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The definition of causality (1)

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The definition of causality (1)

    The definition of causality (1)

    Zhang Nanlun

    nanlunzhang@whut.edu.cn

    The New Control This definition is published as chapter 6 in

    Principle, National Defense Industry Press, China, 2005.

    1.1 Definition of necessary and sufficient symptoms

    In a given causal series, x and y are two events (phenomena), (1) when x appears at the moment k, y will appear at the moment k too, (2) when x does not appear at the moment k, y would not appear at the moment k either; and, (3) when y appears at the moment k, x

    will appear at the moment k, (4) when y does not appear at the moment k, x would not appear at the moment k either.

    The joint truth table is defined as follows:

    x y ”if x then y y x ”if y then x

    1 1 1 1 1 1

    1 0 0 1 0 0

    0 1 0 0 1 0

    0 0 1 0 0 1

    Table 1: definition of necessary and sufficient symptoms

    That is to say:

    (1) If there is the presence of event x (event x true) and there is the presence of y (event y true) then this causal statement is true;

    (2) If there is the presence of event x (event x true) and there is no presence of y (event y false) then this causal statement is false;

    (3) If there is no presence of event x (event x false) and there is the presence of y (event y true) then this causal statement is false;

    (4) If there is no presence of event x (event x false) and there is no presence of y (event y false) then this causal statement is true;

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    (5) If there is the presence of event y (event y true) and there is the presence of x (event x true) then this causal statement is true,

    (6) If there is the presence of event y (event y true) and there is no presence of x (event x false) then this causal statement is false;

    (7) If there is no presence of event y (event y false) and there is the presence of x (event x true) then this causal statement is false;

    (8) If there is no presence of event y (event y false) and there is no presence of x (event x false) then this causal statement is true.

    Conclusion: there is a causality between “there is the presence of event x and there is the presence of y. If there is the presence of event x then there will be the presence of y and if there is the presence of event y then there must be the presence of x.

    Clearly, under this condition there is the presence of event x is

    a necessary and sufficient symptom of there is the presence of y;

    there is the presence of y is also a necessary and sufficient symptom

    of there is the presence of event x.

1.2 Examples

    The examples below are listed in order to show this definition.

Example`1.21

    Causality analysis of there is a force= x and there is a

    . counterforce= y

    In this given causal series, x and y are two events (phenomena), (1) when x appears at the moment k, y will appear at the moment k too, (2) when x does not appear at the moment k, y would not appear at the moment k either; and, (3) when y appears at the moment k, x

    will appear at the moment k, (4) when y does not appear at the moment k, x would not appear at the moment k either.

    The joint truth table is as follows:

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    x y ”if x then y y x ”if y then x

    1 1 1 1 1 1

    1 0 0 1 0 0

    0 1 0 0 1 0

    0 0 1 0 0 1

    Table 2: Causality analysis of example 1.21

    That is to say:

    (1) If there is a force (event x true) and there is a counterforce (event y true) then this causal statement is true;

    (2) If there is a force (event x true) and there is no counterforce (event y false) then this causal statement is false;

    (3) If there is no force (event x false) and there is a counterforce (event y true) then this causal statement is false;

    (4) If there is no force (event x false) and there is no counterforce (event y false) then this causal statement is true;

    (5) If there is a counterforce (event y true) and there is a force (event x true) then this causal statement is true,

    (6) If there is a counterforce (event y true) and there is no force (event x false) then this causal statement is false;

    (7) If there is no counterforce (event y false) and there is a force (event x true) then this causal statement is false;

    (8) If there is no counterforce (event y false) and there is no force

    (event x false) then this causal statement is true.

    Conclusion: there is a causality between “there is a force and there is a counterforce. If there is a force then there will be a

    counterforce and if there is a counterforce then there must be a force.

    Clearly, under certain condition there is a force is a necessary and sufficient symptom of there is a counterforce; there is a counterforce is also a necessary and sufficient symptom of there is a force.

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Example`1.22

    Causality analysis of the light bulb is emitting light= x, and

    the light bulb is emitting heat = y.

    In this given causal series, x and y are two events (phenomena), (1) when x appears at the moment k, y will appear at the moment k too, (2) when x does not appear at the moment k, y would not appear at the moment k either; and, (3) when y appears at the moment k, x

    will appear at the moment k, (4) when y does not appear at the moment k, x would not appear at the moment k either.

