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ParaclitosLogos

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Probabilistic support
« on: August 22, 2015, 11:08:06 pm »
Take with a grain of salt:

Tomoji Shogenji ( the British Journal for the Philosophy of Science 54 (2003), pp. 613-616.) has shown that, even though, "It is well known that probabilistic support is not transitive" , an intermediary proposition screening off the original evidence with respect to the hypothesis in question is a sufficient condition for the transtivity of probabilistic support. This has the consequence that probabilistic support is transitive when the original evidence is testimonial, memorial or perceptual (i.e., to the effect that such and such was testified to, remembered, or perceived), and the intermediary proposition is its representational content (i.e., to the effect that the such and such occurred).


Screning off (Reichenbach):

Definitions:
*:  Causal connection.
-->:  logical implication.

  causal sequence
A     *    B    *    C
B screens off A from C. B screens off C from A.
P( B | C) > P( B)   & P ( A | B )  > P ( A)    -- > P( A | C) > P( A)    by transitivity of support, C suports A. (shogenji)

  common cause
                     B 
              *            *
        A                        C

B screens off A from C. B screens off C from A.
P( B | C) > P( B)   & P ( A | B )  > P ( A)    -- > P( A | C) > P( A)   by transitivity of support, C suports A (shogenji).



Some examples:

Ex. 1
B:( Tratidiontal Cristianity is true. God exists and has instantiated a world with Atonement, according to his divine design of salvation. God resurrected Jesus of Nazareth, the 2nd person of the trinity,  that incarnated, lived among men, suffered, was crucified, and raised by God.)  A  divine bodily resurrected Jesus commissioned the apostles to carry his messages, and, tell of all the events regarding his teaching and atoning sacrifice,  resurrection, and, post assumption continued, appearing on diverse situations and occasions, to many.

A: Reports of bodily appereances and experiences.

C: Reports of non bodily appereances and experiences.

                     B 
              *            *
        A                        C

P( B | C) > P( B)   & P ( A | B )  > P ( A)    -- > P( A | C) > P( A)    by transitivity of support, C suports A.(shogenji).
 


There are other combinations that are also interesting:

like

Ex. 2
B:( Tratidiontal Cristianity is true. God exists and has instantiated a world with Atonement, according to his divine design of salvation. God resurrected Jesus of Nazareth, the 2nd person of the trinity,  that incarnated, lived among men, suffered, was crucified, and raised by God.)  A  divine bodily resurrected Jesus commissioned the apostles to carry his messages, and, tell of all the events regarding his teaching and atoning sacrifice,  resurrection, and, post assumption continued, appearing on diverse situations and occasions, to many.


A: B compatible (causally connected) Event X , with details a,b,c reported

C: B compatible Event (causally connected) Y, with details, d,b,e,f reported

And so on.


P( B | C) > P( B)   & P ( A | B )  > P ( A)    -- > P( A | C) > P( A)    by transitivity of support, C suports A.



Conclusion:

If this is roughly correct, it seems under certain conditions ( screening off ) discrepancies in testimony is not really a problem for it to be mutually confirmatory ( probabilistically support each other ). 
« Last Edit: August 22, 2015, 11:27:28 pm by ontologicalme »

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ParaclitosLogos

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Re: Probabilistic support
« Reply #1 on: August 23, 2015, 12:28:59 pm »
Another example.


B:( Tratidiontal Cristianity is true. God exists and has instantiated a world with Atonement, according to his divine design of salvation. God resurrected Jesus of Nazareth, the 2nd person of the trinity,  that incarnated, lived among men, suffered, was crucified, and raised by God.)  A  divine bodily resurrected Jesus commissioned the apostles to carry his messages, and, tell of all the events regarding his teaching and atoning sacrifice,  resurrection, and, post assumption continued, appearing on diverse situations and occasions, to many.


X1. Paul c. 50 AD -  Jesus has been risen .

X2. Mark c. 70 AD -  Jesus has been bodily resurrected leaving an empty tomb , appereances foreshadowed  "he  is gone ahead to Galielee".

X3. Matthew c. 80 AD -  Empty tomb implied (Jewish polemic) appearances of the risen Jesus narrated.Great commissioning by Jesus (in bodily appereance) to disciple.

X4. Luke 85-90 AD - Empty tomb narrated,  Jesus appearances testified (road to Emaus).Great commissioning by Jesus (in bodily appereance) to disciple. 

X5. Acts : empty tomb refered to, bodily ascension into heaven while people are watching.

X6. John 90-120 AD - Great commissioning by Jesus (in bodily appereance) to disciples.

                       B 
     *        *     *       *       *     *
   X1    X2       X3     X4    X5    X6

B screens off Xm from Xn. B screens off Xn from Xm.
P( B | Xn) > P( B)   & P ( Xm | B )  > P ( Xm)    -- > P( Xm | Xn) > P( Xm)   by transitivity of support, Xn suports Xm (shogenji).



