Download Conditional Statements & Material Implication

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 truthconditions 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
So she’s not allowed on the premises.
premises.

Anatomy of conditionals

If you study hard, then you will pass PHIL0180.
ANTECEDENT
 If
p,
then
CONSEQUENT
q.
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.
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
Intuitive examples of 

True antecedent, true consequent


False antecedent, true consequent


If Khalifa is human, then Khalifa is a mammal.
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


False antecedent, true consequent


If 2+2=4, then Middlebury is in VT.
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
p
q
~p
~p v q
pq
T
T
F
T
T
T
F
F
F
F
F
T
T
T
T
F
F
T
T
T
Exercise A6
(X  Y)  Z
 (F  F)  F
 (T)  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]
STOP & THINK!
{[T] [T]} [F]
{T} [F]
F
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)