    The joint truth table is as follows:

    x y ”if x then y y x ”i f y then x

    1 1 1 1 1 1

    1 0 0 1 0 0

    0 1 0 0 1 0

    0 0 1 0 0 1

    Table 3: Causality analysis of example 1.22

    That is to say:

    (1) If the light bulb is emitting light (event x true) and the light bulb is emitting heat (event y true) then this causal statement is true;

    (2) If the light bulb is emitting light (event x true) and the light bulb is not emitting heat (event y false) then this causal statement is

    false;

    (3) If the light bulb is not emitting light (event x false) and the light bulb is emitting heat (event y true) then this causal statement is

    false;

    (4) If the light bulb is not emitting light (event x false) and the light bulb is not emitting heat (event y false) then this causal statement is true;

    (5) If the light bulb is emitting heat (event y true) and the light

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bulb is emitting light (event x true) then this causal statement is true,

    (6) If the light bulb is emitting heat (event y true) and the light bulb is not emitting light (event x false) then this causal statement is false;

    (7) If the light bulb is not emitting heat (event y false) and the light bulb is emitting light (event x true) then this causal statement is false;

    (8) If the light bulb is not emitting heat (event y false) and the light bulb is not emitting light (event x false) then this causal statement is true.

    Conclusion: there is a causality between “the light bulb is emitting light and the light bulb is emitting heat. If the light bulb is emitting light then the light bulb will be emitting heat and if the

    light bulb is emitting heat then the light bulb must be emitting light.

    Clearly, under certain condition the light bulb is emitting light

    is a necessary and sufficient symptom of the light bulb is emitting heat; the light bulb is emitting heat is also a necessary and sufficient symptom of the light bulb is emitting light.

Example`1.23

    Causality analysis of he is myopic= x, and the parallel rays are focused in front of his retina = y.

    In this given causal series, x and y are two events (phenomena), (1) when x appears at the moment k, y will appear at the moment k too, (2) when x does not appear at the moment k, y would not appear at the moment k either; and, (3) when y appears at the moment k, x

    will appear at the moment k, (4) when y does not appear at the moment k, x would not appear at the moment k either.

    The joint truth table is as follows:

    x y ”if x then y y x ”if y then x

    1 1 1 1 1 1

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    1 0 0 1 0 0

    0 1 0 0 1 0

    0 0 1 0 0 1

    Table 4: Causality analysis of example 1.23

    That is to say:

    (1) If he is myopic (event x true) and the parallel rays are focused in front of his retina (event y true) then this causal statement is true;

    (2) If he is myopic (event x true) and the parallel rays are not focused in front of his retina (event y false) then this causal statement is false;

    (3) If he is not myopic (event x false) and the parallel rays are focused in front of his retina (event y true) then this causal statement is false;

    (4) If he is not myopic (event x false) and the parallel rays are not focused in front of his retina (event y false) then this causal statement is true;

    (5) If the parallel rays are focused in front of his retina (event y

    true) and he is myopic (event x true) then this causal statement is true,

    (6) If the parallel rays are focused in front of his retina (event y

    true) and he is not myopic (event x false) then this causal statement is false;

    (7) If the parallel rays are not focused in front of his retina

    (event y false) and he is myopic (event x true) then this causal statement is false;

    (8) If the parallel rays are not focused in front of his retina

    (event y false) and he is not myopic (event x false) then this causal statement is true.

    Conclusion: there is a causality between “he is myopic and the

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    parallel rays are focused in front of his retina. If he is myopic then the parallel rays will be focused in front of his retina and if the

    parallel rays are focused in front of his retina then he must be

    myopic.

    Clearly, under certain condition he is myopic is a necessary and sufficient symptom of the parallel rays are focused in front of

    his retina; the parallel rays are focused in front of his retina is

    also a necessary and sufficient symptom of he is myopic.

Example`1.24

    Causality analysis of the body is here= x, and the shadow is here = y

    In this given causal series, x and y are two events (phenomena), (1) when x appears at the moment k, y will appear at the moment k too, (2) when x does not appear at the moment k, y would not appear at the moment k either; and, (3) when y appears at the moment k, x

    will appear at the moment k, (4) when y does not appear at the moment k, x would not appear at the moment k either.

