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"The Bayesian Analytic-Synthetic Distinction"[1]
J. T. Allen
The philosophy of science has been dominated
by two traditionally distinct philosophical
viewpoints, viz., philosophy of language
and probability theory. Probability theory
is primarily concerned with investigating
the methodology of scientific reasoning vis-a-vis
the probability calculus. Philosophy of language,
on the other hand, is, for the most part,
interested in applying insights into language,
in general, to the language of science. Both
philosophical vantage points do not necessarily
clash with one another over the analysis
of scientific inference or belief.
Consider the debate over universals. Philosophers
of language have argued that their existence
is not in question since predicates cannot
be quantified over (Quine 1953); '($x)($x)Px'
(where '$' is the sign for existential quantification)
is simply not a well-formed formula. Probability
theorists may very well agree with this result,
but their concern over universals is rather
oriented around the question of whether universals
can be confirmed to exist in virtue of a
probabilistic construal of confirmation.
Both parties tend to agree that universals
are a will-o'-the-wisp, but they tend to
(or can) get there on their own. Sometimes,
however, philosophers of language and probability
theorists outright disagree with one another.
Consider the so-called Quine-Duhem problem.
Probability theories ordinarily have no problem
ascribing to the view that, in the majority
of cases, a hypothesis is explicitly tested
against another hypothesis without testing
the auxiliary assumptions (Sober, 1999; 2008;
Howson and Urbach 1993). Nonetheless, philosophers
of language insist that our decision to change
our beliefs regarding either the hypotheses
or auxiliary assumptions in any given testing
situation is entirely pragmatic. There have
been serious efforts to merge philosophy
of language with probability theory, such
as Carnap's project of logical probability,
but such attempts customarily stand for cautionary
tales rather than praiseworthy philosophical
insights. In the present essay, my goal is
to help change this. Philosophers of language
have, since Quine's "Two Dogmas of Empiricism"
(1951), dispensed with the notion of the
analytic-synthetic distinction (henceforth,
the A-S distinction).
Probability theorists have thus far reserved
probability theory from challenging this
result of the philosophy of language. After
all, the A-S distinction is about meaning,
for which philosophers of language have claimed
ownership. The present suggestion is to merge
points made in the philosophy of language
in regards to the A-S distinction with probability
theory. Thus, the A-S distinction will be
given a probabilistic rendering intended
to further benefit the probability theory
approach to the philosophy of science whilst
overcoming its difficulties in the philosophy
of language.
Though the A-S distinction has been
given
formulations since Leibniz, emphasis
will
be given to its twentieth-century formulation
since it is this construal which philosophers
of language have lobbed their objections
at. Clarifying the earlier conceptions
of
the A-S distinction will serve to show
the
Bayesian A-S distinction to be given
is an
apt explicatum of the explicandum[2]
characterized
by the work of Carnap and Ayer.
As a preliminary remark, the property
of
being analytic or synthetic has been
ascribed
to sentences, statements and propositions.
A sentence, for the purposes of the
present
inquiry, refers to a type of string
of symbols
(a "sign-design" as Carnap
put
it (1942)), that expresses a statement
or
proposition. The only difference I
understand
between statements and propositions
is that
statements purport to be capable of
either
truth or falsity (or probability),
but only
if they do, in fact, possess the significance
of being either true or false (or deserving
of being embedded in a probability
function)
will I refer to those sentences as
propositions.
Ultimately, sentences are used to express
propositions; the only written distinction
I make between them is when sentences
need
to be contrasted with propositions,
in which
I case I explicitly reserve single-quotations
for sentences and nonquoted sentences
for
propositions. Otherwise, comments regarding
the A-S distinction will apply equally
to
sentences as they do propositions,
and single-quotation
will apply to both as a matter of convenience.
In his earlier years, Carnap (1932)
defined
analytic sentences as those which are
true
in virtue of their logical form alone.
For
instance, the sentence 'If apples are
red,
then apples are red' is analytic since,
no
matter what 'apples are red' means,
so long
as it capable of being either true
or false,
the whole sentence is true. Synthetic
sentences,
on the other hand, were considered
true if
and only if they could be reduced to
true
sentences which only make reference
to true
"protocol [propositions]"
or immediate
sensory experience (henceforth, sense-data).
