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Re: A Paradox of Equality?

Newsgroups sci.logic
Date 2023-06-04 12:22 -0700
References (10 earlier) <b20fab3e-3256-4fb5-87da-459884d9f2e3n@googlegroups.com> <6f769c26-e42f-4fdb-a6d8-72181dccad25n@googlegroups.com> <969f397f-1805-4123-b9e5-5710e1ab4d7fn@googlegroups.com> <05a87431-fee8-40ef-82ff-56aa6fc2fabdn@googlegroups.com> <6873ef2b-530f-4b7d-838c-115a6e3f6932n@googlegroups.com>
Message-ID <ef37f091-c8f4-4401-be4b-2c6fad7e7131n@googlegroups.com> (permalink)
Subject Re: A Paradox of Equality?
From Mild Shock <bursejan@gmail.com>

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Omg. 60 Lines. 

Currently you dont have any de Morgan for bi-implicaion.
You only have the one definition of bi-implication,
namely this one:

(A <=> B) <~~> (A => B) & (B => A) 

You could prove it much quicker if you had de Morgan 
for bi-implication. You can derive de Morgan for bi-
implication from this one:

(A <=> B) <~~> (A & B) v (~A & ~B) 

The de Morgan is then:

~ (A <=> B) <~~> (~A v ~B) & (A v B)

You dont take duality in classical logic seriously Dan O Matik.
Your DC Proof is too much leaning towards intuitionistic
logic, or as you say the INTUITIONISTIC FRINGE.

Dan Christensen schrieb am Sonntag, 4. Juni 2023 um 18:33:51 UTC+2:
> On Sunday, June 4, 2023 at 9:47:34 AM UTC-4, Mild Shock wrote: 
> > Or maybe this one: 
> > 
> > ~ (p <=> q) <=> (p <=> ~q)
> '=>' 
> 
> Suppose... 
> 
> 1. ~[P <=> Q] 
> Premise 
> 
> 2. ~[[P => Q] & [Q => P]] 
> Iff-And, 1 
> 
> 3. ~~[~[P => Q] | ~[Q => P]] 
> DeMorgan, 2 
> 
> 4. ~[P => Q] | ~[Q => P] 
> Rem DNeg, 3 
> 
> 5. ~~[P & ~Q] | ~[Q => P] 
> Imply-And, 4 
> 
> 6. P & ~Q | ~[Q => P] 
> Rem DNeg, 5 
> 
> 7. P & ~Q | ~~[Q & ~P] 
> Imply-And, 6 
> 
> Two cases: 
> 
> 8. P & ~Q | Q & ~P 
> Rem DNeg, 7 
> 
> Case 1 
> 
> Suppose... 
> 
> 9. P & ~Q 
> Premise 
> 
> 10. P 
> Split, 9 
> 
> 11. ~Q 
> Split, 9 
> 
> Suppose... 
> 
> 12. P 
> Premise 
> 
> 13. ~Q 
> Copy, 11 
> 
> As Required: 
> 
> 14. P => ~Q 
> Conclusion, 12 
> 
> Suppose... 
> 
> 15. ~Q 
> Premise 
> 
> 16. P 
> Copy, 10 
> 
> As Required: 
> 
> 17. ~Q => P 
> Conclusion, 15 
> 
> 18. [P => ~Q] & [~Q => P] 
> Join, 14, 17 
> 
> 19. P <=> ~Q 
> Iff-And, 18 
> 
> Case 1 
> 
> As Required: 
> 
> 20. P & ~Q => [P <=> ~Q] 
> Conclusion, 9 
> 
> Case 2 
> 
> Suppose... 
> 
> 21. Q & ~P 
> Premise 
> 
> 22. Q 
> Split, 21 
> 
> 23. ~P 
> Split, 21 
> 
> Suppose... 
> 
> 24. P 
> Premise 
> 
> 25. ~P => ~Q 
> Arb Cons, 24 
> 
> 26. ~Q 
> Detach, 25, 23 
> 
> As Required: 
> 
> 27. P => ~Q 
> Conclusion, 24 
> 
> Suppose... 
> 
> 28. ~Q 
> Premise 
> 
> 29. ~Q => P 
> Arb Cons, 22 
> 
> 30. P 
> Detach, 29, 28 
> 
> As Required: 
> 
> 31. ~Q => P 
> Conclusion, 28 
> 
> 32. [P => ~Q] & [~Q => P] 
> Join, 27, 31 
> 
> 33. P <=> ~Q 
> Iff-And, 32 
> 
> As Required: 
> 
> 34. Q & ~P => [P <=> ~Q] 
> Conclusion, 21 
> 
> 35. [P & ~Q => [P <=> ~Q]] & [Q & ~P => [P <=> ~Q]] 
> Join, 20, 34 
> 
> 36. P <=> ~Q 
> Cases, 8, 35 
> 
> '=>' 
> 
> As Required: 
> 
> 37. ~[P <=> Q] => [P <=> ~Q] 
> Conclusion, 1 
> 
> '<=' 
> 
> Suppose... 
> 
> 38. P <=> ~Q 
> Premise 
> 
> 39. [P => ~Q] & [~Q => P] 
> Iff-And, 38 
> 
> 40. P => ~Q 
> Split, 39 
> 
> 41. ~Q => P 
> Split, 39 
> 
> Suppose to the contrary... 
> 
> 42. P <=> Q 
> Premise 
> 
> 43. [P => Q] & [Q => P] 
> Iff-And, 42 
> 
> 44. P => Q 
> Split, 43 
> 
> 45. Q => P 
> Split, 43 
> 
> 46. ~Q => ~P 
> Contra, 44 
> 
> Suppose... 
> 
> 47. P 
> Premise 
> 
> 48. ~Q 
> Detach, 40, 47 
> 
> 49. Q 
> Detach, 44, 47 
> 
> 50. ~Q & Q 
> Join, 48, 49 
> 
> As Required: 
> 
> 51. ~P 
> Conclusion, 47 
> 
> 52. ~P => ~~Q 
> Contra, 41 
> 
> 53. ~P => Q 
> Rem DNeg, 52 
> 
> 54. Q 
> Detach, 53, 51 
> 
> 55. P 
> Detach, 45, 54 
> 
> 56. ~P & P 
> Join, 51, 55 
> 
> As Required: 
> 
> 57. ~[P <=> Q] 
> Conclusion, 42 
> 
> '<=' 
> 
> As Required: 
> 
> 58. [P <=> ~Q] => ~[P <=> Q] 
> Conclusion, 38 
> 
> 59. [~[P <=> Q] => [P <=> ~Q]] 
> & [[P <=> ~Q] => ~[P <=> Q]] 
> Join, 37, 58 
> 
> As Required: 
> 
> 60. ~[P <=> Q] <=> [P <=> ~Q] 
> Iff-And, 59
> Dan 
> 
> Download my DC Proof 2.0 freeware at http://www.dcproof.com 
> Visit my Math Blog at http://www.dcproof.wordpress.com

