Derive_v5_05 / Derive / Users / DiscreteMathematics / MultiValuedLogic

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*           Constructing TRUTH TABLES and comparing propositions            *

*             in a max/min MULTI-VALUED (w-valued) modal LOGIC              *

* (c)Eugenio Roanes-Lozano (Dept. Algebra, Univ. Complutense Madrid) Jan 00 *

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Note: Truth tables are represented as matrices. Big ones appear nicely only

in Dfw but not in the DOS version.





1. REFERENCES



For an introduction to modal multi-valued Logics see (for instance) chapter 3

of:

  R. Turner: Logics for Artificial Intelligence, Ellis Horwood Ltd., 1984.

A similar implementation was developed by this author for the CAS Macsyma and

is included in it (from version 2.3). Details and a Maple version can also be

found in:

  E. Roanes-Lozano: Introducing Propositional Multi-Valued Logics with the

  Help of a CAS. In: Proceedings of ISAAC 97. Kluwer (to appear).





2. GETTING STARTED



We shall refer below to the the DERIVE code in the LOGIC_MU.MTH and .DMO files.



``w" represents the number of truth-values in the Logic. It should be a prime

number and should be assigned before the calculations take place. For instance

for Boolean Logic:



    w:=2



and for 3-valued Logic:



    w:=3



Observe that 0 represents ``false", 1 represents ``true" and intermediate

numerical values represent intermediate degrees of certainty. The truth-values

are:

0=0/(w-1) , 1/(w-1) , 2/(w-1) , 3/(w-1) , ... , (w-2)/(w-1) , (w-1)/(w-1)=1 .



For instance in Kleene's 3-valued Logic the truth-values would be:

0 (false), 1/2 (undecided), 1 (true).





2.1 PROPOSITIONAL VARIABLES



The names of the propositional variables to be used are: p,q,r,s,u,v (plus

tautology and contradiction). More could be added (if necessary) just by adding

a similar line and including its name in the definition of the list ``vp".





2.2 CONNECTIVES



The connectives are (prefix form):

 - negation, possible, necessary (unary)

 - or, and (binary).



Connectives begin with an ``M'' (standing for ``multivalued"), not to interfere

with the built-in boolean connectives. So they are:



   MNEG(a_) , MPOS(a_) , MNEC(a_)

   MOR(a_,b_) , MAND(a_,b_)



Kleene-style conditional and biconditional are represented by MIMP(a_,b_) and

MIFF(a_,b_), respectively.

   



3 TRUTH-TABLES



Truth-tables are constructed as a matrix by function TT(m_,a_,b_).

``m_" indicates the number of different propositional variables that appear

altogether in propositions ``a" and ``b".

They must be the first ``m_" names in the list:  P,Q,R,S,U,V.



For instance to check the commutativity of conjunction in a three valued Logic,

it should be typed:



    w:=3

    TT( 2 , MAND(P,Q) , MAND(Q,P) )



and the two last columns of the matrix should be compared (the two first ones

correspond to the values of P and Q).



Conditional and biconditional are defined in Kleene's style.





4 TAUTOLOGIES



If a proposition ``a_" (depending on ``m_" propositional variables) is a

tautology (i.e., if it is always true) can be checked with:



    ISTAUT(m_,a_)



The answer ``1" correspond to ``YES" and 0 to ``NO".



Observe in the .DMO file how if w>2 this is not intuitive (it works in a very

different way to the Boolean case). For instance ``P OR NOT P" is not a

tautology.





5 TAUTOLOGICAL CONSEQUENCES



If a proposition ``b_" is a tautological consequence of ``a_" (i.e.: if the

consequent is true" "whenever the antecedent is true) can be checked with:



    ISCONSTAUT(m_,a_,b_)



(``m_" indicates the number of different propositional variables that appear

altogether in propositions ``a" and ``b").



The answer ``1" correspond to ``YES" and 0 to ``NO".





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