25. On "LOGIC", by Wilfred Hodges (excerpts) -- basing "likelihoods" on other likelihoods (the trouble with probabilities based on other probabilities)

Excerpts from "Logic",
by Wilfrid Hodges,
Penguin Books,
1977
 
... the
problem
of
evil:
this
involves
three short sentences:
 
God is loving.
God is omnipotent.
God allows people to get hurt.
 
According to
traditional
Christian teaching,
these
three sentences
are not
merely
consistent:
they are all
actually
true.
 
But
can they be?
 
If God
is omnipotent
and
allows people
to get hurt,
we
have to suppose
that
God
doesn't care
whether
people get hurt,
since otherwise
(being omnipotent)
he could have
prevented it.
 
If
God
doesn't care
whether
people get hurt,
can he
really
be described
as "loving"?
 
Humans
who don't care
whether
people get hurt
are
not normally
described
as loving.
 
It could be argued
that God
is
so different
from humans
that
the use of words
like "love" and "care"
in human situations
is
no guide
to
how
they should be used
of
God --
situations
involving God
are
bizarre.
 
This has
often
been suggested.
 
But
it is unconvincing,
for
two reasons.
 
The first
is that
the problem of evil
has been
discussed
for
several centuries
longer
than
the English language
has existed;
bizarre situations
occur
only
when
the speakers
of
a language
have not yet had
a chance
to
determine how
they wish
to
use certain words
to
describe
unusual circumstances.
 
The second
is that,
far from
there being
nothing
to
guide us
in
using words
like
"loving"
of God,
God's
reported
or
observed
behavior
might seem
to make it
impossible
to
describe him
as
straightforwardly
and
constantly
loving.
 
[p. 44 - 45]
 
+++
 
"It seems, then,
that
in order to assess
the
rationality
of
an argument,
we need to
take into account
all
the known facts,
and
not just
the
stated premises.
 
An argument
is
normally deployed
against
a background of
known facts
and
agreed beliefs,
and
the rationality
of
the argument
depends
on
what these facts
and
beliefs
are.
 
All this
makes it
hard to see
how
one
could devise
a simple
and
practical test
of
rationality.
 
Perhaps
also
we are
to
some extent
free to choose
for ourselves
what we count
as
an
adequate reason
for
believing
a thing.
 
By nature
some people
are
more skeptical
than
others.
 
In 1670
Chief Justice
Vaughan
put it
rather well:
"A man
cannot see
by
another's eye,
nor hear
by
another's ear;
no more
can a man
conclude
or
infer
the thing
to be resolved
by
another's
understanding
or
reasoning."
 
[p. 60]
 
+++
 
... one can
easily describe
an "L2 structure"
(that is,
a statement
in
formal language
for
logical analysis)
which
resists
all
systematic methods
for
computing
truth-values
in it;
if there were
a method
for
computing
truth-values
in
this structure
[and thereby
reducing
"Truth"
to
a
formal calculus],
it would
solve
in one blow
some
arithmetical problems
which
have held out
against
three hundred years' battering
by
the
best mathematicians
in
Europe.
[p. 236 - 237]
 
+++
 
The fanatical
formalizer
can
take refuge
in
formal
predicate tableaux
if he wishes,
since they
at least
are
arrays of symbols
which
can be
manipulated
without
any thought
of
meaning
or
interpretation.
 
[...]
 
It seems
to be
a
principle,
at
all levels
of
logic,
that
to justify
a
procedure
(say, tableaux
or truth-tables),
one has to
use methods
which
are
themselves
harder to justify
than
the
procedure
which
they
are
justifying.
 
This looks
crazy,
but
it isn't.
 
[...]
 
The
struggle
for
yet more
global justifications
is
one
of the mainsprings
of
logical research.
 
[p. 237]
 
+++
 
"Improbable
things
do
happen."
[p. 238]
 
+++
 
Imagine
the
shuffled pack
of cards
in front of us.
 
There are
fifty-two cards
arranged
in
four suits.
 
Since the cards
have been
shuffled,
they may be
in any order.
 
We may
therefore
be
in
any one of
n
different situations,
where
n
is
the number
of ways
a pack of cards
can be
ordered.
 
Let us
call these
n
situations
the
allowed
situations.
 
Now
it is
surely true
that :
(42.9)
The top card
is more likely
to be a diamond
than a nine.
 
Why is
(42.9)
true?
 
Because
the
top card
is
a diamond
in a quarter
of the
allowed situations,
but
it is
a nine
in only
a thirteenth
of these situations.
 
[p. 239 - 240]
 
+++
 
This logic
of
Likelihood
is
elegant and convincing;
it
meshes well
with
the mathematical theory
of
probability.
 
