Logic

• Quiz 4 Vocabulary
• Quiz 4 Problems
• Implication
• Translation of English Statements to Symbolic
Logic
• Well-formed Boolean Expressions
• Digital Logic

• Equivalence
• Contradiction
• Conclusion
• Law of excluded middle
• Law of non-contradiction
• Boolean Logic
• Premise
• Proposition
• Syllogism
• Symbolic logic
• Tautology
• Truth table
• Definitions for the new terms are at the end of chapter 3

• Convert binary to base 10
• Convert base 10 to binary
• Convert a sequence of characters to a sequence ASCII codes
(numbers)
• Convert a sequence of numbers representing characters in
ASCII to a sequence of characters
• Create a truth table for a Boolean expression
• Show that 2 Boolean expressions are equivalent
• Translate an English language statement into symbolic logic
• Determine if a expression is a well formed Boolean expression

A
True
True
False
False
B
True
False
True
False
A IMPLIES B
True
False
True
True

• Conditionals
• If P Then Q
– P is the antecedent
– Q is the consequence
• Stoics
– Philo of Megara
• A conditional is false when and only when the antecedent is true and the consequence is false

• Truth-functionality
– The truth value of a compound statement is a
(total) function based only on the truth values of its parts (which must be propositions)
• Frege
• Russell and Whitehead
– P IMPLIES Q is equivalent to (NOT P) OR Q

• References
– www.maa.org/sites/default/files/images/upload_l ibrary/46/Pengelley_projects/truth.pdf
– http://plato.stanford.edu/entries/conditionals/
– http://plato.stanford.edu/entries/dialecticalschool/

• Let A represent the simple statement “Alex is a computer science major”
• Let B represent the simple statement “Alex takes CS 120”
• Let C represent the simple statement “Alex is a
Biology major”
• Let D represent the simple statement “Alex takes BIO 105”

Translate the Following Statements into Symbolic Logic
Expressions and Build the Truth Tables for the
Expressions
• Alex is a computer science major and Alex is a biology major
• Alex is a computer science major or Alex is a biology major
• Alex is not a computer science major
• Alex does not take BIO 105
• If Alex is a computer science major then Alex takes cs
120
• If Alex takes BIO 105 then Alex is a biology major

Translate the Following Statements into Symbolic Logic
Expressions and Build the Truth Tables for the
Expressions
• If Alex takes CS 120 then Alex is a computer science major
• If Alex is a computer science major or Alex is a biology major then Alex takes BIO 105
• If Alex does not take BIO 105 then Alex is not a biology major

• Syntactically correct boolean expressions
• Rule 1
– Each single letter is a well-formed expression
– True is a well-formed expression
– False is a well-formed expression

• Assume P and Q are well-formed expressions
• Rule 2
– P and Q is a well-formed expression
– P or Q is a well-formed expression
– P IMPLIES Q is a well-formed expression
– P ≡ Q is a well-formed expression
– NOT P is a well-formed expression
• Rule 3
– ( P ) is a well-formed expression

• Are the following expressions well-formed expressions?
– P and (NOT Q)
– (A OR B) AND (C OR D)
– (R AND S) (NOT W)
– (NOT (A AND B)) IMPLIES (B OR C)
– (NOT P) IMPLIES (NOT Q)
– NOT IMPLIES B OR C

• Building Blocks Digital Computer Hardware
• Logic Gates
– And gate
– Or gate
– Not gate
• Must be implemented with physical devices
– For example transitors
• The are a low level abstraction

0
0
0
0
0
1
0
1
1
0
1
0
1
1
1
1
+0 +1 +0 +1 +0 +1 +0 +1
00 01 01 10 01 10 10 11

1
1
1
0
0
0
1
C in
0
0
1
1
0
1
1
0 a
0
1
0
1
1
0
1
0 b
0
1
1
1
0
0
1
0 c out
0
0
0
1
1
1
0
1 s
0