Re: what boolean connectives together with -> make a complete system?

From: Owen (oorionus_at_yahoo.com)
Date: 12/20/04 

Date: Mon, 20 Dec 2004 08:23:43 -0500

  I. (nor)

  1.~p =df (p nor p)
  2. p v q =df ~(p nor q)
  3. _T =df p v ~p
  4. F =df ~T
  5. p & q =df ~(~p v ~q)
  6. p nand q =df ~(p & q)
  7. p -> q =df ~p v q
  8. p -|-> q =df ~(p -> q)
  9. p <-> q =df (p -> q)&(q -> p)
  10. p xor q =df ~(p <-> q)

  II. (nand)

  1. ~p =df (p nand p)
  2. p & q =df ~(p nand q)
  3. F =df p & ~p
  4. T =df ~F
  5. p v q =df ~(~p & ~q)
  6. p nor q =df ~(p v q)
  7. p -> q =df ~p v q
  8. p -|-> q =df ~(p -> q)
  9. p <-> q =df (p -> q)&(q -> p)
  10. p xor q =df ~(p <-> q)

  III. (T, -|->)

  1. ~p =df T -|-> p
  2. F =df ~T
  3. p -> q =df ~(p -|-> q)
  4. p v q =df ~p -> q
  5. p nor q =df ~(p v q)
  6. p & q =df ~(~p v ~q)
  7. p nand q =df ~(p & q)
  8. p <-> q =df (p -> q)&(q -> p)
  9. p xor q =df ~(p <-> q)

  IV. (F, ->)

  1. ~p =df p -> F
  2. T=df ~F
  3. p -|-> q =df ~(p -> q)
  4. p v q =df ~p -> q
  5. p nor q =df ~(p v q)
  6. p & q =df ~(~p v ~q)
  7. p nand q =df ~(p & q)
  8. p <-> q =df (p -> q)&(q -> p)
  9. p xor q =df ~(p <-> q)

  V. (~, v)

  1. T =df p v ~p
  2. F =df ~T
  3. p nor q =df ~(p v q)
  4. p -> q =df ~p v q
  5. p -|-> q =df ~(p -> q)
  6. p & q =df ~(~p v ~q)
  7. p nand q =df ~(p & q)
  8. p <-> q =df (p -> q)&(q -> p)
  9. p xor q =df ~(p <-> q)

  VI. (~, &)

  1. F =df p & ~p
  2. T =df ~F
  3. p nand q =df ~(p & q)
  4. p v q =df ~(~p & ~q)
  5. p nor q =df ~(p v q)
  6. p -> q =df ~p v q
  7. p -|-> q =df ~(p -> q)
  8. p <-> q =df (p -> q)&(q -> p)
  9. p xor q =df ~(p <-> q)

  VII. (~, ->)

  1. T =df p -> p
  2. F =df ~T
  3. p -|-> q =df ~(p -> q)
  4. p v q =df ~p -> q
  5. p nor q =df ~(p v q)
  6. p & q =df ~(~p v ~q)
  7. p nand q =df ~(p & q)
  8. p <-> q =df (p -> q)&(q -> p)
  9. p xor q =df ~(p <-> q)

  "|-|erc" <h@r.c> wrote in message news:32n8q4F3onqg5U1@individual.net...
> like NOT and OR is complete.
>
> NOR is complete
>
> -> and ?
>
> Herc
>
> --
> "YOU CAN'T PROVE ME"
> If you prove it's true then it has a proof, which makes it false
> If you don't prove it, then its true
> 10,000 people in sci.math ALL believe this is irrefutable that
mathematics will always be incomplete.
>
>

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