Identity

Consider the sentence,

1. Every officer except Pavel owes money to Hikaru.

Symbolize this sentence, using ‘$F(x)$’ for ‘$\underline{\hskip 15.00002pt}_{x}$ is an officer’. Are you confident that your symbolization is true if, and only if, an exercise given elsewhere in the bookexcept2 is true? What happens if every officer owes money to Hikaru, Pavel does not, but Pavel isn’t an officer?

Explain why:

1. ‘$\exists x\forall y(A(y)\leftrightarrow x=y)$’ is a good symbolization of ‘there is exactly one apple’.

2. ‘$\exists x\exists y\bigl[\neg x=y\wedge\forall z(A(z)\leftrightarrow(x=z\vee y=z)\bigr]$’ is a good symbolization of ‘there are exactly two apples’.