Logical Indefinites

The best extant demarcation of logical constants, due to Tarski, classifies logical con- stants by invariance properties of their denotations. This classification is developed in a framework which presumes that the denotations of all expressions are definite. However, some indefinite expressions, such as Russell’s indefinite description operator η, Hilbert’s ε, and abstraction operators such as ‘the number of’, appropriately in- terpreted, are logical. I generalize the Tarskian framework in such a way as to allow a reasonable account of the denotations of indefinite expressions. This account gives rise to a principled classification of the denotations of logical and non-logical indefi- nite expressions. After developing this classification and its application to particular cases in some detail, I show how this generalized framework allows a novel view of the logical status of certain abstraction operators such as ‘the number of’. I then show how we can define surrogate abstraction operators directly in higher-order languages augmented with an ε-operator.