Aside from the formulation's initial
plausibility,
it suffers serious shortcomings. Such
sentences
as 'If someone is a bachelor, then
someone
is not married', for instance, are
not true
of their logical form alone; they require
a premise which indicates a synonymous
relation
between 'bachelor' as 'unmarried male',
a
fact which is not ordinarily thought
of as
logical. There is great difficulty
met, moreover,
in trying to reduce sentences referring
to
physical objects to sentences referring
only
to sense-data. Such is the problem
of phenomenalism.
Within the same decade, Ayer (1936),
returning
from his visit to Vienna to acquaint
himself
with the growing influence of logical
positivism,
generalized the A-S distinction in
order
to overcome the aforementioned deficiencies.
Analytic sentences became sentences
true
in virtue of the meanings of their
constituent
logical and descriptive signs alone.
Logical
signs include 'not', 'if-then', 'or',
'and',
'some', 'every', etc., and descriptive
signs
include 'Jeff', 'bachelor' and 'is
a redhead'.
If a sentence, for example, 'No bachelor
is married', is true in virtue of what
'no',
'is', 'bachelor' and 'married' mean,
then
'No bachelor is married' is analytic.
If,
on the other hand, we replace the descriptive
sign 'bachelor' with 'hermit', then
the sentence
is no longer analytic. It is not part
of
the meaning of 'hermit' that hermits
are
unmarried—as it happens, some hermits
are
married, some not. In either case,
'No hermit
is married' requires empirical investigation,
whereas 'No bachelor is married' does
not.
Thus, Ayer's A-S distinction is such
that
its extension of analytic sentences
rightly
includes 'No bachelor is married' and
its
extension of synthetic sentences rightly
includes 'No hermit is married' without
commitment
to the view that sentences referring
to physical
objects must be reduced to sentences
referring
only to sense-data.[3]
Before we go any further, it is practical
to give attention to the question:
why give
the A-S distinction in the first place?
Simply
drawing the distinction "is not
enough;
one has to show what its consequences
are—that
is, reveal the problems it allows us
to bring
up" (Proust 1989)[4]. The motivation
for the distinction, i.e., its pragmatic
value, stems from its use by logical
empiricists
such as Carnap, Ayer, Schlick, (early)
Wittgenstein,
Russell, etc. For logical empiricists,
the
distinction was thought to capture
the apparent
epistemological difference between
logical
(and mathematical) propositions and
scientific
(and everyday) propositions[5]. This
difference,
pre-analytically, is best expressed
as a
difference of tenacity. We seem more
ready
to forgo (true) sentences of astronomy
such
as 'The Sun is 93 million miles from
the
Earth' than we are to forgo (true)
sentences
of arithmetic such as '2+2=4', since,
as
the A-S distinction reveals, the latter
is
true merely in virtue of meanings and
the
former requires us to get our hands
dirty
in empirical inquiry. Moreover, the
distinction
appears to solve many prior (in Ayer's
words,
"outstanding") problems of
philosophy
which have traditionally dominated
philosophical
discourse. This is achieved by demoting
these
issues to the status of "pseudo-problems"
in virtue of their putatively problematic
propositions' being neither analytic
nor
synthetic. If, for example, seemingly
unsolvable
problems arise out of inquiry over
the putative
proposition expressed by the sentence
'Objective
moral values exist', then, if the putative
proposition is neither analytic nor
synthetic,
it poses no substantive problem at
all; the
putative statement is a "pseudo-proposition",
a cognitively meaningless string of
symbols
that only appears to have the status
of a
real proposition, viz. being either
true
or false. The A-S distinction, furthermore,
was thought by the logical empiricists
to
account for the a priori-a posteriori
distinction
(henceforth, Prior-Post distinction).
A priori
knowledge is pre-analytically defined
as
knowledge independent of sensory-experience,
and a posteriori knowledge as knowledge
dependent
on sensory-experience. A priori knowledge
was possible, according to the logical
empiricists,
because the propositions that are a
priori
known are all analytic. Likewise, a
posteriori
knowledge was possible because the
propositions
a posteriori known are synthetic. That
a
single distinction so simple could
have such
an overwhelming scope of efficacy on
philosophical
discourse was very attractive for logical
empiricists and comprised, for them,
the
majority of the distinction's high
pragmatic
value.
