 More important for metalogical purposes is the fact that the universal quantifier too is syncategorematic in Leśniewski. This is marked symbolically in that the lower corners which he used to symbolize the universal quantifier are no more than a container for the variables, but more crucially, in that any finite string of different variables may occur in such a quantifier, no matter how assorted their categories. There are some advantages in this flexibility. Leśniewski does not need to give rules for many different kinds of universal quantifier, but gives rules for the one sort in one go. But there are some disadvantages too. In his “official” notation for his logics, Leśniewski has only the universal quantifier, and does not define the particular quantifier or any others in the standard way. This was also Frege’s practice, but whereas in Frege’s case the parsimony seems to have been self-inflicted, in Leśniewski’s case there is a systematic reason. Leśniewski was scrupulous about giving exact rules for admitting new expressions by definition. He gave such rules for Protothetic, and extended them to Ontology. These rules govern only basic and functor category expressions. Quantifiers, being variable binders, are neither basic nor functor expressions, but Leśniewski was unable to come up with acceptable canons of definition for such variable binders. He would have liked to have done so, and indeed offered students any degree they wanted, from Master’s to Habilitation, if they could formulate adequate rules, but no one could. Therefore in the “official” system the universal quantifier remained as a syncategorematic expression, but still entered into legal combinations, which meant that the syntax of his systems was not fully captured by categorial grammar. Such a limitation became apparent to Ajdukiewicz also, who made an unsuccessful attempt to rectify it. Ajdukiewicz noted perceptively that a language containing Russell’s circumflex abstraction operator, which Alonzo Church had notated using the Greek lambda, would be able to express any operator as a combination of an abstraction operator with a functor. Church employed this method to considerable advantage in his logic, but this work came too late to help Leśniewski. In any case, in Church it simply pushed the problem of syncategorematicity over to the lambda operator.