$𝖟𝕷𝖔𝖌𝖎𝖈/𝕾𝖒𝖚𝖑𝖑𝖞𝖆𝖓/𝕾𝖆𝖙𝖆𝖓-𝕮𝖆𝖓𝖙𝖔𝖗-𝖆𝖓𝖉-𝕴𝖓𝖋𝖎𝖓𝖎𝖙𝖞/𝖘𝖆𝖙𝖆𝖓-𝖎-1.𝖘𝖆𝖕𝖕𝖍𝖎𝖗𝖊 
    0-----------------------------|·Satan,·Cantor·&·Infinity·ch·I·pz·1·|-----------------------------
    1--|·Chapter·I·Puzzle·1·is·asked·on·pg·3                                                        
    2                                                                                               
    3--|·A·statement·made·by·a·resident·of·The·Island·of·Knights·and·Knaves·is·true·if·and          
    4--|·only·if·the·resident·is·a·knight.·For·this·first·problem                                   
    5                                                                                               
    6--|·a··Arthur·is·a·kinght···b··Bernard·is·a·knight···c··Charles·is·a·knight                 
    7                                                                                               
    8--|·while··is·used·for·implication,··is·used·instead·of·⇐⇒·for·if·and·only·if,·to            
    9--|·emphasize·equivalence                                                                      
   10                                                                                               
   11--|·The·statement·that·Bernard·and·Charles·are·both·knights·is·translatedd·as·b∧c              
   12                                                                                               
   13--|·Either·Arthur·is·a·knight,·and·so·are·both·Bernard·and·Charles,·which·is·a∧(b∧c)           
   14--|·or·Arthur·is·a·knave,·and·not·both·Bernard·and·Charles·are·knights·is·¬a∧¬(b∧c)            
   15                                                                                               
   16--|·Combined,·a∧(b∧c)∨¬a∧¬(b∧c)                                                                
   17                                                                                               
   18--|·We·can·also·consider,·the·truth·of·a·statement·is·equivalent·to·the·speakers·knightliness. 
   19--|·Symbolically,·a≡b∧c.·The·two·forms·are·equivalent:                                         
   20                                                                                               
   21taut:·a(bc)··¬a¬(bc)··(abc)                                                            
                                  a             │                                    
                                  b             │                                    
                                  c             │                                    
                      ──────────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                    
                                 bc            │                                    
                               a(bc)          │                                    
                                 ¬a             │                                    
                               ¬(bc)           │                                    
                              ¬a¬(bc)         │                                    
                          a(bc)¬a¬(bc)     │                                    
                                abc           │                                    
                      ──────────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                    
                      a(bc)¬a¬(bc)(abc) │                                    
   22                                                                                               
   23--|·Arthur·denies·Bernard·is·a·knight:·a∧¬b··¬a∧¬(¬b),·or·a≡¬b                                
   24                                                                                               
   25taut:·a¬b··¬a¬¬b··(a¬b)                                                                   
                                   a          │                                              
                                   b          │                                              
                           ───────────────────┼─┼─┼─┼─┤                                              
                                   ¬b         │                                              
                                  a¬b        │                                              
                                   ¬a         │                                              
                                  ¬¬b         │                                              
                                 ¬a¬¬b       │                                              
                              a¬b¬a¬¬b     │                                              
                                  a¬b        │                                              
                           ───────────────────┼─┼─┼─┼─┤                                              
                           a¬b¬a¬¬b(a¬b) │                                              
   26                                                                                               
   27--|·we·conclude·Bernard·is·a·knight·and·both·Arthur·and·Charles·are·knaves:·¬a∧b∧¬c            
   28                                                                                               
   29--|·Our·solution·satisfies·Arthur's·first·statement                                            
   30taut:·¬ab¬c··(abc)                                                                        
                                 a        │                                          
                                 b        │                                          
                                 c        │                                          
                          ────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                          
                                ¬a        │                                          
                               ¬ab       │                                          
                                ¬c        │                                          
                              ¬ab¬c     │                                          
                                bc       │                                          
                               abc      │                                          
                          ────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                          
                          ¬ab¬c(abc) │                                          
   31                                                                                               
   32--|·and·Arthur's·second·statement                                                              
   33taut:·¬ab¬c··(a¬b)                                                                         
                                a        │                                           
                                b        │                                           
                                c        │                                           
                          ───────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                           
                                ¬a       │                                           
                               ¬ab      │                                           
                                ¬c       │                                           
                             ¬ab¬c     │                                           
                                ¬b       │                                           
                               a¬b      │                                           
                          ───────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                           
