Confusion on Basic Logic Question about Contrapositive (simple probably)

Question

So, suppose we are told to show the statement: "If $\frac{1}{x}$ is irrational, then $x$ is also irrational."

This seems pretty clearly true, and I can't think of a counterexample, but when we consider the logically equivalent contrapositive: "If $x$ is rational, then $\frac{1}{x}$ is rational." we see that this statement is untrue for $x=0$. Does this imply the original statement is also false?

Answer 1

A statement involving $\frac1x$ is not untrue for $x=0$, it is ill-formed because part of it is not defined (namely $\frac10$). An ill-formed sentence doesn't have a defined truth value.

However, we sometimes interpret a sentence like "if $x$ is rational, then $\frac1x$ is rational" as having an implicit quantifier like "for all real numbers $x\ne0$, ..." so that its truth value doesn't depend on the cases where subexpressions are undefined. Doing this resolves your problem, making the original sentence and the contrapositive version both true.

Answer 2

1. If $\frac{1}{x}$ is irrational, then $x$ is also irrational
2. If $x$ is rational, then $\frac{1}{x}$ is rational

implicitly means:

1. For every $\boldsymbol x,$ if $\frac{1}{x}$ is irrational, then $x$ is also irrational
2. For every $\boldsymbol x,$ if $x$ is rational, then $\frac{1}{x}$ is rational.

But what exactly does “every $x$” mean? Is $7+4i$ a candidate of “every $x$”? Is $0$ a candidate of “every $x$”? Part of the context of every statement is a (generally tacit) discourse domain:

• when the discourse domain is $\boldsymbol{\mathbb R{\setminus}\{0\}},$ then both statements are true;
• when the discourse domain is $\boldsymbol{\mathbb R},$ then $\frac10$ being rational/irrational is non-meaningful, so technically both statements are neither true nor false; alternatively, we can informally say that $\frac10$ being rational/irrational is false, in which case Statement 1 is true while Statement 2 is false;
• when the discourse domain is $\boldsymbol{\mathbb C},$ and when the set of irrationals is defined as $\mathbb R{\setminus}\mathbb Q$ (this is yet another piece of the context), then “$7+4i$ is rational” and “$7+4i$ is irrational” are both false.