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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
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Bertrand Russell was masterful in using expressive
language to make complex subject matter very clear to a careful reader. In the
introductory chapters Russell uses repetition to structure the subject and
impress the importance of each element introduced, and to aid in our learning.
In retrospect I realize my error in condensing the earlier posts, and if the
material is not clearly understood from reading them, I hope these supplemental
posts are useful.
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Script:
Wm M Marston Pencils and Inks: Harry
G. Peter
Wonder Woman™ © Respective
copyright/trademark holders.
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Assertion-sign.
The sign "┣," called the
"assertion-sign," means that what follows is asserted. It is required
for distinguishing a complete proposition, which we assert, from any
subordinate propositions contained in it but not asserted. In ordinary written
language a sentence contained between full stops denotes an asserted
proposition, and if it is false the
book is in error. The sign
“┣"
prefixed to a proposition serves this same purpose in our symbolism. For
example, if "┣ (p ⊃ p)" occurs, it is to be taken as a complete assertion
convicting the authors of error unless the proposition "p ⊃ p" is true (as it is). Also a
proposition stated in symbols without this sign "┣"
prefixed is not asserted, and is merely put forward for consideration, or as a
subordinate part of an asserted proposition.
Inference.
The process of inference is as follows: a proposition "p" is asserted, and a proposition
"p implies q" is asserted, and then as a sequel the proposition "q" is asserted. The trust in
inference is the belief that if the two former assertions are not in error, the
final assertion is not in error. Accordingly whenever, in symbols, where p and q have of course special determinations,
"┣ p" and "┣ (p ⊃ q)"
have occurred, then "┣ q" will occur if it is desired to
put it on record. The process of the inference cannot be reduced to symbols.
Its sole record is the occurrence of "┣ q." It is of course convenient, even
at the risk of repetition, to write "┣ p" and "┣ (p ⊃ q)" in close juxtaposition before
proceeding to "┣ q" as the result of
an inference. When this is to be done, for the sake of drawing attention to the
inference which is being made, we shall write instead
“┣ p ⊃ ┣ q,"
which is to be considered as a mere abbreviation of the
threefold statement
"p"
and "┣ (p
⊃ q)" and "┣ q."
Thus "┣ p ⊃ ┣ q" may be read "p, therefore q," being in fact the same abbreviation, essentially, as this
is; for "p, therefore q" does not explicitly state, what
is part of its meaning, that p
implies q. An inference is the
dropping of a true premise; it is the dissolution of an implication.
The
use of dots. Dots on the line of the symbols have two
uses, one to bracket off propositions, the other to indicate the logical
product of two propositions. Dots immediately preceded or followed by "v" or "⊃"
or "≡" or "┣" or by
"(x)," "(x,y),"
"(x,y,z)"… or "(∃x), "('∃x,y)," "(∃x,y,z)"...
or "[(ιx)
)(φx) ]" or "[R‘y]" or analogous
expressions, serve to bracket off a proposition; dots occurring otherwise serve
to mark a logical product. The general principle is that a larger number of
dots indicates an outside bracket, a smaller number indicates an inside
bracket. The exact rule as to the scope of the bracket indicated by dots is
arrived at by dividing the occurrences of dots into three groups which we will
name I, II, and III. Group I consists of dots adjoining a sign of implication (⊃)
or of equivalence (≡) or of disjunction (v)
or of equality by definition (= Df). Group II consists of dots following
brackets indicative of an apparent variable, such as (x) or (x, y) or (∃x) or (∃x,
y) or [(ιx) )(φx)] or analogous expressions*. Group III consists of dots
which stand between propositions in order to indicate a logical product. Group
I is of greater force than Group II, and Group II than Group III. The scope of
the bracket indicated by any collection of dots extends backwards or forwards
beyond any smaller number of dots, or any equal number from a group of less
force, until we reach either the end of the asserted proposition or a greater
number of dots or an equal number belonging to a group of equal or superior
force. Dots indicating a logical product have a scope which works both
backwards and forwards; other dots only work away from the adjacent sign of
disjunction, implication, or equivalence, or forward from the adjacent symbol
of one of the other kinds enumerated in Group II.
Some examples will serve to illustrate the use of dots.
"p v q .
⊃ .
q v p" means the proposition
"'p or q' implies 'q or p."' When we assert this proposition, instead
of merely considering it, we write
"┣ : p v
q . ⊃ .
q v p,"
where the two dots after the assertion-sign show that
what is asserted is the whole of what follows the assertion-sign, since there
are not as many as two dots anywhere else. If we had written "p: v:
q . ⊃. q v
p," that would mean the
proposition "either p is true,
or q implies 'q or p."' If we
wished to assert this, we should have to put three dots after the assertion-sign.
If we had written "p v q
. ⊃ .
q : v : p," that would
mean the proposition "either ‘p or q’ implies q, or p is true."
The forms "p . v . q
. ⊃ .
q v p" and "p v
q . ⊃ .
q . v . p" have no
meaning.
"p ⊃ q . ⊃ :
q ⊃ r . ⊃ .
p ⊃ r" will mean "if p implies q, then if q implies r, p
implies r." If we wish to assert
this (which is true) we write
"┣ :. p ⊃ q : q ⊃ r.”
Again "p
⊃ q . ⊃ .
q . ⊃ r : ⊃ .
p ⊃ r" will mean "if 'p implies q' implies 'q implies r,' then p implies r." This
is in general untrue. (Observe that "p
⊃ q" is sometimes most conveniently
read as "p implies q," and sometimes as "if p, then q.") "p ⊃ q . q
⊃ r. ⊃. p ⊃ r " will mean "if p implies q, and q implies r, then p implies r." In
this formula, the first dot indicates a logical product; hence the scope of the
second dot extends backwards to the beginning of the proposition. " p ⊃ q: q
⊃ r . ⊃ .
p ⊃ r" will mean "p implies q; and if q implies r, then p implies r." (This
is not true in general.) Here the two dots indicate a logical product; since
two dots do not occur anywhere else, the scope of these two dots extends
backwards to the beginning of the proposition, and forwards to the end.
"p v q
. ⊃ :.
p . v . q ⊃ r : ⊃. p v
r" will mean "if either
p or q is true, then if either p
or 'q implies r' is true, it follows that either p or r is true." If this is
to be asserted, we must put four dots after the assertion-sign, thus:
"┣ :: p v
q . ) :. p . v . q ⊃ r : ⊃ .
p v r."
(This proposition is proved in the body of the work; it
is *2.75.) If we wish to assert (what is equivalent to the above)
the proposition: "if either p or
q is true, and either p or 'q implies r' is true, then either p
or r is true," we write "
“┣ :. p v q : p v q
⊃ r : ⊃ .
p v r."
Here the first pair of dots indicates a logical
product, while the second pair does not. Thus the scope of the second pair of
dots passes over the first pair, and back until we reach the three dots after
the assertion-sign. Other uses of dots follow the same principles, and will be
explained as they are introduced. In reading a proposition, the dots should be
noticed first, as they show its structure. In a proposition containing several
signs of implication or equivalence, the one with the greatest number of dots
before or after it is the principal one: everything that goes before this one
is stated by the proposition to imply or be equivalent to everything that comes
after it.
*
The meaning of these expressions will be explained later, and examples of the use
of dots in connection with them are given in post 12.
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Script: Bill Mantlo Pencils: Sal Buscema Inks: Joe Sinnott The LeaderÔ © Respective copyright/trademark holders.
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective
copyright/trademark holders.
|
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