Selasa, 06 Oktober 2009

Modus Ponen (MP)

1. Modus Ponen (MP)

pq
p
q

Pembuktian:
[(pq)p]q
~[(~pq)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(~qq) (As)
~pT (Komp)
T (Id)

Kesimpulan :
Argumen
pq
p
q
Argumen sah

2. Modus Tollens (MT)

pq
~q
~p

Pembuktian ;
[(pq)~q]~p
~[(~pq)~q]~p (Imp)
[(p~q)q]~p (DM)
[(pq)(~qq)]~p (Dist)
[(pq)T]~p (Komp)
(pq)~p (Id)
(p~p)(q~p) (Dist)
T(q~p) (Komp)
T (Id)

Kesimpulan :
Argumen
pq
~q
~p
Argumen sah

3. Silogisme

pq
qr
pr

Pembuktian :

[(pq)(qr)](pr)
(pq)[(qr)(pr)] (Eksp)
(pq)[(~qr)(~pr)] (Imp)
(pq)[(q~r)(~pr)] (Imp)
(pq)[(q~r)(r~p)] (Kom)
(pq)[(q~r)r]~p (As)
(pq)[(qr)(~rr)]~p (Dist)
(pq)[(qr)T]~p (Komp)
(pq)(qr)~p (Id)
(~pq)qr~p (Imp)
~(~pq)(qr~p) (Imp)
~(~pq)(~pq)r) (As)
Tr (Komp)
T (id)

Kesimpulan :
Argumen
pq
qr
pr
Argumen sah


4. Distruktif Silogisme (DS)

pq
~p
q

Pembuktian :

[(pq)~p]q
~[(pq)~p]q (Imp)
[(~p~q)p]q (DM)
[(~pp)(~qp)]q (Dist)
[T(~qp)]q (Komp)
(~qp)q (Id)
(~qq)p (As)
Tp (Komp)
T (Id)

Kesimpulan :
Argumen
pq
~p
q
Argumen Sah

5. Konstruktif Delema (KD)

pq(rs)
pr
qs

Pembuktian :

{[(pq)(rs)](pr)}qs
[(~pq)(~rs)(pr)]qs (Imp)
[(p~q)(r~s)(~p~r)](qs) (Imp)
[(p~q)(~p~r)(r~s)](qs) (As)
[(p~q)(~p~r)][(r~s)](qs)] (As)
[{(p~q)~p}{(p~q)~r}][(r~s)](qs)] (Dist)
[{(p~q)~p}{(p~q)~r}][{(r~s)]s}q] (As)
[{(p~p)(~q~p)}{(p~r)(~q~r)}][{(rs)(~ss)}q] (Dist)
[{T(~q~p)}{(p~r)(~q~r)}][{(rs)T}q] (Komp)
[{(~q~p){(p~r)(~q~r)}][{(rs)q] (Id)
[{(~q~p){(p~r)(~q~r)}q](rs)] (As)
[{(~q~p)q}{(p~r)q}{(~q~r)q}][(rs)] (Dist)
[{(~qq)~p}(pq~r){(~qq)~r}][(rs)] (As)
[(T~p)(pq~r)(T~r)][(rs)] (Komp)
[T(pq~r)T](rs) (Id)
(pq~r)(rs) (Id)
(r~r)(pqs) (As)
T(pqs) (Komp)
T (Id)

Kesimpulan :
Argumen
pq(rs)
pr
qs
Argumen sah

6. Destruktif Dilema (DD)

pq(rs)
(~q~s)
(~p~r)

Pembuktian:

{[(pq)(rs](~q~s)(~p~r)
[(~pq)(~rs)(~q~s)](~p~r) (Imp)
[(p~q)(r~s)(qs)] (~p~r) (Imp)
[(p~q)(qs)(r~s) (~p~r)] (As)
[(p~q)(qs)][(r~s) (~p~r)] (As)
[{(p~q)q}{(p~q)s}] ∨ [{(r~s)(~p~r)] (Dis)
[{(p~q)q}{(p~q)s}] ∨ [{(r~s)~r}~p] (As)
[{(pq)(~qq)}{(ps)(~qs)}][{(r~r)(~s~r)}~p] (Dis)
[{(pq)T}{(ps)(~qs)}][{T(~s~r)}~p] (Komp)
[(pq)(ps)(~qs)][(~s~r)~p] (Id)
[(pq)(ps)(~qs)p](~s~r) (As)
[{(pq)~p}{(ps)~p}{(qs)~p}](~s~r) (Dis)
[{(p~p)q}{(p~p)s}(qs~p}](~s~r) (As)
[(Tq)(Ts)(qs~p}](~s~r) (Komp)
[(TT(qs~p}](~s~r) (Id)
(qs~p}(~s~r) (Id)
(s~s)(~pq~r) (As)
T(~pq~r) (Komp)
T (Id)

Kesimpulan :
Argumen
pq(rs)
(~q~s)
(~p~r)
Argumen sah.

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