Demuestra en los siguientes argumentos vlidos, por el mtodo de deduccin natural, que las conclusiones se siguen de las premisas. Utiliza todas las reglasde transformacin vistas previamente.

1. P/.: P

{1} 1. P & Q/.: P

{1} 1. P & ( Q S )/.: P

{1} 1. ( P v Q ) & R/.: ( P v Q )

{1} 1. ( S R ) & ( Q P )/.: S R

{1} 1. ( R & Q ) & ( S v T )/.: R

{1} 1. P{2} 2. Q/.: Q & P

{1} 1. P Q{2} 2. ( R & S )/.: ( P Q ) & ( R & S )

{1} 1. P{2} 2. Q{3} 3. R{4} 4. S/.: [( P & Q ) & R ] & S

{1} 1. T v S {2} 2. T/.: S

{1} 1. ( Q & T ){2} 2. ( Q & T ) v S/.: S

{1} 1. S v ( P Q ){2} 2. S/.: ( P Q )

{1} 1. P v Q{2} 2. P/.: Q

{1} 1. R v ( S v T ) {2} 2. R {3} 3. S/.: T

{1} 1. S Q/.: S Q

{1} 1. ( P & S )/.: P & S

{1} 1. Q/.: Q v { P [ P ( P P ) ] }

{1} 1. Q/.: Q v ( P v S )

{1} 1. Q/.: ( Q v P ) v S

{1} 1. P v R{2} 2. P v R/.: ( R P )

{1} 1. P Q{2} 2. Q v S/.: P S

{1} 1. S R{2} 2. Q S/.: ( Q & R )

{1} 1. Q R{2} 2. Q/.: R

{1} 1. Q R{2} 2. Q/.: Q

{1} 1. Q R{2} 2. Q/.: R

{1} 1. Q R

{2} 2. R/.: Q

{1} 1. ( P & Q ){2} 2. S & ( Q v P )/.: Q P

{1}1. R T{2}2. T Q/.: Q R

{1} 1. Q S{2} 2. P Q/.: S P

{1} 1. Q S/.: S Q

{1} 1. Q S/.: S Q

{1} 1. ( Q Q ) Q /.: Q

{1} 1. R ( R R )/.: R

{1} 1. P ( P P ) /.: ( P P )

{1} 1. P v Q{2} 2. P/.: Q v T

{1} 1. R v ( P v S ){2} 2. R & P/.: S

{1} 1. P v ( R v Q ){2} 2. P & R/.: Q

{1} 1. R{2} 2. T/.: ( R v Q ) & ( T v P )

{1} 1. Q

{2} 2. R{3} 3. ( Q & R ) v P/.: P

{1} 1. Q{2} 2. Q v ( P v T ) /.: ( P v T ) & ( Q v R )

{1} 1. R{2} 2. T v ( P v S ){3} 3. Q & (T v R)/.: ( P v S ) & Q

{1} 1. P Q/.: ( P Q )

{1} 1. Q v ( T & S )/.: Q v ( T & S )

{1} 1. T & [( Q & S ) & P ]/.: T

{1} 1. ( P & Q ) & ( R v T )/.: P

{1} 1. Q{2} 2. P{3} 3. R v S/.: ( Q & P ) & ( R v S )

{1} 1. [ P (R P' )] {2} 2. T{3} 3. [( Q R) S ]/.: T & {[ P (R P' )] & [( Q R' ) S ] }

{1} 1. P & ( Q & R ){2} 2. ( Q & R ) & P/.: Q & P

{1} 1. R{2} 2. Q{3} 3. P/.: Q v S

{1} 1. R{2} 2. Q{3} 3. P

/.: [( Q & P ) & R ] v S

{1} 1. Q/.: Q v {[( R S ) T ] v ( P & P' )}

{1} 1. R/.: ( R v Q ) v P

{1} 1. P v (Q v R ){2} 2. P{3} 3. Q/.: R

{1} 1. P v ( Q v R ){2} 2. ( P & R ) {3} 3. Q & S /.: R

{1} 1. ( P v Q ) {2} 2. P/.: P & Q

{1} 1. P v Q{2} 2. P/.: P & Q

{1} 1. ( T & P )/.: P & T

{1} 1. ( P & T ) & Q/.: Q & ( T & P )

{1} 1. Q v ( R v S )/.: ( R v S ) v Q

{1} 1. ( P v Q ) v ( R v T )/.: ( Q v P ) v ( T v R )

{1} 1. [( S R' ) v ( T R )] v [ P' v ( Q v P )]/.: [ P v ( P v Q )] v [( S R' ) v ( T R )]

