Natural Deduction for Predicate Logic

• Bound Variable: A variable within the scope of a quantifier.– (x) Px– (y) (Zy · Uy)– (z) (Mz ~Nz)

• Free Variable: A variable not within the scope of a quantifier.– Px– Py · ~Qy– ~Az Bz

• Universal Instantiation (UI)– Used to remove a universal quantifier.– Consistently replace the bound

variables with ANY free variable or ANY constant.

– For example:• (x) Px

– Px• (y) (~Cy Sy)

– ~Cz Sz

• (z) (Dz ~Tz)

– Da ~Ta

– These uses of UI are invalid because of inconsistent replacements.

• (x) (~Cx Sx)

– ~Cx Sy

• (z) (Dz ~Tz)

– Da ~Tb

• Existential Generalization (EG)

– Used to add an existential quantifier.

– Consistently replace the constants or free variables with ANY bound variable and add (x).

– For example:

• Pa

– (x) Px

• ~Cm Sm

– (y) (~Cy Sy)

• Dx · ~Tx

– (x) (Dx · ~Tx)

– These uses of EG are invalid because of inconsistent replacements.

• ~Ca Sb

– (x) (~Cx Sy)

• Dy ~Tz

– (x) (Dx ~Tx)

• Universal Generalization (UG)– Used to add a universal quantifier.– Consistently replace the free variables with

ANY bound variable and add (x).– For example:

• Px– (x) Px

• ~Cy Sy– (y) (~Cy Sy)

• Dx · ~Tx– (z) (Dz · ~Tz)

– One may not use UG on statements containing constants. (All of these uses of UG are invalid.)

• La

– (x) Lx

• Gb v ~Hb

– (y) (Gy v ~Hy)

• ~Ne Me

– (z) (~Nz Mz)

– These uses of UG are invalid because of inconsistent replacements.

• ~Cx Sy

– (x) (~Cx Sy)

• Dy ~Tz

– (x) (Dx ~Tx)

• Existential Instantiation (EI)– Used to remove an existential quantifier.– Consistently replace the bound variables

with ANY new constant, i.e. any constant that has not been previously used anywhere in the proof.

– For example:

6.) Pa

7.) (x) Qx

8.) Qb 7 EI (valid)

8.) Qa 7 EI (invalid)

1.) Sm v ~Gm

.

.

. / ~Tk · Wk

8.) (y) (Ny · ~My)

9.) Na · ~Ma 8 EI (valid)

9.) Nm · ~Mm 8 EI (invalid)

9.) Nk · ~Mk 8 EI (invalid)

– These uses of EI are invalid because of inconsistent replacements.

• (x) (~Cx Sy)

– ~Ca Sb

• (x) (Dx ~Tx)

– Dn ~Tm

• When one must both EI and UI to the same constant in a proof, do the EI first.

• N. B.: The rules in Section 8.2 may NOT be used on parts of lines.

– All of these moves are INVALID.

• (x) Zx (x) ~Qx

– Zx ~ Qx

• (z) Lz v (z) Pz

– Ln v Pn

• Tm (y) (~Sy Qy)

– Tm (~Sy Qy)

• N. B.: The rules from 7.1 and 7.2 may NOT be used on statements in which the WHOLE statement is quantified

– These moves are INVALID.

• (x) (Ax Bx)

(x) Ax

(x) Bx

• (x) (Cx v Dx)

(x) ~Cx

(x) Dx

• N. B.: The rules from 7.1 and 7.2 MAY be used on statements in which the parts, not the whole, are quantified.

– These moves are VALID.

• (x) Ax (x) Bx

(x) Ax

(x) Bx

• (x) Dx v (x) Cx

~(x) Dx

(x) Cx