Tautology or Open Sentence?

Question

I’m having a hard time determining if this sentence would be a tautology or an open sentence.

x is a multiple of 7 or x is not a multiple of 7.

I’m not sure if this would be a tautology because of the free variable x. For example, the sentence “x is a multiple of 7.” would be an open sentence and therefore not have a truth value. “x is not a multiple of 7” would also be an open sentence and therefore not have a truth table. Therefore I don’t see the sentence shown as a tautology, I see it as an open sentence. If the statement was:

$(\forall x \in \mathbb{R})$ (x is a multiple of 7 or x is not a multiple of 7)

I would see this as a tautology because x was is bounded by the quantifier $\forall$. Can someone explain if I’m wrong? Thank you!

Answer 1

Everybody agrees that A ∨ ¬A is a tautology, but in the context of first-order logic only some authors also call ∀x (Ax ∨ ¬Ax) a tautology. For clarity, we can always say that the latter is a first-order validity but not a propositional-logic tautology.

x is a multiple of 7 or x is not a multiple of 7

I don’t see this sentence shown as a tautology, I see it as an open sentence.

This open formula (i.e., not a sentence, due to the free variable) can be formalised as P or not P, so it is a propositional-logic tautology, so it is also a validity.

1. I thought an atomic statement or atomic open sentence can’t be a tautology. So how is 0=0 a tautology?

Being atomic, it indeed does not make sense for the validities x=x (an open formula) and 0=0 (a sentence) to be propositional-logic tautologies.

2. Does A(x) or not A(x) have a truth value?

∀x∈R (x is a multiple of 7 or x is not a multiple of 7)

I see this sentence as a tautology.

The open formula A(x) or not A(x) is valid, i.e., logically true. So, the sentence ∀x∈R (x is a multiple of 7 or x is not a multiple of 7) is also valid; but it is not a propositional-logic tautology, since in propositional logic it is atomic.

Answer 2

Informally, an open sentence is a formula depending on some variables whose truth value might, but does not necessarily have to, depend on these variables.

For instance, if $A(x)$ is the formula $x=0$, then $A(x)$ is an open sentence whose truth value clearly depends on $x$. However, we might also consider the formula $A(x)$ given by $0=0$. In that case, $A(x)$ is always true, independent of the actual value of $x$. In that case, $A(x)$ is an open sentence, but still a tautology.

For any formula $A$, the formula $B$ given by $A\vee\neg A$ is a tautology. Similarly, for a formula $A(x)$ depending on some variable $x$, also the formula $B(x)$ given by $A(x)\vee\neg A(x)$ is a tautology.