John Rosson

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Survey of Mathematical IdeasMath 100Chapter 3,
Logic

• John Rosson
• Thursday February 15, 2007

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The Lady and the Tiger
p - Lady in room 1 q - Tiger in room 1 r - Lady
in room 2 s - Tiger in room 2
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• Given information
• One sign is true the other false
• The lady and tiger do not share a room

p?s
(p?r) ? (q?s)
Argument If the first sign were true, both p and
s would have to be true. This would mean that
both disjunctions in the second sign would have
to be true and so the sign would be true. But
this contradicts the first piece of information,
so the first sign has to be false. Since they do
not share a room the lady has to be in 2 ( r )
and the tiger in 1 (q).
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Introduction to Logic

• Statements and Quantifiers
• Truth Tables and Equivalent Statements
• The Conditional
• More on the Conditional
• (3.6) Analyzing Arguments using Truth Tables

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The Conditional
Recall the truth table for the conditional
statement.
The conditional statement is false only when the
first statement, called the antecedent, is true
and the second statement, called the consequent,
is false.
A conditional statement is always true when the
antecedent is false and always true when the
consequent is true.
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Calculating Truth Tables
Calculating truth tables involving the
conditional is not difficult. All we have to
remember is that the conditional is false only
when the antecedent is true and the consequent is
false.
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Calculating Truth Tables
Notice that the conditional statement p ? q is
equivalent to the statement p?q.
Since the conditional can be interpreted as
implication, this equivalence can be interpreted
as follows. The claim that p implies q has the
same logical meaning as either p is false or q
is true.
For example, let p be the statement that the
number n is evenly divisible by 4 and let q be
the statement that the number n is evenly
divisible by 2. Now, p implies q since any
number divisible by 4 is divisible by 2. It is
also valid to say that either a number is not
divisible by 4 or it is divisible by 2.
This equivalence also means that any statement
containing a conditional (?) may be logically
replaced by one using only not () and or (?).
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Calculating Truth Tables
Since the conditional can be interpreted as
implication, this equivalence can be interpreted
as follows. The claim that p does not imply q
has the same logical meaning as p is true and q
is false.
Notice that the the negation of the conditional
( p? q) is equivalent to the statement p ? q.
For example, let p be the statement that the
number n is evenly divisible by 2 and let q be
the statement that the number n is evenly
divisible by 4. Now, p does not imply q since
the number 6 is divisible by 2 ( so p is true)
and 6 is not divisible by 4 (so q is false).
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Calculating Truth Tables
This says that if p implies q and p is true then
q has to be true also.
We also get the following tautology.
Most of mathematics and much of Artificial
Intelligence (AI) is founded on this tautology.
Recall
But where do the rules of logic come from?
The argument would go like this All men are
mortal can be express as For all x, if x is a
man then x is mortal. Specializing (another
logical rule) x to Socrates, we have If Socrates
is a man then Socrates is mortal. The first line
becomes Socrates is a man ? Socrates is
mortal . So this tautology is the basis of the
logical rule modus ponens.
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Conditional
The conditional statement is the basic form of
deductive reasoning. It has a direction, from
antecedent to consequent. Since it is so
important, the conditional has many synonyms.

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Relative Forms
Note that the direct and contrapositive
statements are equivalent as are the converse and
the inverse.
Note that this is not the negation of the direct
statement.
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Relative Forms
Let p be the statement you build it and let q
be the statement they will come.

Direct Statement p ? q, If you build it, then
they will come.
Converse q ? p, If they do come, then you did
build it.
Inverse p ? q, If you do not build it, then
they will not come.
Contrapositive q ? p, If they do not come,
then you did not build it.
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Biconditional
Consider the truth table for the conjunction of a
conditional with its converse.
This statement claims that it is true both that p
implies q and conversely that q implies p.
This relationship between p and q is important
enough to get its own symbol called the
biconditional (conditional in both directions).
In words, p ? q means p is true if and only if q
is true. The statement p ? q is true precisely
when p and q have the same truth values. If p and
q are equivalent statements then p ? q is a
tautology.
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Biconditional
Consider the following example.
This statement is a tautology because the two
terms of the biconditional are equivalent (have
the same true table).
It is true that a natural number is even if and
only if it is divisible by 2. It is false that
a natural number is even if and only if it is
divisible by 4.
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Assignments 3.5, 3.6, 4.1, 4.2, 4.3