Mathematical logic to explain paradoxes and find connections between them

[Artem Hevorhian]

Hello everyone!

Happy New Year!

I have a small problem that came into my mind today.

We all know that there is a known paradox of Bertrand Russel about the barber. To recap, here it is. In a city, every man is shaved by the barber, and only barber shaves a man. So does the barber shave himself?

There is also a paradox about Chicken and the Egg. Which is also called Socratic (if I am not mistaken): who came first, the chicken or the egg?

At school, we studied linear temporal logic. Now is it possible to formulate the Socratic paradox in terms of it? If yes, then how about finding paradoxes that are described by non-linear temporal logic? Thanks for any update on this. You can explain the topic to me  if I am mistaken. And in general, is it possible to find paradoxes for a given logical system? First-order predicate calculus, second order etc.

Feel free to refer to any piece of theory that can bring more clarity into this.

Thanks.

[Jamie Oglethorpe]

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I'd consider the Russel Paradox to be a poor specification. If you have a database of the city's residents and selected the people who shaved themselves, and again to select the people shaved by the barber, he would appear in both lists. Implicit in the statement is that all men are shaved, and that the barber is male.

It is a matter of identifying your categories correctly. So, the barber shaves his customers, who don't shave themselves. Everybody else shaves himself. The barber isn't his own customer, so we have no problem.

As for the chicken and egg, the egg always comes first. Darwinian evolution explains the origin of eggs, They are an economical means for multicellular creatures to reproduce. Each egg is created for this purpose. It would have been a puzzle for Socrates who knew nothing about the origin of species.

You can always find contradictions in logical systems. E. W. Dijkstra disliked set theory. He showed that Russel's statement expressed in predicate form simply evaluated to FALSE.  From a programming point of view these are specification errors.

[alexk]

Consider this.
   
    We have in our universe of discourse 'chicken', 'egg', 'first' entity to exist and
    an operator 'before'.
   
    Specificaly:
   
(0)    before.chicken = egg
(1)    before.egg = chicken
(2)    before.first = false  (there is nothing before the first entity)
   
    Now, by case analysis:
   
    egg = first
=>        { Leibniz }
    before.egg = before.first
=        { LHS (1), RHS (2) }
    chicken = false

    By symmetry
   
    egg = false
   
    So neither 'egg' nor 'chicken' do exists (at all).

четверг, 5 января 2023 г. в 07:36:33 UTC+3, Jamie Oglethorpe:

[alexk]

    Actually a should have added a rule:
       
        (3)    before.false = false
       
    and a derivation:
   
    chicken = false
=>        { Leibniz }
    before.chicken = before.false
=         { LHS (0), RHS (3) }
    egg = false

пятница, 6 января 2023 г. в 01:23:34 UTC+3, alexk:

[Diethard Michaelis]

Your "Barber Paradox"

"In a city, every man is shaved by the barber,
and only barber shaves a man.
So does the barber shave himself?"

is yours, not the one incorrectly attributed to Bertrand Russell.
[Russell with double l]. See
https://en.wikipedia.org/wiki/Barber_paradox

For more clarity, see
EWD923a "Where is Russell's paradox?"
https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD923a.html

END OF Barber Paradox

As for "Chicken and Egg"
take the integers (...,-3,-2,-1,0,1,2,3,...):
Every odd is the successor of some even and
every even is the successor of some odd.
Which came first, even or odd?

If you insist on biological chicken see
https://en.wikipedia.org/wiki/Chicken_or_the_egg#Scientific_resolutions

END OF Chicken end Egg

2023 = (9+8)*7*(6+5+4+3-2+1+0)

Diethard.

[Diethard Michaelis]

Russell's Barber once more:

Did not have the time yesterday to check it,
but indeed Eric Hehner has it (not only)
in his "a Theory of Programming" book as an exercise.

Just let Google search for
russell's barber site:www.cs.toronto.edu/~hehner/
and you get the aToP book (-> Exercise 105)
https://www.cs.toronto.edu/~hehner/aPToP/aPToP.pdf
and even Hehner's solution
https://www.cs.toronto.edu/~hehner/aPToP/solutions/Ex105.pdf
and several more of his papers mentioning Russell's barber.

See also https://mathworld.wolfram.com/BarberParadox.html
where it is classified as a pseudoparadox.

END OF Russell's Barber once more

[Artem Hevorhian]

Thanks!

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