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

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Summary[edit]

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

No column/row is empty:
1. \forall x \exist y Lyx:
Everyone is loved by someone.
2. \forall x \exist y Lxy:
Everyone loves someone.
The diagonal is
nonempty/full:
5. \exist x Lxx:
Someone loves himself.
6. \forall x Lxx:
Everyone loves himself.
The matrix is
nonempty/full:
7. \exist x \exist y Lxy:
Someone loves someone.

8. \exist x \exist y Lyx:
Someone is loved by someone.
9. \forall x \forall y Lxy:
Everyone loves everyone.

10. \forall x \forall y Lyx:
Everyone is loved by everyone.
Hasse diagram of the implications
One row/column is full:
3. \exist x \forall y Lxy:
Someone loves everyone.
4. \exist x \forall y Lyx:
Someone is loved by everyone.


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