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# Translating English to Propositional Logic

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### Translating English to Propositional Logic

1. 1. Translating English to Propositional Logic Phil 57 section 3 San Jose State University Fall 2010
2. 2. Does order always matter? <ul><li>John went to school and Mary went to school. </li></ul><ul><li>Mary went to school and John went to school. </li></ul><ul><li>P = John went to school. </li></ul><ul><li>Q = Mary went to school. </li></ul><ul><li>(P  Q) means the same as (Q  P), logically speaking. </li></ul>
3. 3. Does order always matter? <ul><li>John went to school or Mary went to school. </li></ul><ul><li>Mary went to school or John went to school. </li></ul><ul><li>P = John went to school. </li></ul><ul><li>Q = Mary went to school. </li></ul><ul><li>(P  Q) means the same as (Q  P), logically speaking. </li></ul>
4. 4. Conjunction and disjunction are commutative. <ul><li>(P  Q) means the same as (Q  P) </li></ul>P Q (P  Q) (Q  P) T T T T T F F F F T F F F F F F
5. 5. Conjunction and disjunction are commutative. <ul><li>(P  Q) means the same as (Q  P) </li></ul>P Q (P  Q) (Q  P) T T T T T F T T F T T T F F F F
6. 6. Conjunction and disjunction are also associative. <ul><li>((P  Q)  R) means the same as (P  (Q  R)). </li></ul><ul><li>((P  Q)  R) means the same as (P  (Q  R)). </li></ul>
7. 7. With mixed operators, order does matter. <ul><li>P = I like peanut butter. </li></ul><ul><li>Q = I like jelly. </li></ul><ul><li>~(P  Q) means “it is not the case that I like peanut butter and jelly”. </li></ul><ul><li>(~P  ~Q) means “I don’t like peanut butter and I don’t like jelly.” </li></ul>
8. 8. With mixed operators, order does matter. <ul><li>P = Tom will work late. </li></ul><ul><li>Q = Dick will work late. </li></ul><ul><li>R = Harry will call in sick. </li></ul><ul><li>~((P  Q)  R) means “it is not the case that Tom and Dick will work late or that Harry will call in sick.” </li></ul>
9. 9. Translating material conditionals. <ul><li>If [antecedent], then [consequent]. </li></ul><ul><li>P= antecedent </li></ul><ul><li>Q= consequent </li></ul><ul><li>P  Q </li></ul>
10. 10. Material conditionals and necessary conditions. <ul><li>Getting an A on the final exam is a necessary condition for getting an A in the class. </li></ul><ul><li>Necessary because if I don’t meet this condition, I don’t bring about the outcome. </li></ul><ul><li>P= I get an A on the final exam. </li></ul><ul><li>Q= I get an A in the class. </li></ul><ul><li>Q  P </li></ul>
11. 11. Material conditionals and necessary conditions. <ul><li>Getting an A on the final exam is a necessary condition for getting an A in the class. </li></ul><ul><li>P= I get an A on the final exam. </li></ul><ul><li>Q= I get an A in the class. </li></ul>P Q Q  P T T T T F T F T F F F T
12. 12. Material conditionals and sufficient conditions. <ul><li>Getting a B on all the exams is a sufficient condition for getting a B in the class. </li></ul><ul><li>Sufficient because it’s enough to bring the outcome, but it’s not the only way to bring it. </li></ul><ul><li>P= I get a B on all the exams. </li></ul><ul><li>Q= I get a B in the class. </li></ul><ul><li>P  Q </li></ul>
13. 13. Material conditionals and sufficient conditions. <ul><li>Getting a B on all the exams is a sufficient condition for getting a B in the class. </li></ul><ul><li>P= I get a B on all the exams. </li></ul><ul><li>Q= I get a B in the class. </li></ul>P Q P  Q T T T T F F F T T F F T
