1. Section 3.1 Statements and Logical Connectives

2. What You Will Learn Statements, quantifiers, and compound statements
Statements involving the words not, and, or, if … then, and if and only if

3. HISTORY—The Greeks:
Aristotelian logic: The ancient Greeks were the first people to look at the way humans think and draw conclusions. Aristotle ( B.C.) is called the father of logic. This logic has been taught and studied for more than 2000 years.

4. Mathematicians
Gottfried Wilhelm Leibniz ( ) believed that all mathematical and scientific concepts could be derived from logic. He was the first to seriously study symbolic logic. In this type of logic, written statements use symbols and letters.

5. Mathematicians
George Boole (1815 – 1864) is said to be the founder of symbolic logic because he had such impressive work in this area.
Charles Dodgson, better known as Lewis Carroll, incorporated many interesting ideas from logic into his books.

6. Logic and the English Language
Connectives - words such as and, or, if, then
Exclusive or - one or the other of the given events can happen, but not both
Inclusive or - one or the other or both of the given events can happen

7. Statements and Logical Connectives
Statement - A sentence that can be judged either true or false.
Labeling a statement true or false is called assigning a truth value to the statement.

8. Statements and Logical Connectives
Simple Statements - A sentence that conveys only one idea and can be assigned a truth value.
Compound Statements - Sentences that combine two or more simple statements and can be assigned a truth value.

9. Negation of a Statement
Negation of a statement – change a statement to its opposite meaning.
The negation of a false statement is always a true statement.
The negation of a true statement is always a false statement.

10. Quantifiers Quantifiers - words such as all, none, no, some, etc…
Be careful when negating statements that contain quantifiers.

11. Negation of Quantified Statements
Form of statement
All are.
None are.
Some are.
Some are not.
Form of negation
Some are not.
Some are.
None are.
All are.

12. Negation of Quantified Statements
None are.
Some are not.
All are.
Some are.

13. Example 1: Write Negations
Write the negation of the statement.
Some telephones can take photographs.
Solution
Since some means “at least one” this statement is true. The negation is “No telephones can take photographs,” which is a false statement.

14. Example 1: Write Negations
Write the negation of the statement.
All houses have two stories.
Solution
This is a false statement, since some houses have one story, some three or more. The negation “Some houses do not have two stories” or “Not all houses have two stories” or “At least one house does not have two stories” are all true statements.

15. Compound Statements
Statements consisting of two or more simple statements are called compound statements.
The connectives often used to join two simple statements are and, or, if…then…, and if and only if.

16. Not Statements (Negation)
The symbol used in logic to show the negation of a statement is ~. It is read “not”.
The negation of p is: ~ p.

17. And Statements (Conjunction)
⋀ is the symbol for a conjunction and is read “and.”
The conjunction of p and q is: p ⋀ q.
The other words that may be used to express a conjunction are: but, however, and nevertheless.

18. Example 2: Write a Conjunction
Write the following conjunction in symbolic form:
Green Day is not on tour, but Green Day is recording a new CD.

19. Example 2: Write a Conjunction
Solution Let t and r represent the simple statements. t: Green Day is on tour. r: Green Day is recording a new CD. In symbolic form, the compound statement is ~t ⋀ r.

20. Or Statements (Disjunction)
The disjunction is symbolized by ⋁ and read “or.”
In this book the “or” will be the inclusive or (except where indicated in the exercise set).
The disjunction of p and q is: p ⋁ q.

21. Example 3: Write a Disjunction
Let
p: Maria will go to the circus.
q: Maria will go to the zoo.
Write the statement in symbolic form.
Maria will go to the circus or Maria will go the zoo.
Solution
p ⋁ q

22. Example 3: Write a Disjunction
Let
p: Maria will go to the circus.
q: Maria will go to the zoo.
Write the statement in symbolic form.
Maria will go to the zoo or Maria will not go the circus.
Solution
q ⋁ ~p

23. Compound Statements
When a compound statement contains more than one connective, a comma can be used to indicate which simple statements are to be grouped together.
When we write the compound statement symbolically, the simple statements on the same side of the comma are to be grouped together within parentheses.

24. Example 4: Understand How Commas Are Used to Group Statements
Let p: Dinner includes soup.
q: Dinner includes salad.
r: Dinner includes the vegetable of the day
Write the statement in symbolic form.
Dinner includes soup, and salad or vegetable of the day.
Solution p ⋀ (q ⋁ r)

25. Example 4: Understand How Commas Are Used to Group Statements
Let p: Dinner includes soup.
q: Dinner includes salad.
r: Dinner includes the vegetable of the day
Write the statement in symbolic form.
Dinner includes soup and salad, or vegetable of the day.
Solution (p ⋀ q) ⋁ r

26. Example 5: Change Symbolic Statements into Words
Let p: The house is for sale.
q: We can afford to buy the house.
Write the symbolic statement in words.
p ⋀ ~q
Solution
The house is for sale and we cannot afford to buy the house.

27. Example 5: Change Symbolic Statements into Words
Let p: The house is for sale.
q: We can afford to buy the house.
Write the symbolic statement in words.
~p ⋁ ~q
Solution
The house is not for sale or we cannot afford to buy the house.

28. Example 5: Change Symbolic Statements into Words
Let p: The house is for sale.
q: We can afford to buy the house.
Write the symbolic statement in words.
~(p ⋀ q)
Solution
It is false that the house is for sale and we can afford to buy the house.

29. If-Then Statements
The conditional is symbolized by → and is read “if-then.”
The antecedent is the part of the statement that comes before the arrow.
The consequent is the part that follows the arrow.
If p, then q is symbolized as: p → q.

30. Example 6: Write Conditional Statements
Let p: The portrait is a pastel.
q: The portrait is by Beth Anderson.
Write the statement symbolically.
If the portrait is a pastel, then the portrait is by Beth Anderson.
Solution
p → q

31. Example 6: Write Conditional Statements
Let p: The portrait is pastel.
q: The portrait is by Beth Anderson.
Write the statement symbolically.
If the portrait is by Beth Anderson, then the portrait is not a pastel.
Solution
q → ~p

32. Example 6: Write Conditional Statements
Let p: The portrait is pastel.
q: The portrait is by Beth Anderson.
Write the statement symbolically.
It is false that if the portrait is by Beth Anderson, then the portrait is a pastel.
Solution
~(q → p)

33. If and Only If Statements
The biconditional is symbolized by ↔ and is read “if and only if.”
If and only if is sometimes abbreviated as “iff.”
The statement p ↔ q is read “p if and only if q.”

34. Example 8: Write Statements Using the Biconditional
Let
p: Alex plays goalie on the lacrosse team.
q: The Titans win the Champion’s Cup.
Write the symbolical statement in words.
p ↔ q
Solution
Alex plays goalie on the lacrosse team if and only if the Titans win the Champion’s Cup.

35. Example 8: Write Statements Using the Biconditional
Let
p: Alex plays goalie on the lacrosse team.
q: The Titans win the Champion’s Cup.
Write the symbolical statement in words.
q ↔ ~p
Solution
The Titans win the Champion’s cup if and only if Alex does not play goalie on the lacrosse team.

36. Example 8: Write Statements Using the Biconditional
Let
p: Alex plays goalie on the lacrosse team.
q: The Titans win the Champion’s Cup.
Write the symbolical statement in words.
~(p ↔ ~q)
Solution
It is false that Alex plays goalie on the lacrosse team if and only if the Titans do not win the Champion’s Cup.

37. Logical Connectives