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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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This blog sprung up as I was deep in study of Principia Mathematica. Early on, I confess I was a bit at sea with the system, and I stubbornly read each paragraph, each page many times over, and then copied each word in longhand, carefully checking each passage for errors. I created a study guide to refer to, to help me learn the many symbols and notations.
It was for me, a journey from uncertainty to comprehension to clarity; and as I started this blog I forgot the struggle was what, in time, made things clear for me. I jumped out with my study guide, but I now realize that it is inadequate. I hope this and following supplimental posts will prove useful.
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Script: Hannigan Artwork: Perlin & Marcos HellcatÔ & © Respective copyright/trademark holders.
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CHAPTER I.
PRELIMINARY
EXPLANATIONS OF IDEAS AND NOTATIONS. THE notation adopted in
the present work is based upon that of Peano, and the following explanations
are to some extent modeled on those which he prefixes to his Formnulario
Mathematico. His use of dots as brackets is adopted, and so are many of his
symbols.
Variables.
The idea of a variable, as it occurs in the present work, is more general than
that which is explicitly used in ordinary mathematics. In ordinary mathematics,
a variable generally stands for an undetermined number or quantity. In
mathematical logic, any symbol whose meaning is not determinate is called a
variable, and the various determinations of which its meaning is susceptible
are called the values of the variable. The values may be any set of entities,
propositions, functions, classes or relations, according to circumstances. If a
statement is made about "Mr. A and Mr. B," "Mr. A" and
"Mr. B" are variables whose values are confined to men. A variable
may either have a conventionally-assigned range of values, or may (in the
absence of any indication of the range of values) have as the range of its
values all determinations which render the statement in which it occurs
significant. Thus when a text-book of logic asserts that "A is A,"
without any indication as to what A may be, what is meant is that any statement
of the form "A is A" is true. We may call a variable restricted when
its values are confined to some only of those of which it is capable;
otherwise, we shall call it unrestricted. Thus when an unrestricted variable
occurs, it represents any object such that the statement concerned can be made
significantly (i.e. either truly or
falsely) concerning that object. For the purposes of logic, the unrestricted
variable is more convenient than the restricted variable, and we shall always
employ it. We shall find that the unrestricted variable is still subject to
limitations imposed by the manner of its occurrence, i.e. things which can be said significantly concerning a
proposition cannot be said significantly concerning a class or a relation, and
so on. But the limitations to which the unrestricted variable is subject do not
need to be explicitly indicated, since they are the limits of significance of
the statement in which the variable occurs, and are therefore intrinsically
determined by this statement. This will be more fully explained later*1.
Chapter II of the Introduction.
To sum up, the three salient facts connected with the
use of the variable are: (1) that a variable is ambiguous in its denotation and
accordingly undefined: (2) that a variable preserves a recognizable identity in
various occurrences throughout the same context, so that many variables can
occur together in the same context each with its separate identity: and (3)
that either the range of possible determinations of two variables may be the
same, so that a possible determination of one variable is also a possible
determination of the other, or the ranges of two variables may be different, so
that, if a possible determination of one variable is given to the other, the
resulting complete phrase is meaningless instead of becoming a complete
unambiguous proposition (true or false) as would be the case if all variables
in it had been given any suitable determinations.
The
uses of various letters. Variables will be denoted by single
letters, and so will certain constants; but a letter which has once been
assigned to a constant by a definition must not afterwards be used to denote a
variable. The small letters of the ordinary alphabet will all be used for
variables, except p and s after *40, in which constant meanings
are assigned to these two letters. The following capital letters will receive
constant meanings: B, C, D,
E, F, I and J. Among small Greek letters, we shall
give constant meanings to ε, ι,
sand (at a later stage) to η, θ
and ω.
Certain Greek capitals will from time to time be introduced for constants, but
Greek capitals will not be used for variables. Of the remaining letters, p, q,
r will be called propositional letters,
and will stand for variable propositions (except that, from *40 onwards, p must not be used for a variable); ƒ, g,
φ, ψ, χ, θ
and (until *33) F will be called functional letters, and will be used for
variable functions.
