Below are some proofs in predicate logic and some examples of how to show a statement is invalid with a model.

 

 

(x)~Kx, (x)(~Kx→~Sx) ├ (x)(Hx v ~Sx)

 

1††††††††† (1)††††††† (x) ~Kx††††††††††††††††††††††††††††††††††† A

2††††††††† (2)††††††† (x)(~K → ~Sx)†††††††††††† A

1††††††††† (3)††††††† ~Ka††††††††††††††††††††††††††††† 1, UE

2††††††††† (4)††††††† ~Ka → ~Sa††††††††††††††††† 2, UE

1, 2†††††† (5)††††††† ~Sa††††††††††††††††††††††††††††† 3, 4 MPP

1, 2†††††† (6)††††††† Ha v ~Sa††††††††††††††††††††† 5, v-I

1, 2†††††† (7)††††††† (x)(Hx v ~Sx)†††††††††††††† 6, UI

†††††††††††

(x)(Gx → ~Fx), (x)(~Fx → ~Hx) ├ (x)(Gx → ~Hx)

 

††††††††††† 1††††††††† (1)††††††† (x)(Gx → ~Fx)††††††††††††† A

††††††††††† 2††††††††† (2)††††††† (x)(~Fx → ~Hx)†††††††††† A

††††††††††† 1††††††††† (3)††††††† Ga → ~ Fa†††††††††††††††††† 1, UE

††††††††††† 2††††††††† (4)††††††† ~Fa → ~Ha††††††††††††††††† 2, UE

††††††††††† 5††††††††† (5)††††††† Ga††††††††††††††††††††††††††††††† A

††††††††††† 1, 5†††††† (6)††††††† ~Fa††††††††††††††††††††††††††††† 3, 5 MPP

††††††††††† 1, 2, 5†† (7)††††††† ~Ha††††††††††††††††††††††††††††† 4, 6 MPP

††††††††††† 1, 2†††††† (8)††††††† Ga → ~Ha††††††††††††††††††† 5, 7 CP

††††††††††† 1, 2 ††††† (9)††††††† (x)(Gx → ~Hx)†††††††††††† 8, UI

 

($x)~(Cx v ~ Rx) ├ ($x)~Cx

 

††††† 1††††††††† (1)††††††† ($x)~(Cx v ~ Rx)††††††††† A

††††† 2††††††††† (2)††††††† ~(Ca v ~Ra)†††††††††††††††† A

††††† 2††††††††† (3)††††††† ~Ca & Ra†††††††††††††††††††† 2, DeMorganís law

††††† 2††††††††† (4)††††††† ~Ca††††††††††††††††††††††††††††† 3, &-E

††††† 2††††††††† (5)††††††† $x(~Cx)††††††††††††††††††††††† 4, $-I

††††† 1††††††††† (6)††††††† $x(~Cx)††††††††††† ††††††††††† 1, 2, 5, $-E

 

 

$xFx"xGx ├ (x)(Fx→Gx)

 

††††††††††† 1††††††††† (1)††††††† $xFx→"xGx††††††††††††††† A

††††††††††† 2††††††††† (2)††††††† Fa††††††††††††††††††††††††††††††† A

††††††††††† 2††††††††† (3)††††††† $xFx†††††††††††††††††††††††††††† 2, EI

††††††††††† 1,2††††††† (4)††††††† "xGx†††††††††††††††††††††††††† 1, 3 MPP

††††††††††† 1,2††††††† (5)††††††† Ga††††††††††††††††††††††††††††††† 4, UE

††††††††††† 1††††††††† (6)††††††† Fa→Ga††††††††††††††††††††††††††††††††††† 2, 5 CP

††††††††††† 1††††††††† (7)††††††† "x(Fx→Gx)†††††††††††††††† 6, UI

 

 

 

 

(P→ ~"xFx)├ ~"x(P & Fx)

 

††††††††††† 1††††††††† (1)††††††† P→~"xFx††††††††††††††††††† A

††††††††††† 2††††††††† (2)††††††† "x(P & Fx)††††††††††††††††† A

††††††††††† 2††††††††† (3)††††††† P & Fa†††††††††††††††††††††††† 2, UE

††††††††††† 2††††††††† (4)††††††† P††††††††††††††††††††††††††††††††† 3, &-E

††††††††††† 1,2††††††† (5)††††††† ~"xFx††††††††††††††††††††††††† 1, 4 MPP

††††††††††† 2††††††††† (6)††††††† Fa††††††††††††††††††††††††††††††† 3, &-E

††††††††††† 2††††††††† (7)††††††† "xFx††††††††††††††††††††††††††† 6, UI

††††††††††† 1,2††††††† (8)††††††† "xFx & ~"xFx†††††††††††† 5, 7

††††††††††† 1††††††††† (9)††††††† ~"x(P & Fx)††††††††††††††† 2, 8 RAA

 

