─────┬───── $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┊
21┊taut:·a∧(b∧c)·∨·¬a∧¬(b∧c)·≡·(a≡b∧c)
┊ a │✔│✘│✔│✘│✔│✘│✔│✘│
┊ b │✔│✔│✘│✘│✔│✔│✘│✘│
┊ c │✔│✔│✔│✔│✘│✘│✘│✘│
┊ ──────────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ b∧c │✔│✔│✘│✘│✘│✘│✘│✘│
┊ a∧(b∧c) │✔│✘│✘│✘│✘│✘│✘│✘│
┊ ¬a │✘│✔│✘│✔│✘│✔│✘│✔│
┊ ¬(b∧c) │✘│✘│✔│✔│✔│✔│✔│✔│
┊ ¬a∧¬(b∧c) │✘│✘│✘│✔│✘│✔│✘│✔│
┊ a∧(b∧c)∨¬a∧¬(b∧c) │✔│✘│✘│✔│✘│✔│✘│✔│
┊ a≡b∧c │✔│✘│✘│✔│✘│✔│✘│✔│
┊ ──────────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ a∧(b∧c)∨¬a∧¬(b∧c)≡(a≡b∧c) │✔│✔│✔│✔│✔│✔│✔│✔│
22┊
23┊--|·Arthur·denies·Bernard·is·a·knight:·a∧¬b·∨·¬a∧¬(¬b),·or·a≡¬b
24┊
25┊taut:·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
30┊taut:·¬a∧b∧¬c·⇒·(a≡b∧c)
┊ a │✔│✘│✔│✘│✔│✘│✔│✘│
┊ b │✔│✔│✘│✘│✔│✔│✘│✘│
┊ c │✔│✔│✔│✔│✘│✘│✘│✘│
┊ ────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ ¬a │✘│✔│✘│✔│✘│✔│✘│✔│
┊ ¬a∧b │✘│✔│✘│✘│✘│✔│✘│✘│
┊ ¬c │✘│✘│✘│✘│✔│✔│✔│✔│
┊ ¬a∧b∧¬c │✘│✘│✘│✘│✘│✔│✘│✘│
┊ b∧c │✔│✔│✘│✘│✘│✘│✘│✘│
┊ a≡b∧c │✔│✘│✘│✔│✘│✔│✘│✔│
┊ ────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ ¬a∧b∧¬c⇒(a≡b∧c) │✔│✔│✔│✔│✔│✔│✔│✔│
31┊
32┊--|·and·Arthur's·second·statement
33┊taut:·¬a∧b∧¬c·⇒·(a≡¬b)
┊ a │✔│✘│✔│✘│✔│✘│✔│✘│
┊ b │✔│✔│✘│✘│✔│✔│✘│✘│
┊ c │✔│✔│✔│✔│✘│✘│✘│✘│
┊ ───────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ ¬a │✘│✔│✘│✔│✘│✔│✘│✔│
┊ ¬a∧b │✘│✔│✘│✘│✘│✔│✘│✘│
┊ ¬c │✘│✘│✘│✘│✔│✔│✔│✔│
┊ ¬a∧b∧¬c │✘│✘│✘│✘│✘│✔│✘│✘│
┊ ¬b │✘│✘│✔│✔│✘│✘│✔│✔│
┊ a≡¬b │✘│✔│✔│✘│✘│✔│✔│✘│
┊ ───────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ ¬a∧b∧¬c⇒(a≡¬b) │✔│✔│✔│✔│✔│✔│✔│✔│
34┊
35┊--|·and·is·in·fact·logically·equivalent·to·Arthur's·statements,·taken·together
36┊taut:·¬a∧b∧¬c·≡·(a≡b∧c)∧(a≡¬b)
┊ a │✔│✘│✔│✘│✔│✘│✔│✘│
┊ b │✔│✔│✘│✘│✔│✔│✘│✘│
┊ c │✔│✔│✔│✔│✘│✘│✘│✘│
┊ ───────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ ¬a │✘│✔│✘│✔│✘│✔│✘│✔│
┊ ¬a∧b │✘│✔│✘│✘│✘│✔│✘│✘│
┊ ¬c │✘│✘│✘│✘│✔│✔│✔│✔│
┊ ¬a∧b∧¬c │✘│✘│✘│✘│✘│✔│✘│✘│
┊ b∧c │✔│✔│✘│✘│✘│✘│✘│✘│
┊ a≡b∧c │✔│✘│✘│✔│✘│✔│✘│✔│
┊ ¬b │✘│✘│✔│✔│✘│✘│✔│✔│
┊ a≡¬b │✘│✔│✔│✘│✘│✔│✔│✘│
┊ (a≡b∧c)∧(a≡¬b) │✘│✘│✘│✘│✘│✔│✘│✘│
┊ ───────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ ¬a∧b∧¬c≡(a≡b∧c)∧(a≡¬b) │✔│✔│✔│✔│✔│✔│✔│✔│
37┊
38┊taut:·(X⇒A)∧(X⇒B)·≡·X⇒A∧B
┊ X │✔│✘│✔│✘│✔│✘│✔│✘│
┊ A │✔│✔│✘│✘│✔│✔│✘│✘│
┊ B │✔│✔│✔│✔│✘│✘│✘│✘│
