Previous: Clones

Next: Comments on the nature of Boolean algebra

Up: Supplementary Text Topics

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. gif

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.


Previous: Clones

Next: Comments on the nature of Boolean algebra

Up: Supplementary Text Topics

Fri Jan 31 11:14:47 EST 1997
© Stanley Burris