Conditional Statements &

Material Implication

Kareem KhalifaDepartment 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 “.”

“pq” is false if antecedent p is true and consequent q is false; otherwise, true.

p q p q

T T

T F

F T

F F

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 3p q ~p ~p v q p q

T T T

T F F

F T T

F F T

F

F

T

T

T

F

T

T

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

Exercise A22 {[A (BC)] [(A&B) C]} [(YB) (CZ)] {[T (TT)] [(T&T) T]} [(FT) (TF)] {[T (T)] [(T) T]} [(T) (F)] {[T (T)] [(T) T]} [F] {[T] [T]} [F] {T} [F] F

STOP & THINK!

Exercise B11 (P X) (X P) (P F) (F P) (P F) (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

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)