The
small Greek letters not already mentioned will be used for
variables whose values are classes, and will be referred to simply as Greek letters. Ordinary capital letters
not already mentioned will be used for variables whose values are relations,
and will be referred to simply as capital
letters. Ordinary small letters other than p, q, r, s,
ƒ, g will be used for variables whose values are not known to be
functions, classes, or relations; these letters will be referred to simply as
small Latin letters.
After the early part of the work, variable propositions
and variable functions will hardly ever occur. We shall then have three main
kinds of variables: variable classes, denoted by small Greek letters; variable
relations, denoted by capitals; and variables not given as necessarily classes
or relations, which will be denoted by small Latin letters.
In addition to this usage of small Greek letters for
variable classes, capital letters for variable relations, small Latin letters
for variables of type wholly undetermined by the context (these arise from the
possibility of "systematic ambiguity," explained later in the
explanations of the theory of types), the reader need only remember that all
letters represent variables, unless they have been defined as constants in some
previous place in the book. In general the structure of the context determines
the scope of the variables contained in it; but the special indication of the
nature of the variables employed, as here proposed, saves considerable labor of
thought.
The
fundamental functions of propositions. An aggregation of
propositions, considered as wholes not necessarily unambiguously determined,
into a single proposition more complex than its constituents, is a function
with propositions as arguments. The general idea of such an aggregation of
propositions, or of variables representing propositions, will not be employed
in this work. But there are four special cases which are of fundamental
importance, since all the aggregations of subordinate propositions into one
complex proposition which occur in the sequel are formed out of them step by
step.
They are (1) the Contradictory Function, (2) the
Logical Sum, or Disjunctive Function, (3) the Logical Product, or Conjunctive
Function, (4) the Implicative Function. These functions in the sense in which
they are required in this work are not all independent; and if two of them are
taken as primitive undefined ideas, the other two can be defined in terms of
them. It is to some extent-though not entirely-arbitrary as to which functions
are taken as primitive. Simplicity of primitive ideas and symmetry of treatment
seem to be gained by taking the first two functions as primitive ideas.
The
Contradictory Function with argument p, where p is any
proposition, is the proposition which is the contradictory of p, that is, the
proposition asserting that p is not true. This is denoted by ~p. Thus ~p is the contradictory function with p as argument and means the negation of the proposition p. It will also be referred to as the
proposition not-p. Thus ~p means
not-p, which means the negation of p.
The
Logical Sum is a propositional function with two
arguments p and q, and is the proposition asserting p or q disjunctively,
that is, asserting that at least one of the two p and q is true. This is
denoted by p v q. Thus p v q is the logical sum with p
and q as arguments. It is also called
the logical sum of p and q. Accordingly p v q means that at least p
or q is true, not excluding the case
in which both are true.
The
Logical Product is a propositional function with two
arguments p and q, and is the proposition asserting p and q conjunctively,
that is, asserting that both p and q are true. This is denoted by p . q,
or - in order to make the dots act as brackets in a way to be explained
immediately - by p : q, or by p :. q, or by p ::
q. Thus p . q is the logical product with p
and q as arguments. It is also called
the logical product of p and q. Accordingly p . q means that both p and q are true. It is easily seen that this function can be defined in
terms of the two preceding functions. For when p and q are both true it
must be false that either ~p or ~q is true. Hence in this book p . q
is merely a shortened form of symbolism for
~(~p v ~q).
If any further idea attaches to the proposition
"both p and q are true," it is not required here.
The
Implicative Function is a propositional function with two
arguments p and q, and is the proposition that either not-p or q is true, that
is, it is the proposition ~p v q.
Thus if p is true, ~p is false, and
accordingly the only alternative left by the proposition ~p v q is that q is true. In
other words if p and p v
q are both true, then q is true. In this sense the proposition
~p v q will be quoted as stating that p implies q. The idea contained in
this propositional function is so important that it requires a symbolism which
with direct simplicity represents the proposition as connecting p and q without the intervention of ~p.