14. 14. Translating material conditionals. <ul><li>If … , then ... </li></ul><ul><li>It taxes go up, then inflation will rise. </li></ul><ul><li>T= Taxes go up </li></ul><ul><li>R= Inflation will rise </li></ul><ul><li>T  R </li></ul>
15. 15. Translating material conditionals. <ul><li>… only if... </li></ul><ul><li>Iran will supply arms to Syria only if Syria helps Hezbollah. </li></ul><ul><li>R= Iran will supply arms to Syria </li></ul><ul><li>S= Syria helps Hezbollah </li></ul><ul><li>R  S </li></ul>
16. 16. Translating material conditionals. <ul><li>Only if ... will … </li></ul><ul><li>Only if Jenna passes the exam will Jenna get her license. </li></ul><ul><li>P= Jenna passes the exam </li></ul><ul><li>Q= Jenna will get her license </li></ul><ul><li>Q  P </li></ul>
17. 17. Translating material conditionals. <ul><li>… if ... </li></ul><ul><li>I will pass the muffins if you ask me nicely. </li></ul><ul><li>M= I will pass the muffins </li></ul><ul><li>N= You ask me nicely. </li></ul><ul><li>N  M </li></ul>
18. 18. Translating material conditionals. Construction Translation If P, then Q (P  Q ) P, if Q ( Q  P ) P only if Q (P  Q ) Only if P, Q ( Q  P )
19. 19. Translating biconditionals. <ul><li>… if and only if … </li></ul><ul><li>Jill needs a parachute if and only if she is planning to jump from the plane. </li></ul><ul><li>P= Jill needs a parachute. </li></ul><ul><li>Q= Jill is planning to jump from the plane. </li></ul><ul><li>P  Q </li></ul>
20. 20. Translating biconditionals. <ul><li>… just in case … </li></ul><ul><li>Bill will take the geology course just in case it fulfils the science requirement. </li></ul><ul><li>T= Bill will take the geology course </li></ul><ul><li>R= The geology course fulfils the science requirement. </li></ul><ul><li>T  R </li></ul>
21. 21. A biconditional is the conjunction of two material conditionals. <ul><li>Jill needs a parachute if and only if she is planning to jump from the plane. </li></ul><ul><li>If Jill needs a parachute, then she is planning to jump from the plane, </li></ul><ul><li>AND </li></ul><ul><li>If Jill is planning to jump from the plane, then she needs a parachute. </li></ul>
22. 22. Biconditionals are commutative. <ul><li>P  Q is the same as Q  P </li></ul><ul><li>“Bill will take the geology course just in case it fulfils the science requirement” is equivalent to </li></ul><ul><li>“The geology course fulfils the science requirement just in case Bill will take it.” </li></ul><ul><li>Conjunction of two conditionals (and conjunction is commutative) </li></ul>
23. 23. Biconditionals are associative. <ul><li>(P  (Q  R)) is the same as ((P  Q)  R) </li></ul><ul><li>Conjunction of two conditionals (and conjunction is associative) </li></ul><ul><li>But note that material conditionals are neither commutative nor associative. </li></ul><ul><li>(P  Q) ≠ (Q  P) </li></ul><ul><li>(P  (Q  R)) ≠ ((P  Q)  R) </li></ul>
24. 24. Order matters translating conditionals. <ul><li>P = Ben will answer the phones. </li></ul><ul><li>Q = Liz will work out the budget. </li></ul><ul><li>(P  Q): If Ben will answer the phones, then Liz will work out the budget. </li></ul><ul><li>(Q  P): If Liz will work out the budget, then Ben will answer the phones. </li></ul>
25. 25. Order matters translating conditionals. <ul><li>P = Ben will answer the phones. </li></ul><ul><li>Q = Liz will work out the budget. </li></ul>P Q (P  Q) (Q  P) T T T T T F F T F T T F F F T T
26. 26. Translating English to PL. <ul><li>“ Propositional Logic translation guide” on course website </li></ul><ul><li>(http://www.stemwedel.org/logic-and-critical-reasoning/PL-TranslationGuide.pdf) </li></ul><ul><li>Practice (like on HW #7) will help! </li></ul>
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