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7/26/2019 Susan Haack - Justifications of Deduction
1/9
Mind Association
The Justification of DeductionAuthor(s): Susan HaackSource: Mind, New Series, Vol. 85, No. 337 (Jan., 1976), pp. 112-119Published by: Oxford University Presson behalf of the Mind Association
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2/9
The Justificationf Deduction'
SUSAN HAACK
(i)
It is often aken for grantedby writers
who propose-and, for that
matter, y
writerswho
oppose-'justifications'
of nduction,hatdeduction
either
does not
need,
or
can readilybe providedwith, ustification.
he
purpose
of
this
paper is
to
argue that,
ontrary o this commonopinion,
problems analogous to those which, notoriously,
rise in the attempt o
justify nduction, lso arise n the attempt
o ustify eduction.
Hume presented us with
a
dilemma:
we cannot justify nduction
deductively,because to do so would be to show that whenever he
premisses f an inductive rgument re
true, he conclusionmustbe true
too-which
would
be too
strong; nd
we cannot ustify
nduction
in-
ductively, ither, ecause such a 'justification' ould be circular. propose
anotherdilemma: we cannot ustify eduction
nductively, ecause
to do
so
would
be,
at
best,
to
show that usually,when
the
premisses
of a
deductive argument
re
true, the
conclusion
s
true too-which
would
be too
weak;
and we
cannot
ustify
eductiondeductively, ither,
ecause
such
a
justification ould be circular.
The
parallel between he old and
the new dilemmas
an
be illustrated
thus:
Hume's
dilemma
induction
deductive inductive
justification
justification
too
strong
-circular
The new
dilemma
deduction
inductive
deductive
justification justification
-too
weak -circular
(2)
A
necessarypreliminary
o
serious discussion
of
the problems
of
justifyingnduction/deduction
s a
clear statement
f
them.
This means, first, iving some
kind of characterisation f 'inductive
argument'
nd
'deductive
argument'.
This
is
a
more
difficult
ask
than
seems to be generally ppreciated. It will hardly do, for example, to
characterise eductive rguments
s 'non-ampliative' Salmon [i966])
or
I have profited rom ommentsmade when an earlierversionof this
paper
was read to
the Research Students' Seminar in
Cambridge, May
1972.
I
I2
7/26/2019 Susan Haack - Justifications of Deduction
3/9
THE
JUSTIFICATION
OF DEDUCTION
II3
'explicative' Barker
I965]),
and
inductive rguments s 'ampliative'
or
'non-explicative';
for
these
characterisationsre apt to turn out either
false, if the key notion of 'containingnothing n the conclusion not
already
ontained
n the
premisses'
s taken
iterally,
r
trivial,
f t
is
not.
Because of the difficultiesf
demarcating inductive' and 'deductive'
inference,t seems more profitable o
define n argument:
An
argument s a sequence
Al
.
..
An of
sentences (n
> i),
of
which
A1.
. .
An_1
re the premisses
nd
An
is the conclusion
-and then to tryto distinguish
nductivefromdeductive standards
of
a 'good
argument'.
It
is
well known that deductive
standards of validitymay be put
in
either
f
two
ways: syntacticallyr
semantically. o:
D1
An argumentA1 ..
An
1
F An is deductively alid (in
LD)
just
in case the
conclusion,
An,
is
deducible
from
he premisses,
A1
..
An-,,
and the axioms ofLD, if
any,
n virtue f
the
rules
of inference f
LD
(the
syntactic efinition).
D2 An argumentA1 ..
An
1
H
An is deductivelyvalid just in
case it is impossible that the
premisses,
A1
. . .
An-,
should be
true, nd the conclusion,An,
false the semanticdefinition).
Similarly,
we can
express
standards
of
inductive
strength
either
syntactically
r
semantically;
he
syntacticdefinition
would follow D1
butwith LI' for
LD';
thesemanticdefinition ouldfollowD2 butwith
'it
is
improbable, iventhat the
premisses re true, hatthe conclusion s
false'.
The
question
now
arises, which of these kinds of characterisation
should
we
adopt
n our
statement f
the
problems
f
ustifying
eduction/
induction?
This
presents difficulty.f
we adopt semanticaccounts
of
deductive
validity/inductive
trength,
he problem of justificationwill
seem to
have
been
trivialised.
