Exact(11)
Surprisingly, Bolzano lacks a theory of quantification.
(The appendices discuss the theory of quantification, mathematical induction and the axiom of reducibility, and the principle of extensionality respectively).
One reason to take the ramified theory of types seriously (even without accepting propositions) is that it can be usefully incorporated into a substitutional theory of quantification.
This motivation has given rise to a variety of alternatives to the classical theory of quantification and identity that are generally subsumed under the label free logic.
The vocabulary of classical quantificational logic is often supplemented with an identity predicate to yield the classical theory of quantification with identity.
Before we modify the definition of formula for the expanded language, we may give a recursive definition of a term of the language of the classical theory of quantification and identity.
Similar(49)
Our rationale for doing as much is that one will get the technical details, and the particular philosophical issues that motivated them, from looking at specific logics—specific theories of logical consequence (e.g., relevant logics, substructural logics, non-monotonic logics, dynamic logics, modal logics, theories of quantification, and so on).
Like in the case of second-order logic, we may provide at least two sorts of models for the theory of plural quantification.
In particular, second-order logic and the theory of plural quantification will be each closely related to two first-order two-sorted theories, which lack the expressive resources often attributed to each extension of classical quantificational logic.
Several recent commentators have asked what the medieval theory of modes of common personal supposition was: was it a theory of inference, of quantification, of truth-conditions, of fallacies, or what?
With a full testing of the theory and quantification of its parameters, polarimetry has the promise to not only allow efficient and reliable tracing of interstellar and interplanetary magnetic fields, but also to provide new and unique probes of the dust and the interstellar environment.
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