Falsity implies anything

'Are you engaged?'
'I was once. I'm married as a consequence.'

Anthony Powell
The Valley of Bones, p. 15
3rd Movement in A Dance to the Music of Time
University of Chicago Press, 1995

Conditional statements in the form "If A is true then B is true" are called implications and are usually reduced to "If A then B". The notation for this is "A=>B" and is often read as "A implies B" which obviously bears on the terminology.

The statement A=>B may be either true or false depending on the value of A and B. We have to consider four cases that are summarized in the following table

 
A\B 0 1
0 1 1
1 0 1

from which we may conclude several things:

  1. A => B is only false when A is true but B is false.
  2. (Which is the same as 1) If A is false A => B is automatically true.
  3. If B is true then A => B is true whatever A.
  4. If A is true B can't be false

This is a definition and the only criteria to establishment of the falsity or veracity of a particular implication however paradoxical it may sound. For example,

If you are not reading this sentence then I have not written it.

The premise A in this sentence ("you are not reading this sentence") is obviously false or have you managed to skip it? For this reason only the implication is true even though its conclusion B ("I have not written it") is false.

Implications A => B appear as a major premise of the modus ponens. Modus Ponens is one of the syllogisms which are a form of a deductive reasoning. Modus tollens is another.

 
Modus ponensIf A and A => B then B
Modus tollensIf B is false and A => B then A is false

A is the minor premise of the modus ponens. not B (B is false) is the minor premise of the modus tollens. What follows after "then" is called the conclusion.

So falsity implies anything. There are some trivial examples. If 1 = 2 then 5 = 7. Indeed, 1 = 2 implies 2 = 4 (multiply both sides by 2). Add to this the universally valid 3 = 3 to obtain 5 = 7. Some derivations (correct but vacuous as above) require real ingenuity. A story is told that the famous English mathematician G.H. Hardy made a remark at dinner that falsity implies anything. A guest asked him to prove that 2 + 2 = 5 implies that McTaggart is the Pope. Hardy replied, "We also know that 2 + 2 = 4, so that 5 = 4. Subtracting 3 we get 2 = 1. McTaggart and the Pope are two, hence McTaggart and the Pope are one."

An aside

On a mundane level, most of the "logic" in the syllogism is concentrated in the implication A => B while the minor premise is often omitted from consideration. Thus two people may be very consistent and logical in their argument without being able to reach an agreement just because each started with a wrong premise.

Raymond Smullyan gives the following two examples: (Ref. 2)

 

The philosopher Jaako Hintikka makes the delightful argument that one is morally obligated not to do anything impossible. The argument, which ultimately rests on the fact that a false proposition implies any proposition, is this: Suppose Act A is such that it's impossible to perform without destroying the human race. Then surely one is morally obligated not to perform this act. Well, if Act A is an impossible act, then it is indeed impossible to perform it without destroying the human race (since it's impossible to perform it at all!), and therefore one is morally obligated not to perform the act.

But does not the following argument (sic!) show that one is morally obligated to do everything impossible?

Suppose that Act B is such that if one performs it, then the human race will be saved from destruction. Isn't one then morally obligated to perform the act? Now suppose that Act B is impossible to perform. Then it is the case that if one performs Act B, the human race will be saved, because it's false that one will perform this impossible act and a false proposition implies anything. One is therefore morally obligated to perform every impossible act.

Writing about a friend of his in his Autobiography, Bertrand Russell recollects the following episode:

I once devised a test question which I put to many people to discover whether they were pessimists. The question was: "If you had the power to destroy the world, would you do so?" I put the question to him in the presence of his wife and child, and he replied: "What? Destroy my library? - Never!"

Following are several problem from the island of knights and knaves where knights always tell truth whereas knaves always lie.

  1. If anybody on the island says "If I'm a knight then P." then the speaker must be a knight and P is true.
  2. A makes the following statement: "If I am a knight then so is B." What are A and B?
  3. Someone asks A, "Are you a knight?" He replies, "If I am a knight then I'll eat my hat". Must he eat his hat?
  4. A says, "If I am a knight then 2+2=4." Is A a knave or a knight?
  5. A says, "If B is a knight then I am a knave." What are A and B?

References

  1. R. Smullyan, What is the Name of This Book?, Simon&Schuster, NY, 1978.
  2. R. Smullyan, 5000 B.C. and Other Philosophical Fantasies, St. Martin's Press, NY, 1983
  3. I. Stewart, Concepts of Modern Mathematics, Dover, 1995

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