If " predicate variables " are only allowed to be bound to predicate letters of zero arity ( which have no arguments ), where such letters actually represent propositions, then such variables are actually "'propositional variables "', and any predicate logic which allows second-order quantifiers to be used to bind such propositional variables is a second-order predicate calculus, or second-order logic.
42.
As an extension of the case of multi-valued logic, valuations ( \ mu : \ mathit { V } _ o \ to \ mathit { W } ) of propositional variables ( \ mathit { V } _ o ) into a set of membership degrees ( \ mathit { W } ) can be thought of as predicates into fuzzy sets ( or more formally, into an ordered set of fuzzy pairs, called a fuzzy relation ).
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Try proving this : for any wff \ alpha with n logical connectives, if no propositional variables in \ alpha occur more than once, there is at least one truth-value assignment that will cause \ alpha to evaluate to true, and there is at least one truth-value assignment that will cause \ alpha to evaluate to false . ( Math symbols in the original question re-typed . )-- 71.175.24.57 04 : 31, 23 March 2007 ( UTC)
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;update : in this scenario, two satellites, Unit A and Unit B, orbit around Mars; the satellites are programmed to land while transmitting their status to Earth; Earth has received a transmission from one of the satellites, communicating that it is still in orbit; however, due to interference, it is not known which satellite sent the signal; subsequently, Earth receives the communication that Unit A has landed; this scenario can be modeled in the following way; two propositional variables a and b indicate that Unit A and Unit B, respectively, are still in orbit; the initial set of beliefs is \ { a \ vee b \ } ( either one of the two satellites is still in orbit ) and the new piece of information is \ neg a ( Unit A has landed, and is therefore not in orbit ); the only rational result of the update is \ neg a; since the initial information that one of the two satellites had not landed yet was possibly coming from the Unit A, the position of the Unit B is not known;
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