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 Post subject: discrete mathematics "3 value logic" helpPosted: Fri, 8 Jun 2012 05:04:06 UTC
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Joined: Fri, 8 Jun 2012 05:00:12 UTC
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so basically i have a project about 3 valued logic ie truth=1 false = 0, unknown = 1/2

in a previous project I had to come up with formulae for 2 valued logic as follows:

negation
t(~p) = 1-t(p)

Conjunction
T(p^q) = min[t(p), t(q)]

Disjunction
T(p V q) = max[t(p), t(q)]

Conditional
~p -> q === ~p V q => t(p->q) = t[~pVq]
=> max[t(~p), t(q)]
=> max[1-t(p), t(q)]

biconditional
p<->q === (p->q)^(q->p) => t(p->q) = t[(p->q)^(q->p)]
=> min[t(p->q), t(q->p)
=> min[max[1-t(p), t(q)], max[1-t(q), t(p)]]
using this information I have to define the connectives for 3 valued logic. and I dont really know how to do that. this is due tomorrow, please help!!!

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 Post subject: Re: discrete mathematics "3 value logic" helpPosted: Fri, 8 Jun 2012 06:11:24 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6009
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
samaha wrote:
so basically i have a project about 3 valued logic ie truth=1 false = 0, unknown = 1/2

in a previous project I had to come up with formulae for 2 valued logic as follows:

negation
t(~p) = 1-t(p)

Conjunction
T(p^q) = min[t(p), t(q)]

Disjunction
T(p V q) = max[t(p), t(q)]

Conditional
~p -> q === ~p V q => t(p->q) = t[~pVq]
=> max[t(~p), t(q)]
=> max[1-t(p), t(q)]

biconditional
p<->q === (p->q)^(q->p) => t(p->q) = t[(p->q)^(q->p)]
=> min[t(p->q), t(q->p)
=> min[max[1-t(p), t(q)], max[1-t(q), t(p)]]
using this information I have to define the connectives for 3 valued logic. and I dont really know how to do that. this is due tomorrow, please help!!!

Note that there isn't a complemented lattice of 3 elements, so you necessarily don't have double negation (or excluded middle).

On the other hand, the lattice is pseudocomplemented, so ...

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 Post subject: Re: discrete mathematics "3 value logic" helpPosted: Fri, 8 Jun 2012 18:22:34 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7951
Location: Lexington, MA
Hello, samaha!

I've changed the code somewhat.

Quote:
So basically i have a project about 3-valued logic: .true = T, false = F, unknown = 0.

I would construct the basic truth tables, using common sense or baby-talk.

. .

. .

Then apply these to your problems.

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 Post subject: Re: discrete mathematics "3 value logic" helpPosted: Fri, 8 Jun 2012 21:36:42 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6009
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Soroban wrote:

I disagree. , where is our valuation, is our pseudocomplement operator on our pseudocomplemented lattice . In particular, the pseudocomplement of the unique nontrivial element in our 3-element bounded lattice is . See, for example, Heyting algebra example in wikipedia (the third one is what the OP asked for).

A way to understand why we necessary have when is to examine what would you get if you apply your rules for . Since , you must define when . Also, , so cannot be 1/2 or 1 when v(p)=1/2.

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