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

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

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.

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

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

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.

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. $$

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.

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. pq p  q TT TF FT FF T F T T

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.

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.

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.

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 not necessary that this conditional be true.

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.

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.”

More on Response 3 pq~p~p v q p  q TTT TFF FTT FFT F F T T T F T T

Exercise A6  (X  Y)  Z  (F  F)  F  (T)  F FF

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F STOP & THINK!

Exercise B11  (P  X)  (X  P)  (P  F)  (F  P)  (P  F)  (T) TT

Exercise B24  [P  (A v X)]  [(P  A)  X]  [P  (T v F)]  [(P  T)  F]  [P  (T)]  [(T)  F]  [T]  [F] FF

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

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)