Which CSL rules hold in Łukasiewicz's 3-valued logic? CSL is classical logic. So I'm talking about the basic introduction and elimination rules (conditional, biconditional, disjunction, conjunction and negation).

I'm not talking about his infinite-valued logical theory, but the 3-valued one where any atomic sentence can be given T,F or I. (See the Wikipedia article here:

http://en.wikipedia.org/wiki/Three-valued_logic
try it out by 3 valued truthtables :)

out of my head all  except the conjunction introduction , negation elimination and negation introduction rule are valid .
but i can be wrong

added after Comment by Spoonwood 

The problem with the conditional introduction rule is that it is normally a discharging rule and in Lukasiewicz  3 valued logic 

(P -> (P-> Q)) -> (P -> Q) (contraction) is not a tautology in Lukasiewicz  3 valued logic :
(I use i instead of 1/2, the  designated value is 1, * is the problematic line)

 P | Q || (P -&gt; (P -&gt; Q)) -&gt; (P -&gt; Q)  
---|---||-----------------|-----------  
 1 | 1 ||    1     1      1     1  
 1 | i ||    i     i      1     i  
 1 | 0 ||    0     0      1     0  
 i | 1 ||    1     1      1     1  
 i | i ||    1     1      1     1  
 i | 0 ||    1     i      i*    i  
 0 | 1 ||    1     1      1     1  
 0 | i ||    1     1      1     1  
 0 | 0 ||    1     1      1     1  


so you should just not be able to deduce it 
but using the standard conditional introduction rule
you can:

1 ||___ P -&gt; (P -&gt; Q)               Assumption
2 |||__ P                           Assumption
3 |||   P -&gt; Q                      1,2 Conditional elimination
4 |||   Q                           2,3 Conditional elimination
5 ||&lt;   P -&gt; Q                      2-4 Conditional introduction
6 |&lt;    (P -&gt; (P -&gt; Q)) -&gt; (P -&gt; Q) 1-5 Conditional introduction


so the conditional introduction rule as used here is not a valid rule in Lukasiewicz 3 valued logic.

You could have a rule

if P then Q -> P (without discharging an assumption) 
but that is not the standard rule.

While writing this I was also wondering about the or elimination rule, it depends a bit which or elimination rule you use, some are valid some are not, the same applies to the Not introduction rule.

In general non-discharging rules are valid, discharging rules are not
1) I believe that Conditional introduction works: 
     My experience is that the problem lies in getting a getting valid derivation from a set of premises, given the other rules that don't work.

2) Conditional elimination does not.
   This is the chief thing that cripples Lukasiewicz logic as a logic. 
   Modus Ponens and the transitivity of the conditional both fail  for the Lukasiewicz conditional, as they properly should, given that it allows conditionals with a value of I. If your conditionals are doubtful, you should not expect inference using them to be valid. 
    (It is not generally known that this can be remedied: Define Spq as NCCpqNCpq and use that for the conditional)

3) Biconditional introduction works.
4) Biconditional elimination works.
5) Conjunction introduction works.
6) Conjunction elimination works
7) Disjunction introduction works.

8) Disjunction elimination does not work.
9) Negation introduction does not work.
10) Negation elimination does not work. 
    Classically, 9 and 10 depend on the Law of the Excluded Middle, which doesn't hold in Lukasiewicz logic.