Regarding Ayer's particular formulation of
the A-S distinction, notice, however, that
acquisition of empirical facts, in some sense,
is required in order to determine that certain
sentences are analytic. Consider, again,
the sentence 'No bachelor is married'. Ayer
(as well as Carnap) regards this sentence
as analytic in virtue of the fact that 'bachelor'
is synonymous with 'unmarried male'; but
this is neither an axiom nor a theorem of
logic; it demands empirical investigation
of the use of words by their language users.
It would seem, therefore, that the A-S distinction
may have no positive account of the Prior-Post
distinction, since some analytic truths depend
on experience. Nonetheless, according to
Ayer, this does not pose a serious problem
for the A-S distinction, for, "The idea
is that once we grasp what the proposition
is, no further experience is needed to enable
us to know that it is true, or that it is
false" (1973, p. 199; italics: mine).
Carnap, on the other hand, takes a
different
route than Ayer in order to accommodate
sentences
such as 'No bachelor is married' as
analytic
whilst maintaining the distinction's
fidelity
to match the Prior-Post distinction.
Carnap's
idea is that analyticity is better
explicated
in an artificial language, viz. a semantical
system, than in a natural language
such as
English (1938; 1942; 1952). In a semantical
system, rather than a natural language,
meaning
is based on proposals, not usage by
a language's
users. The noteworthy difference is
that
whereas a meaning relation, e.g., synonymy,
is established in English by seeing
how language
users use the language, synonymy in
a semantical
system is a matter of choice. "In
choosing
the rules" for a semantical system,
particularly in regards to meaning
relations,
"we are entirely free" (Carnap
1942). The upshot is that, 'No bachelor
is
married' is analytic not in virtue
of an
empirical fact that 'bachelor' (is
used such
that it) means 'unmarried male'; rather,
it is a matter of convention: 'bachelor'
(is postulated such that it) means
'unmarried
male'. At this point, one might object
that
Carnap's analytic sentences would,
thus,
run the risk of having nothing to do
with
the natural language which led us to
philosophical
ideas of A-S distinction in the first
place.
In defense, Carnap (pp. 13-14) offers
the
following analogy:
[T]he fact that somebody's garden has
the
shape of a pentagon may induce him
to direct
his studies in mathematical geometry
to pentagons,
or rather to certain abstract structures
which correspond in a certain way to
bodies
of pentagonal shape; the shape of his
garden
guides his interests but does not constitute
a basis for the results of his study.
In other words, it is perhaps psychologically
vital that our natural languages lead
us
to analyses of philosophical issues
using
semantical systems; but, the fare of
a philosophical
issue in a semantical system does not
depend
on the semantical system's affinities
with
the natural language from which it
happens
to arise. As in mathematics, we would
not
object to Euclid's axiom of parallel
lines
because we had trouble finding the
preciseness
we get in the abstract models of geometry
in English and the empirical world
of lines.
Though I think Carnap's way of characterizing
analytic propositions shows us something
about language, it does not show us
anything
about truth. Indeed, we are free to
choose
any meaning relation we please between
the
signs of an artificial language. It
is, indeed,
clearly a conventional matter that
the signs
'unmarried male' can take the place
of the
sign 'bachelor' in any true sentence
about
the sign 'bachelor' in some semantical
system
of our choosing, say, L1. However,
suppose
that in L1 'bachelor' means 'unmarried
male'
but in L2 'bachelor' means 'married
male'.
In this case, the following holds true:
'No bachelor is married' is true in
L1
and
'No bachelor is married' is false in
L2.
Now, what about the proposition we
started
with, viz. no bachelor is married?
Is it
at one time both true and false, depending
on the language system we choose to
adopt
to express it? Insofar as the sentences
are
confined to expression within L1 and
L2,
the answer is yes; insofar as the propositions
which the sentences in L1 and L2 express,
i.e., what the sentences say (which
may be
formulated in a metalanguage), however,
the
answer is no. The evaluation of the
proposition
that no unmarried male is married is
based
on logic; it is conventional that this
proposition
is expressed in English as 'No bachelor
is
married', but it doesn't follow that
the
proposition is based on conventions
of language.
