Corsi, Giovanna

Almae Matris Studiorum Campus

Esempi di derivazioni nel calcolo predicativo di deduzione naturale

Esempi di derivazione nel calcolo predicativo diDeduzione NaturaleDecember 7, 2016(1) ∀x(A(x) ∧B(x)) → ∀xA(x) ∧ ∀xB(x)[∀x(A(x) ∧B(x))]1E∀A(c) ∧B(c) ∧EA(c)I∀∀xA(x)[∀x(A(x) ∧B(x))]1E∀A(c) ∧B(c) ∧EB(c)I∀∀xB(x) ∧I∀xA(x) ∧ ∀xB(x)I→1∀x(A(x) ∧B(x)) → ∀xA(x) ∧ ∀xB(x)(2) ∀xA(x) ∧ ∀xB(x) → ∀x(A(x) ∧B(x))[∀xA(x) ∧ ∀xB(x)]1 ∧E∀xA(x)E∀A(c)[∀xA(x) ∧ ∀xB(x)]1 ∧E∀xB(x)E∀B(c) ∧IA(c) ∧B(c)I∀∀x(A(x) ∧B(x))I→1∀xA(x) ∧ ∀xB(x) → ∀x(A(x) ∧B(x))1(3) ∃x(A(x) ∨B(x)) → ∃xA(x) ∨ ∃xB(x)[A(c)]1I∃∃xA(x)∨I∃xA(x) ∨ ∃xB(x)[B(c)]1I∃∃xB(x)∨I∃xA(x) ∨ ∃xB(x) [A(c) ∨B(c)]3E∨1∃xA(x) ∨ ∃xB(x) [∃x(A(x) ∨B(x))]4E∃3∃xA(x) ∨ ∃xB(x)I→4∃x(A(x) ∨B(x))→ ∃xA(x) ∨ ∃xB(x)(4) ∃xA(x) ∨ ∃xB(x) → ∃x(A(x) ∨B(x))[A(c)]1 ∨IA(c) ∨B(c)I∃∃x(A(x) ∨B(x)) [∃A(x)]3E∃1∃x(A(x) ∨B(x))[B(c)]2 ∨IA(c) ∨B(c)I∃∃x(A(x) ∨B(x)) [∃B(x)]3E∃2∃x(A(x) ∨B(x)) [∃xA(x) ∨ ∃xB(x)]4E∨3∃x(A(x) ∨B(x))I→4∃xA(x) ∨ ∃xB(x) → ∃x(A(x) ∨B(x))(5) ∃x(A(x) ∧B(x)) → ∃xA(x) ∧ ∃xB(x)[A(c) ∧B(c)]1 ∧EA(c)I∃∃xA(x) [∃x(A(x) ∧B(x))]3E∃1∃xA(x)[A(d) ∧B(d)]2 ∧EB(d)I∃∃xB(x) [∃x(A(x) ∧B(x))]3E∃2∃xB(x) ∧I∃xA(x) ∧ ∃xB(x)I→3∃x(A(x) ∧B(x)) → ∃xA(x) ∧ ∃xB(x)(6) ∀xA(x) ∨ ∀xB(x) → ∀x(A(x) ∨B(x))2[∀xA(x)]1E∀A(c) ∨IA(c) ∨B(c)I∀∀x(A(x) ∨B(x))[∀xB(x)]1E∀B(c) ∨IA(c) ∨B(c)I∀∀x(A(x) ∨B(x)) [∀xA(x) ∨ ∀xB(x)]2E∨∀x(A(x) ∨B(x))I→2∀xA(x) ∨ ∀xB(x) → ∀x(A(x) ∨B(x))(7) A ∧ ∃xB(x) → ∃x(A ∧B(x)) per x non libera in A[A ∧ ∃xB(x)]2 ∧EA [B(c)]1A ∧B(c)I∃∃x(A ∧B(x))[A ∧ ∃xB(x)]2 ∧E∃xB(x)E∃1∃x(A ∧B(x))I→2A ∧ ∃xB(x) → ∃x(A ∧B(x))(8) ∀x(A ∨B(x)) → A ∨ ∀xB(x)) per x non libera in A∅...B ∨ ¬BB1I∨∀xA ∨B[¬B]1[∀x(A(x) ∨B)]2E∀A(x) ∨Bmodus tollendo ponensA(x)I∀∀xA(x)I∨∀xA ∨BE1∨∀xA ∨BI→2∀x(A(x) ∨B) → (∀xA ∨B)(9) ∀x(A→ B(x)) → (A→ ∀xB(x)) per x non libera in A3[∀x(A→ B(x))]2E∀A→ B(b) [A]1E→B(b)I∀∀xB(x)I→1A→ ∀xB(x)I→2∀x(A→ B(x)) → (A→ ∀xB(x))(10) (A→ ∀xB(x)) → ∀x(A→ B(x)) per x non libera in A[A]1 [A→ ∀xB(x)]2E→∀xB(x)E∀B(b)I→1A→ B(b)I∀∀x(A→ B(x))I→2∀x(A→ B(x)) → (A→ ∀xB(x))(11) ∃x(A→ B(x)) → (A→ ∃xB(x)) per x non libera in A[A]2 [A→ B(c)]1E→B(c)I∃∃xB(x) [∃x(A→ B(x))]3E∃1∃xB(x)I→2A→ ∃xB(x)I→3∃x(A→ B(x)) → (A→ ∃xB(x))(12) (A→ ∃xB(x)) → ∃x(A→ B(x)) per x non libera in A4A ∨ ¬A[¬A]3 [A]1 ¬iB(c)I→1A→ B(c)I∃∃x(A→ B(x))[B(d)]2I→A→ B(d)I∃∃x(A→ B(x))[A]3 [A→ ∃xB(x)]4E→∃xB(x)E∃2∃x(A→ B(x))E3∨∃x(A→ B(x))I→4(A→ ∃xB(x)) → ∃x(A→ B(x))(13) ∃x(A(x) → B) → (∀xA(x) → B) per x non libera in B[∀xA(x)]2E∀A(c) [A(c) → B]1E→B [∃x(A(x) → B)]3E∃1BI→2∀xA(x) → BI→3∃x(A(x) → B) → (∀xA(x) → B)(14) ∀x(A(x) → B) → (∃xA(x) → B) per x non libera in B[∀x(A(x) → B)]3E∀A(c) → B [A(c)]1E→B [∃xA(x)]2E∃1BI→2∃xA(x) → BI→3∀x(A(x) → B) → (∃xA(x) → B)(15) (∃xA(x) → B) → ∀x(A(x) → B) per x non libera in B5[A(d)]1I∃∃xA(x) [∃xA(x) → B]2E→BI→1A(d) → BI∀∀x(A(x) → B)I→2(∃xA(x) → B) → ∀x(A(x) → B)(16) ∀x(A(x) ↔ B(x)) → (∀xA(x) ↔ ∀xB(x))[∀x(A(x) ↔ B(x))]3E↔∀x(A(x) → B(x)E∀A(c) → B(c)[∀xA(x)]1E∀A(c)E→B(c)I∀∀xB(x)I→1∀xA(x) → ∀xB(x)[∀x(A(x) ↔ B(x))]3E↔∀x(B(x) → A(x)E∀B(c) → A(c)[∀xB(x)]2E∀B(c)E→A(c)I∀∀xA(x)I→2∀xB(x) → ∀xA(x)I↔∀xA(x) ↔ ∀xB(x)I→3∀x(A(x) ↔ B(x)) → (∀xA(x) ↔ ∀xB(x))(17) ∀x(A(x) ↔ B(x)) → (∃xA(x) ↔ ∃xB(x))[∀x(A(x)↔ B(x))]5E∀A(c)↔ B(c)E↔A(c)→ B(c) [A(c)]1E→B(c)I∃∃xB(x) [∃xA(x)]2E∃1∃xB(x)I→2∃xA(x)→ ∃xB(x)[∀x(A(x)↔ B(x))]5E∀A(c)↔ B(c)E↔B(c)→ A(c) [B(c)]3E→A(c)I∃∃xA(x) [∃xA(x)]4E∃3∃xA(x)I→4∃xB(x)→ ∃xA(x)I↔(∃xA(x)↔ ∃xB(x))I→5∀x(A(x)↔ B(x))→ (∃xA(x)↔ ∃xB(x))6(18) ∃x∃y(x, y) → ∃y∃x(x, y)[A(c, d)]1I∃∃xA(x, d)I∃∃y∃xA(x, y) [∃yA(c, y)]2E∃1∃y∃xA(x, y) [∃x∃yA(x, y)]3E∃2∃y∃x(x, y)I→3∃x∃y(x, y) → ∃y∃x(x, y)(19) ∃x∀yA(x, y) → ∀y∃xA(x, y)[∀yA(c, y)]1E∀A(c, d)I∃∃xA(x, d) [∃x∀yA(x, y)]2E∃1∃xA(x, d)I∀∀y∃xA(x, y)I→2∃x∀yA(x, y) → ∀y∃xA(x, y)(20) ∀x(A(x) → B(x)) → (∀xA(x) → ∀xB(x))[∀x(A(x) → B(x))]2E∀A(d) → B(d)[∀xA(x)]1E∀A(d)E→B(d)I∀∀xB(x)I→1∀xA(x) → ∀xB(x)I→2∀x(A(x) → B(x)) → (∀xA(x) → ∀xB(x))(21) ∀x(A(x) → B(x)) → (∃xA(x) → ∃xB(x))7[∀x(A(x) → B(x))]3E∀A(c) → B(c) [A(c)]1E→B(c)I∃∃xB(x) [∃xA(x)]2E∃1∃xB(x)I→2∃xA(x) → ∃xB(x)I→3∀x(A(x) → B(x)) → (∃xA(x) → ∃xB(x))(22) ¬∀xA(x) → ∃x¬A(x)[¬A(c)]1I∃∃x¬A(x) [¬∃x¬A(x)]2 ¬k1A(c)I∀∀xA(x) [¬∀xA(x)]3 ¬k2∃x¬A(x))I→3¬∀xA(x) → ∃x¬A(x)(23) Interdefinibilita` dei quantificatori 1. ∀xP (x) → ¬∃x¬P (x)[¬P (a)]2[∀xP (x)]1I∀P (a) ¬m1¬∀xP (x) [∃x¬P (x)]3E∃2¬∀xP (x)I→3∃¬P (x) → ¬∀xP (x)Cp∀xP (x) → ¬∃x¬P (x)(24) Interdefinibilita` dei quantificatori 2. ¬∃x¬P (x) → ∀xP (x)8[¬P (a)]1I∃∃x¬P (x) [¬∃x¬P (x)]2 ¬k1P (a)I∀∀xP (x)I→2¬∃x¬P (x) → ∀xP (x)(25) ¬∀x¬A(x) → ∃xA(x)[A(c)]1I∃∃xA(x) [¬∃xA(x)]2 ¬m1¬A(c)I∀∀x¬A(x) [¬∀x¬A(x)]3 ¬k2∃xA(x)I→3¬∀x¬A(x) → ∃xA(x)(26) ∃xA(x) → ¬∀x¬A(x)[∀x¬A(x)]1I∀¬A(c) [A(c)]2 ¬m1¬∀x¬A(x) [∃xA(x)]3E∃2¬∀x¬A(x)I→3∃xA(x) → ¬∀x¬A(x)(27) ¬∃xA(x) → ∀x¬A(x)9[A(c)]1I∃∃xA(x) [¬∃xA(x)]2 ¬m1¬A(c)I∀∀x¬A(x)I→2¬∃xA(x) → ∀x¬A(x)(28) ∀x¬A(x) → ¬∃xA(x)[∀x¬A(x)]1E∀¬A(c) [A(c)]2 ¬m1¬∀x¬A(x) [∃xA(x)]3E∃2¬∀x¬A(x) [∀x¬A(x)]4 ¬m3¬∃xA(x)I→4∀x¬A(x) → ¬∃xA(x)(29) ∃x¬A(x) → ¬∀xA(x)[∀xA(x)]1E∀A(c) [¬A(c)]2 ¬m1¬∀xA(x) [∃x¬A(x)]3E∃2¬∀xA(x)I→3∃x¬A(x) → ¬∀xA(x)(30) (∀xA(x) → B) → ∃x(A(x) → B) per x non libera in B10[¬A(c)]2 [A(c)]1 ¬iBI→1A(c) → BI∃∃x(A(x) → B)[∀xA(x) → B]4¬B → ¬∀xA(x) [¬B]3E→¬∀xA(x)Ded(22)∃x¬A(x)E∃2∃x(A(x) → B)[B]3I→A(d) → BI∃∃x(A(x) → B) ¬B ∨BE3∨∃x(A(x) → B)I→4(∀xA(x) → B) → ∃x(A(x) → B)(31) ∃x(A→ ∀xA) Esistenza dell’ultimo∀xA ∨ ¬∀xA[∀xA]5I→A(a/x)→ ∀xAI∃∃x(A→ ∀xA)[¬A(a/x)]3I∃∃x¬A [¬∃x¬A]4¬k3A(a/x)I∀∀xA [¬∀xA]5¬k4∃x¬A[¬A]2 [A]1¬i∀xAI→1A→ ∀xAI∃∃x(A→ ∀xA)E∃2∃x(A→ ∀xA)E∃5∃x(A→ ∀xA)(32) ∀xA→ ¬∃x¬A∃x¬A3∀xA1E∀A ¬A2 ¬m1¬∀xAE∃2¬∀xA ¬m3¬∃x¬AI→1∀xA→ ¬∃x¬A(33) (∀xA(x) → B) → ∃x(A(x) → B) per x non libera in B.11Prima dimostrazione∃x¬A(x) ∨ ¬∃x¬A(x)[A(c)]1[¬A(c)]2¬i∀xA(x) [∀xA(x)→ B]4E→BI→1A(c)→ BI∃∃x(A(x)→ B) [∃x¬A(x)]3E∃2∃x(A(x)→ B)[¬A(c)]5I∃∃x¬A(x) [¬∃x¬A(x)]3¬k5A(c)I∀∀xA(x) [∀xA(x)→ B]6E→BI→A(c)→ BI∃∃x(A(x)→ B)E3∨∃x(A(x)→ B)I→6(∀xA(x)→ B)→ ∃x(A(x)→ B)Seconda dimostrazioneB ∨ ¬B[B]5I→A(c)→ BI∃∃x(A(x)→ B)[¬A(c)]1I∃∃x¬A(x) [¬∃x¬A(x)]2¬k1A(c)I∀∀xA(x)[∀xA(x)→ B]6 [∀xA(x)]7E→B [¬B]5¬m7¬∀xA(x)¬k2∃x¬A(x)[¬A(c)]4 [A(c)]3¬iBI→3A(c)→ BI∃∃x(A(x)→ B)E∃4∃x(A(x)→ B)E5∨∃x(A(x)→ B)I→6(∀xA(x)→ B)→ ∃x(A(x)→ B)(34) ∃xA(x) → ¬∀x¬A(x)[∀x¬A(x)]1E∀¬A(c) [A(c)]2 ¬m1¬∀x¬A(x) [∃xA(x)]3E∃2¬∀x¬A(x)I→3∃xA(x) → ¬∀x¬A(x)12(35) ` ∀x(A(x) → B) → (∃xA→ B) ove x non occorre in B∃xA[A(c)]1∀x(A(x) → B)E∀A(c) → BE→BE∃1BI seguenti teoremi della logica classica NON sono teoremi della logicaintuizionista¬∀xA→ ∃x¬A¬∀x¬A→ ∃xA∀x(A(x) ∨B) → ∀xA(x) ∨B, dove x non occorre in B(A→ ∃xB(x)) → ∃x(A→ B(x)), dove x non occorre in A(∀xA(x) → B) → ∃x(A(x) → B) , dove x non occorre in B∃x(A(x) → ∀yA(y))¬¬∃xA(x) → ∃x¬¬A(x)∀x¬¬A(x) → ¬¬∀xA(x)13

http://campus.unibo.it/261608/1/nat_deduz_pred_esempi.pdf

Esempi di derivazioni nel calcolo predicativo di deduzione naturale

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Almae Matris Studiorum Campus

Almae Matris Studiorum Campus