A long genotype contains enough information for one to know the outer message (that is, how the phenotype can be derived from it). A prosaic isomorphism is one in which the corresponding parts of each side of the mapping are trivial to detect.

He refuses to accept the rule of joining.

1. <p ⊃ <q ⊃ p>>

   If this mind is Buddha then: if the moon is shining on the lake then this mind is Buddha.

   [
     p
     [
       q
       p
     ]
     <q⊃p>
   ]
   <p⊃<q⊃p>>
   

2. <p ⊃ <q ∨ ~q>>

   If this mind is Buddha then: either the moon is shining on the lake or it is not.

   [
     p
     [
       ~q
     ]
     <~q⊃~q>
     <q∨~q>
   ]
   <p⊃<q∨~q>>
   

3. <<p ∧ ~p> ⊃ q>

   If this mind is Buddha and this mind is not Buddha then: the moon is shining on the lake.

   [
     <p∧~p>
     p
     ~p
     [
       ~q
       p
       ~~p
     ]
     <~q⊃~~p>
     <~p⊃q>
     q
   ]
   <<p∧~p>⊃q>
   

4. <<p ∧ ~p> ⊃ ~q>

   If this mind is Buddha and this mind is not Buddha then: the moon is not shining on the lake.

   [
     <p∧~p>
     p
     ~p
     [
       q
       p
     ]
     <q⊃p>
     <~p⊃~q>
     ~q
   ]
   <<p∧~p>⊃~q>
   

5. <<p ⊃ q> ∨ <~q ∧ p>>

   Either (1) this mind is Buddha implies the moon is shining on the lake, or (2) the moon is not shining on the lake and this mind is Buddha.

   [
     ~<p⊃q>
     ~<~~p⊃q>
     ~<~p∨q>
     <~~p∧~q>
     ~~p
     ~q
     p
     <~q∧p>
   ]
   <~<p⊃q>⊃<~q∧p>>
   <<p⊃q>∨<~q∧p>>
   

1. 8 is not the square of 3.
   It is not the case that 8 is equal to the square of 3
   ~SSSSSSSS0=(SSS0•SSS0)
2. 8 is not the square of any number.
   There does not exist a number a such that 8 is the square of a
   It is not the case that there exists an a such that 8 is the square of a
   ~∃a:SSSSSSSS0=(a•a)
3. 53 is odd.
   There exists a number a such that the successor of 2•a is 53
   ∃a:S(SS0•a)=S530
4. Multiplying a number by 1 gives you that number.
   For every number a, a•1=a
   ∀a:(a•S0)=a
5. There are infinitely many prime numbers.
   For every number, there is a prime number that is larger
   For every a, there exists a number greater than a which is prime
   For every a, there exists a number greater than a which is not the product of two numbers both greater than 1
   ∀a:∃b:~∃c:∃d:(a+Sb)=(SSc•SSd)
6. Every number has a successor.
   For every a there exists a b which is the successor of a
   ∀a:∃b:Sa=b
7. There is a least number.
   There exists a number which is not the succesor of any number
   There exists an a such that a is not the successor of any number
   There exists an a such that for every b, a is not the successor of b
   ∃a∀b:~a=Sb
8. Every even prime is the sum of two numbers.
   For every even prime a, there exists b and c such that a=b+c
   For every a such that a is even and a is prime, there exists b and c such that a=b+c
   ∀a:<<∃d:a=(d•SS0)∧~∃e:∃e':a=(SSe•SSe')>⊃∃b:∃c:a=(b+c)>
9. Every even number is the sum of two primes.
   For every number a, if a is even, then there exist two primes b and c such that a=b+c
   ∀a:<∃a':a=(SS0•a')⊃∃b:∃c:<<~∃b':∃b'':b=(SSb'•SSb'')∧~∃c':∃c'':c=(SSc'•SSc'')>∧a=(b+c)>>
10. There is a number n such that 3n+1 is prime.
    There exists an a such that 3a+1 is not the product of two numbers both greater than 1
    ∃a:~∃b:∃c:((SSS0•a)+S0)=(SSb•SSc)
11. (EXTRA CREDIT) There are no solutions to aⁿ + bⁿ = cⁿ
    Heh, did you expect me to answer this???

Argh, this was a terribly phrased question. I meant to ask: "For each, identify the preconceptions you are being drawn to discard while practicing the kōan." In any case, answers vary, and saying anything about the way that everyday logic and inference is twisted or ignored, or the way in which words do not take on their usual meaning, should be sufficient for full credit.

1. Are you still beating your wife?
   無 is an appropriate answer if you have no wife or have a wife which you have never beaten. The question is loaded, and provided you are innocent, is best unasked.
2. Is the proposition "this sentence is true" true?
   Whether you answer yes or no, you are not creating any contridictions. If you require that all yes-no questions must have either the answer "yes" or the answer "no" then weaseling out with 無 can be justified. But a more modern view is that the answer "You can take it as true or false without creating any contrictions, it's up to you" works fine.
3. Who is the King of France?
   無 is an appropriate because there is no King of France, so no answer that names a King of France can be truthfully given. Responding to the question by saying "There is no King of France" is exactly unasking the question, which is the function of the answer 無.
4. Where did you hide the murder weapon?
   無 is an appropriate if there is no murder weapon or if there is a murder weapon and you did not hide it. This question is also loaded, as any direct answer implies a murder took place and you were at least an accomplice.
5. Has a dog Buddha-nature or not?
   無 is an appropriate answer here because that is the answer Jōshū gave. ☺

Recursiveness. Completeness.

First, the Tortoise essentially asks whether the set of strings with Buddha-nature is recursive. He's directly asking whether there is a way to make strings without Buddha-nature, i.e., whether the set of strings without Buddha-nature are recursively enumerable. Since the Art of Zen Strings is an r.e. way to make strings with Buddha-nature, an affirmative answer to whether there was a way to make the complement would imply the Budda-nature strings formed a recursive set. (Note that the answer "inconsistency" is incorrect here because Achilles states the given rules can only make strings with Buddha-nature.)

Second, if there were strings with Buddha-nature that could not be made by the rules, then the rules would be incomplete.

If the master answered "no," he may simply have been wrong. For example, the set of possible teachings may be infinite, defined by a recursive set of rules much like the set of English sentences. But if he's right, then the number of true teachings is finite and they have all already been conveyed, though this does not mean there is no reason to keep teaching because new people are constantly being born and there are students that have not been taught everything anyway. Now, would it have made a difference if he answered "no"? Well, on the logical level, yes, the story would have gone differently, because the student would not have asked what the as-yet-untaught teaching was. However, the student may have instead asked "how many teachings are there?" or "is there a teaching that subsumes all others?" and, this being a kōan, the master might have given then same answer he gave to the followup question to his original positive answer. In that sense you could argue it would not have mattered.

We can certainly imagine such a state, and even describe it with words, and Jill Bolte Taylor has given a description of the experience of that state—attained through losing most of the functionality of her left hemisphere. Still, the notion of dissolving the boundaries between the self and all else leads to very interesting questions, among them: if you were in that state, what would it mean to be you? Being one with the universe means (by definition of the term "universe") that there is nothing outside you, nothing that isn't you, if you were indeed you. From a formal perspective, the self-symbol and the universe-symbol would be identical.