{1} 1. P ( R Q ) {2} 2. S/.: ( P & P ) [ R (Q v Q )]

{1} 1. P S/.: [( P v P ) & ( P v P )] S

{1} 1. [( P & P ) v ( P & P )] v [( P & P ) v ( P & P )]/.: P

{1} 1. ( R & P ) & Q /.: R & ( P & Q )

{1} 1. ( R & P ) & P /.: R & P

{1} 1. ( P v Q ) v ( R v S )/.: P v [ Q v ( R v S ) ]

{1} 1. ( P v Q ) v ( R v S )/.: [ ( P v Q ) v R ] v S

{1} 1. P & [ ( Q & S ) & R]/.: [ P & (Q & S ) ] & R

{1} 1. [ ( Q & R ) & P ] & S/.: ( Q & R ) & ( P & S )

{1} 1. [ ( Q & R ) & P ] & S/.: Q & [ R & ( P & S ) ]

{1} 1. Q v ( P v Q) /.: P v Q

{1} 1. P & ( P & R )/.: R & P

{1} 1. ( P v Q ) v ( Q v R )/.: P v ( Q v R )

{1} 1. ( P v Q ) v ( Q v P )/.: P v Q

{1} P v ( Q v R )/.: ( P v Q ) v ( R & R )

{1} 1. [( P v P ) & ( P v P )] v S/.: ( S & S ) v P

{1} 1. ( P v Q ) v P /.: Q v ( Q v P )

{1} 1. P S/.: ( P S )

{1} 1. Q v ( T & S )/.: Q v ( T & S )

{1} 1. T & [( Q & S ) & P ]

/.: T

{1} 1. ( P & Q ) & ( R v T )/.: P

{1} 1. Q{2} 2. P{3} 3. T v S/.: ( Q & P ) & ( T v S )

{1} 1. ( R T ){2} 2. P{3} 3. T' S /.: P & [( R T ) & ( T' S )]

{1} 1. S & ( Q & R ){2} 2. ( Q & R ) & S/.: Q & S

{1} 1. T{2} 2. Q{3} 3. P/.: Q v S

{1} 1. T{2} 2. Q{3} 3. P/.: [( Q & P ) & T ] v S

{1} 1. Q/.: Q v {[( T S ) R ] v ( P & S' )}

{1} 1. T/.: ( T v Q ) v P

{1} 1. P v ( Q v R ){2} 2. P{3} 3. Q/.: R

{1} 1. P v ( Q v R ){2} 2. P & R {3} 3. Q & S /.: R

{1} 1. P v Q {2} 2. P/.: P & Q

{1} 1. P v Q

{2} 2. P/.: Q v S'

{1} 1. ( P v Q ) v R/:. R v ( Q v P )

{1} 1. [( P T ) v R ] & S/:. S & [ R v ( P T )]

{1} 1. ( R v T ) v ( S v Q )/:. ( Q v S ) v ( T v R )

{1} 1. P & [ Q & ( T & S )]/:. [( S & T ) & Q ] & P

{1} 1. [( Q & S ) & T ] & [ S' v ( R v P )]/:. [( P v R ) v S' ] & [ T & ( S & Q )]

{1} 1. { P T ( S v P )]} & [( Q P ) ( T v R )] /:. [( Q P ) R v T )] & {P ( P v S )]}

{1} 1. [( T & P ) & Q ] & ( R & S )/:. ( S & R ) & [ Q & ( P & T )]

{1} 1. [( Q v Q ) & ( Q v Q )]/:. Q

{1} 1. R ( P Q )/:. R ( P v P ) ( Q & Q )

{1} 1. ( R v S ) v ( T & P )/:. [( R v R ) v S ] v [( T & P ) v ( T & P )]

{1} 1. [( P Q ) & ( S v S )] R/:. [(P Q) & S] R

{1} 1. [( T S ) v ( T S )] & [ P v ( Q v Q )]/:. ( T S ) & ( P v Q )

{1} 1. [( Q v Q ) v ( R & R )] & [( Q v Q ) v ( R & R )]/:. Q v R

{1} 1. ( P v Q ) v (S v T)/:. [( P v Q ) v S ] v T

{1} 1. R v [( T P ) v Q ]/:. [ R v ( T P )] v Q

{1} 1. ( P & T ) & ( T R )/:. P & [ T & (T R )]

{1} 1. { P [( Q v R ) v ( S v T )]} T''/:. { P [ Q v [( R v S ) v T ]]} T''

{1} 1. [ Q & ( R & T )] & [ S & ( P & Q' )]/:. { Q & [ R & [ T & ( S & P )]]} & Q'