The
justification roblem will reappear,
however,
n a
disguisedform, s the question Are there ny deductively
valid/inductivelytrongarguments?'. f,
on
the other hand, we adopt
syntacticccountsofdeductivevalidity/inductivetrength, he natureof
the
justification roblem
is
clear: to
show that argumentswhich
are
deductively valid/inductively trong
are also truth-preserving/truth-
preservingmost
of
the
time i.e.
deductively alid/inductivelytrong
n
the semantic
ccounts).
On
the other
hand, there s
the
difficultyhat
we
must
somehow
specify
which
systems
re
possible values
of
'LD'
and
'LI',
and
this will
presumably equire
appeal
to
inevitably ague
con-
siderations
oncerning
he intentions
f
the authors
of
a formal
ystem.
A
convenient
ompromise s this. There are certain orms
f
nference,
such as
the rule:
RI From: m/nfall observedA's havebeen B's
to
infer:
m/n
f
all
A's are
B's
which are
commonly taken
to
be
inductively trong, and similarly,
certain
forms
f
inference, uch as
MPP
From:
A
- B,A
to infer:
B
7/26/2019 Susan Haack - Justifications of Deduction
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114
SUSAN
HAACK:
which
re
generally
aken o be
deductively
alid.
Analogues
of
the
general
justification
roblems
an
now
be set
up
as
follows:
theproblemof the ustificationfinduction: howthat RI is truth-
preserving
most of
the
time.
the
problem of the
justification
f
deduction:
show
that MPP
is
truth-preserving.
My procedurewill
be,
then,
o
show
that
difficulties
rise
n the
attempt
to
ustify
MPP which
re
analogous
to
notorious
ifficulties
rising
n
the
attempt
o
ustify
RI.
(3)
I
consider
first he
suggestion
hat
deduction
needs
no
ustification,
that
the
call
for a
proof
that MPP
is
truth-preserving
s
somehow
mis-
guided.
An
argument
orthis
position
might
go
as
follows:
It is
analytic hat
a
deductively
alid
argument s
truth-preserving,
for
by
valid'
we mean
argument
whose
premisses ould
not
be
true
without ts
conclusion
being
true
too'. So
there
can
be no
serious
question
whether
deductively
alid
argument
s
truth-preserving.
It
seems
clear
enough that
anyone who
argued ike this
would
be the
victim of a
confusion.
Agreed,
if we
adopt
a semantic
definition
f
'deductively
alid'
it
follows
mmediately
hat
deductively
alid
arguments
aretruth-preserving.uttheproblemwas,to showthat particular orm
of
argument,
form
deductively
valid in the
syntactic
ense,
is truth-
preserving;
nd this
s a
genuine
problem,
which
has
simply een
evaded.
Similar
arguments
how the
claim,
made
e.g.
by
Strawson n
[I952],
p.
257,
that
nduction
needs no
justification,
o be
confused.
(4)
I
argued
in
Section
i) that
justifications' f
deduction
re
liable
either
o
be
inductive nd too
weak,
or to
be
deductive
nd
circular.The
former,
nductive
kind
of
ustification as
enjoyed ittle
opularity
except
with
the
Intuitionists? f.
Brouwer
I952]).
But
arguments f
the
second
kind
are not hard
to
find.
(a) Considerthefollowingttempt o ustifyMPP:
Ai
Suppose that A'
is
true,
nd
that A
D
B'
is
true.
By
the
truth-
table for
D',
if
A' is true and
'A
D
B'
is
true,
hen
B'
is true
too.
So 'B'
must be true
too.
This
argument
as
a
serious
drawback: t
is of the
very
form
which
t
is
supposed
to
ustify.
For
it
goes:
Ai'
Suppose
C
(that
A'
is
true
nd thatA
D
B'
is
true). f
C then
D
(if
'A'
is
true and
'A
D
B'
is
true,
B'
is
true).
So,
D
('B'
is true
too.
The
analogy
with
Black's
'self-supporting' rgumentfor
induction
[I954]
is
striking. lack
proposes
to
support
nduction
by
means of
the
argument:
A2
RI
has
usually
been
successful
n
observed
nstances.
.'.
RI
is
usually
successful.
7/26/2019 Susan Haack - Justifications of Deduction
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THE JUSTIFICATION
OF
DEDUCTION
II5
He defendshimself gainst
the charge of circularity
y pointingout
that this argument s not a simple
case of question-begging:
t does not
contain ts conclusionas a premiss. t might, imilarly, e pointedout
thatAi' is not
a simple case
of question-begging: or t does not
contain
its conclusion
s
a
premiss, ither.
