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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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It will be seen that, according to the previous post, a
judgment does not have a single object, namely the proposition, but has several
interrelated objects. That is to say, the relation which constitutes judgment
is not a relation of two terms, namely the judging mind and the proposition,
but is a relation of several terms, namely the mind and what are called the
constituents of the proposition. That is, when we judge (say) "this is
red," what occurs is a relation of three terms, the mind, and "this,"
and red. On the other hand, when we perceive
"the redness of this," there is a relation of two terms, namely the
mind and the complex object "the redness of this." When a judgment
occurs, there is a certain complex entity, composed of the mind and the various
objects of the judgment. When the judgment is true, in the case of the kind of
judgments we have been considering, there is a corresponding complex of the objects of the judgment alone.
Falsehood, in regard to our present class of judgments, consists in the absence
of a corresponding complex composed of the objects alone. It follows from the
above theory that a "proposition," in the sense in which a
proposition is supposed to be the object of a judgment, is a false abstraction,
because a judgment has several objects, not one. It is the several-ness of the
objects in judgment (as opposed to perception) which has led people to speak of
thought as "discursive," though they do not appear to have realized
clearly what was meant by this epithet.
Owing to the plurality of the objects of a single
judgment, it follows that what we call a "proposition" (in the sense
in which this is distinguished from the phrase expressing it) is not a single
entity at all. That is to say, the phrase which expresses a proposition is what
we call an "incomplete" symbol (More on this to come); it does not
have meaning in itself, but requires some supplementation in order to acquire a
complete meaning. This fact is somewhat concealed by the circumstance that
judgment in itself supplies a sufficient supplement, and that judgment in
itself makes no verbal addition to
the proposition. Thus " the proposition 'Socrates is human'" uses
"Socrates is human" in a way which requires a supplement of some kind
before it acquires a complete meaning; but when I judge "Socrates is
human," the meaning is completed by the act of judging, and we no longer
have an incomplete symbol. The fact that propositions are "incomplete
symbols" is important philosophically, and is relevant at certain points
in symbolic logic.
The judgments we have been dealing with hitherto are
such as are of the same form as judgments of perception, i.e. their subjects are always particular and definite. But there
are many judgments which are not of this form. Such are "all men are
mortal," "I met a man," "some men are Greeks." Before
dealing with such judgments, we will introduce some technical terms.
We will give the name of "a complex" to any such object as "a
in the relation R to b" or "a having
the quality q," or "a
and b and c standing in the relation S."
Broadly speaking, a complex is anything which occurs in the universe and is not
simple. We will call a judgment elementary when it merely asserts such things
as "a has the relation R to b," "a has the quality q" or
"a and b and c stand in the relation S." Then an elementary judgment is true when there is a corresponding complex,
and false when there is no corresponding complex.
But take now such a proposition as "all men are
mortal." Here the judgment does not correspond to one complex, but to many, namely "Socrates is mortal,"
" Plato is mortal," "Aristotle is mortal," etc. (For the
moment, it is unnecessary to inquire whether each of these does not require
further treatment before we reach the ultimate complexes involved. For purposes
of illustration, "Socrates is mortal" is here treated as an
elementary judgment, though it is in fact not one, as will be explained later.
Truly elementary judgments are not very easily found.) We do not mean to deny
that there may be some relation of the concept man to the concept mortal which
may be equivalent to "all men are mortal," but in any case this
relation is not the same thing as what we affirm when we say that all men are
mortal. Our judgment that all men are mortal collects together a number of
elementary judgments. It is not, however, composed of these, since (e.g.) the fact that Socrates is mortal
is no part of what we assert, as may be seen by considering the fact that our
assertion can be understood by a person who has never heard of Socrates. In
order to understand the judgment "all men are mortal," it is not
necessary to know what men there are. We must admit, therefore, as a radically
new kind of judgment, such general assertions as "all men are
mortal." We assert that, given that x
is human, x is always mortal. That
is, we assert "x is mortal"
of every x which is human. Thus we
are able to judge (whether truly or falsely) that all the objects which have
some assigned property also have some other assigned property. That is, given
any propositional functions φx^
and yx^,
there is a judgment asserting yx with every x for which we have φx.
Such judgments we will call general
judgments.
It is evident (as explained above) that the definition
of truth is different in the case of
general judgments from what it was in the case of elementary judgments. Let us
call the meaning of truth which we gave for elementary judgments "elementary
truth." Then when we assert that it is true that all men are mortal, we
shall mean that all judgments of the form "x is mortal," where x
is a man, have elementary truth. We may define this as "truth of the
second order" or "second-order truth." Then if we express the
proposition "all men are mortal" in the form
(x) . x is mortal, where x is a man,"
and call this judgment p, then "p is true"
must be taken to mean "p has
second-order truth," which in turn means
"(x) . 'x is mortal' has elementary truth, where
x is a man."
In order to avoid the necessity for stating explicitly
the limitation to which our variable is subject, it is convenient to replace
the above interpretation of "all men are mortal" by a slightly
different interpretation. The proposition "all men are mortal" is
equivalent to "'x is a man'
implies 'x is mortal,' with all
possible values of x." Here x is not restricted to such values as
are men, but may have any value with which "'x is a man' implies 'x is
mortal'" is significant, i.e. either true or false. Such a
proposition is called a "formal implication." The advantage of this
form is that the values which the variable may take are given by the function
to which it is the argument: the values which the variable may take are all
those with which the function is significant.
We use the symbol "(x). φx"
to express the general judgment which asserts all judgments of the form "φx." Then the
judgment "all men are mortal" is equivalent to
"(x) . 'x is a man' implies 'x is a mortal,"
i.e.
(in virtue of the definition of implication) to
"(x) . x
is not a man or x is mortal."
