If
Pigs Could Fly
Material Implication
(Last updated 20230509)
You can truthfully say, “If pigs could fly, then I am the
King France.” You cannot, however, infer from this true statement that you are
the King France because, well… pigs can’t really fly!
This is an application of material
implication, a logical operator characterized by the following truth table
for any logical propositions A and B:
Truth Table for IMPLIES

A 
B 
A => B 
1 
T 
T 
T 
2 
T 
F 
F 
3 
F 
T 
T 
4 
F 
F 
T 
Let
us now consider a somewhat more “realworld” application: If it is raining (R), then it is cloudy (C). In symbolic logic, can write:
R
=> C
What does this
actually mean? It does not mean that
rain causes cloudiness. It also does
not mean that it is always cloudy
when it is raining, as during a socalled “sun shower.” It simply rules out the
possibility that it is currently raining AND not cloudy. This is entirely
consistent with the usual truth table:

R 
C 
R => C 
1 
T 
T 
T 
2 
T 
F 
F 
3 
F 
T 
T 
4 
F 
F 
T 
Note the
following:
·
When R is true and R => C is true (line
1), C will also be true. (The Rule of
Detachment)
·
When R is false (lines 34), R => C will be true regardless of the
truth value of C. (The
Principle of Vacuous Truth) This form of argument is rarely if ever used in
daily discourse since we are not usually interested in the implications of propositions
known to be false. It is, however, routinely used in very technical arguments,
e.g. mathematical proofs.
A Formal Development of
the Truth Table
We
can derive each line of the truth table using the following, more fundamental
rules of logic:
Conclusion
Rules
Direct proof
(Introducing =>): In
a mathematical proof, if we assume A
is true, and, without introducing any intervening assumptions, we can
subsequently prove that B is also
true, then we can infer that A => B
is true. We can still infer that A =>
B if all intervening assumptions had previously been discharged and deactivated.
(Note that in DC Proof, an assumption is called a premise.)
Proof By Contradiction
(Introducing ~): In
a mathematical proof, if we assume A
is true, and, without introducing any intervening assumptions, we can
subsequently prove that both B is
true and B is false, thus obtaining a
contradiction B & ~B, then we can
infer that A is false (i.e ~A is true)
As above, we can still infer that A
is false if all intervening assumptions had previously been discharged and deactivated.
Conjunction Rules
Join (Introducing &):
If both A and B are true, then A & B
is true
Split
(Eliminating &): If A
& B is true, then both A and B are true.
Detachment Rule
(Eliminating =>)
If both A => B and A are true,
then we can infer that B is also
true.
Double
Negation Rule (Eliminating ~~)
If
~~A is true, then so is A.
Following
are formal proofs (in the DC
Proof format) deriving each line of the above truth table for A => B:
A & B => [A => B] (Truth table, line 1) Formal Proof (6 lines)
A & ~B
=> ~[A => B] (Truth table,
line 2) Formal Proof (8 lines)
~A => [A => B] (Truth table, lines 34) Formal Proof (8 lines)
Other useful results:
A => [~A => B] (The Principle of Vacuous Truth) Formal Proof (8 lines)
[A => B]
<=> ~[A & ~B] (Often
given as the definition of IMPLIES) Formal
Proof (19 lines)