As Sober (2000) succinctly puts it:
A real analytic truth is true, and
what it
says typically does not depend for
its truth
on our choice of language...Genuine...analytic
truths express truths, and the propositions
those sentences express are language-independent.[6]
Regardless of the misgivings of Carnap's
conventionalist understanding of analyticity,
there is another, more general formulation
that Carnap gives that is worth mentioning.
Carnap (1954, p. 16) suggests that
the A-S
distinction be characterized vis-a-vis
the
fashion by which we establish statements'
truth-values. Analytic statements,
for instance,
are true in virtue of the fact they
require
only that we know the meanings of their
constituent
signs (both logical and descriptive),
whereas
synthetic sentences require more, i.e.,
a
correspondence (or lack thereof) between
what the sentence says and how the
world
is, factually speaking. Therefore,
Carnap
establishes a general understanding
of analytic
statements which leaves room to argue,
as
Ayer does, that, though we may take
meaning-relations
to be "facts", they are nonetheless
of an entirely different type of "facts"
than those involved in establishing
the truth
(or falsity) of synthetic statements.[7]
Turning now to the criticisms of the
A-S
distinction, one is naturally led to
the
work of Quine; in particular, his "Two
Dogmas". Quine's primary complaint
is
that the A-S distinction cannot be
clarified
without recourse to equally unclear
notions,
such as synonymy (i.e., meaning), definition,
or necessity (Sober 2000).
In order for Quine's criticisms to
be successful,
he must suppose that A-S distinction
advocates
accept the following set of statements:
A statement is analytic (and true or
false)
if and only if its truth (or falsity)
is
not based on any matters of fact.
If a statement is syntactically true
or false,
then it is not so based on any matters
of
fact.
The reason for (i) is that, as Quine
tacitly
assumes, an A-S distinction would be
useless
without entirely coinciding with the
Prior-Post
knowledge distinction. Quine, less
than actually
accepting it, concedes (ii), as a practice
of shallow analysis, perhaps[8].
Quine, rightfully, places paramount
importance
on the A-S distinction's construal
of synonymy.
For if analytic statements are true
in virtue
of the meanings of their constituent
terms,
then, since the meanings of constituent
terms
consists of their synonyms, synonymy
plays
an indispensable role in understanding
the
A-S distinction. Subsequently, Quine
embarks
on a mission with the sole objective
of clarifying
synonymy. Throughout "Two Dogmas"
there are three criteria of synonymy
that
Quine deals with: behavioral-synonymy
(C-S1),
substitutivity-synonymy (C-S2) and
necessity-substitutivity-synonymy
(C-S3)[9], each of which Quine deems
inadequate
for a genuine explication of analyticity.
Quine's suspicion of the A-S distinction
stems from the transformation of 'No
bachelor
is married' to 'No unmarried male is
married',
a problem putatively solved by the
aforementioned
revision of the A-S distinction given
by
Ayer and Carnap by tacit recourse to
synonymy.
Naturally, in order to understand and
evaluate
this transformation, one must have
some understanding
of synonymy. The first criterion of
synonymy
to consider is C-S1:
Criterion of Synonymy (behavior-synonymy).
Two terms are synonymous if and only
if behavioral
patterns B1, ... , Bn involving the two
terms
are true.
B1, ... , Bn is not, by Quine's own tacit
standards,
clear and it may be objected that Quine
needs
to explicate the nature of the putative
behavioral
patterns in order to demoralize C-S1's
fare
as a suitable criterion of synonymy.
Quine's
retort, however, is that even if we
could
identify a precise behavioral pattern
which
indicated two terms were synonymous,
condition
(i) would be violated since the terms'
being
synonymous with each other would be
based
on matters of fact, viz. the behavior
of
language users. Thus, regardless of
what
B1, ... , Bn may turn out to be, it is,
in
principle, unable to deliver what the
A-S
distinction needs.
More promising, perhaps, is C-S2:
Criterion of Synonymy (substitutivity-synonymy).
