4. Truth Values and Open Sentences • A statement’s Truth Value is whether it is true (T) or false (F) • So P 1 : Lansing is the Capitol of Michigan has a truth value of true (T) • While P 2 : All swimming pools are rectangles, has a truth value of false (F) • Open sentence – a sentence whose truth value depends on the value of some variable. • Example: - 3x = 12; is a open math sentence.
5. Truth Tables • Truth Tables are a way of organizing the possible truth values of a statement or series of statements F T P F T Q F F T F F T T T Q P
6. Negation – “Not statements” • Negation – Changing a statement so that it has the opposite meaning and truth values - We generally do this by inserting the word ‘NOT’ - The symbol for negation is ‘~’ and is read “Not” - So if we have a statement P: five plus two is seven; the negation of that would be ~P: five plus two is not seven • Example: P: There is snow on the ground ~P: There is not snow on the ground
16. Example of an “If-Then” (Cont.) 2) P is true, but Q is false - The student got an ‘A’ on the exam and then did not receive an ‘A’ in the class - Therefore, I was not telling the truth about the student’s final grade - What I said was false, which agrees with the 2 nd row of the truth table
17. Example of an “If-Then” (Cont.) 3) P is false and Q is true - The student did not get an ‘A’ on the exam (say they got a ‘B’) and then received an ‘A’ in the class - I did not lie when I spoke with the student initially, so I was telling the truth
18. Example of an “If-Then” (Cont.) 4) Both P and Q are false - The student did not get an ‘A’ on the exam and did not get an ‘A’ in the class - I only promised an ‘A’ in the class if the student got an ‘A’ on the exam, so again I was telling the truth, which agrees with the last row in the truth table.
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21. Truth Tables for Biconditional - We will work out the 1 st truth table in order to complete the bottom one - Note: A Biconditional is only true when the truth values of ‘P’ and ‘Q’ are the same T F F F T F F F T T T T P<=>Q Q P
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