$x(P v Fx) ├ ~P→$xFx

 

††††††††††† 1††††††††† (1)††††††† $x(P v Fx)††††††††††††††††††† A

††††††††††† 2††††††††† (2)††††††† P v Fa††††††††††††††††††††††††† A

††††††††††† 3††††††††† (3)††††††† P††††††††††††††††††††††††††††††††† A

††††††††††† 3††††††††† (4)††††††† ~~P††††††††††††††††††††††††††††† 3, DN

††††††††††† 3††††††††† (5)††††††† ~P→$xFx†††††††††††††††††††† Negated antecedent (51)

††††††††††† 6††††††††† (6)††††††† Fa††††††††††††††††††††††††††††††† A

††††††††††† 6††††††††† (7)††††††† $xFx†††††††††††††††††††††††††††† 6, E-I

††††††††††† 6††††††††† (8)††††††† ~P→$xFx†††††††††††††††††††† Affirmed consequence (50)

††††††††††† 2††††††††† (9)††††††† ~P→$xFx†††††††††††††††††††† 2, 3, 5, 6, 8 v-elim

††††††††††† 1††††††††† (10)††††† ~P→$xFx†††††††††††††††††††† 1,2,9 $-elim

 

~P→$xFx ├ $x(P v Fx)

 

††††††††††† 1††††††††† (1)††††††† ~P→$xFx†††††††††††††††††††† A

††††††††††† 2††††††††† (2)††††††† ~$x(P v Fx)††††††††††††††††† A

††††††††††† 2††††††††† (3)††††††† "x~(P v Fx)†††††††††††††††† 2, QE

††††††††††† 2††††††††† (4)††††††† ~(P v Fa)††††††††††††††††††††† 3, UE

††††††††††† 2††††††††† (5)††††††† ~P & ~Fa†††††††††††††††††††† 4, De Morgan

††††††††††† 2††††††††† (6)††††††† ~P††††††††††††††††††††††††††††††† 5, &-E

††††††††††† 1, 2†††††† (7)††††††† $xFx†††††††††††††††††††††††††††† 1, 6 MPP

††††††††††† 2††††††††† (8)††††††† ~Fa††††††††††††††††††††††††††††† 5, &-E

††††††††††† 2††††††††† (9)††††††† "x~Fx††††††††††††††††††††††††† 8, UI

††††††††††† 2††††††††† (10)††††† ~$xFx†††††††††††††††††††††††††† 9, QE

††††††††††† 1, 2†††††† (11)††††† $xFx & ~$xFx††††††††††††† 7, 10 &-I

††††††††††† 1††††††††† (12)††††† $x(P v Fx)††††††††††††††††††† 2, 11 RAA

 

†††††††††††

 

 

 

 

 

 

 

 

 

Note: To show a sequent is invalid, one needs to find an interpretation where the interpretation makes the premises true but the conclusion false.

 

$x)Fx Fm

 

Let:††† U = N

††††††††††† Fx = x is odd

††††††††††† m = 4

Under this interpretation, ($x)Fx is true (for example 3 satisfies it), but Fm is false, since 4 is not odd. Therefore, this interpretation shows that the sequent is invalid.

 

 

(x)(Fx→Gx), (x)(Hx→Gx) ($x)(Hx & Fx)

 

Let:††††† U = N

††††††††††† Fx = x < 5

††††††††††† Hx = 5 < x < 10

††††††††††† Gx = x < 10

 

Under this interpretation, clearly, (x)(Fx → Gx) and (x)(Hx → Gx) are true, since, for all x, if x is less than 5, its obviously less than 10 and for all x greater than 5 and less than 10, it is true that x is also less than 10.But, there is no x, such that x is both 5<x<10 and x<5, hence ($x)(Hx & Fx) is false under this interpretation.So, we have found an interpretation that makes (x)(Fx → Gx) and (x)(Hx → Gx) true, but ($x)(Hx & Gx) false, in other words we have shown that the sequent is invalid.

 

 

(x)($y)~Lxy (x)~Lxx

 

Let:††††† U = N

††††††††††† Lxy = x is equal to y

 

With this interpretation, it is obvious that (x)($y)~Lxy is true, since, for any x, there is a y (say let y =x+1) such that x is not equal to y.But, (x)~Lxx is false, since (x)~Lxx is equalivent to ~($x)Lxx and we know that there is a natural number (all of them in fact) such that x=x, other words we know that ($x)Lxx is true, so (x)~Lxx must be false. Thus, with this interpretation the sequent is shown to be invalid.