31. Aspects of Truth and Falsehood in terms of Negation and Disjunction

Harold R. (Hal) Foster’s Prince Valiant

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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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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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Harold R. (Hal) Foster’s Prince Valiant

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30. Systematic Ambiguity - A Poem in the System

Harold R. (Hal) Foster’s Prince Valiant

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 Systematic Ambiguity
(A N Whitehead and B Russell)

The universe consists

of objects which occur

having various qualities

and standing in

various relations.

Some of the objects

which occur in

the universe are

complex.

 

When an object

is complex, it

consists of

interrelated parts.

 

Let us consider

a complex object

composed of two parts

a and b standing

to each other in

the relation R.

The complex object

“a-in-the-relation-R-to-b”

may be capable

of being perceived;

when perceived,

it is perceived

as one object

Attention may show

that it is complex;

we then judge

that a and b

stand in the

relation R.

Such a judgment,

being derived from

perception by

mere attention,

may be called a

“judgment of perception.”

This judgment of

perception, considered

as an actual occurrence,

in a relation

of four terms,

namely: a and b

and R and the percipient.

The perception,

on the contrary,

is a relation

of two terms, namely:

“a-in-the-relation-R-to-b.”

and the percipient.

 

Since an object

of perception

cannot be nothing,

we cannot perceive

“a-in-the-relation-R-to-b,”

unless a is in the

relation R to b.

 

Hence a judgment

of perception,

according to

the above definition,

must be true.

This does not

mean that, in a

judgment which

appears to us

to be one of

perception, we

are sure of

not being in error,

since we may err

in thinking that

our judgment has

really been derived

merely by analysis

of what was perceived.

But if our

judgment has

been so derived,

it must be true.

 

In fact

we may define truth,

where such judgments

are concerned, as 

consisting in the fact

that there is a

complex corresponding

to the

discursive thought

which is the judgment.

 

That is,

when we judge

“a has the relation

R to b,” our

judgment is said

to be true when

there is a complex

“a-in-the-relation-R-to-b,”

and is said

to be false when

this is not the case.

 

This is a

definition of truth

and falsehood

in relation to

judgments of

this kind.

 
When I read the first two sentences in the above paragraph I read it as amazing poetry, and copied it down in my notebook broken down as presented above. I saw this analysis of truth and falsehood in terms of complexity and relations as something that relates to everyone and everything. This poem has been an aspect of Principia Mathematica that I can share with other persons and that they can understand in some fashion, and even if parts seem unclear to them, at least everyone derives some perception of love poetry.

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Harold R. (Hal) Foster’s Prince Valiant

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29. Definition and Systematic Ambiguity of Truth and Falsehood

Harold R. (Hal) Foster’s Prince Valiant

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III    Definition and Systematic Ambiguity of Truth and Falsehood.

 

Since "(x) . φx" involves the function φx^, it must, according to our principle, be impossible as an argument to φ. That is to say, the symbol "φ{(x) . φx}" must be meaningless. This principle would seem, at first sight, to have certain exceptions. Take, for example, the function "p^ is false," and consider the proposition "(p). p is false." This should be a proposition asserting all propositions of the form "p is false." Such a proposition, we should be inclined to say, must be false, because "p is false" is not always true. Hence we should be led to the proposition

"{(p). p. is false} is false,"

i.e. we should be led to a proposition in which "(p) . p is false" is the argument to the function "p^ is false," which we had declared to be impossible. Now it will be seen that "(p) . p is false," in the above, purports to be a proposition about all propositions, and that, by the general form of the vicious-circle principle, there must be no propositions about all propositions. Nevertheless, it seems plain that, given any function, there is a proposition (true or false) asserting all its values. Hence we are led to the conclusion that "p is false" and "q is false" must not always be the values, with the arguments p and q, for a single function "p^ is false." This, however, is only possible if the word "false" really has many different meanings, appropriate to propositions of different kinds.

That the words " true" and "false" have many different meanings, according to the kind of proposition to which they are applied, is not difficult to see. Let us take any function φx^, and let φa be one of its values. Let us call the sort of truth which is applicable to φa "first truth." (This is not to assume that this  would be first truth in another context: it is merely to indicate that it is the first sort of truth in our context.) Consider now the proposition (x) . φx. If this has truth of the sort appropriate to it, that will mean that every value φx has "first truth." Thus if we call the sort of truth that is appropriate to (x) . φx "second truth," we may define "{(x) . φx} has second truth" as meaning "every value for φx^ has first truth," i.e. "(x). (φx has first truth)." Similarly, if we denote by "(∃x). φx" the proposition "φx sometimes," i.e. as we may less accurately express it, "φx with some value of x," we find that (∃x) . φx has second truth if there is an x with which φx has first truth; thus we may define " {(∃x) . φx} has second truth" as meaning "some value for φx^ has first truth," i.e. "(∃x) . (φx has first truth)." Similar remarks apply to falsehood. Thus "{(x) . φx} has second falsehood" will mean "some value for φx^ has first falsehood," i.e. "(∃x) . (φx has first falsehood)," while "{(∃x). φx} has second falsehood" will mean "all values for φx^ have first falsehood," i.e. "(x) . (φx has first falsehood)." Thus the sort of falsehood that can belong to a general proposition is different from the sort that can belong to a particular proposition.

 Applying these considerations to the proposition "(p). p is false," we see that the kind of falsehood in question must be specified. If, for example, first falsehood is meant, the function "p^ has first falsehood" is only significant when p is the sort of proposition which has first falsehood or first truth. Hence "(p). p is false" will be replaced by a statement which is equivalent to "all propositions having either first truth or first falsehood have first falsehood." This proposition has second falsehood, and is not a possible argument to the function "p^ has first falsehood." Thus the apparent exception to the principle that "φ{(x). φx}" must be meaningless disappears.

 

Similar considerations will enable us to deal with "not-p" and with "p or q." It might seem as if these were functions in which any proposition might appear as argument. But this is due to a systematic ambiguity in the meanings of "not" and "or," by which they adapt themselves to propositions of any order. To explain fully how this occurs, it will be well to begin with a definition of the simplest kind of truth and falsehood. 

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 It is not difficult to see how this perception of a graduated value system of truth and falsehood benefits ambiguity and belies structured systems of laws and religious dogma. The truth of these assertions is also clear and we plainly deal with them constantly in common and extraordinary circumstances; e.g. a twelve year old girl is not allowed to drive the car by her parents or the police authorities. While on a family camp-out, her parents are hurt in a fall in a remote area and she hikes back to the car and drives to a place with cell service and calls for help. First truth in this example, law and custom forbid unlicensed people from driving; second truth is this young girl can’t be responsible or capable of driving. Conversely in this example the second truth corresponds to the first falsehood, for the young girl could and did drive responsibly; also the first truth is the converse of the second falsehood in that license laws can fade in importance in a case of emergency. 

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Harold R. (Hal) Foster’s Prince Valiant

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