Falsity implies anything
'Are you engaged?'
'I was once. I'm married as a consequence.'
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:
- A => B is only false when A is true but B is false.
- (Which is the same as 1) If A is false A => B is automatically true.
- If B is true then A => B is true whatever A.
- 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.
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
On a mundane level, most of the "logic" in the syllogism is concentrated in the implication
Raymond Smullyan gives the following two examples: (Ref. 2)
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.