1. Chapter 1 Section 1.2 Symbolic Logic

2. Sentences vs Statements A truth value is one of two words either true (T) or false (F). A statement is a particular type of sentence whose truth value can always be determined. This means it is always possible by using some method (however impractical) to determine if the sentence is true or false. Here are some examples, can you tell if they are statements or not (i.e. just sentences): 1. It is hot out today. 2. The high temperature is over 80  today. 3. My taxes are too high. 4. I paid over $5,000 in taxes in 2007. 5. Turn off the lights. 6. 5 + 7 = 75 7. x + 3 = 10 1. Not a statement 2. Statement 3. Not a statement 4. Statement 5. Not a statement (Command) 6. Statement 7. Not a statement

3. Negation of a Statement The negation of a statement is the statement whose truth value is opposite of what it was originally. If the statement was true its negation would be false and if the statement were false it negation would be true. In a particular statement in English this is most often done by either inserting or removing the word “not” after the verb. Statement 1. A mustang is made by Ford. 2. The president does not support raising taxes. 3. Laura Bush votes republican Negation 1. A mustang is not made by Ford. 2. The president does support raising taxes. 3. Laura Bush does not vote republican. The negation of a universal statement is an existential statement and the negation of an existential statement is a universal statement. Statement: All cars are made by Ford. Negation: Not all cars are made by Ford. Equivalent: Some cars are not made by Ford. Statement: Some doctors make house calls. Negation: It is not true some doctors make house calls. Equivalent: No doctors make house calls.

4. Truth Variables and Truth Tables We let a variable s stand for the truth value of a statement. This is not a variable in the way you usually think of it as standing for a number, but either true (T) or false (F). s : Dr. Daquila is the instructor for this class.( s = true (T)) s : Dr. Daquila teaches music classes. ( s = false (F)) All the different values the variables can have for a statement can be consolidated into something called a truth table. This shows if the variables start out with one truth value what they will be in the statement. The symbol ~ is used to stand for the negation of the variable (i.e. its corresponding statement) s ~s~s TF FT s ~s~s ~(~s)~(~s) TFT FTF The truth table above to the right shows what happens with a double negative. The column for s and ~(~ s ) are the same. This means the statement and it double negative will always have the same truth value. We call statements that have the same truth values logically equivalent.

5. Representing Complicated Statements Simple statement can be put together to make more complicated statements. This gets to be long and cumbersome in English so we letters stand for the truth values of certain statements. (Generally we use the letters p, q and r.) Example: p : The low today is less than 60 . q : I will wear a coat today. p Λ qp Λ q The low today is less than 60  and I will wear a coat today. p V qp V q The low today is less than 60  or I will wear a coat today. Either the low today is less than 60  or I will wear a coat today. (~ p ) Λ q The low today is not less than 60  and I will wear a coat today. p V (~ q ) The low today is less than 60  or I will not wear a coat today. p Λ (~ q ) The low today is less than 60  and I will not wear a coat today. The two most common ways to combine statements are using the “and” symbol Λ (conjunction) and the “or” symbol V (disjunction).

6. Conditional Statements One of the most important statements that is used in logic is that of a conditional statement (sometimes called an implication). It provides a link between a condition and a consequence. This is most often written using the “if-then” sentence construction, but it does take other forms. Example: If you pay your electric bill on time then the power company will keep your electricity on. This can be seen as a combination of two statements: p : You pay your electric bill on time. q : The power company will keep your electricity on. Symbolically this is represented in the following way: p → q (read “ p implies q ”) The statement p is called the hypothesis (sometimes called the premise) and the statement q is called the conclusion. The hypothesis occurs after the word “if” in the sentence no matter where it occurs. Unlike the conjunction and disjunction the statement that is the hypothesis and the statement is the conclusion determine the truth value of the conditional.

7. Universal and Conditional Statements Universal statements can be thought of (or rewritten) as conditional statements. If you remember the universal statements can be pictured using a Venn Diagram. This helps you picture what is the hypothesis and what is the conclusion in the statement. Universal Statement: All people who pay their electric bill on time the power company will keep their electricity on. Conditional Statement: If you pay your electric bill on time then the power company will keep your electricity on. people with electricity people who pay on time This can be seen as a combination of two statements: p : You pay your electric bill on time. q : The power company will keep your electricity on. Symbolically this is represented in the following way: p → q (read “ p implies q ”)

8. Universal Statement: All mustangs are Ford products. Conditional Statement: If it is a mustang then it is a Ford product. Ford Products Mustangs Universal Statement: No democrats support the war in Afghanistan. Conditional Statement: If you are a democrat then you do not support the war in Afghanistan. democratswar supporters p : It is a Mustang. q : It is a Ford product. How is this represented symbolically? p → q p : You are a democrat. q : You support the war in Afghanistan. How is this represented symbolically? p → (~ q )