# Template:Predicate logic; 2 variables; single; example matrices

## SummaryEdit

There are 10 sentences with 8 different meanings, using the loving-relation Lxy and the quantifiers ∀ and ∃:

 No column/row is empty: 1. ${\displaystyle \forall x\exists yLyx}$ : Everyone is loved by someone. 2. ${\displaystyle \forall x\exists yLxy}$ : Everyone loves someone.
 The diagonal is nonempty/full: 5. ${\displaystyle \exists xLxx}$ : Someone loves himself. 6. ${\displaystyle \forall xLxx}$ : Everyone loves himself.
 The matrix is nonempty/full: 7. ${\displaystyle \exists x\exists yLxy}$ : Someone loves someone. 8. ${\displaystyle \exists x\exists yLyx}$ : Someone is loved by someone. 9. ${\displaystyle \forall x\forall yLxy}$ : Everyone loves everyone. 10. ${\displaystyle \forall x\forall yLyx}$ : Everyone is loved by everyone.

Hasse diagram of the implications
 One row/column is full: 3. ${\displaystyle \exists x\forall yLxy}$ : Someone loves everyone. 4. ${\displaystyle \exists x\forall yLyx}$ : Someone is loved by everyone.

Watchduck (a.k.a. Tilman Piesk)

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