[FOM] Some informative questions about intuitionistic logic and mathematics
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Tue Nov 8 18:54:53 EST 2005
On Mon, 7 Nov 2005, Thomas Forster wrote:
> On the subject of intuitionists not denying bivalence ...
But they can and do. An intuitionist can assert "it is not the case that
every statement is either true or false" without being committed to the
(strictly classical) conclusion "some statement is neither true nor
false". Indeed (as pointed out by Mark van Atten, the intuitionist will
also assert "no statement is neither true nor false".
> ... it may be worth noting that
> constructive propositional logic explicitly denies that there are
> more than two truth values. The sequent
>
>
> -(A <--> B), -(B <--> C), -(C <--> A) |-
>
> (which says roughly that A, B and C cannot all have different truth-values)
>
> is constructively correct..... but hard! It's guaranteed to
> keep even quite good students out of mischief for several hours
Actually, 33 milliseconds---see below ... I called your problem "thomas"
and fed it to my least talented student.
Neil Tennant
____________________________________________________________________
Script started on Tue Nov 08 18:45:10 2005
NU-Prolog 1.6.5
1?- [irf].
Loading /n/mercutio/0/neilt/prooffinders/irf.no.
Consulting /n/mercutio/0/neilt/prooffinders/irpp.nl.
Consulting /n/mercutio/0/neilt/prooffinders/irfmanager.nl.
Warning in op(1200, xfx, (?-)) -- changing a system-defined operator.
Loading /n/mercutio/0/nuprolog/lib/lib.10605/osets.no.
Consulting /n/mercutio/0/neilt/prooffinders/syntax.nl.
Consulting /n/mercutio/0/neilt/prooffinders/writeproblem.nl.
done
true.
2?- t(thomas).
PROBLEM NUMBER 1:
[not(and(if(a, b), if(b, a))),
not(and(if(a, c), if(c, a))),
not(and(if(b, c), if(c, b)))] ?- # .
The length of this problem is 25
I take only 33 ms to find the following PROOF in IR,
which has 43 steps of inference:
| | | | | | | | >i| b
| | | | | | | | >cb
| | | | | | | | >i| c
| | | | | | | &i| >bc
| | | | | | | &>bc>cb
| | | | | | ~e| ~&>bc>cb
| | | | | >i| #
| | | | | >ca
| | | | | | | | >i| a
| | | | | | | | >ba
| | | | | | | | >i| b
| | | | | | | &i| >ab
| | | | | | | &>ab>ba
| | | | | | ~e| ~&>ab>ba
| | | | | >i| #
| | | | &i| >ac
| | | | &>ac>ca
| | | ~e| ~&>ac>ca
| | >i| #
| | >ba
| | | | | | | | >i| a
| | | | | | | | >ca
| | | | | | | | >i| c
| | | | | | | &i| >ac
| | | | | | | &>ac>ca
| | | | | | ~e| ~&>ac>ca
| | | | | >i| #
| | | | | >cb
| | | | | | | | >i| a
| | | | | | | | >ba
| | | | | | | | >i| b
| | | | | | | &i| >ab
| | | | | | | &>ab>ba
| | | | | | ~e| ~&>ab>ba
| | | | | >i| #
| | | | &i| >bc
| | | | &>bc>cb
| | | ~e| ~&>bc>cb
| | >i| #
| &i| >ab
| &>ab>ba
~e| ~&>ab>ba
#
where all the given premisses have been used
and the conclusion is what was sought
Total worktime spent so far finding 1 proofs is 33 ms
Total worktime spent so far reaching 0 rejections is 0 ms
Total worktime so far on 1 problems is 33 ms
Thank you for your attention.
true.
3?- ^DEnd of Session
% Time was 0.434s
[7mmercutio[m:~/prooffinders [2] exit
script done on Tue Nov 08 18:45:30 2005
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