                          ¬ab¬c(a¬b) │                                           
   34                                                                                               
   35--|·and·is·in·fact·logically·equivalent·to·Arthur's·statements,·taken·together                 
   36taut:·¬ab¬c··(abc)(a¬b)                                                                 
                                 a            │                                      
                                 b            │                                      
                                 c            │                                      
                       ───────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                      
                                 ¬a           │                                      
                                ¬ab          │                                      
                                 ¬c           │                                      
                              ¬ab¬c         │                                      
                                bc           │                                      
                               abc          │                                      
                                 ¬b           │                                      
                                a¬b          │                                      
                           (abc)(a¬b)     │                                      
                       ───────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                      
                       ¬ab¬c(abc)(a¬b) │                                      
   37                                                                                               
   38taut:·(XA)(XB)··XAB                                                                      
                                 X          │                                        
                                 A          │                                        
                                 B          │                                        
                        ────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                        
                                XA         │                                        
                                XB         │                                        
                            (XA)(XB)     │                                        
                           (XA)(XB)X    │                                        
                                AB         │                                        
                        ────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                        
                        ((XA)(XB)X)AB │                                        
   39taut:·(XA)(XB)··XAB                                                                      
                                 X          │                                        
                                 A          │                                        
                                 B          │                                        
                        ────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                        
                                XA         │                                        
                                XB         │                                        
                            (XA)(XB)     │                                        
                           (XA)(XB)X    │                                        
                                AB         │                                        
                        ────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤                                        
                        ((XA)(XB)X)AB │                                        
   40                                                                                               
   41                                                                                               
   42proof:·abc·a¬b··¬ab¬c                                                                    
   43A··A····················[sp]···--|·Simple·Rewrite·via·Sapphire·Normal·Form                  
   44P(PQ)··Q··············[mp]···--|·Modus·Ponens                                             
   45¬AB·⏴⏵·AB··············[imp]··--|·Implication                                              
   46XAB·⏴⏵·(XA)(XB)·····[cic]··--|·Composition·of·Implication·over·Conjunction              
   47AB·⏴⏵·¬B¬A·············[cp]···--|·Contrapositive                                           
   48                                                                                               
   49(AB)(A¬B)·⏴⏵·¬A·······[cdct]·--|·Contradiction/Explosion                                  
   50                                                                                               
   51¬(AB)·⏴⏵·¬A¬B··········[dna]··--|·Distribution·of·negation·over·and                        
   52                                                                                               
   53                                                                                               
   54a¬b·····················[anb] - from assumptions                                            
   55¬ab·········cp········[nab] - AB ⏴⏵ ¬B¬A A¬b Ba                                     
   56                                                                                               
   57abc····················[abc] - from assumptions                                            
   58ab··········cic·            - XAB ⏴⏵ (XA)(XB) Xa Ac Bb                          
   59¬a···········cdct·anb··[na]  - (AB)(A¬B) ⏴⏵ ¬A Aa B¬b                               
   60b············mp··nab···[b]   - P(PQ) ⏵ Q P¬a Qb                                      
   61                                                                                               
   62¬a¬(bc)····cp·abc - AB ⏴⏵ ¬B¬A Abc Ba                                              
   63¬(bc)·······mp··na - P(PQ) ⏵ Q P¬a Q¬(bc)                                          
   64¬b¬c········dna·   - ¬(AB) ⏴⏵ ¬A¬B Ac Bb                                            
   65b¬c·········imp·   - ¬AB ⏴⏵ AB B¬c Ab                                               
   66¬c···········mp·b·  - P(PQ) ⏵ Q Pb Q¬c                                               
   67¬ab¬c······sp·na·b· - A ⏵ A Ab¬a¬c                                                  
   68                                                                                               
   69                                                                                               
   70taut:···(A⇒B)∧(A¬B)··¬A                                                                      
                                   A        │                                                
                                   B        │                                                
                            ────────────────┼─┼─┼─┼─┤                                                
                                  AB       │                                                
                                  ¬B        │                                                
                                 A¬B       │                                                
                             (AB)(A¬B)   │                                                
                                  ¬A        │                                                
                            ────────────────┼─┼─┼─┼─┤                                                
                            (AB)(A¬B)¬A │