    The joint truth table is as follows:

    x y ”if x then y y x ”if y then x

    1 1 1 1 1 1

    1 0 0 1 0 0

    0 1 0 0 1 0

    0 0 1 0 0 1

    Table 5: Causality analysis of example 1.24

    That is to say:

    (1) If the body is here (event x true) and the shadow is here

    (event y true) then this causal statement is true;

    (2) If the body is here (event x true) and the shadow is not here

    (event y false) then this causal statement is false;

    (3) If the body is not here (event x false) and the shadow is here

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(event y true) then this causal statement is false;

    is not (4) If the body is not here (event x false) and the shadow here (event y false) then this causal statement is true;

    (5) If the shadow is here (event y true) and the body is here (event x true) then this causal statement is true,

    (6) If the shadow is here (event y true) and the body is not here

    (event x false) then this causal statement is false;

    (7) If the shadow is not here (event y false) and the body is here

    (event x true) then this causal statement is false;

    (8) If the shadow is not here (event y false) and the body is not here (event x false) then this causal statement is true.

    Conclusion: there is a causality between “the body is here and

    the shadow is here. If the body is here then the shadow will be here

    and if the shadow is here then the body must be here.

    Clearly, under certain condition the body is here is a necessary and sufficient symptom of the shadow is here; the shadow is here

    is also a necessary and sufficient symptom of the body is here.

Example`1.25

    Causality analysis of the water is boiling= x, and the water is

     heated to 100?= y.

    In this given causal series, x and y are two events (phenomena), (1) when x appears at the moment k, y will appear at the moment k too, (2) when x does not appear at the moment k, y would not appear at the moment k either; and, (3) when y appears at the moment k, x

    will appear at the moment k, (4) when y does not appear at the moment k, x would not appear at the moment k either.

    The joint truth table is as follows:

    x y ”if x then y y x ”if y then x

    1 1 1 1 1 1

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    1 0 0 1 0 0

    0 1 0 0 1 0

    0 0 1 0 0 1

    Table 6: Causality analysis of example 1.25

    That is to say:

    (1) If the water is boiling (event x true) and the water is heated to 100? (event y true) then this causal statement is true;

    (2) If the water is boiling (event x true) and the water is not heated to 100? (event y false) then this causal statement is false;

    (3) If the water is not boiling (event x false) and the water is heated to 100? (event y true) then this causal statement is false;

    (4) If the water is not boiling (event x false) and the water is not heated to 100? (event y false) then this causal statement is true;

    (5) If the water is heated to 100? (event y true) and the water is boiling (event x true) then this causal statement is true,

    (6) If the water is heated to 100? (event y true) and the water is not boiling (event x false) then this causal statement is false;

    (7) If the water is not heated to 100? (event y false) and the water is boiling (event x true) then this causal statement is false;

    (8) If the water is not heated to 100? (event y false) and the water is not boiling (event x false) then this causal statement is true.

    Conclusion: there is a causality between “the water is boiling

    and the water is heated to 100?. If the water is boiling then the water will be heated to 100? and if the water is heated to 100? then

    the water must be boiling.

    Clearly, under certain condition the water is boiling is a

    necessary and sufficient symptom of the water is heated to 100?;

    the water is heated to 100? is also a necessary and sufficient symptom of the water is boiling.

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Example`1.26

    Causality analysis of the speed of flight is faster than the speed

    of sound= x, and the sound barrier occurs= y.

    In this given causal series, x and y are two events (phenomena), (1) when x appears at the moment k, y will appear at the moment k too, (2) when x does not appear at the moment k, y would not appear at the moment k either; and, (3) when y appears at the moment k, x

    will appear at the moment k, (4) when y does not appear at the moment k, x would not appear at the moment k either.

    The joint truth table is as follows:

    x y ”if x then y y x ”if y then x

    1 1 1 1 1 1

    1 0 0 1 0 0

    0 1 0 0 1 0

    0 0 1 0 0 1

    Table 7: Causality analysis of example 1.26

    That is to say:

    (1) If the speed of flight is faster than the speed of sound (event x

    true) and the sound barrier occurs (event y true) then this causal statement is true;

    (2) If the speed of flight is faster than the speed of sound (event x

    true) and the sound barrier does not occur (event y false) then this causal statement is false;

    (3) If the speed of flight is not faster than the speed of sound

    (event x false) and the sound barrier occurs (event y true) then this causal statement is false;

    (4) If the speed of flight is not faster than the speed of sound

    (event x false) and the sound barrier does not occur (event y false) then this causal statement is true;

    (5) If the sound barrier occurs (event y true) and the speed of

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