Simple probability question: Suppose P(B|A)=1. Does that mean that P(A|B)=1?

[sysprog]

But actually, if $A \rightarrow B$ AND $C \rightarrow NOT(B)$, then I wonder if C=True results in A=NOT True (or, of course, A=True in C=NOT True).

We can do a short proof here:

To be proven: $C \rightarrow \neg A$

$1: A \rightarrow B$ assumption 1
$2: C \rightarrow \neg B$ assumption 2
$3: C$ assumption 3
$4: \neg B$ modus ponens 2,3
$5: \neg A$ modus tollens 1,4
$6: C \rightarrow \neg A$ hypothethical syllogism (1,2),3,(4),5

$-$ and there you have it $\dots$

 

[Demystifier]

No, $A \rightarrow B$ is equivalent to $NOT(B) \rightarrow NOT(A)$.

But actually, if $A \rightarrow B$ AND $C \rightarrow NOT(B)$, then I wonder if C=True results in A=NOT True (or, of course, A=True in C=NOT True).

I always wondered why probability and logic are in the same forum. Now I see why.

 

[entropy1]

We can do a short proof here:

To be proven: $C \rightarrow \neg A$

$1: A \rightarrow B$ assumption 1
$2: C \rightarrow \neg B$ assumption 2
$3: C$ assumption 3
$4: \neg B$ modus ponens 2,3
$5: \neg A$ modus tollens 1,4
$6: C \rightarrow \neg A$ hypothethical syllogism (1,2),3,(4),5

$-$ and there you have it $\dots$

So does that mean that one of (1) or (2) gets "reversed"? Can we then speak of retrocausality? (reversed causality?)

 

[Dale]

Can we then speak of retrocausality? (reversed causality?)

Please do not ask this question again without a clear and exact definition of retrocausality. Preferably one from the professional literature.

To all other participants: please do not respond to this question without such a definition.

 

[Dale]

At this point we will go ahead and close this thread. I strongly recommend studying the existing literature in this topic, perhaps including the time symmetric formulation of quantum mechanics. It is best to use definitions from the literature as they are more likely to have addressed some of the basic issues mentioned so far.

For any future threads on this topic please start with a professional scientific reference that can serve as the basis of discussion