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Thread

A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 10:33 -0700
  Re: A Paradox of Equality? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-03 11:02 -0700
    Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 11:17 -0700
      Re: A Paradox of Equality? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-03 11:47 -0700
        Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 11:57 -0700
        Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 12:37 -0700
  Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 11:13 -0700
    Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 11:20 -0700
      Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 11:42 -0700
        Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 12:14 -0700
      Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 11:56 -0700
        Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 11:58 -0700
          Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 12:08 -0700
            Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 12:25 -0700
    Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 11:26 -0700
      Re: A Paradox of Equality? Mild Shock <janburse@fastmail.fm> - 2023-06-03 21:12 +0200
        Re: A Paradox of Equality? Mild Shock <janburse@fastmail.fm> - 2023-06-03 21:20 +0200
        Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 12:27 -0700
          Re: A Paradox of Equality? Mild Shock <janburse@fastmail.fm> - 2023-06-03 21:41 +0200
          Re: A Paradox of Equality? Mild Shock <janburse@fastmail.fm> - 2023-06-03 21:41 +0200
            Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 13:02 -0700
              Re: A Paradox of Equality? Mild Shock <janburse@fastmail.fm> - 2023-06-03 22:07 +0200
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 13:27 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 13:49 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 13:56 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 14:12 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-03 14:24 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-03 19:24 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-04 06:46 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-04 06:47 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-04 06:51 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-04 09:33 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-04 12:22 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-04 13:48 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-04 14:38 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-04 15:29 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-04 17:20 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-04 18:23 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 01:31 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 02:55 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 08:22 -0700
                Re: A Paradox of Equality? Mild Shock <janburse@fastmail.fm> - 2023-06-05 19:47 +0200
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 11:59 -0700
                Re: A Paradox of Equality? Mild Shock <janburse@fastmail.fm> - 2023-06-05 21:49 +0200
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 13:39 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 12:32 -0700
                Re: A Paradox of Equality? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-05 12:39 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 13:03 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 13:09 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 13:14 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 14:34 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 15:04 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 15:14 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 15:26 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 16:33 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 14:22 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 15:06 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 17:13 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 17:26 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 17:36 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 17:47 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 17:54 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 18:00 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 18:02 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 18:05 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 18:35 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 22:38 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 22:47 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-05 22:58 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-06 07:35 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-06 09:11 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-06 10:08 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-08 01:25 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 08:09 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-08 11:09 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 11:35 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-08 11:36 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-09 12:51 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-09 13:12 -0700
                Re: A Paradox of Equality? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-05 18:17 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 18:44 -0700
                Re: A Paradox of Equality? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-05 14:09 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 14:32 -0700
                Re: A Paradox of Equality? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-05 17:42 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 18:32 -0700
                Re: A Paradox of Equality? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-05 19:04 -0700
                Re: A Paradox of Equality? Mild Shock <bursejan@gmail.com> - 2023-06-16 05:53 -0700
                Re: A Paradox of Equality? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-05 19:15 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 20:39 -0700
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-04 08:43 -0700
                Re: A Paradox of Equality? Mild Shock <janburse@fastmail.fm> - 2023-06-04 17:55 +0200
                Re: A Paradox of Equality? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-04 09:40 -0700
                Re: A Paradox of Equality? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-04 10:42 -0700

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