If
Pigs Could Fly
Photo credit: Lost at E Minor
Material Implication
(Last updated 20180521)
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.
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 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, see below) Formal
Proof (19 lines)
For proofs in standard FOL, see my
MSE posting at Deriving the Truth Table for Material Implication.
What “Paradoxes?”
The paradoxes of
material implication [sic] are
a group of formulae that are truths of classical logic but
are intuitively problematic. The root of the paradoxes lies in a mismatch between
the interpretation of the validity of logical implication in natural
language, and its formal interpretation in classical logic, dating back
to George Boole's algebraic logic. In classical
logic, implication describes conditional ifthen statements using
a truthfunctional interpretation, i.e. "A implies B"
is defined to be "it is not the case that A is
true and B false".… This truthfunctional
interpretation of implication is called material implication or material
conditional.
– “The Paradoxes of Material Implication,” Wikipedia
Paradoxes? What “paradoxes?” As we
have seen here, the usual “definition” A
=> B <=> ~[A & ~B] can be derived from the selfevident, elementary notions of
classical logic.
The above "definition" of material implication
would seem to fully capture the essence of the natural language
"ifthen" construct provided, of course, that we are talking about
logical propositions are unambiguously either true or false. With that proviso,
there is no "mismatch between the interpretation of the validity of
logical implication in natural language, and its formal interpretation in
classical logic."
From the above definition, we see that the logical
expressions A => B and ~[A & ~B] are logically interchangeable, always having the same truth value
for any logical expressions A and B. Using this interchangeability (the
ImplyAnd Rule in DC Proof), each of the above mentioned “paradoxes” can be
trivially resolved based on the same selfevident, elementary notions of
classical logic (see proofs below).
Each of the following is a proof by
contradiction, the first line being a negation of the require result. Each
implication is then replaced by the required negation.
1.
~P & P => Q Formal Proof
(9 lines)
2.
P => [Q => P] Formal Proof (12 lines)
3.
~P => [P => Q] Formal Proof (12 lines)
4.
P => Q  ~Q Formal Proof (9 lines)
5.
[P => Q]  [Q => R] Formal Proof (16 lines)
6.
[P => Q] => P & ~Q Formal Proof (7 lines)
These “paradoxes” are the inevitable, if somewhat
counterintuitive results of the selfevident, elementary notions of classical
logic as discussed above. It is not the rules of logic that must be changed to
reflect usage in natural language, but the other way around. Implication should
not be seen in natural language as having to do with causality, but simply with
a coincidence of a pair of possibly unrelated truth values in the moment.
Logical Literacy
Summary of what students should now about logical implication:
1.
A
=> B has meaning only for logical
propositions A and B that are unambiguously either true or
false. These propositions need not be related in any way. A => B does not mean
that A causes B, or that B follows A over time.
2.
Do either of the following to prove that A => B is true:
a. Assume A is true, then prove that B is true.
b.
Assume B is false, then prove that A is false. (Proving the
contrapositive.)
c.
Prove that both A and B are true.
d.
Prove that it is not the case that
A is true and B is false.
e.
Prove that B is true. (From all propositions will follow that which is
true.)
f. Prove that A is false. (From a falsehood, all propositions will follow.)
3.
How to prove A => B is false: Prove
A is true and B is false.