Prove that if 𝐗 and 𝐲 are distinct individual variables such...

Prove that if $\mathbf{X}$ and $\mathbf{y}$ are distinct individual variables such that $y$ is not free in $\mathbf{C}$ and $\mathbf{y}$ is free for $\mathbf{x}$ in $\mathbf{C}$, then $S_{\mathbf{x}}^y S_y^x \mathbf{C}=\mathbf{C}$.

Key Concepts

Substitution in First-Order Logic

This concept refers to the operation of systematically replacing each free occurrence of a variable in a logical formula with a term or another variable. Substitution is central in formal logic for manipulating formulas, especially during proofs and transformations. It requires careful handling to ensure that the semantic content of the formula is preserved.

Free and Bound Variables

A variable in a formula can occur either as free or bound. Free variables are those not quantified over, and they stand for arbitrary elements of the domain, while bound variables are quantified and have their scope limited. Understanding this distinction is crucial when performing substitutions, as operations like variable replacement must avoid inadvertently altering the roles of variables or causing unintended interactions between free and bound occurrences.

Substitutability and the 'Free for' Condition

The 'free for' condition specifies when a variable can be safely substituted for another without risking variable capture, where the substituting variable might become unintentionally bound. This condition is essential in ensuring that the substitution preserves the logical structure and meaning of the formula, as it prevents the accidental entanglement of free and bound variables.

Avoiding Variable Capture

Variable capture occurs when a substitution causes a free variable to become bound by a quantifier in the formula, altering its intended interpretation. To avoid this problem, substitution rules typically include conditions (such as the 'free for' condition) that must be checked before performing a substitution. This ensures that the logical equivalence of formulas is maintained during variable replacement operations.

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