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

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,

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.

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)

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

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