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Table 1:

Natural logic relations (NatOps) and their set theoretic definitions.

NatOP: Name	Definition
⥮: Alternation	x ∩ y = ⊘ ∧ x ∪ y ≠ U
⌣ : Cover	x ∩ y ≠ ⊘ ∧ x ∪ y = U
≡: Equivalence	x = y
$⊑$ ⁠: Forward Entailment	x ⊂ y
$⋏$ ⁠: Negation	x ∩ y = ⊘ ∧ x ∪ y = U
$⊒$ ⁠: Reverse Entailment	$x \supset y$
⋕: Independence	All other cases

NatOP: Name	Definition
⥮: Alternation	x ∩ y = ⊘ ∧ x ∪ y ≠ U
⌣ : Cover	x ∩ y ≠ ⊘ ∧ x ∪ y = U
≡: Equivalence	x = y
$⊑$ ⁠: Forward Entailment	x ⊂ y
$⋏$ ⁠: Negation	x ∩ y = ⊘ ∧ x ∪ y = U
$⊒$ ⁠: Reverse Entailment	$x \supset y$
⋕: Independence	All other cases