TUGAS 3 PDM

BUKTI KESAHAN ARGUMEN

1) Modus Ponens (MP)
[(p → q) ∧ p] → q
≡ ∼[(∼p ∨ q) ∧ p] ∨ q (Imp)
≡ [(p ∧ ∼q) ∨ ∼p] ∨ q (Komp. DM)
≡ [(p ∨ ∼p) ∧ (∼p ∨ ∼q)] ∨ q (Dist)
≡ [T ∧ (∼p ∨ ∼q)] ∨ q (Komp)
≡ (∼p ∨ ∼q) ∨ q (Id)
≡ ∼p ∨ (∼q ∨ q) (Asso)
≡ ∼p ∨ T (Komp)
≡ T (Id)


2) Modus Tolens (MT)
[(p → q) ∧ ∼q] → ∼p
≡ [(∼p ∨ q) ∧ ∼q] → ∼p (Imp)
≡ (∼p ∧ ∼q) ∨ (q ∧ ∼q) → ∼p (Dist)
≡ (∼p ∧ ∼q) ∨ F → ∼p (Komp)
≡ (∼p ∧ ∼q) → ∼p (Id)
≡ ∼(∼p ∧ ∼q) ∨ ∼p (Imp)
≡ p ∨ q ∨ ∼p (DM)
≡ p ∨ (q ∨ ∼p) (Asso)
≡ p ∨ (∼p ∨ q) (Komu)
≡ (p ∨ ∼p) ∨ q (Asso)
≡ T ∨ q (Komp)
≡ T (Id)

3) Silogisme
[(p → q) ∧ (q → r)] → (p → r)
≡ (p → q) → [(q → r)→ (p → r)] (Eksp)
≡ (p → q) → [(∼q ∨ r)→ (∼p ∨ r)] (Imp)
≡ (p → q) → [(q ∧ ∼r) ∨ (∼p ∨ r)] (Imp)
≡ (p → q) → [(q ∧ ∼r) ∨ (r ∨ ∼p)] (Kom)
≡ (p → q) → [(q ∧ ∼r) ∨ r] ∨ ∼p (Asso)
≡ (p → q) → [(q ∨ r) ∧ (∼r ∨ r)] ∨ ∼p (Dist)
≡ (p → q) → [(q ∨ r) ∧ T] ∨ ∼p (Komp)
≡(p → q) → (q ∨ r) ∨ ∼p (Id)
≡(∼p ∨ q) → q ∨ r ∨ ∼p (Imp)
≡∼(∼p ∨ q) ∨ (q ∨ r ∨ ∼p) (Imp)
≡∼(∼p ∨ q) ∨ (∼p ∨ q ) ∨ r (Asso)
≡ T ∨ r (komp)
≡ T (id)


4) Distruktif Silogisme (DS)
(p ∨ q) ∧ ∼p → q
≡ (p ∧ ∼p) ∨ (q ∧ ∼p) → q (Dist)
≡ F ∨ (q ∧ ∼p) → q (Komp)
≡ (q ∧ ∼p) → q (Id)
≡ ∼(q ∧ ∼p) ∨ q (Imp)
≡ (∼q ∨ p) ∨ q (DM)
≡ ∼q ∨ (p ∨ q) (Komu)
≡ (∼q ∨ q) ∨ p (Asso)
≡ T ∨ p (Komp)
≡ T (Id)

