Conditional Statements & Material Implication

1. Conditional Statements & Material Implication Kareem Khalifa Department of Philosophy Middlebury College

2. Overview • Why this matters • Anatomy of a conditional statement • Some nuances in translating conditionals • Truth-conditions for  • Weirdness with  • Possible solutions: Your first foray into philosophy of logic! • Sample Exercises

3. Why this matters • Conditional statements are the most fundamental logical connectives, so understanding their truth-conditions is necessary for analyzing and criticizing many arguments. • A “cheap trick” for making any argument valid. • Sally is under 18. • If Sally is under 18, then she’s not allowed on the premises. • So she’s not allowed on the premises.

4. Anatomy of conditionals • If you study hard, then you will pass PHIL0180. • If p, then q. CONSEQUENT ANTECEDENT

5. Some nuances in translating conditionals • “If p then q”can also be expressed in the following ways: • If p, q • q, if p • p only if q • p is sufficient for q • q is necessary for p • p requires q • p entails q • p implies q • p renders (yields, produces, etc.) q • In case of p, q • Provided that p, q • Given that p, q • On the condition that p, q

6. Examples • If Khalifa is human, then Khalifa is a mammal. • Khalifa’s being human suffices for his being a mammal. • Khalifa’s being a mammal is necessary for his being human. • Khalifa’s humanity requires that he be a mammal. • Khalifa’s humanity entails that he is a mammal.

7. More examples • Washing your hands decreases the chance of infection. • If you wash you your hands, then the chance of infection decreases. • Paying off the professor will produce the desired effect. • If the professor is paid off, then the desired effect will be produced. $ $

8. Truth conditions for conditionals • Recall: A logical connective is a piece of logical syntax that: • Operates upon propositions; and • Forms a larger (compound) proposition out of the propositions it operates upon, such that the truth of the compound proposition is a function of the truth of its component propositions. • Today, we’re looking at “IF…THEN...” • The truth of the whole “if-then” statement is a function of the truth/falsity of the antecedent and consequent.

9. Truth-conditions for  • In logic, we represent “if p then q” as “p  q.” This is called material implication. • Alternatively, “” may be represented as “.” • “pq” is false if antecedent p is true and consequent q is false; otherwise, true. T F T T

10. Intuitive examples of  • True antecedent, true consequent • If Khalifa is human, then Khalifa is a mammal. • False antecedent, true consequent • If Khalifa is a dog, then Khalifa is a mammal. • False antecedent, false consequent • If Khalifa is a dog, then Khalifa is a canine.

11. Weirdness with  • True antecedent, true consequent • If 2+2=4, then Middlebury is in VT. • False antecedent, true consequent • If the moon is made of green cheese, then 2+2 =4. • False antecedent, false consequent • If Khalifa is a dog, then the moon is made of green cheese.

12. More weirdness: the paradoxes of material implication • The following are both valid arguments • B, so A  B • Ex. 2+2=4, so if unicorns exist, then 2+2=4. • ~A, so A  B • Ex. The moon is not made of green cheese, so if the moon is made of green cheese, then Khalifa is a lizard.

13. Different responses to the weirdness • Response 1: Logic must be revised! • The English “If p then q” is just elliptical for “Necessarily, if p then q.” 2+2 = 4 doesn’t necessitate anything about Middlebury, nor does the moon’s green cheesiness necessitate anything about arithmetic, etc. • Ex. Although it is actually the case that 2+2 = 4 and Midd is in VT, it is possible that 2+2=4 and Midd is not in VT. • Thus it is notnecessary that this conditional be true.

14. Response 2 (Copi & Cohen’s) • “If … then…” statements in English express several different relationships: • Logical: If either Pat or Sam is dating Chris and Sam is not dating Chris, then Pat is dating Chris. • Definitional: If a critter is warm-blooded, then that critter has a relatively high and constant internally regulated body temperature relatively independent of its surroundings. • Causal: If I strike this match, then it will ignite. • Decisional: If the median raw score on the exam is 60, then I should institute a curve. • Each of these if-then statements is false when the antecedent is true and the consequent is false. • This is exactly what material conditionals state, and thus they capture the “core” of all conditional statements. The rest is an issue of context.

15. Response 3 • Suppose that the English “If p then q” is true. • Either ~p is true or p is true. • In the first case, ~p v q is true. • In the second case, q is true by modus ponens. • Thus, in either case ~p v q is true. • Since ~p v q is equivalent to p  q, the latter is a fair interpretation of “If p then q.”

16. More on Response 3 F T F F T T T T

17. Exercise A6 • (X  Y)  Z • (F  F)  F • (T)  F • F

18. Exercise A22 • {[A (BC)] [(A&B) C]} [(YB) (CZ)] • {[T (TT)] [(T&T) T]} [(FT) (TF)] • {[T (T)] [(T) T]} [(T) (F)] • {[T (T)] [(T) T]} [F] • {[T] [T]} [F] • {T} [F] • F STOP & THINK!

19. Exercise B11 • (P  X)  (X  P) • (P  F)  (F  P) • (P  F)  (T) • T

20. Exercise B24 • [P  (A v X)]  [(P  A)  X] • [P  (T v F)]  [(P  T)  F] • [P  (T)]  [(T)  F] • [T]  [F] • F

21. Exercise C22 • Argentina’s mobilizing is a necessary condition for Chile to call for a meeting of all the Latin American states. • C  A

22. Exercise C25 • If neither Chile nor the DR calls for a meeting of all the Latin American states, then Brazil will not protest to the UN unless Argentina mobilizes. • (~C & ~D)  (~B v A)