Re: Contradiction or paradox

On Apr 24, 8:16 pm, Charlie-Boo <shymath...@xxxxxxxxx> wrote:
On Apr 21, 6:33 pm, translogi <wilem...@xxxxxxxxxxxxxx> wrote:





On Apr 21, 12:59 am, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

On Apr 19, 9:03 pm, G. Frege <nomail@invalid> wrote:

On 19 Apr 2007 12:43:03 -0700, LauLuna <laureanol...@xxxxxxxx>
wrote:

(1) X & -X
(2) X <-> -X

(1) and (2) are equivalent in propositional logic.

Right.

One direction ((1) => (2)) is immediate:

P & Q |- P <-> Q

1 (1) P & Q A
2 (2) P A
1 (3) Q 1 &E
1 (4) P -> Q 3 ->I (2)
5 (5) Q A
1 (6) P 1 &E
1 (7) Q -> P 6 ->I (5)
1 (8) P <-> Q 4,7 <->I

Substitution instance: X & -X |- X <-> -X

The other direction ((2) => (1)) is slightly more involved:

X <-> -X |- X & -X

1 (1) X <-> -X A
(2) X v -X TND
3 (3) X A

You'll have to do better than that, Frege. Each line in a formal
proof must be justified as being an axiom, premise of the theorem
being proven, or deduced from earlier lines via a rule of inference.
You didn't do that for line (3).

Can you fix it?

C-B

1 (4) X -> -X 1 <->E
1,3 (5) -X 3,4 ->E
1,3 (6) X & -X 3,5 &I
7 (7) -X A
1 (8) -X -> X 1 <->E
1,7 (9) X 7,8 ->E
1,7 (10) X & -X 7,9 &I
1 (11) X & -X 2,3,6,7,10 vE

F.

--

E-mail: info<at>simple-line<dot>de- Hide quoted text -

- Show quoted text -- Hide quoted text -

- Show quoted text -

Dear Charlie Boo

Did you ever study natural deduction?
like 3 and line 7 are assumptions
line 11 discharges them both
that is how vE works

No "Just following orders" defense, please. Axiomatic theorem proving
is the presentation of a linear representation of a proof tree, where
each node is an axiom at a leaf or a rule applied to its children, and
the root is the theorem. Do you agree?

I ask again, can you make it in that format? Are you in fact being
superfluous when you go outside of the classic model?

You are using reductio ad absurdum to produce a proof within a proof.
Proof trees can be transformed because they are not the normal form of
clauses.

If you don't know what I mean, please don't jump to the conclusion
that it is meaningless nor be afraid to ask. I can give you examples
of transforming a proof tree.

C-B



See one of the many books on natural deduction.- Hide quoted text -

- Show quoted text -- Hide quoted text -

- Show quoted text -- Hide quoted text -

- Show quoted text -

If you realy want to do it that way:
Right from the master himself
Principa Mathematica, Vol 1, pag 113 theorem *3.44
I don't understand the principa Mathematica but it looks OK
Maybe you can explain it to me.

* 3.44 |- :. q > p . r > p .>: q v r . > p
represents the horseshoe (not an ascii symbol)

I saw an earlier post that would give you a link to fully worked out
proofs of the principa mathematica but i couldn't find it anymore
Sorry.

.

Relevant Pages

  • Re: Contradiction or paradox
    ... proof must be justified as being an axiom, ... or deduced from earlier lines via a rule of inference. ... of transforming a proof tree. ...
    (sci.logic)
  • Re: Contradiction or paradox
    ... proof must be justified as being an axiom, ... of transforming a proof tree. ... quote a 1,000 page formalization? ... primitives along the way and take up 1,000 pages doing so. ...
    (sci.logic)