5) Konstruktif Dilema (KD)
{[(p → q) ∧ (r → s)] ∧ (p ∨ r)} → (q ∨ s)
≡ [(∼p ∨ q) ∧ (∼r ∨ s)] ∧ (p ∨ r)} → (q ∨ s) (Imp)
≡ [(p ∧ ∼q) ∨ (r ∧ ∼s)] ∨ (∼p ∧ ∼r) ∨ (q ∨ s) (Imp)
≡ [(p ∧ ∼q) ∨ (∼p ∧ ∼r)] ∨ (r ∧ ∼s) ∨ (q ∨ s) (Asso)
≡ [(p ∧ ∼q) ∨ (∼p ∧ ∼r)] ∨ [(r ∧ ∼s) ∨ (q ∨ s)] (Asso)
≡ [{(p ∧ ∼q) ∨ ∼p }∧ {(p ∧∼q) ∨ ∼r}] ∨ [(r ∧ ∼s) ∨ (q ∨ s)] (Dist)
≡ [{(p ∧ ∼q) ∨ ∼p }∧ {(p ∧∼q) ∨ ∼r}] ∨ [{(r ∧ ∼s) ∨ s} ∨ q)] (Asso)
≡ [{(p ∨ ∼p) ∧ (∼q ∨ ∼p)}∧ {(p ∨∼r) ∧ (∼q ∨∼r)}] ∨[{(r ∨ s) ∧ (∼s ∨ s)} ∨ q] (Dist)
≡ [{T ∧ (∼q ∨ ∼p)}∧ {(p ∨∼r) ∧ (∼q ∨∼r)}] ∨[{(r ∨ s) ∧ T} ∨ q] (Komp)
≡ [{(∼q ∨ ∼p)}∧ {(p ∨∼r) ∧ (∼q ∨∼r)}] ∨[{(r ∨ s)} ∨ q] (Id)
≡ [{(∼q ∨ ∼p)}∧ {(p ∨∼r) ∧ (∼q ∨∼r)}∨ q] ∨ (r ∨ s) (Asso)
≡ [{(∼q ∨ ∼p)}∨ q} ∧ {(p ∨∼r) ∨ q} ∧ {(∼q ∨∼r)∨ q}] ∨ (r ∨ s) (Dist)
≡ [{(∼q ∨ q) ∨ ∼p} ∧ (p ∨ q) ∨ ∼r) ∧ {(∼q ∨ q) ∨ ∼r}] ∨ (r ∨ s) (Asso)
≡ [T ∨ ∼p ∧ (p ∨ q) ∨ ∼r) ∧ (T ∨ ∼r)] ∨ (r ∨ s) (Komp)
≡ [T ∧ (p ∨ q ∨ ∼r) ∧ T] ∨ (r ∨ s) (Id)
≡ (p ∨ q ∨ ∼r) ∨ (r ∨ s) (Id)
≡ (r ∨ ∼r) ∨ (p ∨ q ∨ s) (Asso)
≡ T ∨ (p ∨ q ∨ s) (Komp)
≡ T (Id)

6) Distruktif Dilema (DD)
{[(p → q) ∧ (r → s)] ∧ (∼q ∨ ∼s)} → (∼p ∨ ∼r )
≡ [(∼p ∨ q) ∧ (∼r ∨ s)] ∧ (∼q ∨ ∼s)] → (∼p ∨ ∼r ) (Imp)
≡ [(p ∧ ∼q) ∧ (r ∧∼s) ∨ (q ∧ s)] ∨ (∼p ∨ ∼r ) (Imp)
≡ [(p ∧ ∼q) ∧ (q ∧ s) ∨ (r ∧ ∼s) ∨ (∼p ∨ ∼r )] (Asso)
≡ [(p ∧ ∼q) ∧ (q ∧ s)] ∨ [(r ∧ ∼s) ∨ (∼p ∨ ∼r )] (Asso)
≡ [{(p ∧ ∼q) ∨ q} ∧ {(p ∧ ∼q) ∨ s)}] ∨ [(r ∧ ∼s) ∨ (∼p ∨ ∼r )] (Dist)
≡ [{(p ∧ ∼q) ∨ q} ∧ {(p ∧ ∼q) ∨ s)}] ∨ [{(r ∧ ∼s) ∨∼r}∨∼p] (Asso)
≡ [{(p ∨ q) ∧ (∼q ∨ q)} ∧ {(p ∨ s) ∧ (∼q ∨ s)}] ∨ [{(r ∨∼r) ∧ (∼s ∨∼r)}∨∼p] (Dist)
≡ [{(p ∨ q) ∧ T} ∧ {(p ∨ s) ∧ (∼q ∨ s)}] ∨ [{T ∧ (∼s ∨∼r)}∨∼p] (Komp)
≡ [(p ∨ q) ∧ (p ∨ s) ∧ (∼q ∨ s)] ∨ [(∼s ∨∼r) ∨∼p] (Id)
≡ [(p ∨ q) ∧ (p ∨ s) ∧ (∼q ∨ s) ∨ ∼p] ∨ (∼s ∨∼r) (Asso)
≡ [{(p ∨ q) ∨ ∼p} ∧ {(p ∨ s) ∨ ∼p} ∧ {(∼q ∨ s) ∨ ∼p}] ∨ (∼s ∨∼r) (Dist)
≡ [{(p ∨ ∼p) ∨ q} ∧ {(p ∨ ∼p)∨ s} ∧ (∼q ∨ s ∨∼p)] ∨ (∼s ∨∼r) (Asso)
≡ [(T ∨ q) ∧ (T∨ s) ∧ (∼q ∨ s ∨∼p)] ∨ (∼s ∨∼r) (Komp)
≡ [T ∧ T ∧ (∼q ∨ s ∨∼p)] ∨ (∼s ∨∼r) (Id)
≡ (∼q ∨ s ∨∼p) ∨ (∼s ∨∼r) (Id)
≡ ( s ∨∼s) ∨ (∼p ∨∼q ∨∼r) (Asso)
≡ T ∨ (∼p ∨∼q ∨∼r) (Komp)
≡ T (Id)