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‘The’ can be a quantifier, e.g. ‘the square is broken’. How to formalise it?

The square is broken \\ ⫤⊨ There is exactly one square and it is broken

Recall that we can translate `There is exactly one square' as:

\hspace{5mm} ∃x ( Square(x) ∧ ∀y ( Square(y) → x=y ) )

So `There is exactly one square and it's broken':

\hspace{5mm} ∃x ( Sqr(x) ∧ ∀y ( Sqr(y) → x=y ) ∧ Broken(x) )

There is an end, and all ends are this end, and it is near.

∃x ( End(x) ∧ ∀y ( End(y) → x=y ) ∧ Near(x) )

14.26, 14.28

14.2

14.4, 14.5

14.10, 14.11

14.26, 14.28