One is inclined
to feel that
A2
is objectionably
ircular,
n spite
of
Black's defence;
nd this ntuition
an be supportedby an argument,
ike
Salmon's [I966],
to show that
f
A2
supports RI, an exactly
analogous
argumentwould
support counter-inductiveule,
say:
RCI
From: most observedA's have not been
B's
to infer:
most A's
are
B's.
Thus:
A3 RCI
has
usually been
unsuccessful n thepast.
RCI is
usually
successful.
In a similarway, one can support
the intuition
hat there s something
wrong
with
Ai',
in
spite
of
its
not being straightforwardlyuestion-
begging, by showing that if
Ai'
supports MPP,
an exactly analogouis
argument
would
support
deductively
nvalid
rule, say:
MM
(modus
morons);
From:
A
D)
B
and
B
to
infer:A.
Thus:
A4 Supposing that
A
D
B'
is true and
'B' is true, A
D
B'
is true
v 'B' is true.
Now, by the truth-table
or
::',
if A' is true, hen, f A
D
B'
is
true,
B'
is
true.
Therefore,
A'
is true.
This
argument,
ike
Ai,
has the
very
ormwhich t s
supposed
to
ustify.
For it
goes:
A4' Suppose
D
(if A
D
B'
is true,
B'
is true).
If C, thenD (if A' is true, hen, f A
D
B' is true, B' is true).
So,
C
('A'
is
true).
It
is no
good
to
protest
hat
A4'
does
not
ustify
modusmorons
ecause
it
uses
an invalidrule of
nference,
hereas
A4'
does
ustify
modus
onens,
because
it uses
a
valid rule
of
nference-for
o
ustify
ur
conviction
hat
MPP
is
valid
and
MM is
not
is
precisely
what s at issue.
Neither
s
it any
use
to protest
hat
Ai'
is
not
circularbecause it
is an
argument
n the
meta-language,
whereas
he
rule which
t
is
supposed
to
justify
s a
rule
in the
object
language.
For
the attemptto save
the
argument
orRI
by taking
t
as
a
proof,
n
level2,
of
a rule of evel i,
also
fallsprey o the difficultyhatwe could withequal justice give a counter-
inductive
rgument,
n level
2,
forthe
counter-inductive
ule at
level i.
And
similarly,
f
we
may give
an
argumentusing MPP,
at
level
2,
to
support
the
rule MPP at
level
i,
we
could, equally,
give
an
argument,
using MM,
at
level 2,
to
support
he rule
MM at level i.
(b)
Another
way
to
try
to
justify MPP, which promises
not to be
7/26/2019 Susan Haack - Justifications of Deduction
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II6 SUSAN
HAACK:
vulnerableto the difficultyhat, f it is
acceptable, so is an
analogous
justification f
MM, is suggestedby Thomson's discussion I963] of the
Tortoise's argument.Carroll's tortoise, n
[i895],
refuses o draw the
conclusion,
B',
from A - B' and 'A',
insisting hat a new premiss,
'A
( (A -
B) - B)' be added; and when thatpremiss s grantedhim,
will stillnot
draw the conclusion, ut insists n a further remiss, nd so
ad indefinitum.homson argues hatAchilles
hould neverhave
conceded
that
n
extra
premisswas needed; for,he argues, f the original nference
was
valid
(semantically) he added premiss
s true but not needed,
and
if the original
nferencewas invalid
semantically) he added premiss s
needed
but false.There is an analogy,here,
gain,with ttempts o ustify
inductionby
appending
premiss-something, sually, o the effect hat
'Nature is uniform'-which turns nferencesn accordancewith RI into
deductively alid inferences. he required
premisswould, presumably,
be true but not
needed if RI were
deductively
alid,
false but needed
if
it
is not.
Thomson's
idea suggeststhat we should contrast his picture n
the
case of MPP:
A5
(i)
A
D
((A
D
B)
D
B)
(true
but
superfluous
remiss)
(2)
A
assumption
(3) (A
B)
B
I,
2,
MPP
(4)
A
B
assumption
(5) B 3, 4, MPP
withthis picture n the case of MM:
A6 (i)
B
D
( (A
D
B)
D
A) (false but
needed premiss)
(2)
B
assumption
(3) (A B) A I,
2,
MPP
(4)
A
B
assumption
(5)
A
3,
4,
MPP
Thomson's
point s that n As premiss
i)
is
a tautology, o true; but it
is not
needed, since
ines
2),
(4)
and
(5)
alone constitute valid argument.
