The document discusses translating statements from English to propositional logic, including: - Conjunction and disjunction are commutative but order matters for statements with mixed operators - How to translate conditional statements like "if P then Q" and biconditionals like "P if and only if Q" - Necessary and sufficient conditions and how they relate to conditionals - Examples of translating various English language statements into propositional logic statements

1. Translating English to Propositional Logic Phil 57 section 3 San Jose State University Fall 2010

2. Does order always matter? John went to school and Mary went to school. Mary went to school and John went to school. P = John went to school. Q = Mary went to school. (P  Q) means the same as (Q  P), logically speaking.

3. Does order always matter? John went to school or Mary went to school. Mary went to school or John went to school. P = John went to school. Q = Mary went to school. (P  Q) means the same as (Q  P), logically speaking.

4. Conjunction and disjunction are commutative. (P  Q) means the same as (Q  P) P Q (P  Q) (Q  P) T T T T T F F F F T F F F F F F

5. Conjunction and disjunction are commutative. (P  Q) means the same as (Q  P) P Q (P  Q) (Q  P) T T T T T F T T F T T T F F F F

6. Conjunction and disjunction are also associative. ((P  Q)  R) means the same as (P  (Q  R)). ((P  Q)  R) means the same as (P  (Q  R)).

7. With mixed operators, order does matter. P = I like peanut butter. Q = I like jelly. ~(P  Q) means “it is not the case that I like peanut butter and jelly”. (~P  ~Q) means “I don’t like peanut butter and I don’t like jelly.”

8. With mixed operators, order does matter. P = Tom will work late. Q = Dick will work late. R = Harry will call in sick. ~((P  Q)  R) means “it is not the case that Tom and Dick will work late or that Harry will call in sick.”

9. Translating material conditionals. If [antecedent], then [consequent]. P= antecedent Q= consequent P  Q

10. Material conditionals and necessary conditions. Getting an A on the final exam is a necessary condition for getting an A in the class. Necessary because if I don’t meet this condition, I don’t bring about the outcome. P= I get an A on the final exam. Q= I get an A in the class. Q  P

11. Material conditionals and necessary conditions. Getting an A on the final exam is a necessary condition for getting an A in the class. P= I get an A on the final exam. Q= I get an A in the class. P Q Q  P T T T T F T F T F F F T

12. Material conditionals and sufficient conditions. Getting a B on all the exams is a sufficient condition for getting a B in the class. Sufficient because it’s enough to bring the outcome, but it’s not the only way to bring it. P= I get a B on all the exams. Q= I get a B in the class. P  Q

13. Material conditionals and sufficient conditions. Getting a B on all the exams is a sufficient condition for getting a B in the class. P= I get a B on all the exams. Q= I get a B in the class. P Q P  Q T T T T F F F T T F F T

14. Translating material conditionals. If … , then ... It taxes go up, then inflation will rise. T= Taxes go up R= Inflation will rise T  R

15. Translating material conditionals. … only if... Iran will supply arms to Syria only if Syria helps Hezbollah. R= Iran will supply arms to Syria S= Syria helps Hezbollah R  S

16. Translating material conditionals. Only if ... will … Only if Jenna passes the exam will Jenna get her license. P= Jenna passes the exam Q= Jenna will get her license Q  P

17. Translating material conditionals. … if ... I will pass the muffins if you ask me nicely. M= I will pass the muffins N= You ask me nicely. N  M

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. Translating biconditionals. … if and only if … Jill needs a parachute if and only if she is planning to jump from the plane. P= Jill needs a parachute. Q= Jill is planning to jump from the plane. P  Q

20. Translating biconditionals. … just in case … Bill will take the geology course just in case it fulfils the science requirement. T= Bill will take the geology course R= The geology course fulfils the science requirement. T  R

21. A biconditional is the conjunction of two material conditionals. Jill needs a parachute if and only if she is planning to jump from the plane. If Jill needs a parachute, then she is planning to jump from the plane, AND If Jill is planning to jump from the plane, then she needs a parachute.

22. Biconditionals are commutative. P  Q is the same as Q  P “Bill will take the geology course just in case it fulfils the science requirement” is equivalent to “The geology course fulfils the science requirement just in case Bill will take it.” Conjunction of two conditionals (and conjunction is commutative)

23. Biconditionals are associative. (P  (Q  R)) is the same as ((P  Q)  R) Conjunction of two conditionals (and conjunction is associative) But note that material conditionals are neither commutative nor associative. (P  Q) ≠ (Q  P) (P  (Q  R)) ≠ ((P  Q)  R)

24. Order matters translating conditionals. P = Ben will answer the phones. Q = Liz will work out the budget. (P  Q): If Ben will answer the phones, then Liz will work out the budget. (Q  P): If Liz will work out the budget, then Ben will answer the phones.

25. Order matters translating conditionals. P = Ben will answer the phones. Q = Liz will work out the budget. P Q (P  Q) (Q  P) T T T T T F F T F T T F F F T T

26. Translating English to PL. “ Propositional Logic translation guide” on course website (http://www.stemwedel.org/logic-and-critical-reasoning/PL-TranslationGuide.pdf) Practice (like on HW #7) will help!