6.1 Logic 6.1 Logic

Logic is not only the foundation of mathematics, but Logic is not only the foundation of mathematics, but also is important in numerous fields including law, also is important in numerous fields including law, medicine, and science. Although the study of medicine, and science. Although the study of logic originated in antiquity, it was rebuilt and logic originated in antiquity, it was rebuilt and formalized in the 19formalized in the 19thth and early 20 and early 20thth century. century. George Boole (Boolean algebra) introduced George Boole (Boolean algebra) introduced mathematical methods to logic in 1847 while mathematical methods to logic in 1847 while Georg Cantor did theoretical work on sets and Georg Cantor did theoretical work on sets and discovered that there are many different sizes of discovered that there are many different sizes of infinite sets. infinite sets.

Statements or PropositionsStatements or Propositions

A proposition or statement is a declaration which is either true or false. A proposition or statement is a declaration which is either true or false. Some examples: Some examples:

2+2 = 5 2+2 = 5 is a statement because it is a false is a statement because it is a false declaration. declaration.

Orange juice contains vitamin C Orange juice contains vitamin C is a statement that is a statement that is true. is true.

Open the door. Open the door. This is not considered a statement This is not considered a statement since we cannot assign a true or false value to this since we cannot assign a true or false value to this sentence. It is a command, but not a statement or sentence. It is a command, but not a statement or proposition. proposition.

Negation Negation

The negation of a statement, p , is The negation of a statement, p , is “not p” and is denoted by ┐ p“not p” and is denoted by ┐ p

Truth table: Truth table: p ┐ p p ┐ p TT FF FF TT If p is true, then its negation is false. If p is false, then its negation If p is true, then its negation is false. If p is false, then its negation

is true.is true.

DisjunctionDisjunction

A disjunction is of the form p V q and is read A disjunction is of the form p V q and is read p or q. p or q. Truth table for disjunction: Truth table for disjunction: pp qq p V q p V q TT TT TT TT FF TT FF TT TT FF FF FF

A disjunction is true in all cases except when A disjunction is true in all cases except when both p and q both p and q are false.are false.

Conjunction Conjunction

A conjunction is A conjunction is only trueonly true when both p and q are true. when both p and q are true. Otherwise, a conjunction of two statements will be false: Otherwise, a conjunction of two statements will be false:

Truth table: Truth table:

pp qq p qp q

TT TT T T

TT FF F F

FF TT F F

FF FF F F

Conditional statementConditional statement

To understand the logic behind the truth table for the conditional statement, To understand the logic behind the truth table for the conditional statement, consider the following statement. consider the following statement.

““If you get an A in the class, I will give you five bucks.” If you get an A in the class, I will give you five bucks.” Let p = statement “ Let p = statement “ You get an A in the classYou get an A in the class” ” Let q = statement “ Let q = statement “ I will give you five bucksI will give you five bucks.” .” Now, if p is true (you got an A) and I give you the five bucks, the truth value of Now, if p is true (you got an A) and I give you the five bucks, the truth value of p qp q is true. The contract was satisfied and both parties fulfilled the is true. The contract was satisfied and both parties fulfilled the

agreement. agreement. Now, suppose p is true (you got the A) and q is false (you did not get the five Now, suppose p is true (you got the A) and q is false (you did not get the five

bucks). You fulfilled your part of the bargain, but weren’t rewarded with the five bucks). You fulfilled your part of the bargain, but weren’t rewarded with the five bucks.bucks.

So So p qp q is false since the contract was broken by the other party. is false since the contract was broken by the other party. Now, suppose p is false. You did not get an A but received five bucks anyway. (q is Now, suppose p is false. You did not get an A but received five bucks anyway. (q is

true) No contract was broken. There was no obligation to receive 5 bucks, so truth true) No contract was broken. There was no obligation to receive 5 bucks, so truth value of value of p qp q cannot be false, so it must be true. cannot be false, so it must be true.

Finally, if both p and q are false, the contract was not broken. You did not receive Finally, if both p and q are false, the contract was not broken. You did not receive the A and you did not receive the 5 bucks. So the A and you did not receive the 5 bucks. So p qp q is true in this case. is true in this case.

Truth table for conditionalTruth table for conditional

pp qq p qp qTT TT TT

TT FF FF

FF TT TT

FF FF TT

Variations of the conditionalVariations of the conditional

Converse: Converse: The converse of p q is The converse of p q is q pq p

Contrapositive: Contrapositive: The contrapositive of p q The contrapositive of p q is is

┐┐q ┐pq ┐p

ExamplesExamples

Let p = you receive 90% Let p = you receive 90% Let q = you receive an A in the course Let q = you receive an A in the course p q ? p q ? If If you receive 90%,you receive 90%, then then you will receive an A in the you will receive an A in the

coursecourse.. Converse: q p Converse: q p

If you If you receive an A in the coursereceive an A in the course, then , then you receive 90%you receive 90% Is the statement true? No. What about the student who Is the statement true? No. What about the student who

receives a score greater than 90? That student receives an receives a score greater than 90? That student receives an A but did not achieve a score of A but did not achieve a score of exactly 90%. exactly 90%.

Example 2Example 2

State the contrapositive in an English sentence: State the contrapositive in an English sentence: Let p = you receive 90% Let p = you receive 90% Let q = you receive an A in the course Let q = you receive an A in the course p q ? p q ? If If you receive 90%,you receive 90%, then then you will receive an A in the courseyou will receive an A in the course

┐┐q ┐pq ┐p If you If you don’t receive an A in the coursedon’t receive an A in the course, then you , then you didn’t didn’t

receive 90%.receive 90%. The contrapositive is true not only for these particular The contrapositive is true not only for these particular

statements but for all statements , p and q. statements but for all statements , p and q.

Logical equivalent Logical equivalent statements statements

Show that is logically equivalent to Show that is logically equivalent to

We will construct the truth tables for both sides and We will construct the truth tables for both sides and determine that the truth values for each statement are determine that the truth values for each statement are identical. identical.

The next slide shows that both statements are logically The next slide shows that both statements are logically equivalent. The red columns are identical indicating the equivalent. The red columns are identical indicating the final truth values of each statement. final truth values of each statement.

p q p q