L1.2
Notes: Logic (continued)
Compound propositions may have specific properties of interest, and some
of these are defined here.
TAUTOLOGY:
A compound proposition that is TRUE for all
combinations of its atomic propositions
E.g. (p
∧
q)
→
(p
∨
q)
Truth Table:
p
q
p
∧
q
p
∨
q
(p
∧
q)
→
(p
∨
q)
T
T
T
T
T
T
F
F
T
T
F
T
F
T
T
F
F
F
F
T
Note the bold, italicized T’s in the final column. That is the “signature” of a
tautology. A tautology is a logical way of combining propositions so that a
false result is impossible. It will be the basis of proofs because proofs are
intended to ensure that something is true in all circumstances.
Another interesting tautology is [(p
→
q)
∧
(
¬
p
→
q)]
→
q
This is a symbolic representation of the logic of sentences like “If I do my
homework (p) my parents yell at me (q) and if I don’t do my homework my
parents yell at me, therefore my parents will yell at me.”
1
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CONTRADICTION:
A compound proposition that is FALSE for all
combinations of its atomic propositions.
Like
tautology
the term
contradiction
has a specific meaning in logic and is
not to be confused with other commone usages of the terms.
E.g.
¬
p
∧
(p
∧
q)
p
q
p
∧
q
¬
p
¬
p
∧
(p
∧
q)
T
T
T
F
F
T
F
F
F
F
F
T
F
T
F
F
F
F
T
F
Note the bold, italicized F’s in the final column. That is the “signature” of a
contradiction. A contradiction cannot be true under any circumstances (i.e.,
under any assignment of truth values to its atomic propositions). The classic
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 Spring '08
 WATKINS
 Logic, Atlas Shrugged, atomic propositions

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