Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark holders.
Formal implication and formal equivalence
When an implication, say φx . É . y x, is said to hold always, i.e. when (x) : φx . É . yx, we shall say that
φx formally implies yx; and propositions
of the form "(x): φx . É .
yx"
will be said to state formal
implications. In the usual instances of implication, such as
"'Socrates is a man' implies 'Socrates is mortal,"' we have a
proposition of the form "φx . É .
yx "
in a case in which "(x): φx . É .
yx"
is true. In such a case, we feel the implication is a particular case of a
formal implication. Thus it has come about that implications which are not
particular cases of formal implications have not been regarded as implications
at all. There is also a practical ground for the neglect of such implications,
for, speaking generally, they can only be known when it is already known either
that their hypothesis is false or that their conclusion is true; and in neither
of these cases do they serve to make us know the conclusion, since in the first
case the conclusion need not be true, and in the second it is known already.
Thus such implications do not serve the purpose for which implications are chiefly
useful, namely that of making us know, by deduction, conclusions of which we
were previously ignorant. Formal implications, on the contrary, do serve this
purpose, owing to the psychological fact that we often know "(x) : φx . É .
yx"
and yy, in
cases where by (which follows from these premises) cannot easily be known
directly.
These reasons, though they do not warrant the complete
neglect of implications that are not instances of formal implications, are
reasons which make formal implication very important. A formal implication
states that, for all possible values of x,
if the hypothesis φx is true, the
conclusion yx
is true. Since "φx . É .
yx"
will always be true when φx is false, it is only the values of x that make φx true that are important
in a formal implication; what is effectively stated is that, for all these
values, yx
is true. Thus propositions of the form "all a
is b," "no a
is b" state formal implications, since the
first (as appears by what has just been said) states
(x) : x is an a .
É . x
is a b,
while the second states
(x) : x is an a .
É . x
is not a b.
And any formal implication "(x): φx . É .
yx" may be interpreted as: "All
values of x which satisfy* φx satisfy
yx,"
while the formal implication "(x)
: φx . É .
~yx"
may be interpreted as: "No values of x
which satisfy φx satisfy yx."
We have similarly for "some a
is b " the formula
($x) . x is an a .
x is a b,
and for "some a
is not b" the formula
($x). x is an a .
x is not a b.
Two functions φx,
yx
are called formally equivalent when
each always implies the other, i.e.
when
(x) : φx º .
yx,
and a proposition of this form is called a formal equivalence. In virtue of what
was said about truth-values, if φx
and yx
are formally equivalent, either may replace the other in any truth-function.
Hence for all the purposes of mathematics, or of the present work, φz^ may
replace yz^ or vice versa in any proposition with
which we shall be concerned. Now to say that φx and yx
are formally equivalent is the same thing as to say that φz^ and yz^
have the same extension, i.e. that any value of x which satisfies either satisfies the
other. Thus whenever a constant function occurs in our work, the truth-value of
the proposition in which it occurs depends only upon the extension of the
function. A proposition containing a function φz^ and having this property (i.e. that its truth-value depends only
upon the extension of φz^) will be called an extensional function of φz^. Thus
the functions of functions with which we shall be specially concerned will all
be extensional functions of functions.
What has just been said explains the connection (noted
above) between the fact that the functions of propositions with which
mathematics is especially concerned are all truth-functions and the fact that
mathematics is concerned with extensions rather than intensions.
* A value of x is said to satisfy φx or yx
when φx is true for that value of x.
Walt Kelly’s Songs of the Pogo©
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When contemplating a complex problem of
any sort, the answers are very often unknowable by empirical means. This
conundrum however is not an insurmountable obstacle to finding a solution to
any such problem by means of tools including formal implication and formal
equivalence.
These same tools are paramount to the
Indian’s perceptual toolbox. Instead of expecting a new object, creature,
person or idea to fit into her collective experience, she will seek
implications and equivalences to reach some means to bring the unknown into a
level of common knowledge.
We use the tools of implication and
equivalence to find meaning in many ways, including in this beautiful song by
Jackson Browne and performed with David Lindley, with vocals by renowned Spanish
songbird Luz Casal.
Script: Bob Haney Pencils and inks: John Rosenberger
Superman™ Wonder Woman™ © Respective
copyright/trademark holders.
hanBLOGlaka
Logical Philosophy In American
Indian thought and
Perception
Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.
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