In the first type, a single class of lexemes is used in more than one

propositional function.

In 'F(F(fx))', the first 'F' and the second will not have the same meaning, since, to use Russellian terminology, the first 'F' ranges over

propositional functions of type n, while the second ranges over functions of type n + 1" (Ostrow 2002: 66-67).

As a result the latter is a propositional, or logical function, of the form '[lambda]z[lambda]t(z > t)[y][x]', and identifying the appropriate variables in it merely turns this into another

propositional function with two subjects, '[lambda]z[lambda]t(z > t)[x][x]', not a function of one variable '[lambda]z(z > z)[x]' as with 'x-x'.

Modality: Both verbal and nonverbal modalities are responsible for carrying out the interactional and

propositional functions.

In his original substitutional theory, Russell applies the notion of "incomplete symbol," which he had recently introduced in OD, to expressions for

propositional functions and classes: even though these expressions can occur in meaningful sentences, there really are no

propositional functions or classes corresponding to them.

However, their model "can be related in several ways to the general methodology of semantic maps" (Hengeveld and van Lier 2010: section 6), and they highlight three respects in which the implicational map of PoS can push the theory of semantic maps a step further: it shows (i) that the analytical primitives can consist of

propositional functions, (ii) that semantic maps may have a high predictive power if they include a "hierarchy of hierarchies" like the one at issue, (iii) that, if semantic maps are implicational in nature, they can make predictions about the frequency with which specified constructions for the mapped functions are in fact attested across languages.

Moreover, when one says, "Some tame tigers growl," this asserts that the

propositional function x is a tame tiger and growls has a value which is a true proposition.

Similarly, it is reasonable to suppose that every

propositional function (or, in ordinary language, every term that one might use in a sentence, such as the world man) determines a class (in our example, all of the male members of the species homo sapiens).

The connection between this principle and the notion of

propositional function is not hard to trace.

Geach is right, and attempts to say what exactly a Russellian

propositional function is, or is supposed to be, are bound to end in frustration.

IFF is construed in a manner parallel to CONJ and DISJ, and Tr is to be the

propositional function that takes an entity and returns the proposition with respect to it that it is true.

The idea of the

propositional function is obviously rooted in mathematics, where statements like [x.

By a

propositional function, I mean an [infinity]-valued [read: indefinitely-valued: RPP] statement, containing one or more variables, such that when single values are assigned to these variables the expression becomes, in principle, a one-valued proposition.

The reader is subsequently plunged into a discussion of Russell's theory of descriptions (lectures 6-8) and Russell's notion of a

propositional function (lectures 9-23).

Bertrand Russell introduced the term

propositional function concerning which Cassius J.