{1} 1. P & [Q v (S R)]/:. (P & Q) v [P & (S R)]

{1} 1. [ P v (S R )] & [ P v (T Q )]/:. P v [( S R ) & ( T Q )]

{1} 1. R v ( S & P )/:. ( R v S ) & ( R v P )

{1} 1. ( P v Q ) v R/:. ( P v R ) v ( Q v P )

{1} 1. ( P & S ) & ( R & S )/:. P & ( R & S )

{1} 1. P/:. ( P v S ) & ( P v T )

{1} 1. ( P v R ) v S{2} 2. P/:. R v S

{1} 1. [( R v T ) v S ] & Q{2} 2. T/:. (Q & R) v (Q & S)

{1} 1. ( Q S ) S'{2} 2. Q S/.: S'

{1} 1. P ( R & T ){2} 2. P {3} 3. ( R & T ) S/.: S

{1} 1. ( P Q ) ( S R ) {2} 2. S ( P Q ) {3} 3. S/.: R

{1} 1. Q { P [ Q ( P T )]}{2} 2. Q

{3} 3. P/.: T

{1} 1. ( Q P ) T {2} 2. T{3} 3. Q ( Q P )/.: Q

{1} 1. R Q{2} 2. Q/ .: R

{1} 1. [( P Q ) v R ]{2} 2. S [( P Q ) v R ]/.: S

{1} 1. ( S & T ){2} 2. [( P v Q ) R ] ( S & T )/ .: [( P v Q ) R ]

{1} 1. S' (P v T ){2} 2. ( P v T ){3} 3. ( P v T ) S'/ .: ( P v T )

{1} 1. P Q{2} 2. Q R {3} 3. R S{4} 4. S P/ .: P P

{1} 1. ( Q & P ) R{2} 2. R S {3} 3. S T{4} 4. T ( Q & P )/ .: R R

{1} 1. T ( P Q ){2} 2. ( P Q ) ( T v R ){3} 3. P' T {4} 4. [( T v R ) & S ] P'/ .: [( T v R ) & S ] ( T v R )

{1} 1. P Q{2} 2. R S{3} 3. Q v S/.: P v R

{1} 1. P v Q{2} 2. P' Q

{3} 3. P'' P / .: P'' v P'

{1} 1. T ( Q & S ){2} 2. S v S{3} 3. ( Q & S ) S{4} 4. R P {5} 5. P S / .: 7. T v R

{1} 1. R /.: ( P & Q ) S R )

{1} 1. Q{2} 2. R S{3} 3. Q S/.: R

{1} 1. P ( S T ){2} 2. P{3} 3. T Q/.: S Q

{1} 1. ( P Q ) ( S R' ) {2} 2. R T{3} 3. T ( P Q ){4} 4. R{5} 5. ( S R' ) P/ .: P

{1} 1. P ( S v R ){2} 2. R T{3} 3. P{4} 4. S Q'/.: Q' v T

{1} 1. P Q {2} 2. S P {3} 3. Q T{4} 4. T R{5} 5. S v Q/ .: Q v R

{1} 1. P Q{2} 2. S R''{3} 3. P v S{4} 4. Q T{5} 5. R'' ( R & Q' ) / .: T v ( R & Q' )

{1} 1. P Q{2} 2. S S {3} 3. Q T{4} 4. P v S/.: T v S

{1} 1. P v S{2} 2. S S'{3} 3. S' Q'{4} 4. R T{5} 5. P R/.: T v Q'

{1} 1. P T {2} 2. S Q{3} 3. T v Q{4} 4. R P{5} 5. T''' S/.: R v T'''

{1} 1. ( R'' & T ) v Q{2} 2. R S{3} 3. S Q{4} 4. P ( R'' & T ) {5} 5. Q' P/.: Q' v R

{1} 1. P T {2} 2. S Q{3} 3. T v Q{4} 4. P R{5} 5. S R''/.: R v R''

{1} 1. T ( Q & S ){2} 2. S v S{3} 3. ( Q & S ) S{4} 4. R P {5} 5. P S /.: T v R

{1} 1. Q'' R{2} 2. (P & Q ) v S {3} 3. S ( T & R' ){4} 4. ( T & R' ) T{5} 5. ( P & Q ) R{6} 6. S'' T/.: S'' v Q''

{1} 1. P ( Q R )

{2} 2. R {3} 3. P/.: [( P Q ) R] [ P ( R Q )]

{1} 1. Q S{2} 2. T{3} 3. Q v P/.: S v T

{1} 1. P{2} 2. ( R & S ) Q{3} 3. P v Q/.: P v ( R & S )

