Logic ChAPTER 3 1

Truth Tables and Validity of Arguments 3.6 2

Problem Solving and Arguments Determining Validity 1.Write each premise on a separate line. 2.Write the conclusion after the premises and separate it by a horizontal line. 3.Make a truth table using a column for each premise and a column for the conclusion. 4.Check only the rows in which all the premises are true. For the argument to be valid, the conclusion must be also valid. 3

Problem Solving and Arguments 4 Symbolize each argument using the suggested abbreviations. In each case, determine the validity of the given argument.

Problem Solving and Arguments 5 If you study logic(s), mathematics is easy(e). Mathematics is not easy.. Therefore, you did not study logic. Determine whether the argument is valid.

Problem Solving and Arguments 6 ses → e~ e~ s TTTFF TFFTF FTTFT FFTTT VALID 1 2 c 1 2 c

Problem Solving and Arguments 7 You will be eligible for a grant(e) if you meet all the criteria(m). You do not meet all the criteria.. ∴ So you are not eligible for a grant. Determine whether the argument is valid.

Problem Solving and Arguments 8 mem → e~ m~ e TTTFF TFFFT FTTTF FFTTT INVALID 1 2c 1 2 c

Problem Solving and Arguments 9 pqp → qpq TT TF FT FF VALID Determine whether the argument is valid. T F T T T T F F T F T F 2 1 c 1 2 c

Problem Solving and Arguments 10 Determine whether each argument is valid.

11 SOLUTION: pqrp → qq → r ~ r~ p ~ r → ~ p TTT TTF TFT TFF FTT FTF FFT FFF T T F F T T T T T F T T T F T T V A L I D F T F T F T F T F F F F T T T T T F T F T T T T c 1 2

SOLUTION: pr~r~r~ p~ p~ r → ~ p~ r → ~ p T T F F T F T F F T F T F F T T T F T T prp → r T T F F T F T F T F T T Equivalent 12

pqrp → qq → rp → r TTT TTF TFT TFF FTT FTF FFT FFF T F T F T T T T Problem Solving and Arguments 13 VALID T T F F T T T T T F T T T F T T 12c 1 2 c

Problem Solving and Arguments Find valid conclusions using all the premises. 14 c

pqrsp → qq → r~ s~ r~ s → ~ rp TTTT TTTF TTFT TTFF TFTT TFTF TFFT TFFF FTTT FTTF FTFT FTFF FFTT FFTF FFFT FFFF T T T T F F F F T T T T T T T T T T F F T T T T T T F F T T T T F T F T F T F T F T F T F T F T F F T T F F T T F F T T F F T T T F T T T F T T T F T T T F T T T T T T T T T T F F F F F F F F s T F T F T F T F T F T F T F T F valid c 4

Valid Argument Forms Modus Ponens Modus Tollens Hypothetical Disjunctive Syllogism 16

Problem Solving and Arguments 17 Select the conclusion that will make each entire argument valid. If I drive to work, then I will not be late. If I am not late, then I do not lose any pay. a. If I am late, then I drive to work. b. If I do not lose any pay, then I drive to work. c. If I drive to work, then I do not lose any pay. d. If I do not drive to work, then I lose some pay.

SOLUTION: c. If I drive to work, then I will not lose any pay. p → r valid If I drive to work, then I will not be late. p → q If I am not late, then I do not lose any pay.. q → r a. If I am late, then I drive to work. ~q ~q → p invalid Hypothetical Syllogism b. If I do not lose any pay, then I drive to work. r → p invalid If I drive to work, then I will not lose any pay. p → r d. If I do not drive to work, then I lose some pay. ~p ~p →~r invalid 18 END