In the chapter XV of the Intro. to Philosophical Math, Russell says that every propositional function (PF) of the form:
- "$\phi x$ implies $\psi x$" is always true.
Russell gives the following example:
- "If $x$ is human, x is mortal" is always true, no matter if x is human or not.
How should I understand that?. He does not give any concrete example for "$x$ is not human". For example if I replace "x" with "the god Zeus" in the above statement as follow:
"if the god Zeus is human, the god Zeus is mortal"
Why should that statement be true?
In other section of the same chapter he says:
"All S is P" = " '$\phi x$ implies $\psi x$' is always true"
Here S is defined by $\phi x$ and P is defined by $\psi x$
Again, he states that no matter if x is an S or not, the implication is always true. In fact, he argues that the reductio ad adsurdum would not be possible without that assertion. I really dont understand what he is trying to mean.
What about if $\phi x$ is true and $\psi x$ is false? Why should $\phi x \Rightarrow \psi x$ be true?. According to the truth-table of the logical implication it must be false.