There are 10 sentences with 8 different meanings, using the lovingrelation Lxy and the quantifiers ∀ and ∃:
No column/row is empty: 
1. $\forall x\exists yLyx$:
Everyone is loved by someone.

2. $\forall x\exists yLxy$:
Everyone loves someone.


The diagonal is
nonempty/full: 
5. $\exists xLxx$:
Someone loves himself.

6. $\forall xLxx$:
Everyone loves himself.


The matrix is
nonempty/full: 
7. $\exists x\exists yLxy$:
Someone loves someone.
8. $\exists x\exists yLyx$:
Someone is loved by someone.

9. $\forall x\forall yLxy$:
Everyone loves everyone.
10. $\forall x\forall yLyx$:
Everyone is loved by everyone.


Hasse diagram of the implications

One row/column is full: 
3. $\exists x\forall yLxy$:
Someone loves everyone.

4. $\exists x\forall yLyx$:
Someone is loved by everyone.


Watchduck (a.k.a. Tilman Piesk)
Public domainPublic domainfalsefalse 

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