If Pigs Could Fly

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Material Implication

(Last updated 2018-03-26 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 self-evident rules of detachment and conclusion. We will also require the self-evident 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 3-4)                                     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.

 

 

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