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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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