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.) Qb7 EI(valid) 8.) Qa7 EI(invalid)

1.) Sm v ~Gm.../ ~Tk · Wk 8.) (  y) (Ny · ~My) 9.) Na · ~Ma8 EI(valid) 9.) Nm · ~Mm8 EI(invalid) 9.) Nk · ~Mk8 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