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PLAN OF CA PREREQUISITES
ca1 introduction to propositional calculus
ca2 introduction to predicate logic
caa 2D exact propositional calculus
cab ND exact propositional calculus
cac implication
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IMPLICATION (Exact 2D Propositional Calculus)
Let's consider two statements
p: "it has been raining over the street"
q: "the street is wet"
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.
Implication in metalanguage:
"If it has been raining over the street then the street is
wet"
or "it has been raining over the street implies that the
street is wet"
Implication in Calculus:
case. imp(pq)
1: 1 11
2: 0 10
3: 1 01
4: 1 00
Let imp(pq) axiom of theory T.
We shall discuss:
A. Application of T
AA.Deductive
AB. Inductive
B. Research on T
A. APPLICATION OF T (Application is based upon belief that
imp(pq) holds).
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, e.g.
clean the gutters.
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.
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 case 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.
It's surprising to find that it took 2000 years to formulate
this Principle, when it is obviously inherent to Implication
which was known to Aristoteles.
EXERCISE
Let: p: "it has been raining over the
street" q: "the street is wet"
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.
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.
