Quantifier

logic

Universal

Definition

Universal Quantifier

The universal quantifier $\forall x\phi$ is true under an interpretation $(\mathcal{M},\ell )$ if $\phi$ holds for every re-assignment of $x$:

$$(\mathcal{M},\ell )\models \forall x\phi ⟺(\mathcal{M},\ell [d/x])\models \phi \text{for all }d\in \mathcal{D}$$

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Existenial

Definition

Existential Quantifier

The existential quantifier $\exists x\phi$ is true under an interpretation $(\mathcal{M},\ell )$ if $\phi$ holds for some re-assignment of $x$:

$$(\mathcal{M},\ell )\models \exists x\phi ⟺(\mathcal{M},\ell [d/x])\models \phi \text{for some }d\in \mathcal{D}$$

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