- (x)(Ax ⊃Bx) ≡ ¬ (∃x)(Ax • ¬ Bx)
- ¬ (∃x)(Ex • ¬ Cx)
- ¬ (∃x)Ex
- ∴(x)[Ex ⊃(Ax • Bx • Cx)]
- * Ex ......... ACP
- * (x) ¬ Ex ......... 3 QC
- * ¬ Ex ......... 6 UI
- * ¬ Ex ∨(Ax • Bx • Cx) ......... 7 Add.
- * Ax • Bx • Cx ......... 8,5 DS
- Ex ⊃(Ax • Bx • Cx) ......... 5-9 CP
- (x)[Ex ⊃(Ax • Bx • Cx)] ......... 10 UG
Wednesday 27 April 2011
Predicate Logic, Howard Pospesel, Prentice Hall, 2003, Chpt. 14, problem 25, p. 213
Thursday 21 April 2011
Symbolic Logic, D. Jacquette, Wadsworth, 2001, Chpt. 8, IV(13), p.435
- (x)(Txl ⊃Hx)
- ∴ (x)(¬ Hx ⊃¬ Txl) • (x)(¬ Txl ∨Hx)
- (x)(¬ Hx ⊃¬ Txl) ......... 1 Contrap.
- (x)(¬ Txl ∨Hx) ......... 1 MI
- (x)(¬ Hx ⊃¬ Txl) • (x)(¬ Txl ∨Hx) ......... 3,4 Conj.
Thursday 14 April 2011
The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.6E, 4(b), p. 572
- (∃x)(Ix • Khx)
- j = h
- Dj
- ∴(∃x)[Dx • (∃y)(Iy • Kxy)]
- Im • Khm ......... 1 EI x/m
- Khm ......... 5 Simp.
- Kjm ......... 3,6 Id
- Im ......... 5 Simp.
- Im • Kjm ......... 7,8 Conj.
- (∃y)(Iy • Kjy) ......... 9 EG
- Dj • (∃y)(Iy • Kjy) ......... 3,10 Conj.
- (∃x)[Dx • (∃y)(Iy • Kxy)] ......... 11 EG
Thursday 7 April 2011
The 'undisciplined man' argument
In Plato’s Gorgias, Socrates expounds on the virtues of temperance, discipline and order to Callicles. At one point, the argument proceeds along these lines:
- An undisciplined man is not a fellow of any other man.
- If a man is not a fellow of any other, then he is not their friend either.
- If a man is not a friend of any other man, then he is not a friend of God.
- Therefore, an undisciplined man is neither a friend of any other man nor a friend of God.
- (x)(y){[Ux • ¬ (x = y)] ⊃¬ Exy}
- (x)(y){[¬ (x = y) • ¬ Exy] ⊃¬ Rxy}
- (x){(y)[¬ (x = y) ⊃¬ Rxy] ⊃¬ Rxg}
- ∴ (x){Ux ⊃ ¬ {(∃y)[ ¬ (x = y) • Rxy] ∨ Rxg}}
The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.6E, 3(d), p. 572
- * x = y • y = z ......... ACP
- * x = y ......... 1 Simp.
- * y = z ......... 1 Simp.
- * x = z ......... 2,3 Id
- (x = y • y = z)⊃ x = z ......... 1-4 CP
- (z)[(x = y • y = z)⊃ x = z] ......... 5 UG
- (y)(z)[(x = y • y = z)⊃ x = z] ......... 6 UG
- (x)(y)(z)[(x = y • y = z)⊃ x = z] ......... 7 UG