Solutions to Chapter 26 Identity

A. Consider the sentence,

1. 19.
   ​

   Every officer except Pavel owes money to Hikaru.

Symbolize this sentence, using ‘$F(x)$’ for ‘blank _x is an officer’. Are you confident that your symbolization is true if, and only if, sentence 19 is true? What happens if every officer owes money to Hikaru, Pavel does not, but Pavel isn’t an officer? Most people, including most linguists thinking about ‘except’, read sentence 19 as entailing all three of the following:

1. 20.
   ​

   Every officer who is not Pavel owes money to Hikaru.

2. 21.
   ​

   Pavel does not owe money to Hikaru.

3. 22.
   ​

   Pavel is an officer.

So it can be symbolized as ‘$\forall x((F(x)\wedge \neg x=p)\to O(x,h))\wedge \neg O(p,h)\wedge F(p)$’.

B. Explain why:

1. 1.
   ​

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

2. We might naturally read this in English thus:

   • ‣
     ​

     There is something, $x$, such that, if you choose any object at all, if you chose an apple then you chose $x$ itself, and if you chose $x$ itself then you chose an apple.

   The $x$ in question must therefore be the one and only thing which is an apple.

3. 2.
   ​

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

4. Similarly to the above, we might naturally read this in English thus:

   • ‣
     ​

     There are two distinct things, $x$ and $y$, such that if you choose any object at all, if you chose an apple then you either chose $x$ or $y$, and if you chose either $x$ or $y$ then you chose an apple.

   The $x$ and $y$ in question must therefore be the only things which are apples, and since they are distinct, there are two of them.