Two terms are synonymous if and only
if they
are interchangeable salva veritate
in all
purely referential contexts.[10]
Before we proceed to give Quine's criticism
of C-S2, let’s examine the right-hand
side
of the biconditional more closely.
To say
that two terms are "interchangeable
salva veritate" is to say that
those
terms are interchangeable in sentences
without
changing the sentence's truth-value.
"Contexts",
moreover, are schemata for statements,
e.g.
'... is ---', a schemata for the statement
'Jeff is a redhead'. To say a context
is
purely referential is to say that its
corresponding
full (and true) sentence has no instance,
when combined with a true identity
statement
whose constituents include one of the
sentence's
terms, of a false deduction. This last
amendment
is devised in order to avoid easily
spotted
counterexamples such as "'Bachelor'
has eight letters" which, when
combined
with 'bachelor = unmarried male', entails
the false conclusion "'unmarried
male'
has eight letters"--'...has eight
letters'
is simply not purely referential, and
so
it may be ignored when investigating
the
synonymy between two terms. Quine's
issue
with this route is that it is too weak.
For
consider the two terms 'the number
of planets'
and '9'. At this point in time, whatever
is true of one term is true of the
other.
However, suppose the number of planets
is
discovered by astronomers to be '8'.
Since
it would be absurd to subsequently
suppose
'9=8', we say that they are true of
the same
objects (for now), but they differ
in meaning,
i.e., they are not synonymous. Because
two
terms may not be synonymous yet be
interchangeable
salva veritate in all purely referential
contexts, C-S2 does not suffice as
a criterion
of synonymy.
Lastly, Quine considers a way to make
interchangeability
a strong enough notion of synonymy,
C-S3:
Criterion of Synonymy (necessity-substitutivity-synonymy).
Two terms are synonymous if and only
if it
is necessary that one sentence is true
if
and only if another, alike except with
an
occurrence of the other term, is true.
Suppose, for example, that we have
the following
sentences:
The number of planets is greater than
7 if
and only if 9 is greater than 7,
and
3^2 is greater than 5 if and only if
9 is
greater than 5.
Is (3) necessary? No. For it is possible
that 'the number of planets = 7' yet
'9 >
7'. (4), on the other hand, would seem
to
involve two synonymous terms, viz.
'3^2'
and '9', since it is presumably necessary
that (4) is true, i.e., it is not possible
that '3^2 > 5', yet '9 ? 5'. Quine's
objection
to this way of explicating synonymy,
however,
is that it presupposes the very notion
we're
trying to explicate.
The above [criterion] supposes we are
working
with a language rich enough to contain
the
adverb 'necessarily', this adverb being
so
construed as to yield truth when and
only
when applied to an analytic statement.
But
can we condone a language which contains
such an adverb? Does the adverb really
make
sense? To suppose that it does is to
suppose
that we have already made satisfactory
sense
of 'analytic'.
Seeing as though these beforehand likely
contenders for a clarification of synonymy
fail to help explicate analyticity,
Quine
deems synonymy a dead-end. Again, Quine's
fundamental complaint is that if we
accept
(i) and (ii) as legitimate desiderata,
we
cannot include such statements as 'No
bachelor
is married' as analytic, since it is
transformable
into a syntactically true sentence
only if
we consult matters of fact regarding
the
synonymy relation between 'bachelor'
and
'unmarried male'. And in that case,
calling
a sentence an "analytic truth"
adds nothing to our saying it is a
"syntactical
truth", which is bound to have
the serious
shortcoming of dividing sentences such
that
'No unmarried male is married' and
'No bachelor
is married' have nothing of epistemological
interest in common, though 'No bachelor
is
married' and 'The Sun is 93, 000, 000
miles
away from Earth' do.
The Bayesian A-S distinction offers
a fresh
solution to Quine's criticisms, but
it requires
a definition of the A-S distinction
to be
probabilistic. As this might appear
prima
facie strange, I will take great care
to
ensure that the explicatum is related
enough
to the explicandum to qualify the Bayesian
A-S distinction as a legitimate explication
of Carnap's and Ayer's A-S distinction.
Let me begin with a few preliminary
remarks.
