SENTENTIAL LOGIC
CORRESPONDING CONDITIONALS
Ch2 Gustason
Every argument has a corresponding conditional. The corresponding conditional of any
argument is the conditional formed by:
 An antecedent consisting of the conjunction of its premises and
 A consequent consisting of its conclusion.
For example, the corresponding conditional of the argument with premises P1 through Pn,
with conclusion C is as follows (where n > 0):
(P1 & P2 & … Pn)  C
EXAMPLE:
p
~r
rq
p&q
Corresponding Conditional:
[p & (~r & (r  q))]  (p & q)
The corresponding conditional of an argument is useful as a way of representing that
argument because it is a conditional. When we evaluate the form of an argument we are
looking for a certain relationship between the premises and the conclusion. More
specifically, for an argument to be VALID, the truth of the premises must force the truth of
the conclusion, such that we can say that the premises IMPLY the conclusion (which we
can represent using the corresponding conditional).
A conditional sentence is true if and only if either
(1) its consequent is true or
(2) its antecedent is false.
The corresponding conditional of an argument will be true if and only if
(1) the conclusion of the argument is true, making the consequent true, or
(2) one of the premises is false, making the antecedent false.
The corresponding conditional of an argument might be true under certain scenarios and
false under others.
EXAMPLE:
P1
P2
Premise 3
Corresponding Conditional
p | q |
r | ~r | r  q | ~r & (r  q) | p & (~r & (r  q)) | p & ~r & (r  q)]  (p & q)
T | T |
T | F |
T
|
F
|
F
|
T
T | T |
F | T |
T
|
T
|
T
|
T
T | F |
T | F |
F
|
F
|
F
|
T
T | F |
F | T |
T
|
T
|
T
|
F
F | T |
T | F |
T
|
F
|
F
|
T
F | T |
F | T |
T
|
T
|
F
|
T
F | F |
T | F |
F
|
F
|
F
|
T
F | F |
F | T |
T
|
T
|
F
|
T
The truth table of this argument shows that its corresponding conditional is false on row 4.
This means that the corresponding conditional is not a tautology. This same row also
shows that the argument is not valid.