Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
Classes
and relations
A class (which is the same as a manifold or aggregate) is
all the objects satisfying some propositional function. If a
is the class composed of the objects satisfying φx, we shall say that a
is the class determined by φx^. Every propositional function thus determines a class, though if the
propositional function is one which is always false, the class will be null, i.e. will have no members. The class determined by the function φx^ will be represented by z^ (φz)*. Thus for example if φx is an
equation, z^ (φz) will be the class
of its roots; if φx is " x has
two legs and no feathers," z^ (φz)
will be the class of men; if φx is
" 0 < x < 1," z^ (φz)
will be the class of proper fractions, and so on.
It is obvious that the same class of objects will have
many determining functions. When it is not necessary to specify a determining
function of a class, the class may be conveniently represented by a single
Greek letter. Thus Greek letters, other than those to which some constant
meaning is assigned, will be exclusively used for classes.
There are two kinds of difficulties which arise in
formal logic; one kind arises in connection with classes and relations and the
other in connection with descriptive functions. The point of the difficulty for
classes and relations, so far as it concerns classes, is that a class cannot be
an object suitable as an argument to any of its determining functions. If a
represents a class and φx one of its
determining functions [so that a =
z^ (φz)], it is not sufficient that φa
be a false proposition, it must be nonsense.
Thus a certain classification of what appear to be
objects into things of essentially different types seems to be rendered
necessary. This whole question is discussed in Chapter II, on the theory of
types, and the formal treatment in the systematic exposition, which forms the
main body of this work, is guided by this discussion. The part of the
systematic exposition which is especially concerned with the theory of classes
is *20, and in this Introduction it is discussed in Chapter III. It is
sufficient to note here that, in the complete treatment of *20, we have avoided
the decision as to whether a class of things has in any sense an existence as
one object. A decision of this question in either way is indifferent to our logic,
though perhaps, if we had regarded some solution which held classes and
relations to be in some real sense objects as both true and likely to be
universally received, we might have simplified one or two definitions and a few
preliminary propositions. Our symbols, such as "x^ (φx)" and a
and others, which represent classes and relations, are merely defined in their
use.
The result of our definitions is that the way in which
we use classes corresponds in general to their use in ordinary thought and
speech; and whatever may be the ultimate interpretation of the one is also the
interpretation of the other. Thus in fact our classification of types in
Chapter II really performs the single, though essential, service of justifying
us in refraining from entering on trains of reasoning which lead to
contradictory conclusions. The justification is that what seem to be a
proposition, are really nonsense.
* Any other letter may be used instead of z.
Script: Stan Lee Pencils: Jack Kirby Inks: Joe
Sinnott Fantastic Four © Respective copyright/trademarkholders.
Here we come to the
heart of the system’s value in accessing the usefulness or error of our complex
hypothesizes. It is not impossible to mix different groups and types of data
within a proposition, but if one is not careful of their properties and natures,
the results may arrive as nonsense.
Pencils and Inks: Pat Boyette © Respective copyright/trademark holders.
Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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