]> This page is written neither in HTML, nor for that matter in XHTML, but in pure XML. You might say it is written in a simple markup language that is derived from XML, and designed expressly for constructing logical derivations and proofs. Call it XGBXLNML! ("XGB Extensible Logical Notation Markup Language", if you will.) If anything, this exercise demonstrates that HTML is doomed to shrivel away in time, as might XHTML, except for marking up the most basic structure of a generic document. In the future, people may craft their own markup languages using the elegantly eXtensible Markup Language that is XML, style them with CSS, and post their results freely on the Web. This page opens in a new window. If you're not using a tabbed browser (Firefox, Netscape, Opera), check applications at bottom to return to home page. 1. P Premise 2. (P → Q) Premise 3. Q 1, 2, MP 4. ~~Q 3, DNI 5. ~~~~Q 4, DNI 6. ~~P 1, DNI 7. (~~~~Q ^ ~~P) 5, 6, Conj ________________________________________________________________ 1. (P ↔ (~Q ^ R)) Premise 2. Q Premise 3. ~~P Hypothesis 4. P 3, DNE 5. (P → (~Q ^ R)) 1, BCE 6. (~Q ^ R) 4, 5, MP 7. ~Q 6, Simp 8. Q 2, Reit 9. ~P 3-8, RAA ________________________________________________________________ 1. (Fa v Ga) Hypothesis 2. ~Fa Hypothesis 3. Ga 1, 2, DM 4. (~Fa → Ga) 2-3, CD 5. ~Ga Hypothesis 6. Fa 1, 5, DM 7. (~Ga → Fa) 5-6, CD 8. ((~Fa → Ga) ^ (~Ga → Fa)) 4, 7, Conj 9. ((Fa v Ga) → ((~Fa → Ga) ^ (~Ga → Fa))) 1-8, CD 10. ((~Fa → Ga) ^ (~Ga → Fa)) Hypothesis 11. ~(Fa v Ga) Hypothesis 12. ~Fa 11, DM 13. (~Fa → Ga) 10, Simp 14. Ga 12, 13, MP 15. ~Ga 11, DM 16. (Fa v Ga) 11-15, RAA 17. (((~Fa → Ga) ^ (~Ga → Fa)) → (Fa v Ga)) 10-16, CD 18. ((Fa v Ga) ↔ ((~Fa → Ga) ^ (~Ga → Fa))) 9, 17, BCI 19. (x) ((Fx v Gx) ↔ ((~Fx → Gx) ^ (~Gx → Fx))) 18, UG ________________________________________________________________ 1. ~(Fa → Gb) Hypothesis 2. ~(Fa ^ ~Gb) Hypothesis 3. Fa Hypothesis 4. ~Gb Hypothesis 5. ~Fa 2, 4, CS 6. Fa 3, Reit 7. Gb 4-6, RAA 8. (Fa → Gb) 3-7, CD 9. ~(Fa → Gb) 1, Reit 10. (Fa ^ ~Gb) 2-9, RAA 11. (~(Fa → Gb) → (Fa ^ ~Gb)) 1-10, CD 12. (Fa ^ ~Gb) Hypothesis 13. ~~(Fa → Gb) Hypothesis 14. (Fa → Gb) 13, DNE 15. Fa 12, Simp 16. Gb 14, 15, MP 17. ~Gb 12, Simp 18. ~(Fa → Gb) 13-17, RAA 19. ((Fa ^ ~Gb) → ~(Fa → Gb)) 12-18, CD 20. (~(Fa → Gb) ↔ (Fa ^ ~Gb)) 11, 19, BCI 21. (y) (~(Fa → Gy) ↔ (Fa ^ ~Gy)) 20, UG 22. (x)(y) (~(Fx → Gy) ↔ (Fx ^ ~Gy)) 21, UG ________________________________________________________________