A logical implication in which both sides are unrelated

Question

Is this proposition true or not:

$$ \forall x(x=x) \rightarrow \forall x( x^2 -a^2 = (x+a)(x-a))$$

The left side $x=x$ does not have to do anything with the right side $x^2 - a^2 = (x+a)(x-a),$ but they are both true in any case; does this mean that a tautology is implying a tautology, and thus this statement is true?

Answer 1

That the left-hand side has "nothing to do" with the right-hand side does not matter. Classical logic only cares about truth values, not contextual relevance or causal relation or anything.

The left-hand side is, as you suspect, tautological: There is no way to make it false, no matter which domain of discourse the quantifier ranges over.

The right-hand side is not always true. It is true with the standard interpretation of the symbols $+, -, \cdot, ²$, but false if we e.g. were to assign the function symbol $²$ the meaning "square root". When checking for tautologicity, such non-standard interpretations have to be considered as well!

An implication with a tautological antecedent is logically equivalent to the succedent, that is, if the left-hand side is a tautology, the implication as a whole will take the truth value of the right-hand side. Since $\forall x(x=x)$ is tautological, and $\forall x( x^2 -a^2 = (x+a)(x-a))$ is true in the "real world" but false for different interpretations of the symbols, the implication comes out as true, but not tautological.

Answer 2

Some definitions $\large($tautology $\subseteq$ validity $\subseteq$ truth$\large)$:

• a tautology is a formula that is true in its truth-functional form regardless of interpretation;
• a validity is a formula that is true regardless of interpretation;
• a (synthetic or analytic) truth is a formula that is true in a particular interpretation.

Thus in the given statement $$\forall x \left(x=x\right)\rightarrow \forall x\forall y \left(x^2 -y^2 = (x+y)(x-y)\right),$$

• the antecedent (call it $L$) is valid (and therefore true in the standard interpretation), but not tautological;
• the consequent (call it $R$) is true in the standard interpretation, but neither valid nor tautological;
• the statement as a whole is true in the standard interpretation $\left(\text{so we can write }\,L\Rightarrow R\,\right),$ but neither valid nor tautological.

P.S. I prefer to call $\,\large\rightarrow\,$ the material conditional and $\,\large\implies\,$ the implication symbol. (Why I don't call $\,\rightarrow\,$ “implication”.)