But "implies" as used here expresses nothing else than the connection
between p and q also expressed by the disjunction "not-p or q." The symbol
employed for "p implies q," i.e. for "p v q,"
is "p ⊃ q." This
symbol may also be read "if p, then q." The association of
implication with the use of an apparent variable produces an extension called
"formal implication." This is explained later: it is an idea
derivative from "implication" as here defined. When it is necessary
explicitly to discriminate "implication"from "formal
implication," it is called "material implication." Thus
"material implication" is simply "implication" as here
defined. The process of inference, which in common usage is often confused with
implication, is explained immediately.
These four functions of propositions are the
fundamental constant (i.e. definite) propositional functions with propositions as arguments, and all other
constant propositional functions with propositions as arguments, so far as they
are required in the present work, are formed out of them by successive steps.
No variable propositional functions of this kind occur in this work.
Equivalence.
The simplest example of the formation of a more complex function of
propositions by the use of these four fundamental forms is furnished by
"equivalence." Two propositions p
and q are said to be "equivalent"
when p implies q and q implies p. This relation between p and q is denoted by "p ≡
q." Thus "p ≡ q"
stands for "(p ⊃ q). (q
⊃ p)." It is easily seen that' two
propositions are equivalent when, and only when, they are both true or are both
false. Equivalence rises in the scale of importance when we come to
"formal implication" and thus to "formal equivalence." It
must not be supposed that two propositions which are equivalent are in any
sense identical or even remotely concerned with the same topic. Thus
"Newton was a man" and "the sun is hot" are equivalent as
being both true, and "Newton was not a man" and "the sun is
cold" are equivalent as being both false. But here we have anticipated
deductions which follow later from our formal reasoning. Equivalence in its
origin is merely mutual implication as stated above.
Truth-values.
The "truth-value" of a proposition is truth if it is true, and falsehood
if it is false*2. It
will be observed that the truth-values of p
v q, p . q, p
⊃ q, ~p,
p ≡ q depend only upon those of p
and q, namely the truth-value of
"p v q" is truth if the
truth-value of either p or q is truth, and is falsehood otherwise;
that of "p . q" is truth if that of both p and q is truth, and is falsehood otherwise; that of "p ⊃ q"
is truth if either that of p is falsehood or that of q is truth; that of "~p" is the opposite of that of p;
and that of "p ≡ q" is truth if p and q have the same
truth-value, and is falsehood otherwise. Now the only ways in which
propositions will occur in the present work are ways derived from the above by
combinations and repetitions. Hence it is easy to see (though it cannot be
formally proved except in each particular case) that if a proposition p occurs in any proposition ƒ(p) which we shall ever have occasion to
deal with, the truth-value of ƒ(p) will
depend, not upon the particular proposition p,
but only upon its truth-value; i.e.
if p ≡ q, we shall have ƒ(p) ≡ ƒ(q). Thus whenever two propositions are
known to be equivalent, either may be substituted for the other in any formula
with which we shall have occasion to deal.
We may call a function ƒ(p) a "truth-function "when its argument p is a proposition, and the truth-value
of ƒ(p) depends only upon the
truth-value of p. Such functions are
by no means the only common functions of propositions. For example, "A believes p" is a function of p
which will vary its truth-value for different arguments having the same
truth-value: A may believe one true
proposition without believing another, and may believe one false proposition
without believing another. Such functions are not excluded from our
consideration, and are included in the scope of any general propositions we may
make about functions; but the particular functions of propositions which we
shall have occasion to construct or to consider explicitly are all
truth-functions. This fact is closely connected with a characteristic of
mathematics, namely, that mathematics is always concerned with extensions
rather than intensions. The connection, if not now obvious, will become more so
when we have considered the theory of classes and relations.
*
Post 26.
*2
This phrase is due to Frege.
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Harold R. (Hal)
Foster’s Prince Valiant
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copyright/trademark holders.
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