If
Pigs Could Fly
Photo credit: Lost at E Minor
Material Implication
(Last updated 20180326 4:45 PM)
As a general rule, all things follow from a falsehood. You can
truthfully say, “If pigs could fly, then I am King France.” (Note: Pigs really
cannot fly! And I am really not the King of France.) This is an application of material implication, a logical operator
characterized by the following truth table.
Truth Table for IMPLIES

A 
B 
A => B 
1 
T 
T 
T 
2 
T 
F 
F 
3 
F 
T 
T 
4 
F 
F 
T 
We
can derive each line of the truth table using well known properties of material
implication, i.e. those given by the selfevident rules of detachment and conclusion.
We will also require the selfevident double
negation rule.
We
assume here that A and B are logical propositions that are
known to be unambiguously either true or false. This does not mean that you have to know ahead of
time which propositions are true and which are false. The rules of logic allow
you to consider all possibilities and combinations.
Detachment Rule
If both A => B and A are true,
then we can infer that B is also
true.
Conclusion
Rule
Direct proof: 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.)
By Contradiction: 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.
Double
Negation Rule
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 and the most
commonly used “definition” of material implication.
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)
A => [~A
=> B] (From a falsehood, all things follow) Formal Proof (8 lines)
[A => B]
<=> ~[A & ~B] (Often
given as a definition) Formal Proof (19 lines)
For proofs in standard FOL, see my
MSE posting at Deriving the Truth Table for Material Implication.