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Exact 2D Prop Calc Implication
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Let's consider two statements 

p: "it has been raining over the street"
q: "the steet is wet"

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NOTE: By p and q we mean that: 
K.The rain was sufficient to wet the street.
L.It's been raining recently and the street had no time to dry,
M.It's the same street in p and q.
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Implication in metalanguage:
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"If it has been raining over the street then the steet is wet"
or
"it has been raining over the street implies that the steet is wet"
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Implication in Calculus:
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case.imp(pq)
1:...1...11
2:...0...10
3:...1...01
4:...1...00
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Let imp(pq) axiom of theory T.
 
We shall discuss:
A.Application of T
AA.Deductive
AB.Inductive
B.Research on T

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A.Application of T
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Application is based upon belief that imp(pq) holds.
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AA.Deductive 
------------
Meteo forcasts rain in concerned area.
We deduce from case 1:

case.imp(pq)
1:...1...11

that the street will soon be wet and act accordingly, eg. clean the gutters.
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AB.Inductive
------------
Meteo stated that it has been raining in concerned area, but we state that our 
street is dry. Upon our belief in imp(pq) holding, we induce from case 4:

case.imp(pq)
4:...1...00

that p=0, i.e. that it has not been raining in concerned area and we inform 
meteo system that it has a bug.

On the other hand, if we state that our street is wet, we induce from case 1:

case.imp(pq)
1:...1...11 

that meteo was right and keep happy and quiet.
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B.Research on T
---------------
We gather FACTUAL INFORMATION, i.e. OBSERVATIONS of p and q to see how they fit 
pertinent cases of the axiom: "imp(pq)":

case.imp(pq)
1:...1...11
2:...0...10
3:...1...01
4:...1...00   

Let's note that T CAN BE DISPROVED by a single observation fitting case 2.

On the contrary, it cannot be PROVEN. Indeed, no matter how many observations 
may fit cases 1,3,4, they don't prove that some day we will not observe the 
cas 2.

The more observations fit 1,3,4, and the stronger gets our PRAGMATIC BELIEF 
in imp(pq) holding, but no matter how strong our belief, it is not a PROOF.

Billions of observations per minute fit the "gravity theory", making us believe 
so strongly in gravity, that we take it for obvious and granted. Still, gravity 
is not PROVEN and while we believe that it will be there to morrow, there is no 
logical reason to be 100% certain that it will not cease in next second.

We encounter here a central premise in the philosophy of science,  the Principle 
of Falsifiability, first formally discussed by Karl Popper. This principle states 
that in order to be useful (or even scientific at all), a scientific statement 
('fact', theory, 'law', 'principle', etc) must be falsifiable, i.e. able to be 
proven wrong. Without this property, it would be difficult (if not impossible) 
to test a scientific statement against the evidence.
 
COMMENT: It's surprising to find that it took 2000 years to formulate this 
Principle, when it is obviously inherent to Implication which has been known 
to Aristoteles.

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EXERCISE
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Let:
p: "it has been raining over the street"
q: "the steet is wet"   

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NOTE: By p we mean additionally that:
K.The rain was sufficient to wet the street.

L.It's been raining recently and the street
had no time to dry,

M.The street in p is the same as in q.
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Let imp(pq) axiom of theory T.

case.imp(pq)
1:...1...11
2:...0...10
3:...1...01
4:...1...00 

1.Try to find factual examples for case 3.
and explain why they do not refute T.

2.Explain why case 1. does not prove T.

3.Try to refute T.
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