{1} 1. P ( Q R ){2} 2. R {3} 3. P/.: R Q

{1} 1. P ( Q R){2} 2. R {3} 3. P/.: P ( R Q )

{1} 1. T v S{2} 2. S P{3} 3. T Q /.: Q v R

{2} 1. ( Q v R ) & P/.: [( Q v R ) v P ]

{1} 1. ( Q v R ) v P/.: ( Q & R ) P

{1} 1. ( Q v R ) & P/.: [ (Q & R) v P ]

{1} 1. ( S v Q ){2} 2. ( P v R )/.: S & [( Q & P ) & R]

{1} 1. P v Q{2} 2. P v Q/.: ( Q P )

{1} 1. P Q{2} 2. Q v R/.: P R

{1} 1. S R{2} 2. Q S/.: ( Q & R )

{1} 1. Q R{2} 2. Q/.: R

{1} 1. Q P{2} 2. Q/.: Q

{1} 1. Q P{2} 2. Q/.: P

{1} 1. Q R{2} 2. R/.: Q

{1} 1. ( P & Q ){2} 2. S & ( Q v P )/.: Q P

{1} 1. R T{2} 2. T Q/.: Q R

{1} 1. Q S{2} 2. P Q/.: S P

{1} 1. Q S/.: S Q

{1} 1. Q S/.: S Q

{1} 1. ( Q Q ) Q/.: Q

{1} 1. R ( R R )/.: R

{1} 1. P ( P P )/.: ( P P )

{1} 1. Q{2} 2. P/.: ( R v R ) & Q

{1} 1. P{2} 2. Q /.: ( P v P ) & Q

{1} 1. S ( Q Q ) {2} 2. R S/.: R

{1} 1. ( P & P ) v R{2} 2. R Q /.: Q

{1} 1. Q/.: ( T S ) v ( T S )

{1} 1. R & Q/.: R & {[( S & P ) v T ] & [( S & P ) v T ]}

{1} 1. ( P v R ) v S{2} 2. R{2} 3. P Q/.: S Q

{1} 1. ( R v T ){2} 2. R ( Q S ) /.: S Q

{1} 1. R{2} 2. P{3} 3. S ( R v P )/.: S v Q

{1} 1. ( Q v T ) ( T v Q ){2} 2. S Q/.: S

{1} 1. R v Q'{2} 2. S{3} 3. P{4} 4. ( P' & T ) & Q{5} 5. P' v [ S v ( P v R)]/ Q'

{1} 1. ( P R ){2} 2. T{3} 3. (Q & S )/ ( S & T ) & ( P R )

{1} 1. ( P & Q ) & R{2} 2. T & S/ [ T v ( P' Q' )] & ( P v Q' )

{1} 1. Q S{2} 2. T R{3} 3. Q v T/ ( Q v T ) ( R v T )

{1} 1. P ( T & R ){2} 2. ( R & T ) v ( S v Q )/ P ( Q S )

{1} 1. T R{2} 2. Q S{3} 3. R v S/ T v Q

{1} 1. ( P v T ) ( S & T ){2} 2. ( S & T ) ( Q R ){3} 3. ( S & T ) v ( P v T )/ Q v T

{1} 1. P/ ( P v S ) & ( P v Q )

{1} 1. ( Q v R ) v S{2} 2. Q/ R v S

{1} 1. [( R v T ) v S ] & Q{2} 2. T/ ( Q & R ) v ( Q & S )

{1} 1. P ( S Q ){2} 2. ( S Q ) R{3} 3. T Q'{4} 4. P & Q'/ R & T

{1} 1. R & (S Q)/ ( P v R ) & [ P v ( S v Q )]

{1} 1. R v T {2} 2. R / T

{1} 1. ( P & S )

{2} 2. P/ S

{1} 1. P v Q {2} 2. P & ( R & S){3} 3. R/ Q & S

{1} 1. ( R v Q ) & P{2} 2. ( P v R )/ Q

{1} 1. [( Q & S ) & ( R v T )]{2} 2. ( S v Q )/ R T

{1} 1. P{2} 2. Q/ P & ( Q v S)

{1} 1. ( P v R ) v S{2} 2. R{3} 3. P Q/ S Q

{1} 1. R{2} 2. P ( Q T ){3} 3. R v S/ ( Q T ) v S

{1} 1. R{2} 2. T{3} 3. S ( R v T )/ S v Q

{1} 1. ( R v T ){2} 2. R ( Q S ) / S Q

{1} 1. ( Q v T ) ( T v Q ){2} 2. S Q/ S

{1} 1. P [ S ( R v Q )]{2} 2. (Q S)/:. P v (S & R)

