statement ("if p then q") is true, then q must be true whenever p is true; the events p and q do not need to have any actual relations at all. The truth of that statement is only violated when p is true and q is false. If p is false, the whole statement is true regardless of whether q is true or false, as the statement is not violated. Logically, "if p then q" is equivalent to "~p or q", where the "~" symbol means "not".

A rule of inference is composed of any number of premises and a conclusion. It is valid if and only if the truthfulness of its conclusion is guaranteed by the truthfulness of all of its premises. In other words, given the premises a and b, and the conclusion c, the statement "if a and b then c" is a tautology (i.e. a statement that is always true regardless of the truth values of the variables inside it). Even in this case, the truthfulness of the premises do not physically "cause" the conclusion to be true. The conclusion is only true because of certain rules we've defined in our system of logic; in a way we've defined that the conclusion is true whenever the premises are true. Otherwise our rules will be violated, and we can't have that. The statement "if p then q" is not a rule of inference, because it is not composed of premises and a conclusion.

There is no room for physical causality in logic. To assert that p physically causes q, we'll have to work within a model of the physical universe where causality like such is intrinsic in the rules governing how the model functions. Causality is in no way a necessary ingredient of all possible models.

If you're talking about causality in real life, you're out of luck. In special relativity, events simultaneous in one frame of reference may not be simultaneous in another frame of reference.  And it gets even more tangled up when you have uncertainty on a quantum level. We can't prove causality in real life because we don't yet fully know how real life works, and that's what many physicists are working on right now.

Necro'd, because I neglected to thank you for this.  I consider myself well answered; the question was, indeed, poorly formed.