8.2 Using the Rule of Inference

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Presentation transcript:

8.2 Using the Rule of Inference Predicate Logic 8.2 Using the Rule of Inference

Using Rules of Inference (x)(Ax → Bx) (x)(Bx → Cx) / (x)(Ax → Cx) Ax → Bx 1, UI Bx → Cx 2, UI Ax → Cx 2,3, HS (x)(Ax → Cx) 5, UG

Using Rules of Inference (x)(Bx → Cx) (Ex)(Ax ● Bx) / (Ex)(Ax ● Cx) Am ● Bm 2, EI Bm → Cm 1, UI Am 3, Simp Bm ● Am 3, Com Bm 6, Simp Cm 4,7, MP Am ● Cm 5,8, Conj (x)(Ax ● Cx) 9, UG

Using Rules of Inference (x)(Ax → Bx) ~Bm / (Ex)~Ax Am → Bm 1, UI ~Am 2,3, MT (Ex)~Ax 4, EG

Using Rules of Inference (x)[Ax → (Bx v Cx)] Ag ● ~Bg / Cg Ag → (Bg v Cg) 1, UI Ag 2, Simp Bg v Cg 3,4, MP ~Bg ● Ag 2, Com ~Bg 6, Simp Cg 5,7, DS

Using Rules of Inference (x)[Jx → (Kx ● Lx)] (Ey)~Ky / (Ez)~Jz ~Km 2, EI Jm → (Km ● Lm) 1, UI ~Km v ~Lm 3, Add ~(Km ● Lm) 6, DM ~Jm 4,6, MT (Ez)~Jz 7, EG

Using Rules of Inference (x)[Ax → (Bx v Cx) (Ex)(Ax ● ~Cx) / (Ex)Bx Am ● ~Cm 2, EI Am → (Bm v Cm) 1, UI Am 3, Simp Bm v Cm 4,5, MP Cm v Bm 6, Com ~Cm ● Am 3, Com ~Cm 8, Simp Bm 7,9, DS (Ex)Bx 10, EG

Using Rules of Inference (x)(Ax → Bx) Am ● An / Bm ● Bn Am → Bm 1, UI Am 2, Simp Bm 3,4, MP (Ex)Bx 5, EG Bn 6, EI Bm ● Bn 5,7, Conj

Using Rules of Inference (x)(Ax → Bx) Am v An / Bm v Bn Am → Bm 1, UI An → Bn 1, UI (Am → Bm) ● (An → Bn) 2,3, Conj Bm v Bn 2,5, CD

Using Rules of Inference (x)(Bx v Ax) (x)(Bx → Ax) / (x)Ax Bx v Ax 1, UI Bx → Ax 2, UI Ax v Bx 3, Com ~~Ax v Bx 5, DN ~Ax → Bx 6, DM ~Ax → Ax 4,7, HS ~~Ax v Ax 8, DM Ax v Ax 9, DN Ax 10, Taut (x)Ax 11, UG

Using Rules of Inference (x)[(Ax ● Bx) → Cx] (Ex)(Bx ● ~Cx) / (Ex)~Ax Bm ● ~Cm 2, EI (Am ● Bm) → Cm 1, UI ~Cm ● Bm 3, Com ~Cm 5, Simp ~(Am ● Bm) 4,6, MT Bm 3, Simp ~Am v ~Bm 7, DM ~~Bm 8, DN ~Bm v ~Am 9, Com ~Am 10,11, DS (Ex)~Ax 12, EG

Using Rules of Inference (Ex)Ax → (x)(Bx → Cx) Am ● Bm / Cm Am 2, Simp (Ex)Ax 3, EG (x)(Bx → Cx) 1,4, MP Bm → Cm 5, UI Bm ● Am 2, Com Bm 7, Simp Cm 6,8, MP

Using Rules of Inference (Ex)Ax → (x)Bx (Ex)Cx → (Ex)Dx An ● Cn / (Ex)(Bx ● Dx) An 3, Simp (Ex)Ax 4, EG (x)Bx 1,5, MP Cn ● An 3, Com Cn 7, Simp (Ex)Cx 8, EG (Ex)Dx 2,9, MP Dm 10, EI (Choosing m because n is taken, and don’t know that D can be n) Bm 6, UI (B can be n or m, as both appear above it, but m is convenient to get the conclusion) Bm ● Dm 11,12, Conj (Ex)(Bx ● Dx) 13, EG

Using Rules of Inference (Ex)Ax → (x)(Cx → Bx) (Ex)(Ax v Bx) (x)(Bx → Ax) / (x)(Cx → Ax) Am v Bm 2, EI Bm → Am 3, UI ~Am → Bm 4, DM ~Am → Am 5,6, HS Am v Am 7, DM Am 8, Taut (Ex)Ax 9, EG (x)(Cx → Bx) 1,10, MP Cm → Bm 11, UI (B taking m is okay, but is C?) Cm → Am 5,12, HS (x)(Cx → Ax) 13, UG

Using Rules of Inference (Ex)Ax → (x)(Bx → Cx) (Ex)Dx → (Ex)~Cx (Ex)(Ax ● Dx) / (Ex)~Bx Am ● Dm 3, EI Dm ● Am 4, Com Dm 5, Simp (Ex)Dx 6, EG (Ex)~Cx 2,7, MP Am 4, Simp (Ex)Ax 9, EG (x)(Bx → Cx) 1,10, MP Bm → Cm 11, UI (Again, ,,, ~Cm 8, EI ~Bm 12,13, MT (Ex)~Bx 14, EG