Nevertheless
our account
of it
has
one major flaw
which
we
must correct.
 
The flaw
is a
matter
of
interpretation.
 
At the beginning
of
our discussion
we imagined
a shuffled pack
of cards,
and
we
noted that
as
far as
the
order of the pack
was concerned,
we
could be
in
any one
of
several possible
situations.
 
Without saying so
explicitly,
we assumed that
every one
of
these situations
was
equally likely
to be
the
actual situation.
 
If some
orderings
of
the pack
had been
more likely
than others --
if,
say,
there was
a suspicion
that
a clever shuffler
had
brought all the nines
to
the top --
then
(42.9,
"The top card
is more likely
to be a diamond
than a nine.")
need
no longer
be true.
 
In
a similar way,
the
restrictions
to
"allowed" situations
merely excluded
those situations
which
we
thought
were
too unlikely
to
take seriously.
 
For example,
if there
was a chance
that
someone
had quietly removed
the diamonds
from
the pack,
we
should have to
take this
into account
when
we assessed
the
truth
of (42.9 --
"The top card
is more likely to be
a diamond
than a nine.").
 
For reasons
such as these,
we are forced
to
reinterpret
probability
measures
as showing ,
not
what proportion
of
the
allowed situations
assign
the
indicated truth-values,
but
how likely it is
that
the actual situation
is
one
which
assigns them.
 
Once this
reinterpretation
is made,
our calculations
can
no longer be justified
by
an
appeal to
(42.11--
"There are
at least
as many
allowed situations
in which it's true
[that
the top card
will be a diamond]
as
there are
allowed situations
in which
it's true
[that
the card
will be a nine]").
 
Instead
we
have to
base them
on
our
intuitive
understanding
of
probability.
 
In
other words,
our calculus
only
tells us
how
to
deduce likelihoods
from
other likelihoods.
 
It
is
not clear
how far
likelihoods
can be
deduced
from
anything else.
 
Certainly
we all do
estimate
likelihoods
all the time --
what chance
I'll reach
the shops
before
closing time?
 
Might James
take offense?
 
Will
another
drink
make me
feel sick?
 
Is
the ladder
safe?
 
Apparently
we do it
on the basis
of
the
facts
we know.
 
But
nobody
has yet provided
a complete
and
convincing
account
of
how
to deduce
a likelihood
from
brute facts
alone.
 
Maybe
it can't
be done,
and
estimating
likelihoods
is
fundamentally different
from
deducing
them.
 
Maybe
it can,
but
only
by arguments
which are
too long
to set down
on paper.
 
Here
is an example
of
the difficulties.
 
It used to
be thought
that
if every
"S"
so far
discovered
is a
"P",
then
the discovery
of one more
"S"
which
was a
"P"
would
make it
more likely
that
every "S"
is
a "P".
 
But
in fact
this is
not so.
 
For instance,
take the question
whether
recognizably
human
animals
existed
three and a half million years ago.
 
Every
human being
discovered
so far
is
less than
three and a half million years old.
 
On
the earlier view,
the discovery
of
another skull
less than
three and a half million years old
should make it
less likely
that humans
go back
three and a half million years.
 
But
in fact
if a
decidedly
human skull
was found
to be
three and a quarter million years old,
this would
make it
more likely
that
our
human ancestors
go back
still
further.
 
Likewise,
if
both
the known sufferers
from
a
rare bone disease
are
called
John,
then
the discovery
of
a
third John
with
the disease
would make it
more likely
that
there are
people
not called John
who
have
the disease.
 
[Drawn from pages 239 -244]
 
+++
 
... many
scientific
concepts
are
defined
in terms of
subjunctive
conditionals.
 
To say
that a substance
is a gas
is
to say that
it would contract
in
a certain way
if
increased pressure
were applied
to it.
 
To say
that a substance
is
poisonous
is to say
that
it would cause harm
if
it were administered
to
organisms
in
certain ways.
 
As
another example,
English Law
has
been known
to
decide questions
of
reasonableness
by
appealing
to
what the man
on the
Clapham omnibus
would have
thought
if
he had been
a witness.
 
This
legal criterion
is
admittedly
something
of a
joke --
there are
after all
many different men
on
Clapham omnibuses.
 
But courts
have used it
for
several decades
as
a rough and ready
test.
 
[p. 248]
 
+++
 
... the
calculations
might be
inordinately long.
Indeed
they might be
infinite,
so that
only
an angel,
or
a
retired god
could
complete them.
 
[p. 250]
 
+++

Many people
believe
inconsistent things --
perhaps we all do.
 
[p. 252]
 
+++