1. Validity: Long and short truth tables Sign In! Week 10! Homework Due Review: MP,MT,CA Validity: Long truth tables Short truth table method Evaluations! For Next Time: Read Chapter 9 pages 325-334

2. Review We ended last time by looking at three valid argument forms: Modus Ponens Modus Tollens Chain Argument We used truth tables to show that each argument type was valid

3. Modus Ponens 1. P > Q 2. P 3. :. Q Why is this argument form always valid? What about affirming the consequent? 1. P > Q 2. Q 3. :. P PQP > Q TTT TFF FTT FFT

4. Modus Tollens 1. P > Q 2. ~Q 3. :. ~P Why is Modus Tollens a valid argument form? What about denying the antecedent? 1. P > Q 2. ~P 3. :. ~Q PQP > Q TTT TFF FTT FFT

5. Chain Argument We also said that chain arguments are valid argument forms: Every chain argument has two conditional premises where the consequent of one conditional premise is the antecedent of the other 1. P > Q 2. Q > R 3. :. P > R

6. Chain Argument PQRP > QQ > RP > R TTT T T T TTF T F F TFT F T T TFF F T F FTT T T T FTF T F T FFT T T T FFF T T T

7. Invalid Conditional Arguments What's wrong with the following conditional argument? 1. P > Q 2. R > Q 3. :. P > R This is an invalid argument form, but why? Here's a hint: recall the relationship that must hold between the consequent and antecedent of conditionals in a chain argument

8. Proving Invalidity PQRP > QR > QP > R TTT T T T TTF T T F TFT F F T TFF F T F FTT T T T FTF T T T FFT T F T FFF T T T

9. Practice Given the following argument, can you derive R (by itself)? Hint: do not use a truth table, use only MP, MT, and/or CA 1. ( P v Q) > ( A > B) 2. P & A 3. ~(A > B) 4. ~(P v Q) > R

10. Practice We can prove that R follows by using Modus Tollens and Modus Ponens: 1. ( P v Q) > ( A > B) 2. P & A 3. ~(A > B) 4. ~(P v Q) > R 5. ~(P v Q) 1, 3 MT 6. R4,5 MP

11. Proving Invalidity We have been using truth tables to prove that arguments were valid and invalid (MP, MT, CA) How did we do that? We plotted out all the possible truth values for the premises and checked to see if a row existed where the premises were true and the conclusion was false If this kind of row exists then the argument is invalid If this kind of row does not exist then the argument is valid

12. Examples Is the following argument valid or invalid? Prove this using a truth table 1. A > (B & C) 2. ~B v ~C 3. :. ~A Your first step should be to construct a truth table Your second step should be to plot all of the truth values into the table Finally, check to see if there is a row where the premises are true and the conclusion is false

13. Practice ABCB & CA > (B & C)~B v ~C~A TTT T T F F TTF F F T F TFT F F T F TFF F F T F FTT T T F T FTF F T T T FFT F T T T FFF F T T T

14. Short Truth Table Method We could construct a truth table for any argument in order to determine whether the argument is valid or invalid Constructing entire truth tables can be time consuming however Thankfully there is a faster way to figure out whether an argument is valid or invalid using a truth table We could use the short truth table method

15. Short Truth Table An argument is invalid when we find a row where the premises are true and the conclusion is false When we construct a short truth table we are looking to find only the row that invalidates In order to do this, we first assume that the conclusion is false (assign it an F) and then see if it is possible to construct a row where the premises are still true If we can do this then the argument is invalid

16. Example Let's construct a short truth table for the following argument: 1. A > B 2. ~B > C 3. :. ~A > C The first thing to do is to make (~A > C) false When is the only time that conditionals are false?

17. Example ABCA > B~B > C~ A > C FF F Conditionals are only false when the antecedent is true and the consequent false This means that ~A must be true and C must be false What about the second premise (~B > C)? If C is false what must ~B be in order for the entire conditional to come out true?

18. Example ABCA > B~B > C~ A > C FTF T T F If we must make B true then how does this affect our first premise: A > B? If B is true and A is false then the first premise is true We therefore have created a row on the truth table where the premises are true but the conclusion is false This argument is invalid

19. Practice Construct a short truth table to prove whether the following argument is valid or invalid: 1. A & (B v C) 2. C > D 3. A > E 4. :. D & E This is is tricky because the conclusion is a conjunction, there are three possible ways it can be false Try to make the premises true first, some of the truth values are 'forced' on us and that makes things easier

20. Practice ABCDE TTFFT 1. A & (B v C) 2. C > D 3. A > E 4. :. D & E If we know that E must be true then we know that D must be false if the conclusion is false If D must be false then C must be false and if C must be false then B must be true This argument is INVALID A must be true because the first premise is a conditional and in order for a conditional to be true both conjuncts must be true If A must be true then we know that E must be true as well in order for premise 3 to be true

21. For Next Time For Next Time: Read Chapter 9 pages 330-334 Bring your books Wednesday!