Appendix D. Reflections on
Law of the Excluded Middle
Origins, Benefits, Limitations
In writing a story or developing a theory, there may be some benefit in
be able to say for a possibility A, that
A or NOT A
holds - or is true. The assumption that A or NOT A holds provides the law
of the excluded middle in logic, one that may be used in mathematics to
arrive at conclusions.
In talking about this law of excluded middle, I will play the role of a
devil`s advocate.
A material implication rule A if B is vacuously true when the situation B
never occurs. In the formulation of a theory, consistency requires that
the situation
A and NOT A
never occur. That in turn implies
A or NOT A
in the exclusive OR sense. But the requirement for consistency does not
tell us which one of the two, A and NOT A occurs. If the elements of a
story, play or theory so far written are unconnected to the occurrence of
A or its negation NOT A, then whether or not A occurs (provided it is
internally consistent) does affect the consistency of the existing part
of the story, play or theory. With respect to the latter, the situation A
and its negation NOT A unreachable and informally at least, undecidable.
Thus the law of excluded middle A or NOT A holds in a
vacuous manner. On the other hand, if one of the two
statements A and NOT A is within the reach of (decidable, proveable)
within the existing story or play or theory then consistency requires
exactly one be implied, and the law of the excluded middle holds for A
provided the reach is consistent. Thus the law of excluded middle can be
assumed without loss of consistency.
Remark: The discussion of logic in mathematics is a
delicate manner. There is a risk here and in the following page of making
some naive mistakes.
Law of the Excluded Middle: A or Not A.
Let A be the statement that some situation occurs. Then a story or theory
that suggests a statement A is both true and false is inconsistent. So
for the sake of consistency in our present and further reason, we may
require and assume the statement
A AND Not A
to be false - NEVER TO OCCUR. So in our story or theory in its present
and further development, we require
A OR not A
to be true but not both at any instance (except during a brief transition
period).
So A requires not (not A) for consistency with A AND not A, and not (not
A) requires A at any instance (except during a brief transition period).
Remark: The discussion of transition time
suggests the law of excluded middle might be broken momentarily when
situations are time-dependent or place dependent. For example, in
counting people in a room that has a door, we cannot say a person is
all in or all out because of the middle possibility of a person being
part in and part out. So a person has three static states namely, in,
out and partly both, and two transition state namely, going from in to
out, and going from out to in. During these transitions, the middle
state of partly in and partly out occurs for a short or long period of
time.
Law of the Excluded Middle
essay Dec 1, 2008
(To do - keep, rewrite or cut, what is best here?)
Logic I of Partial Inclusion : Let B be a
region in a one occupant house. We say B is true when the
occupant is partially in B. Likewise we say Not B is true when
the occupant is partially in the rest of the house. Now for that
occupant, the statement
B or Not B
holds, but the two events B and Not B may occur
simultaneously. That is the assertion
B and Not B
may be true. They are not mutually exclusive.
Whence to say B holds is not to say Not B does not, and
vice-versa.
Logic II of Full Exclusion: Again, let B be
a region in a one occupant house. We say B is true when the
occupant is fully in B. Likewise we say Not B is true when the
occupant is fully in the rest of the house. Now for that
occupant, the statements B and Not B are mutually exclusive.
Thus
B and Not B never both hold.
However the statement
B or Not B
fails when the occupant is partially in
both.
A More Careful Logic III: Yet again, again,
let B be a region in a one occupant house. But this time, let A
be the statement that
the occupant is partially in B.
Then Not A would be the statement
the occupants is not partially in B - the
occupant is fully out of it.
Then statement A and the Not A are mutually
exclusive: The statement
A and NOT A
can never hold. Moreover, the statement
A or Not A
will be hold as well. So Not (Not A) implies A and
A implies Not (Not A). That be said, even though the latter
holds, we still may be a state of ignorance which one occurs and
when.
The Law of Excluded Middle. This law holds
when the statement when an assertion C and Not C are (a) mutually
exclusive and (b) at least one of the two statement C or Not C
occurs. The law of excluded middle fails for logic I and II, but
holds for logic III.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
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Road
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See too, the BBC-Belgium story Texting and
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The Logic of Injustice:
How Texas sent
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by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
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Work Booklets for ages 3+ to 13 Use these or others to check
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the choice is theirs. But in retrospect, the selection does not
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Calculus Starter Lessons
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Flash
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They cover basic topics in ways likely to complement your
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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