Logic
CSLU 1100.003
Fall 2007
Cameron McInally
[email protected]
Fordham University
Logic
A statement can be either True or False.
Logic
• Who is telling the truth and who is lying?
– Person A: “We are both lying”.
– Person B: “Person A is lying”.
• Who is telling the truth and who is lying?
– Person A: “At least one of us is lying”.
– Person B: “Person A and C are lying”.
– Person C: “Only one of the other two are lying”.
Logic
• Some statements are obviously True:
o “All men are mortal”.
o “Socrates is a man”.
• Some statements are obviously False:
o “All men are Socrates”.
Logic
• Some statements, we cannot tell if it is True or
False.
o “Peter Piper picked a peck of pickled
peppers”.
But, we do know that it must one of the two.
Logic
• Some statements are too hard to determine
the correct answer…
– “Seinfeld is the best show ever”.
So,
We don’t care about these.
Logic
• Mathematicians like to use symbols to
represent things.
• For example,
o p Ξ Peace
o q Ξ Quiet
• So,
o p and q Ξ Peace and Quiet
Logic
• So, if we wanted “Peace and Quiet”…
o We could say we want “p and q” which is easier
and quicker to write.
Logic
• We can associate either a True or False value
with a symbol.
– What is the value of q, if q = Quiet?
• It is Quiet?
– This implies that q = True.
• It is not Quiet?
– This implies the q = False.
Logic
• There are also the operations And and Or…
– And (Λ)
• p and q = p Λ q
– Or (V)
• p or q = p V q.
• These operations allow us to combine more
than one statement.
Logic
• First, lets look at And (Λ)
– So, for p Λ q, we can have
•
•
•
•
p Ξ True Λ q Ξ True
p Ξ True Λ q Ξ False
p Ξ False Λ q Ξ True
p Ξ False Λ q Ξ False
Logic
• Next, lets look at Or (V)
– So, for p V q, we can have
•
•
•
•
p Ξ True V q Ξ True
p Ξ True V q Ξ False
p Ξ False V q Ξ True
p Ξ False V q Ξ False
Make sure that you see both And and Or can have
the same combinations of True and False.
Logic
• The combinations of True and False values, for
a set of statements, can be represented as a
Truth Table:
p pΛqΛr
– For example,
q
r
Combination #1
True
True
True
Combination #2
True
True
False
Combination #3
True
False
True
Combination #4
True
False
False
Combination #5
False
True
True
Combination #6
False
True
False
Combination #7
False
False
True
Combination #8
False
False
False
Logic
• Who is telling the truth and who is lying?
– Person A: “We are both lying?”
– Person B: “Person A is lying?”
a
True
True
False
False
b
True
False
True
False
Logic
• What does this Truth Table imply…
– Person A: “We are both lying?”
– Person B: “Person A is lying?”
Person A
Person B
True
True
False
False
True
False
True
False
Statement Statement
a is
b is
False
False
False
False
False
True
True
True
←
Logic
• Who is telling the truth and who is lying?
– Person A: “At least one of us is lying”.
– Person B: “Person A and Person C are lying”.
– Person C: “Only one of the other two are lying”.
You try it!!!
Logic
• What did you come up with?
a
b
c
p
q
r
True
True
True
False
False
False
True
True
False
True
False
False
True
False
True
True
False
True
True
False
False
False
False
False
False
True
True
True
False
True
False
True
False
True
False
True
False
False
True
True
False
False
False
False
False
True
True
False
←
Logic
• Logical Connectives
– We need a way to combine logical statements, e.g.
• “I am going to the movies”.
• “I am going to the mall”.
– Both of these are correct. But, what if we want to
do both, one or the other, or neither? What if we
want to say that one will cause the other? How do
we represent this?
Logic
• How do we say…
– “I am not going to the movies”.
– “I am going to the movies” and “I am going to the
mall”.
– “I am going to the movies” or “I am going to the
mall”.
– If “I am going to the movies”, then “I am going to
the mall”.
Logic
• The connective Not (¬)
– Not reverses the meaning of a statement.
p
¬p
True
False
False
True
Logic
• The connective And
• Works the same as in the English language.
p
True
True
False
False
q
True
False
True
False
pΛq
True
False
False
False
Logic
• The connective Or
• Does NOT work the same as in the English
Language.
p
True
True
False
False
q
True
False
True
False
pVq
True
True
True
False
Logic
• The connective Implies (→)
• If the first statement happens, the second
statement will follow.
p
True
True
False
False
q
True
False
True
False
p→q
True
False
True
True
Logic
• Lets make a Truth Table for…
¬ (p Λ q) → (p V q)
p
q
(p Λ q)
¬(p Λ q)
(p V q)
¬ (p Λ q) → (p V q)
True
True
True
False
True
False
False
True
True
True
True
True
False
False
True
False
False
False
True
True
True
False
True
False
Logic
• Tautologies
– This happens when every combination of values in
a Truth Table leads to a True result.
• Contradiction
– This happens when every combination of values in
a Truth Table leads to a False result.
Logic
• Hints:
– Break each column of you table into dealing with
only one logical connective at a time… this will
reduce logic errors
– Use the parentheses as your guide for how to
break the statement down.
– Do not try to perform any transformations on the
logic statement outside of those that have been
taught.
Logic
• People and problems often use the phrase
“show two statements are equivalent”
• All this means is that when you complete the
truth table for both of them then the have the
same values all the way down the column in
the truth table.
Logic
• Practice
( p  a)
Logic
• Practice
( p  (r  a ))
Logic
• Practice
(a  b)  [b  (c  a)]
Logic
• A few more practice problems:
– Truth Table Practice
• http://storm.cis.fordham.edu/~kinley/classes/summer0
6/cseu1100/flash/ch1/sec1_3/truthtables/tt_control.ht
ml
– Equivalences Practice
• http://storm.cis.fordham.edu/~kinley/classes/summer0
6/cseu1100/flash/ch1/sec1_3/equivalence/tteq_contro
l.html
Logic
Homework
(Always Due in One Week)
• Read Section 3.1 to 3.6
• Complete Section 3.8 pages 49-50:
1a, 1b(i-v), 1c(i,ii)