{1} 1. Q v ( R & S ){2} 2. R/:. [ T ( S R )] v Q

{1} 1. P & Q{2}. Q v ( R v S )/:. ( R S) T

{1} 1. ( Q & R ) v ( R Q ){2} 2. [P (Q R)] v T/:. P T

{1} 1. [( P P ) ( Q Q )]{2} 2. [( P & R ) v ( P & R )]/ R Q

{1} 1. ( Q v S ) & ( R T ){2} 2. ( T v P' ) v ( P & S ){3} 3. ( P v S ) & R / P'

{1} 1. (P' & Q' ) v ( Q' P' ){2} 2. [P ( P' Q' )] v R'/ P R'

{1} 1. P [ S ( R v S' )]{2} 2. (S' S)/ P v ( S & R )

{1} 1. S' v ( R & P' ){2} 2. R/ [ R' ( P' R )] v S'

{1} 1. P & R'{2} 2. R' v (R v S)/ ( R S) T

{1} 1. ( P v R ) v S{2} 2. R{3} 3. P Q/ S Q

{1} 1. ( R v T ){2} 2. R ( Q S ) / S Q

{1} 1. R{2} 2. T'{3} 3. S ( R v T' )/ S v Q

{1} 1. ( P v T ) ( T v P ){2} 2. S P

/ S

{1} 1. P ( P P ){2} 2. Q ( Q Q ){3} 3. P ( P P ){4} 4. Q ( Q Q ){5} 5. R ( P Q ) / R

{1} 1. P v ( R v Q ){2} 2. T R{3} 3. S v ( P ( P P )){4} 4. (S T)/ Q

{1} 1. Q v P {2} 2. R v P'{3} 3. S v T{4} 4. R v T{5} 5. Q v P'/ P v S

{1} 1. Q v ( P v S ){2} 2. [ S v ( R' P' )] v T{3} 3. [(S' R) v S' ]{4} 4. {[ S' ( S' & R )] T } & Q/ R' v P'

{1} 1. P{2} 2. {[ P ( S v P' )] & [( Q' & T ) v ( S v P' )]} v {[ P & ( R & R )] v [ P ( R v R )]}/:. ( Q' & T ) v P

{1} 1. P/:. P ({{( Q Q ) & [( T v T ) & P ]} v ( P & P )} v {{ ( Q Q ) & [( T v T ) & ( R v S )]} v P })

{1} 1. P' Q'{2} 2. R' & S'{3} 3. P'' v P'{4} 4. T T'{5} 5. ( Q' R ) ( P' & P' ){6} 6. P'' P{7} 7. S Q''{8} 8. R' T'/:. { T' & [ R' v ( P R )]} & { S' & [ R' v ( T v S )]}

{1} 1. S [( P Q ) & ( R & P' )]{2} 2. ( R v P ) & ( P v R ){3} 3. Q { P [ ( P P ) v { [ Q & ( R v P' )]}]}

{4} 4. {[( Q P ) v ( Q R )] v ( P R )} Q/:. ( R v P' ) & ( P & Q )

{1} 1. [( P' & P ) v T ] [ R ( S & P' )]{2} 2. ( P' v P ) T{3} 3. [ P v ( S & P' ) ] v [ ( P' & T ) v P ) ]{4} 4. ( P' v S )/:. ( R P ) & [ { P & [ T & ( S & P' )]} v { ( S v P' ) & [( P v P ) & P ]}]

{1} 1. S P'{2} 2. ( R S) ( T' & P'' ){3} 3. ( P v T ){4} 4. P' R{5} 5. ( Q' v Q'' ) ( P v T ){6} 6. R' S'/:. ( R S ) & {[( P v T ) v ( T' & P'' ) ] & [( Q' v Q'' ) & ( R'' v S'' )]}

{1} 1. P T{2} 2. Q P{3} 3. ( P' & T ){4} 4. P v R{5} 5. S R{6} 6. P' v Q'{7} 7. Q v P'/:. ( P' & S ) v {[( Q & Q ) ( T & P ) ] & ( T v Q' )}

{1} 1. [( P & Q ) ( R S )] & R {2} 2. S & {( P Q ) [ P ' v ( T v P' )]}{3} 3. ( Q' R ' ) & ( R' & Q' )/ T & Q'

{1} 1. P' S'{2} 2. Q & P''{3} 3. T' v P'{4} 4. T R'{5} 5. ( S' R ) ( T' & P' ){6} 6. T' P{7} 7. S Q''{8} 8. Q R'/:. { R' & [ Q v ( P R )]} & { P'' & [ Q v ( T v S )]}