In the paper On the logic of quantification Prof. W. V. Quine showed that for the theory of quantification we have a mechanical process to determine whether or not a monadic expression is a valid logical formula, and that to deduce the polyadic theory from the monadic theory we need only the generalized modus ponens, which reads:
If a conditional is valid, and its antecedent consists of zero or more quantifiers followed by a valid schema, then its consequent is valid, i.e.:
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200098637/resource/name/S0022481200098637_eqnU1.gif?pub-status=live)
That we have a mechanical process to determine the validity of a monadic expression was pointed out by Löwenheim long ago; Quine's method, however, has the merit that it is very simple and hence practical. The present paper is to show that we may deduce the polyadic theory by means of only a mechanical process and the ordinary modus ponens alone, i.e., the schema,
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or even by means of only the weakened modus ponens, i.e., the schema,
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200098637/resource/name/S0022481200098637_eqnU3.gif?pub-status=live)
For, if an expression becomes monadic when we omit the initial all-quantifiers (i.e., those preceding every existence-quantifier) and regard the corresponding variables as constants, then evidently we may determine by a mechanical process whether or not it is a valid formula by examining the resulting monadic expression. For example, the expression
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is of the same validity as the expression
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200098637/resource/name/S0022481200098637_eqnU5.gif?pub-status=live)
and hence as the expression
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200098637/resource/name/S0022481200098637_eqnU6.gif?pub-status=live)