In A6, by contrast, remiss
i)
isnot tautology; ut t sneeded, because
lines
2),
(4)
and
(5)
alone
do
not constitute valid
argument.
ut
this s
to assume
that
MPP,
which
s
the rule
of
nference n virtue
f
which n
As (2) and (4) yield
(5),
is valid; whereas
MM, which is the rule
of
inference n
virtueof which, n A6,
(2)
and
(4)
would yield
(5),
is not
valid.
But
this s
just
what
was to have been
shown.
If
As justifiesMPP, which,after ll, it
uses, then the following rgu-
ment
equally ustifiesMM:
A7
(i)
(A- B)
(A
:D
B) (true
but superfluous remiss)
(2)
A
B
assumption
(3)
A
B
I, 2,
MM
(4)
B
assumption
(5)
A
3,
4,
MM
In
A7
as in
As
the
first remiss s a
tautology, o true,but it is super-
fluous, ince ifMM is accepted) ines
2),
(4)
and
5)
alone onstitute
valid
argument.
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THE
JUSTIFICATION
OF DEDUCTION
I17
(c)
Nor will it do to
argue that
MPP is, whereasMM
is not,
ustified
'in virtue
f
the
meaning
of
D
'.
For how
is the meaningof
D
'
given?
There are threekinds of answer commonly iven: that the meaningof
the
connectives s given
by the rules
of inference/axioms
f the
system
in which
theyoccur;
thatthe meaning
s given
by the interpretation,
r,
specifically,
he
truth-table, rovided;
that the
meaning s given
by the
English
readings
of the connectives.
Well,
if
D'
is supposed
to be at
least
partially
defined by the rules
of inference
governing entences
containing
t (cf.
Prior
[I960], [I964]) then
MPP and MM
would be
exactly
on a
par.
In a
system
containingMPP the
meaning
of
'2)'
is
partially
definedby the
rule, from A
=)
B'
and 'A',
to infer B'.
In a
systemcontaining
MM
the meaning of
D
'
is partially
definedby the
rule,fromA - B' and B', to inferA'. In either ase the rule nquestion
would be justified n virtue
of the
meaning of
'-', finally,
ince the
meaning
of
D'
would be
given by
the rule. If, on the
other
hand, we
thought
of
'v'
as partially
defined
by its truth-table
cf. Stevenson
[i96i1]),
we are in the difficulty
iscussed earlier (a)
above)
that argu-
mentsfrom he
truth-table
o the ustification
f a rule
of inference re
liable to employ
the rule
in question. Nor
would it do
to appeal to the
usual
reading
of
D'
as
'if
..
then
. .
',
not ust because
the
propriety
of
that
reading
has been doubted,
but also
because the question,
why
'B'
follows
rom
if
A
then
B'
and 'A' but
not A' from
if A then
B'
and
'B',
is
precisely
nalogous
to the question
at
issue.
(d) Our arguments against attempted ustificationof MPP have
appealed
to the fact that
analogous
procedureswould
justify
MM. So
at
this point it
might be
suggested that we
can produce
independent
arguments gainst
MM.
(Compare attempts
o diagnose
ncoherence n
RCI.)
In
particular,
t
might
be supposed
that it
is
a relatively
imple
matter
o
show that MM cannot
be truth-preserving,
ince with
MM
at
our disposal we
could argue
as follows:
A8
(i) (P
&-P)
=D
(Pv-P)
(2)
Pv -P
(3) P&-P
1,2MM
So
that a
system
ncluding
MM
would be
inconsistent.
This
idea
is
suggestedby
Belnap's paper
on
'tonk'.)
However,
this
argument
s inconclusive
because
it
depends
upon
certain ssumptions
bout what
else we
have
in the
system
o which
MM
is appended-in
particular,
hat
i)
and
(2)
are theorems.
Now
certainly
if
a
system
ontained
i)
and
(2)
as
theorems,
hen
3)
could
be
derived
by
MM,
and the
system
would
be
inconsistent;
ut a
system
llowing
MM
can
hardly
be assumed
to
be
otherwise
onventional.
After
ll, many
systems ack 'Pv
-
P' as a theorem, nd minimal ogic also lacks P
D
(-P
Q)'.)
