[Math] Representing a logic puzzle with mathematical symbols

logic

Consider the following logic puzzle, which is one of many created by Lewis Carroll, the author of Alice in Wonderland.

No birds, except ostriches, are 9 feet high.
There are no birds in this aviary that belong to anyone but me.
No ostrich lives on mince pies.
I have no birds less than 9 feet high.

Prove that these premises imply the following conclusion:

Any bird in this aviary does not live on mince pies.

Use the following symbols to represent statements:

H:  Height of the bird is not less than nine feet.
O:  The bird is an ostrich.
M:  The bird lives on mince pies.
I:  I own the bird.
A:  The bird is in this aviary.
  1. Show the premises as logical formulas represented using these symbols.
    $$O \rightarrow H$$
    $$A \rightarrow I$$
    $$O \rightarrow \neg M$$
    $$I \rightarrow H$$
  2. Show the conclusion as a logical formula represented using these symbols.
    $$A \rightarrow \neg M$$
  3. Show the negation of the conclusion using these symbols.
    $$\neg (A \rightarrow \neg M)$$
  4. Show all premises and the negation of the conclusion as a set of clauses.
    $$A \wedge M, \neg O \vee H, \neg A \vee I, \neg O \vee \neg M, \neg I \vee H$$
  5. Use the resolution method for your proof, and show for each resolution step which formulas are involved as parents and what the resolvent is.
    ???

Best Answer

Your formulation of the first hypothesis, "$O\to H$", says that an ostrich is necessarily at least 9 feet high. The first hypothesis as stated by Carroll, however, says that no other birds are that tall. If you formalize that information, I think you'll find the problem quite easy.

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