┊ ────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ X⇒A │✔│✔│✘│✔│✔│✔│✘│✔│
┊ X⇒B │✔│✔│✔│✔│✘│✔│✘│✔│
┊ (X⇒A)∧(X⇒B) │✔│✔│✘│✔│✘│✔│✘│✔│
┊ (X⇒A)∧(X⇒B)≡X │✔│✘│✘│✘│✘│✘│✘│✘│
┊ A∧B │✔│✔│✘│✘│✘│✘│✘│✘│
┊ ────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ ((X⇒A)∧(X⇒B)≡X)⇒A∧B │✔│✔│✔│✔│✔│✔│✔│✔│
39┊taut:·(X⇒A)∨(X⇒B)·≡·X⇒A∨B
┊ X │✔│✘│✔│✘│✔│✘│✔│✘│
┊ A │✔│✔│✘│✘│✔│✔│✘│✘│
┊ B │✔│✔│✔│✔│✘│✘│✘│✘│
┊ ────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ X⇒A │✔│✔│✘│✔│✔│✔│✘│✔│
┊ X⇒B │✔│✔│✔│✔│✘│✔│✘│✔│
┊ (X⇒A)∨(X⇒B) │✔│✔│✔│✔│✔│✔│✘│✔│
┊ (X⇒A)∨(X⇒B)≡X │✔│✘│✔│✘│✔│✘│✘│✘│
┊ A∨B │✔│✔│✔│✔│✔│✔│✘│✘│
┊ ────────────────────┼─┼─┼─┼─┼─┼─┼─┼─┤
┊ ((X⇒A)∨(X⇒B)≡X)⇒A∨B │✔│✔│✔│✔│✔│✔│✔│✔│
40┊
41┊
42┊proof:·a≡b∧c·a≡¬b·∴·¬a∧b∧¬c
43┊∙∙A·⏵·A····················[sp]···--|·Simple·Rewrite·via·Sapphire·Normal·Form
44┊∙∙P∧(P⇒Q)·⏵·Q··············[mp]···--|·Modus·Ponens
45┊∙∙¬A∨B·⏴⏵·A⇒B··············[imp]··--|·Implication
46┊∙∙X⇒A∧B·⏴⏵·(X⇒A)∧(X⇒B)·····[cic]··--|·Composition·of·Implication·over·Conjunction
47┊∙∙A⇒B·⏴⏵·¬B⇒¬A·············[cp]···--|·Contrapositive
48┊
49┊∙∙(A⇒B)∧(A⇒¬B)·⏴⏵·¬A·······[cdct]·--|·Contradiction/Explosion
50┊
51┊∙∙¬(A∧B)·⏴⏵·¬A∨¬B··········[dna]··--|·Distribution·of·negation·over·and
52┊
53┊
54┊∙∙a≡¬b·····················[anb] - from assumptions
55┊∙∙¬a⇒b········∵·cp·⇑·······[nab] - A⇒B ⏴⏵ ¬B⇒¬A ⟪A→¬b B→a⟫
56┊
57┊∙∙a≡b∧c····················[abc] - from assumptions
58┊∙∙a⇒b·········∵·cic·⇑ - X⇒A∧B ⏴⏵ (X⇒A)∧(X⇒B) ⟪X→a A→c B→b⟫
59┊∙∙¬a··········∵·cdct·anb·⇑·[na] - (A⇒B)∧(A⇒¬B) ⏴⏵ ¬A ⟪A→a B→¬b⟫
60┊∙∙b···········∵·mp·⇑·nab···[b] - P∧(P⇒Q) ⏵ Q ⟪P→¬a Q→b⟫
61┊
62┊∙∙¬a⇒¬(b∧c)···∵·cp·abc - A⇒B ⏴⏵ ¬B⇒¬A ⟪A→b∧c B→a⟫
63┊∙∙¬(b∧c)······∵·mp·⇑·na - P∧(P⇒Q) ⏵ Q ⟪P→¬a Q→¬(b∧c)⟫
64┊∙∙¬b∨¬c·······∵·dna·⇑ - ¬(A∧B) ⏴⏵ ¬A∨¬B ⟪A→c B→b⟫
65┊∙∙b⇒¬c········∵·imp·⇑ - ¬A∨B ⏴⏵ A⇒B ⟪B→¬c A→b⟫
66┊∙∙¬c··········∵·mp·b·⇑ - P∧(P⇒Q) ⏵ Q ⟪P→b Q→¬c⟫
67┊∙∙¬a∧b∧¬c·····∵·sp·na·b·⇑ - A ⏵ A ⟪A→b∧¬a∧¬c⟫
68┊
69┊
70┊taut:···(A⇒B)∧(A⇒¬B)·≡·¬A
┊ A │✔│✘│✔│✘│
┊ B │✔│✔│✘│✘│
┊ ────────────────┼─┼─┼─┼─┤
┊ A⇒B │✔│✔│✘│✔│
┊ ¬B │✘│✘│✔│✔│
┊ A⇒¬B │✘│✔│✔│✔│
┊ (A⇒B)∧(A⇒¬B) │✘│✔│✘│✔│
┊ ¬A │✘│✔│✘│✔│
┊ ────────────────┼─┼─┼─┼─┤
┊ (A⇒B)∧(A⇒¬B)≡¬A │✔│✔│✔│✔│
─────┴────────────────────────────────────────────────────────────────────────────────────────────────