Conclusion
Each combinatorial set of lines of evidence, minus one, confirm the one (left out of the set).
« Last Edit: August 23, 2015, 05:31:38 pm by ontologicalme »

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phidiasv

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Re: Probabilistic support
« Reply #2 on: August 24, 2015, 02:59:43 pm »
It isn't clear to me that your examples hold to the screen-off condition that Shogenji lays out in his paper: Y screens off X with respect to Z when P(Z|X&Y)=P(Z|Y). Shogenji argues that this screen-off condition holds when Y is the representational content of the testimony, memory or perception that is X. But, in your examples, it doesn't seem that B is any such thing. It doesn't seem that your examples meet the screen-off condition, as your examples of testimony have different content than the screen-off proposition B.

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ParaclitosLogos

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Re: Probabilistic support
« Reply #3 on: August 24, 2015, 04:23:50 pm »
It isn't clear to me that your examples hold to the screen-off condition that Shogenji lays out in his paper: Y screens off X with respect to Z when P(Z|X&Y)=P(Z|Y). Shogenji argues that this screen-off condition holds when Y is the representational content of the testimony, memory or perception that is X. But, in your examples, it doesn't seem that B is any such thing. It doesn't seem that your examples meet the screen-off condition, as your examples of testimony have different content than the screen-off proposition B.



Thanks for the answer:

The concept of screening off comes from Reichenbach.


Other´s and Shogenji´s own examples are quite open, if you ask me, but nevertheless.

This is B: A  divine bodily resurrected Jesus commissioned the apostles to carry his messages, and, tell of all the events regarding his teaching and atoning sacrifice,  resurrection, and, post assumption continued, appearing on diverse situations and occasions, to many.

And B can be a common cause of all the testimony I have described.


X1. Paul c. 50 AD -  Jesus has been risen .

X2. Mark c. 70 AD -  Jesus has been bodily resurrected leaving an empty tomb , appereances foreshadowed  "he  is gone ahead to Galielee".

X3. Matthew c. 80 AD -  Empty tomb implied (Jewish polemic) appearances of the risen Jesus narrated.Great commissioning by Jesus (in bodily appereance) to disciple.

X4. Luke 85-90 AD - Empty tomb narrated,  Jesus appearances testified (road to Emaus).Great commissioning by Jesus (in bodily appereance) to disciple. 

X5. Acts : empty tomb refered to, bodily ascension into heaven while people are watching.

X6. John 90-120 AD - Great commissioning by Jesus (in bodily appereance) to disciples.




The proposition B =   P1 & P2 & P3 & P4 & P5 & P6


If you take for example X1, it suports

P1: Jesus is bodily resurrected

B   *  P1 * X1 

by the same kind of screening off : P1 screens off B from X1 and viceversa.

And the support given to P1 by X1 is transitive to B, through the support given to B , by P1.

The same goes for P2,P3,P4,P5,P6 and X2, X3,X4,X5,X6.

« Last Edit: August 24, 2015, 04:38:50 pm by ontologicalme »

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phidiasv

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Re: Probabilistic support
« Reply #4 on: August 24, 2015, 10:24:27 pm »
It isn't clear to me that your examples hold to the screen-off condition that Shogenji lays out in his paper: Y screens off X with respect to Z when P(Z|X&Y)=P(Z|Y). Shogenji argues that this screen-off condition holds when Y is the representational content of the testimony, memory or perception that is X. But, in your examples, it doesn't seem that B is any such thing. It doesn't seem that your examples meet the screen-off condition, as your examples of testimony have different content than the screen-off proposition B.


Thanks for the answer:

The concept of screening off comes from Reichenbach.


Other´s and Shogenji´s own examples are quite open, if you ask me, but nevertheless.

This is B: A  divine bodily resurrected Jesus commissioned the apostles to carry his messages, and, tell of all the events regarding his teaching and atoning sacrifice,  resurrection, and, post assumption continued, appearing on diverse situations and occasions, to many.

And B can be a common cause of all the testimony I have described.


X1. Paul c. 50 AD -  Jesus has been risen .

X2. Mark c. 70 AD -  Jesus has been bodily resurrected leaving an empty tomb , appereances foreshadowed  "he  is gone ahead to Galielee".

X3. Matthew c. 80 AD -  Empty tomb implied (Jewish polemic) appearances of the risen Jesus narrated.Great commissioning by Jesus (in bodily appereance) to disciple.

X4. Luke 85-90 AD - Empty tomb narrated,  Jesus appearances testified (road to Emaus).Great commissioning by Jesus (in bodily appereance) to disciple. 

X5. Acts : empty tomb refered to, bodily ascension into heaven while people are watching.

X6. John 90-120 AD - Great commissioning by Jesus (in bodily appereance) to disciples.




The proposition B =   P1 & P2 & P3 & P4 & P5 & P6


If you take for example X1, it suports

P1: Jesus is bodily resurrected

B   *  P1 * X1 

by the same kind of screening off : P1 screens off B from X1 and viceversa.

And the support given to P1 by X1 is transitive to B, through the support given to B , by P1.

The same goes for P2,P3,P4,P5,P6 and X2, X3,X4,X5,X6.



Ah, did not realize B was meant to be merely such a conjunction. Cheers.