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Comparing the expressive power of the propositional logic with the calculus of classes

The universal Aristotelian statements can be expressed by propositional formulas as follows:

Using this we have expressed the lengthy argument of Lewis Carroll in the propositional calculus since all the statements are universal in Example 2.7.11 of LMCS.

Unfortunately we do not have a translation of I,O statements into propositional formulas. The simplest ``upgrade'' of the propositional calculus which is adequate to handle the I,O statements is the monadic predicate calculus which deals with quantified first-order statements about unary predicates.

We can translate propositional logic into the Calculus of Classes by letting be the conversion of propositional formulas into Calculus of Classes terms obtained by simply replacing with , with , and with '; and then observing that an argument

is valid in the Propositional Logic iff

is valid in the Calculus of Classes.

Conversely, given an equational argument in the Calculus of Classes we can assume that it is in the form

and reversing our translation we have a corresponding argument in the propositional logic.

In summary we have a translation of arguments in the propositional calculus into arguments in equations in the calculus of classes, and conversely. Hence they can be thought of as equivalent. Both are adequate to handle the universal Aristotelian statements. To strengthen the Calculus of Classes to handle I and O statements we only need to add . No such easy strengthening is available for the propositional logic.

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Fri Jan 31 11:14:47 EST 1997