LEARNING OBJECTIVES
Learning Objectives
- Simple statements
- Compound statements
- Negation
- Translating English to Symbolic Logic
Statements
In the English language there are many types of sentences. A few of the types are questions, exclamations and commands. In our study of logic we will only consider a declarative sentence.
Statement
A statement is a declarative sentence that is either true or false.
Examples
Examples of sentences that are statements:
- George Washington is a man.
- A triangle has three sides.
- Denver is the capital of Colorado.
Examples of sentences that are not statements:
- Who are you?
- Broccoli tastes good.
- Run for your life!
- Front Range Community College is the best.
Simple and Compound Statements
In logic statements can be classified as simple or compound.
A simple statement contains only one idea while a compound statement is two or more simple statements joined together with a connective.
Connectives
There are 4 basic connectives used in logic
- Conjunction (the word “and”)
- Disjunction (the word “or”)
- Conditional (if…then)
- Biconditional (if and only if)
Examples
Here are some examples of compound statements.
- I went to Egypt and I rode a camel. (Conjunction)
- I will read my textbook or watch a reality show. (Disjunction)
- If the mountains are covered in clouds, then it will snow soon. (Conditional)
- I will graduate on time if and only if I take 15 credits each semester (Biconditional)
Negation
Negation tells us, “It is not the case that… ”
For example if we negate the statement “It is snowing”, we would say “It is not the case that it is snowing”. Or in more simple terms, “It is not snowing”.
Symbolic Notation
The main goal in the study of logic is to be able to objectively evaluate logical arguments. In order to do this, we will need to translate English statements into symbolic form. We will use symbols to represent the negation and the connectives and, or, if…then and if and only if. Simple statements in logic are usually denoted by the lowercase letters p, q, and r.
Symbols
The symbol [latex]\wedge[/latex] is used for and (also called a conjunction): A and B is notated [latex]A\wedge{B}[/latex].
The symbol [latex]\vee[/latex] is used for or (also called a disjunction): A or B is notated [latex]A\vee{B}[/latex]
The symbol [latex]\sim[/latex] is used for not (also called a negation): not A is notated [latex]\sim{A}[/latex]
The symbol → is used for if … then (also called a conditional)
The symbol ↔ is used for if and only if (also called the biconditional)
Examples
Translate each statement into symbolic notation. Let p represent “I like Pepsi” and let q
represent “I like Coke”.
a. I like Pepsi or I like Coke.
b. I like Pepsi and I like Coke.
c. I do not like Pepsi.
d. It is not the case that I like Pepsi or Coke.
e. I like Pepsi and I do not like Coke.
f. If I like Pepsi, then I don’t like Coke.
g. I will like Coke if and only if I don’t like Pepsi
Solution:
a. p ⋁ q
b. p ⋀ q
c. ~p
d. ~(p ⋁ q)
e. p ⋀ ~q
f. p → ~q
g. q ↔ ~p
Logical Order of Operations
We often use parenthesis in logical statements when more than one connective is involved in order to specify the order.
Examples
Let p represent “The resort is in Mexico” and let q represent “The resort is all inclusive”. Translate the following statements.
a) (~p) ⋀ q
b) ~ (p V q)
Solution:
a) The resort is not in Mexico and it is all inclusive.
b) It is not the case that the resort is in Mexico or it is all inclusive
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