Chapter 19, Elements of Logic
Logos is a Greek word for thought.
Previous algebra and symbol free chapters on reason showed how
implication rules can be directly used or chained together to arrive at
conclusions. In daily life with the exception perhaps of detective
stories, the direct use of rules and patterns is usually sufficient
(enough).
Yet in mathematics, direct and indirect chains of reasoning appear. The
study of logic, that is, methods or laws for rule- and pattern-based
thought, has been motivated by the need in mathematics to reach
conclusions. In particular, proofs based on (1) mathematical induction,
(2) the contrapositive, and (3) proof by contradiction all stem or
originate from the conclusion-reaching needs of mathematics.
The chapter Direct and Indirect Reason below, will
describe methods (2) and (3). Suggestion: try to read this last chapter
to see how much can be immediately understood. Its exposition is
independent of the discussion of occurrence and truth tables.
The subject of logic as it is studied within mathematics courses, is
often presented as an algebraic (or symbolic) perspective of the methods
of reason. The next chapters present the algebraic perspective. They with
the earlier algebra-free discussion of implication rules and chains of
reason give some preparation for the description of the indirect methods
(2) and (3) for in the last chapter Direct and Indirect
Reason
The algebraic description of logic also has a role in the design and
simplification of electrical controls and computing circuits.
The algebraic description of logic further allows algebraic methods for
arriving at conclusions, in particular mathematical induction, to be
applied to the drawing conclusions about rule-based reason and logic.
The algebraic description of logic provides models of mathematical
logic. Conclusions drawn about the models then reflect on the
limitations and reach of logical or rule-based thought in mathematics.
1. About the Next Chapters
The next five chapters continue the description of logic.
- Shorthand or Pronouns in Logic<
- Occurrence Tables
- The Contrapositive
- Truth Tables and
- Direct and Indirect Reason
The occurrence (or obedience) tables invented and introduced below
identify those situations in which implication rules are obeyed,
disobeyed or not disobeyed. The latter notions are intended to simplify
the explanation of truth tables. An implication rule is said to be true
in the case when it is obeyed or it is at least not disobeyed. An
implication rule is said to be false or not true when it is disobeyed.
[1]
[1] The language previously used to explain and justify the entries of
truth tables overuses the word true. The introduction of the three
notions of an implication rule if A then B being
obeyed, disobeyed or not disobeyed
aims to avoid this situation. Such implication rule is said to be false
in situations where it is disobeyed, and it is said to hold (or be
true) in those situations where it is obeyed or at least not disobeyed.
Finally, the implication rule is said to be always true in the
circumstances of interest provided it is never disobeyed in those
circumstance. See the text for further explanation.
The chapter The Contrapositive shows the equivalence of
an implication rule with its contrapositive formulation. The analysis is
based on the three notions of a rule being obeyed, disobeyed or not
disobeyed.
The chapter Direct and Indirect Reason describes and
explains direct and indirect methods for reaching or proving conclusions.
Among the indirect methods, this chapter describes in particular, how an
implication rule can be shown to always hold by (a) showing its
contrapositive form always hold, or by (b) looking for absurdities that
would occur if the implication rule did not hold. The second method (b)
is more indirect than the first method (a).
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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
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Secondary
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Road
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The Logic of Injustice:
How Texas sent
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first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
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
and some in Pattern
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May 2012, Composition Starting:
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Parent-friendly
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
cover integral and fractions liquid weights and measures - ask
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Calculus Starter Lessons
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They cover basic topics in ways likely to complement your
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if one or more explanations is not to liking, try another. It may
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
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calculus and more generally in the first year of college. Bon
Appetite.
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