Chapter 24, Direct and
Indirect Reason
To prove a statement is to show that it must hold. To refute a statement
is to show that it cannot hold. In between proof and refutation1 lies
uncertainty or We don`t know.
PS. The concept of proof comes perversely from the idea of
testing. The first question can we test if a statement is false? If a
consequence of a statement or assertion fails to be true, then the
statements or assertion is false or refuted.
1 Proof and Refutation
Methods
Proof and Refutation was the title of a work by
the philosopher Latakos, Cambridge University Press, 1976. More recent
editions are available.
In a given situation:
- what evidence or proof do we need to say a rule if A then B is
never disobeyed?
- what evidence or proof do we need to say the rule is false?
False here means sometimes disobeyed. The proof or disproof techniques
described next are usually employed in a context where some implication
rules are assumed to be never disobeyed. The first question (1) then
becomes what further implications rules are never disobeyed. The
techniques described next provide answers in some but not all
circumstances.
Refutation: Proving Falseness
To show that a rule if A then B is false is simple, see if we can
find (or show there exists or must be) a situation in which A
occurs and B does not. Then the implication rule A implies
B is false. In other words, it is not always obeyed or it is
sometimes disobeyed.
A Direct Approach: Applying Implication Rules
PS: A proof in mathematics consists of a chain of reason, direct or
indirect, but carefully done, which implies that a statement or
assertion is a consequence of an assumption or set of assumptions. Then
if the assumptions and chains of reason are not false, the statement or
assertion must hold (we hope).
You may show that the rule If A then B is always true by showing
that when situation A occurs so must B. Such a proof could
employ a chain of reason (deduction) using trusted implication rules:
rules that are not disobeyed in the situation at hand. Such a proof shows
that when A occurs, so must B. This says and shows that the
situation A and NOT B never occurs. See the chapter Chains of
Reason for an illustration of this approach.
A Somewhat Indirect Approach: Proving The Contrapositive Method )
Alternatively we can show the (contrapositive) rule If NOT B then NOT
A is always true by showing that when situation NOT B occurs
then so must NOT A. Such a proof again shows the situation A
and not B never occurs. That is what we need. (See the chapter
Chapter 22, The Contrapositive, or see Chapter 4, Implication
Rules.
A Fully Indirect Approach: The Contradiction Method
The aim here is to find a situation C with the following
properties:
- the situation C does not occur (is obviously false) and
-
(A and NOT B) implies situation C.
These properties tell us that the situation (A and NOT B) cannot
occur - why? So that the implication rule A implies B is never
disobeyed. Next are a few words to explain why:
The contrapositive form of the implication rule if (A and NOT B) then
C is what we need. It says if NOT C then NOT (A and NOT B).
But the situation NOT C, by chance or discovery, occurs. Thus the
situation (A and not B) can never occur. And that is what we mean
when we say that the rule A implies B always holds.
Remark (a-looking we will go). This proof by
contradiction method can be applied without knowing in advance what
the situation C will be. We search for it. That is, for the sake
of finding such a situation C, we assume the situation A and
not B occurs. After this we follow whatever chains of reason we can
to reach a conclusion that an absurd (or obviously false) situation
C occurs.
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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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May 2012, Composition Starting:
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Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
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the choice is theirs. But in retrospect, the selection does not
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Arithmetic
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Algebra
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Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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Flash
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They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
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calculus and more generally in the first year of college. Bon
Appetite.
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