MGF 1106 Section 3.7 Arguments and Truth Tables p q p → q (p ...

MGF 1106Section 3.7 Arguments and Truth TablesObjective 1The learner will use truth tables to determine validity.Argument: Consists of two parts : the premise(s) and the conclusionExample 1If we close the door, there is less noise.We close the door.premisesTherefore, there is less noise.conclusionValid: the conclusion is true whenever the premises are assumed to be true.Invalid: not valid, also known as a fallacyTruth tables can be used to test validity.An argument is valid if and only if all final truth values are true.The argument above seems obvious (and it is) but let’s write it in symbolic form and use a truth table todetermine the validity of the argument.p: We close the door.q: There is less noise.Premise 1: If we close the door, then there is less noise.Premise 2: We close the door.Conclusion: There is less noise.p → qp-------∴ qTruth Table:p q p → q (p → q)∧ p [(p → q)∧ p] → qT TT FF TF FThis particular type of argument is called direct reasoning and we just proved it is always valid.

Example 2: Use a truth table to determine whether the symbolic form of the argument is valid or invalid.a) p → q~p-------∴ ~qp q p → q ~p (p → q)∧ ~p ~q [(p → q)∧ ~p] → ~qT TTFFFTFb) p → qq--------∴ pp q p → q (p → q)∧ q [(p → q)∧ q] → pT TTFFFTFc) p → q~q----------∴ ~pp q p → q ~q (p → q)⋀~q ~p [(p → q)⋀~q]→~pT TTFFFTF

d) p → qq→ r-------∴ p → rp q r p → q q → r (p → q)⋀ (q → r) p → r [(p → q)⋀ (q → r)]→ (p →r)T T TTTTT FF TF FF T TF T FF F TF F FExample 3If I tell you I cheated, I’m miserable. If I don’t tell you I cheated, I’m miserable.∴ I’m miserable.p: I tell you I cheatedq: I am miserableSymbolic Form:Truth Tablep q p → q ~p ~p → q (p → q) ⋀(~p → q) [(p → q) ⋀(~p → q)]→qT TTFFFTF

Objective 2The learner will recognize and use forms of valid and invalid arguments.ValidInvalidDirectp → qp------∴ qFallacy of the Conversep → qq------∴ pContrapositivep → q~q------∴ ~pFallacy of the Inversep → q~p------∴ ~qDisjunctivep ∨ q~p-------∴ q(also valid for ~q, p)Misuse of Disjunctivep ∨ qp-------∴ ~q(also invalid for q, ~p)Transitivep → qq→ r-------∴ p → r(also valid for ~r → ~p)Misuse of Transitivep → qq→ r-------∴ r →p(also invalid for ~p → ~r)

Example 4: Determine if each argument is valid or invalidArgumentIf we’re out of gas, then the car won’t start.If the car won’t start, then we’ll miss the party.∴ If we’re out of gas, then we’ll miss the party.If we’re out of gas, then the car won’t start.If the car won’t start, then we’ll miss the party.∴ If we miss the party, then we’re out of gas.If I am happy, I paint with purple.I am not happy.∴ I did not paint with purple.If I am happy, I paint with purple.I did not paint with purple.∴ I am not happy.We study longer, or we don’t understand the material.We do not study longer.∴ We don’t understand the material.We study longer, or we don’t understand the material.We study longer.∴ We understand the material.If all people obey the law, then no jails are needed.Some jails are needed.∴ Some people do not obey the law.If all people obey the law, then no jails are needed.Some people do not obey the law.∴ Some jails are needed.SymbolicFormValidor Invalid?ReasonExample 5: Form a valid conclusion based on the given premises.a) If a painter mixes red and blue, then he’ll make purple.A painter doesn’t make purple.Therefore …b) If a person breaks the law, then the person will be arrested.If a person is arrested, then the person will go to jail.A person broke the law.Therefore …

Example 6Write each argument symbolically. Then write a valid conclusion symbolically and in words.p: He is a painter. m: He mixes yellow and blue.g: He makes green. R: He is Rembrandt.1. He is Rembrandt or he does not mix yellow and blue.He is not Rembrandt.∴2. If he makes green, then he is not Rembrandt.He is Rembrandt.∴3. If he is Rembrandt, then he does not make green.If mixes yellow and blue, then he does make green.∴4. If a painter mixes yellow and blue, then he’ll make green.If he doesn’t make green, then he is not a painter.If he is Rembrandt, then he is a painter.∴