How is "Soulical" Derived?

The reason that "natural" in 1 Cor 15:44 corresponds to "soulical" is because the Greek root of that word is psuche which is the Greek word for soul, which occurs about 105 times in the New Testament.  According to the note in the Companion Bible, the full word then is "pseudikos". The word that contrasts it in that verse is "spiritual" or Greek - "pneumatikos" which has the same suffix tacted onto the  Greek word for spirit - "pneumo." Here is the interlinear Bible's copy of that verse which is basically the same with only a slight change in the suffix presentation.

 
1 Cor 15:44
44 It is sown a body;     natural   it is raised a    body.        spiritual  
     

     4687           4983      5591           1453           4983            4152        
     peíretai     soóma   psuchikón  egeíretai     soóma      pneumatikón
                                                                               
     There is a body, a natural       there is  and 
     1487 207 6983      5591         2076     2532
     Ei   éstin    soóma  psuchikón   éstin     kaí  
                                                     
     spiritual  
     4152       
     pneumatikón
               

(Interlinear Transliterated Bible. Copyright (c) 1994 by Biblesoft)

    There is a fairly in-depth look at the whole issue of the Biblical usage of the Hebrew and Greek words for soul including this verse in the last few pages of a larger study called "The State of the Dead". You can also look at Appendix 110 of the Companion Bible which is available on-line in its entiret. There is also a brief reference to this in Volume 7, pg. 260-262 in the article entitled "Spiritual" in the Alphabetical Analysis by C.H. Welch. There is also an exposition of 1 Cor. 15 by Welch called the "Hope of the Resurrection", W-248-251 in MP 3 format. Hope this helps.