If Pigs Could Fly

Material Implication

(Last updated 2023-05-09)

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 “real-world” 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 so-called “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 3-4), 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 3-4) 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)

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