(5)
It
might
be
suggested
at
this
point
that to direct
our
search
for
justification
o
a
form
f
argument,
r
argument
chema,
uch
as
MPP,
is
misguided,
hat
the
ustification
f
the
schema
lies in
the
validity
f
its
instances.
o the answerto the
question,
What
ustifies
he conclusion?'
7/26/2019 Susan Haack - Justifications of Deduction
8/9
I
I8
SUSAN HAACK:
is simply The
premisses'; and
the
answerto the further
uestion,
what
justifies
he argument
chema, s
simply hat ts instances
re valid.
This suggestion s unsatisfactoryor everalreasons.First, tshifts he
justification
roblem
from
he
argument chema to its
nstances,
without
providing ny
solution o the
problem f the
ustification
f the nstances,
beyond the bald assertion hat
they
re
ustified.The
claim
that
one can
just see
that the
premisses ustify he
conclusion s implausible
n the
extreme n view
of
the
fact hat
people can
and do
disagree bout which
arguments re
valid. Second,
there s
an
imiplicit
enerality
n
the
claim
that a particular
rgument s valid. For to
say that
an
argument s valid
is not
ust
to
say
that
ts
premisses
nd
its
conclusion are true-for that
is neithernecessary
nor
sufficient
or
semantic)
validity.
Rather, t is to
say that tspremisses ouldnot be truewithout tsconclusionbeingtrue
also, i.e. that
there s
no
argumentf
that ormwith rue
remissesnd
false
conclusion.
ut if the claim
that
a
particular
rgument
s
valid
is to be
spelled
out
by appeal
to
other
arguments
f
that
form,
t is
hopeless
to
try to justifythat form
of
argumentby appeal
to
the
validity
of
its
instances. Indeed, it
is not
a
simple matter
o
specify
f
what schema
a
particular rgument
s an
instance.
Our
decision about what the
logical
formof an
argument s may
depend upon our
view
about whether he
argument
s
valid.)
Third,
since
a
valid schema
has
infinitelymany
instances, f
the
validity f the schema
were
to
be
provenon the basis of
the validity f ts
instances, he
ustification f theschema wouldhave tobe inductive,and would in
consequence
inevitably
fail to
establish
a
resultof the desired
strength.
Cf. Section .)
In
rejecting
his
suggestion
do
not,
of
course,
deny
the
genetic
point,
that
the
codification
f
valid
forms
of
inference,
he
construction
f a
formal
ystem,may
proceed n
partvia
generalisation ver
cases-though
in
part,
I
think,
he
procedure
may
also
go
in
the
opposite
direction.
(This
geneticpoint s, I think,
elated
o
the one Carnap [I968] is
making
when he
observes hat
we could not
convince
man
who
is
'deductively
blind'
ofthe
validity
f
MPP.)
But
I do
claimthat he
ustification
f form
of nference
annot
derive from ntuition f
the
validity f its
instances.
(6) What I havesaid in thispapershould, perhaps,be alreadyfamiliar
-it is
foreshadowedn Carroll I895],
and more
or
less
explicit n
Quine
[1936] and Carnap I968
('.
.
. the
epistemological ituation n
inductive
logic .
.
.
is not
worse than that
in
deductive
logic,
but
quite
analogous
to
it', p. 266).
But
the
point
does
not
seem
to
have been
taken.
The
moral
of the
paper might
e
put,
pessimistically,s that
deduction
is no
less in need of
ustification
han
induction; or,
optimistically,
s
that induction s
in no
more need
of
justification
han
deduction.
But
howeverwe put t,the
presumption, hat
nduction s shaky
but deduction
is
firm,
s
impugned.
And this
presumption
s
quite crucial,
e.g.
to
Popper's proposal
[I959]
to replace inductivism y deductivism.Those
of us who
are
sceptical about the
analytic/syntheticistinction
will,
no
doubt,
find
these
consequences
less
unpalatable
than
will
those
who
accept it.
And those of
us who
take
a
tolerant ttitude o
nonstandard
logics-who regard
ogic
as a
theory, evisable, ike other
heories,
n
the
light
of
experience-may
even find
hese
consequences welcome.
7/26/2019 Susan Haack - Justifications of Deduction
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THE JUSTIFICATION
OF
DEDUCTION
II9
BIBLIOGRAPHY
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[ig6i].
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and
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I96I,
and in Philo-
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[I954].
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1954.
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icket', Analysis, 1960,
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o
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UNIVERSITY OF WARWICK