Exercises for first-order logic  (Page 3/5)

For the sentence $\forall x\colon \forall y\colon (A(x)\land B(x, y))\implies A(y)$ state whether it is true or false, relative to the following interpretations.If false, give values for $x$ and $y$ witnessing that.

1. The domain of the natural numbers, where $A$ is interpreted as

   even?

   , and $B$ is interpreted as

   equals

2. The domain of the natural numbers, where $A$ is interpreted as

   even?

   , and $B$ is interpreted as

   is an integer divisor of

3. The domain of the natural numbers, where $A$ is interpreted as

   even?

   , and $B$ is interpreted as

   is an integer multiple of

4. The domain of the Booleans, $\{\mbox{true}, \mbox{false}\}$ , where $A$ is interpreted as

   false?

   , and $B$ is interpreted as

   equals

5. The domain of WaterWorld locations in the particular boardwhere locations $Y$ and $Z$ contain pirates, but all other locations are safe, the relation symbol $A$ is interpreted as

   unsafe?

   , and $B$ is interpreted as

   neighbors

6. All WaterWorld boards, where $A$ is interpreted as

   safe?

   , and $B$ is interpreted as

   neighbors

   . (That is, is the formula valid for WaterWorld?)

Translate the following conversational English statements into first-order logic, using the suggested predicates,or inventing appropriately-named ones if none provided. (You may also freely use $$ which we'll choose to always interpret as the standard equality relation.)

1. All books rare and used

   . This is claimed by a local bookstore;what is the intended domain?Do you believe they mean to claim

   all books rare or used

   ?

2. Everybody who knows that UFOs have kidnapped people knows that Agent Mulder has been kidnapped.

   (Is this true, presuming that no UFOs have actually visited Earth…yet?)

Write a formula for each of the following. Use the two binary relations $\mathrm{isFor}$ and $\mathrm{isAgainst}$ and domain of all people.

• All for one, and one for all!

  We'll take

  one

  to mean

  one particular person

  , and moreover, that both

  one

  s are referring the same particular person,resulting in

  There is one whom everybody is for, and that one person is for everybody.

  Dumas' original musketeers presumably meant something different: that each one of them was for each (other) one of the them,making the vice-versa clause redundant. But this is boring for our situation, so we'llleave that interpretation to Athos, Porthos, and Aramis alone.)

• If you're not for us, you're against us.

  In aphorisms,

  you

  is meant to be an arbitrary person; consider using the word

  one

  instead. Furthermore, we'll interpret

  us

  as applying to everybody. That is,

  One always believes that `if one is not for me, then one is against me'

  .

• The enemy of your enemy is your friend.

  By

  your enemy

  we mean

  somebody you are against

  , and similarly,

  your friend

  will mean

  somebody you are for

  . (Be carefule! This may be differentthan

  somebody who is against/for you

  ).

• Somebody has an enemy.

  (We don't know of an aphorism expressing this. None of the following quite capture it:

  Life's not a bed of roses

  ;

  It's a dog-eat-dog world

  ;

  Everyone for themselves

  ;

  You can't please all the people all the time

  . )