predicate quantifier

Question

• a)There is a tree in the back yard.
• b)If the tree in the back yard is an elm or an oak, then the treasure is in the kitchen and not in the garage.
• c)If this house is made of bricks or the tree in the back yard is an oak, then the treasure is not in the kitchen.
• d)The tree in the back yard is an elm or the treasure is buried under the mailbox.
• e)All houses are made of bricks.
• f)If the tree in the back yard is an oak, then the treasure is in the garage.

Hello, i am given the above problem to find where is the treasure hidden. and i have currently come up with the following premises:

• ∃x(Tx):There is a tree in the back yard.
• T(e):The tree in the back yard is an elm.
• T(o):The tree in the back yard is an oak.
• K(t):The treasure is in the kitchen.
• G(t):The treasure is in the garage.
• B(h)The house is made of bricks.
• ∀x(Bx):All houses are made of bricks.

• M(t):The treasure is buried under the mailbox

  1. ∃x(Tx) , a)
  2. T(e) v T(o) -> K(t) ^ ~G(t) , b)
  3. B(h) v T(o) -> ~K(t) , c)
  4. T(e) v M(t) , d)
  5. ∀x(Bx) , e)
  6. T(o) -> G(t) , f)
  7. B(h) ,5,universal instantiation
  8. B(h) v T(o) ,7,addition
  9. ~K(t) ,3,8,modus ponens
  10. T(e) v T(o) -> K(t) , 2,simplification
  11. ~(T(e) v T(o)) 10,9, modus ponens
  12. ~T(e) ^ ~ T(o) , 11,demorgan
  13. ~T(e) ,12,simplification
  14. M(t) , 4,13,Modus tollen

Answer 1

Caution: Why in (7) from the fact that there exists a tree in the front yard it follows this tree is an oak? It could be an elm, orange or even a sakura tree! Recall in the existential intantiation we should always use a new variable In order to not to make incorrect conclusions from previous assumptions. You can see Velleman's How to Prove It (p.115) for a longer discussion on this subject.

Now let's discuss you Treasure Hunter problem (I particularly enjoyed it very much!). Let's analyze the atomic formulae first, hoping that it would be easier to draw conclusions from molecular ones later.

All houses are made of brick.

$\forall x (Bx)$

Now by universal instantiation we have that the house h is made of bricks:

$B(h)$

Then from your statement (c) we conclude that the treasure is not in the kitchen:

$((B(h) \vee T(o)) \rightarrow \neg K(t)), B(h) \vdash \neg K(t)$

Now, from (b) we conclude that the tree is not an elm and not an oak:

$ (T(e) \vee T(o)) \rightarrow (K(t) \wedge \neg G(t)), \neg K(t) \vdash (\neg T(e) \wedge \neg T(o))$

And since the tree is not an elm, from (d) we conclude that the treasure is buried under the mailbox:

$(\neg T(e)), (T(e) \vee M(t)) \vdash M(t)$

Can you get the deduction steps? A good question, let's keep working!

Answer 2

The treasure is buried under the mailbox.

Since all houses are made of bricks, 'the house' is made of bricks and therefore the treasure is not in the kitchen. (E, C)

Since the treasure is not in the kitchen, the tree in the front yard cannot be either an oak or an elm. (B)

Since the tree in the front yard is not an elm, it must be buried under the mailbox. (D)