Natural Negations


Given a setting of natural deduction.

natural negation (two equivalent systems)
(1) add logical constant f
define ~p as p -> f

(2) add single place logical operator ~
add axiom
p -> q -> (p -> ~q) -> ~p
define f as ~(p -> p)

Positive Logic. natural negation + Pierce's axiom
p -> q -> p -> p

Intuitionistic negation (two equivalent systems)
natural negation (1) with axiom f -> p,

natural negation (2) with one added axiom
~(p -> p) -> q
or
p -> (~p -> q)

Classical negation (two equivalent systems)
natural negation (1) with axiom ~~p -> p

add single place logical operator ~
add axiom
~p -> q -> (~p -> ~q) -> p
define f as ~(p -> p)

Natural Negation gives
p -> ~p -> ~p
Positive Logic gives
~p -> p -> p
Positive Logic and Intuitionistic negation give classical negation

NN
PL IN
CN

Any other variants of negation?
What good are these variants?

Given ND and PA are there any truth table valid statements
without ~ or f that can't be proven or derived?

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Relevant Pages

  • Just say no, naturally.
    ... natural negation ... Natural Negation gives ... Positive Logic and Intuitionistic negation give classical negation ...
    (sci.logic)
  • Re: Natural Negations
    ... So you add the following axiom ... deduction to natural deduction, see also: ... Intuitionistic negation (two equivalent systems) ...
    (sci.logic)
  • Re: Just say no, naturally.
    ... > Positive Logic and Intuitionistic negation give classical negation ... > Any other variants of negation? ...
    (sci.logic)