# Confusion on Basic Logic Question about Contrapositive (simple probably)

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?

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.

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.