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Natural Deduction for Predicate Logic

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)

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Natural Deduction for Predicate Logic

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  1. 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

  2. Universal Instantiation (UI) • Used to remove a universal quantifier. • Consistently replace the bound variables with ANYfreevariable or ANY constant. • For example: • (x) Px • Px • (y) (~Cy  Sy) • ~Cz  Sz

  3. (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

  4. Existential Generalization (EG) • Used to add an existential quantifier. • Consistently replace the constants or free variables with ANYboundvariable and add (x). • For example: • Pa • (x) Px • ~Cm  Sm • (y) (~Cy  Sy)

  5. 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)

  6. Universal Generalization (UG) • Used to add a universal quantifier. • Consistently replace the free variables with ANYboundvariable and add (x). • For example: • Px • (x) Px • ~Cy  Sy • (y) (~Cy  Sy) • Dx · ~Tx • (z) (Dz · ~Tz)

  7. 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)

  8. These uses of UG are invalid because of inconsistent replacements. • ~Cx  Sy • (x) (~Cx  Sy) • Dy ~Tz • (x) (Dx  ~Tx)

  9. 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)

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

  11. 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.

  12. 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)

  13. 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

  14. 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

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