Equality Axioms

Equality Axioms

Suppose $u, v, w$ are variables. Then $u = u, u = v \to v = u, (u = v \land v = w) \to u = w,$ are equality axioms.

Suppose $u_{1}, ..., u_{n}$ and $v_{1}, ..., v_{n}$ are variables and $f$ is an n-ary function symbol. Then $(u 1 = v 1) \land \cdot\cdot\cdot \land (u_{n} = v_{n}) \to f (u_{1}, ..., u_{n}) = f (v_{1}, ..., v_{n})$ is an equality axiom.

Suppose $u_{1}, ..., u_{n}$ and $v_{1}, ..., v_{n}$ are variables and $r$ is an n-ary relation symbol. Then $(u 1 = v 1) \land \cdot\cdot\cdot \land (u_{n} = v_{n}) \to (r (u_{1}, ..., u_{n}) \to r (v_{1}, ..., v_{n}))$ is an equality axiom.

Remark

Here the equality is like an equivalence relation.

Quartz 4

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Equality Axioms

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