As we have just seen, the meaning of truth which is applicable to this
proposition is not the same as the meaning of truth which is applicable to "x is a man" or to "x is mortal." And generally, in any
judgment (x). φx,
the sense in which this judgment is or may be true is not the same as that in
which φx
is or may be true. If φx
is an elementary judgment, it is true when it points to a corresponding complex. But (x) . φx does not point to a
single corresponding complex: the corresponding complexes are as numerous as
the possible values of x.
It follows from the above that such a proposition as
"all the judgments made by Epimenides are true" will only be prima
facie capable of truth if all his judgments are of the same order. If they are
of varying orders, of which the nth
is the highest, we may make n
assertions of the form "all the judgments of order m made by Epimenides are true," where m has all values up to n.
But no such judgment can include itself in its own scope, since such a judgment
is always of higher order than the judgments to which it refers.
Let us consider next what is meant by the negation of a
proposition of the form "(x) . φx." We observe,
to begin with, that "φx in
some cases," or "φx sometimes,"
is a judgment which is on a par with "φx
in all cases," or "(x
always." The judgment "φx
sometimes" is true if one or more values of x exist for which φx
is true. We will express the proposition "φx sometimes" by
the notation "(∃x) .
φx," where
"∃"
stands for "there exists," and the whole symbol may be read "
there exists an x such that φx."
We take the two kinds of judgment expressed by "(x). φx"
and "(∃x). φx"
as primitive ideas. We also take as a primitive idea the negation of an elementary proposition. We can then
define the negations of (x). φx and (∃x). φx. The negation of
any proposition p will be denoted by
the symbol "~p." Then the
negation of (x) . φx will be defined as
meaning
"(∃x) .
~φx,"
and the negation of (∃x) . φx will be defined as meaning "(x) . ~φx" Thus, in the traditional language of formal
logic, the negation of a universal affirmative is to be defined as the
particular negative, and the negation of the particular affirmative is to be
defined as the universal negative. Hence the meaning of negation for such
propositions is different from the meaning of negation for elementary
propositions.
An analogous explanation will apply to disjunction.
Consider the statement "either p,
or φx always." We
will denote the disjunction of two propositions p, q by "p v q."
Then our statement is "p . v . (x). φx."
We will suppose that p is an
elementary proposition, and that φx is always an
elementary proposition. We take the disjunction of two elementary propositions
as a primitive idea, and we wish to define the disjunction
"p. v . (x). φx."
This may be defined as "(x). p v φx,"
i.e. "either p is true, or φx is always
true" is to mean "'p or φx' is always
true." Similarly we will define
"p. v. (∃x) . φx "
as meaning "(∃x).
p v φx,"
i.e. we define "either p is true or there is an x for which φx is true" as
meaning " there is an x for
which either p or φx
is true." Similarly we can define a disjunction of two universal
propositions: "(x). φx. v . (y). yy"
will be defined as meaning "(x, y) . φx v
yy," i.e. "either φx
is always true or yy
is always true" is to mean "'φx
or yy'
is always true." By this method we obtain definitions of disjunctions
containing propositions of the form (x)
. φx or (∃x) . φx in terms of
disjunctions of elementary propositions; but the meaning of
"disjunction" is not the same for propositions of the forms (x). φx,
(∃x)
. φx,
as it was for elementary propositions.
Similar explanations could be given for implication and
conjunction, but this is unnecessary, since these can be defined in terms of
negation and disjunction.
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Script: Jack Schiff Pencils and Inks: Sheldon
Moldoff © Respective copyright holders.
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To proceed with our examination of the nature of complexity
and truth and falsehood we must, as the authors warn us, make a philosophical
leap that is important to recognize; because in taking this into our conscious
calculations in life we are altered and separated from our family, peers and
culture. It is a trade-off of clear perception of fine distinctions in
propositions of any complexity and the ability to analyze them to isolate
aspects in error. To analyze in this context constitutes division. The conflict
arises in our common cultural acceptance of the status quo as a baseline for
normal happiness, and the fact that the way things are in general is riddled
with error, chief among which is the strong belief in a constant unchanging
concept of right and wrong.
The logical existence of differing values of truth and
falsehood is unsettling taken out of context, and to many the very statement of
first-truth and second-truth and falsehood is enough to cause violent rejection
of the logical system, and perhaps the speaker herself. So much of many
cultures are based upon morals and perceptions of ‘this being true’ and ‘that
being false,’ that any advanced metaphysical thought or system threatens their
foundations. There is also a real element of separation due to achieved
perception derived from our system.
Bertrand Russell writes in his autobiography:
“The strain of unhappiness combined with very severe
intellectual work, in the years from 1902 till 1910, was very great. At the
time I often wondered whether I should ever come out at the other end of the
tunnel in which I seemed to be. I used to stand on the footbridge at
Kennington, near Oxford, watching the trains go by, and determining that
tomorrow I would place myself under one of them. But when the morrow came I
always found myself hoping that perhaps Principia Mathematica would be finished
some day. Moreover the difficulties appeared to me in the nature of a challenge,
which it would be pusillanimous not to meet and overcome. So I persisted, and
in the end the work was finished, but my intellect never quite recovered from
the strain. I have been ever since definitely less capable of dealing with
difficult abstractions than I was before. This is part, though by no means the
whole, of the reason for the change in the nature of my work.”
I was tickled by this excerpt from a letter in the
above book:
“My dying words to you are “Say good-bye to
mathematical logic if you wish to preserve your relations with concrete realities!”
Truly yours,
Wm. James
October
4, 1908
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| Script: Gerry Conway Pencils and inks: Patton/Machlan Wonder Woman™ © Respective
copyright/trademark holders. |
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
|
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