Chapter 2: The Logic of Compound Statements
2.1: Logical Forms and Logical Equivalence
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Statements
Definition
A statement (or proposition) is a sentence (or assertion) that is true or
false but not both. We typically use letters like p, q to denote propositions.
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Examples
Examples
1
Washington, DC, is the capital of United States.
2
Annapolis is the capital of United States.
3
It is snowing.
4
I made a mistake in signing up for this course.
5
1+1=2
All these statements are simple statements.
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Examples
Examples
1
Washington, DC, is the capital of United States.
2
Annapolis is the capital of United States.
3
It is snowing.
4
I made a mistake in signing up for this course.
5
1+1=2
All these statements are simple statements.
Food for thought
Am I saying the truth in the following statement? “I am lying now.”
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Compound Statements
Definition
A combination of two or more simple statements is a compound
statement.
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Examples
Examples
1
Washington, DC, is the capital of United States and it is snowing.
2
Washington, DC, is the capital of United States or it is snowing.
3
It is not snowing.
4
I did not make a mistake in signing up for this course or 1 + 1 6= 2.
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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The negation symbol
Definition
The symbol ∼ denotes not. Given a statement p, the sentence ∼ p is read
not p.
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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The negation symbol
Definition
The symbol ∼ denotes not. Given a statement p, the sentence ∼ p is read
not p.
Examples
1
p: “It is snowing”. Then ∼ p: “It is not snowing”
2
q: 1 + 1 = 2. Then ∼ q: 1 + 1 6= 2.
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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The or and and symbols
Definition
Let p and q be two statements.
1
The conjunction of p and q is p ∧ q and is read “p and q”
2
The disjunction of p and q is p ∨ q and is read “p or q”
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Example
Example (Kemeny, Shell, Thompson)
Let p: “Fred likes George” and q: George likes Fred. Write the following
statements in symbolic form:
1
Fred and George like each other.
2
Fred and George dislike each other.
3
Fred likes George, but George does not reciprocate.
4
George is liked by Fred, but Fred is disliked by George.
5
Neither Fred nor George dislike each other.
6
It is not true that Fred and George dislike each other.
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Truth Tables
Fact
The truth value of a compound statement is determined by the truth
value of its component.
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth Tables
Fact
The truth value of a compound statement is determined by the truth
value of its component.
A very convenient way of tabulating this dependency is by means of a
truth table.
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Truth table for ∼ p
Definition
The truth table for negation is:
p
T
F
∼p
F
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for ∼ p
Definition
The truth table for negation is:
p
T
F
∼p
F
T
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for conjunction
Definition
The truth table for conjunction is:
p
T
q
T
p∧q
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for conjunction
Definition
The truth table for conjunction is:
p
T
T
q
T
F
p∧q
T
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for conjunction
Definition
The truth table for conjunction is:
p
T
T
F
q
T
F
T
p∧q
T
F
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for conjunction
Definition
The truth table for conjunction is:
p
T
T
F
F
q
T
F
T
F
p∧q
T
F
F
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for conjunction
Definition
The truth table for conjunction is:
p
T
T
F
F
q
T
F
T
F
p∧q
T
F
F
F
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for disjunction
Definition
The truth table for disjunction is:
p
T
q
T
p∨q
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for disjunction
Definition
The truth table for disjunction is:
p
T
T
q
T
F
p∨q
T
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for disjunction
Definition
The truth table for disjunction is:
p
T
T
F
q
T
F
T
p∨q
T
T
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for disjunction
Definition
The truth table for disjunction is:
p
T
T
F
F
q
T
F
T
F
p∨q
T
T
T
Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalenc
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Truth table for disjunction
Definition
The truth table for disjunction is:
p
T
T
F
F
q
T
F
T
F
p∨q
T
T
T
F
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