Is x*(1/x)=1 propositional logic?

Question

My book says variables cannot be used in propositional logic. Is this an exception? It can't be false so I know it is a tautology, but my book doesn't categorize tautology into a type of logic.

Answer 1

There are multiple things going on here, so I'm going to unpack this.

1. $x \cdot (\frac{1}{x}) = 1$ is not necessarily true. If $x = 0$, then the left-hand side is nonsense.
2. When developing basic propositional logic, numbers don't yet exist. The numbers are (traditionally) built out of sets or directly out of axioms, but both of these approaches are dependent on logic. We have to define logic first and then define the numbers on top of it.

Of course, once the numbers are defined, you can "go back" and mix basic logic with the "new" machinery of numbers, so that it is perfectly reasonable to write $x < 3 \vee x > 7$, for example. But then you're not working in "pure" propositional logic any more, because you've added axioms describing the numbers.

Answer 2

No, this is not a formula of propositional logic.

In propositional logic, the formulas are always either propositional letters or built from smaller formulas using logical connectives. That doesn't match here because your formula is something different.

(Note that in the jargon of mathematical logic, something like $x\cdot(1/x)$ is not a "formula" but a "term". Formulas are just things that can be true or false).

Answer 3

This is not a propositional logic. It would be a propositional logic if it was: $$x\neq0 \Rightarrow x\frac{1}{x} = 1$$