# Why implication ($\phi x \Rightarrow \psi x$) is always true according to Russell?

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.

Evidently, some propositional functions are always true, others are true in some instances and not others, some are never true. Russell is entirely clear about this. E.g.

When we say "some men are Greeks" that means that the propositional function "$x$ is a man and a Greek" is sometimes true. (p. 159)

Russell clearly holds the same about propositional functions of the form: "$\varphi(x)$ implies $\psi(x)$". Some are always true, others are true in some instances and not others, some are never true.

He does indeed say

For example, "if $a$ is human, $a$ is mortal" is true whether $a$ is human or not; in fact, every proposition of this form is true. Thus the propositional function 'if $x$ is human, $x$ is mortal" is "always true" or "true in all cases." (p. 158)

But this is a particular case, not an illustration of something which is supposed to be true of every propositional function.

As to why, e.g. "if the god Zeus is human, the god Zeus is mortal" is true, it is true because for Russell the conditional here is the truth-functional material conditional. The antecedent of the conditional is false, so the whole is true. Likewise for any non human denoted by "$a$". And if "$a$" denotes a human, the conditional is still true. So it is indeed true in all cases.

• Hi @PeterSmith. If I understand T(a) = "if a is human, a is mortal" is a PF of the form $\phi x \Rightarrow \psi x$. Now what Russell says is that T(a) is always true and he is not talking about all of the PFs of the form $\phi x \Rightarrow \psi x$. Am I right? – Harold Jan 23 '13 at 19:24
• No. "If a is human, a is mortal" is a proposition containing the name "a", not a propositional function expressed using a variable. – Peter Smith Jan 23 '13 at 19:45

If there is a specific $x$ for whic $\phi x$ is true and $\psi x$ is false, then $\phi x\Rightarrow\psi x$ is false. If you find a specific John Doe such that "John Doe is human" is true, but "John Doe is mortal" is false, then the statement "All humans are mortal" is false.