If Pigs Could Fly

Photo credit: Lost at E Minor

Material Implication

Assume, for the sake of argument, that pigs cannot actually fly.  Then, regardless of whether or not you are the King of France, you could truthfully say, “If pigs could fly, then I’d be the King 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

 

As a general rule, all things follow from a falsehood, e.g. from the falsehood that pigs can fly (in the above example, anyway). For any true-or-false propositions A and B, we can infer that A implies B knowing only that A is false. The proposition B need not even be true.  (See lines 3 and 4 of the truth table.) 

We can derive each line of the truth table using well known properties of material implication, i.e. those given by the quite intuitive rules of detachment and conclusion (deduction).

Detachment Rule: If both A => B and A are true, then we can infer that B is also true.

Conclusion (deduction) Rule: In a mathematical proof, if we assume A is true, and, without making any further assumptions, we can subsequently determine that B is also true, then we can infer that A => B is true. If we can also determine that B is false (obtaining a contradiction), we can also infer that A is false.

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 3-4)            Formal Proof (8 lines)

 

[A => B] <=> ~[A & ~B] (Often given as a definition) Formal Proof (19 lines)

 

It should be emphasized that material implication has nothing to do with cause and effect, or the passage of time. As in all of mathematics, we are simply looking at ordinary, true-or-false propositions in the moment.

 

 

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