I hold that prior probabilities are
functions
of subjective degrees of belief, constrained
by (at least) coherency and, if one
is feeling
generous, other pragmatic factors such
as
simplicity, conservatism, modesty,
and fruitfulness
(Quine 1970; Salmon 2001). Moreover,
if a
prior probability is equal to 1 or
0, then
it is either a syntactical truth or
a syntactical
falsity, respectively. Keeping these
things
in mind, I now propose the following
Bayesian
empiricist formulation of the A-S distinction:
(A) H is analytic if and only if ($t)[Pr(at
t)(H | AA1 & ... & AAn) = 1 or
0].
(S) H is synthetic if and only if (t)[1
>
Pr(at t)(H | AA1 & ... & AAn)
>
0]
Here, t is time (thus, '($t)(...(at t)...)'
means
'there is a time t such that...(at t)...',
H
is any hypothesis, and AA1, ... , AAn
are any
auxiliary assumptions, so long as they
are
suitable, i.e. they are true independent
of the evidence or hypothesis in question
(Sober 1999; 2000; 2007; 2008). To
make the
utility of this explicatum as clear
as possible,
it is easiest to first go through the
most
controversial case which it is designed
to
accommodate: 'No bachelor is married'.
To
simplify matters, I will reserve the
time
index notation for the generalized
definition
of the A-S distinction; for the following
examples, it suffices to use Pr(...)
and Pr*(...),
where the latter is the updated probability
of the former given an addition of
the total
evidence between times in which both
probabilities
are calculated. The auxiliary assumptions,
AA1, ... , AAn, will also be referred
to by
the letter B. The A-S distinction,
in these
simplifying terms, may be rewritten
as:
(A') H is analytic if and only if Pr*(H
|
B) = 1 & [(Pr(H | B) = 1) or (1
>
Pr(H | B) > 0)].
(S') H is synthetic if and only if
1 >
Pr*(H | B) > 0.
Suppose we take H to be the following
hypothesis:
H: No bachelor is married.
This, as Quine points out and Carnap
and
Ayer would concede, is not a syntactical
truth; i.e., it is not true given the
syntactical
rules of pure logic or the meanings
of logical
and (uninterpreted) descriptive signs
alone.
I will likewise concede this, which
leads
to the following result regarding H's
prior
probability:
1 > Pr(H | B) > 0.
What if, however, the following evidence
was available?
E: 'bachelor' is synonymous with 'unmarried
male'
I allow that E may be based on behavior
of
language users, a concession in line
with
what Quine (italics: mine) deems acceptable
for any notion of synonymy:
Just what it means to affirm synonymy,
just
what the interconnections may be which
are
necessary and sufficient in order that
two
linguistic forms be properly describable
as synonymous is far from clear; but,
whatever
these interconnections may be, ordinarily
they are grounded in usage. Definitions
reporting
selected instances of synonymy come
then
as reports of usage.
I proceed, unlike Quine, however, to
show
that given this, H is still analytic.
Let's
consider, first, the following set
of auxiliary
assumptions, B:
AA1: 'No unmarried male is married'
is a
syntactical truth and therefore the
prior
probability that no unmarried male
is married
is 1.
AA2: 'Some unmarried male is married'
is
a syntactical falsity and therefore
the prior
probability that some unmarried male
is married
is 0.
AA3: 'No...is---' is a purely referential
context,
where '...' and '---' are purely referential
general terms.
AA4: 'Some...is---' is a purely referential
context, where '...' and '---' are purely
referential
general terms.
AA5: If two signs (or string of signs)
are
synonymous, then they are interchangeable
salva veritate in all purely referential
contexts.
Given this, (5) is still true; B does
not
entail H (or not-H). But this is not
the
end of the story. In order to determine
the
posterior probability of H, two other
values
must be taken into account, viz. H
and not-H's
likelihoods.
Beginning with H's likelihood, we need
to
ask: what is the expectancy of E given
H
and B; that is, what is the expectancy
that
'bachelor' is synonymous with 'unmarried
male' given the noted auxiliary assumptions
and the hypothesis that no bachelor
is married?
As Quine notes in his criticism of
C-S2,
interchangeability salva veritate (in
all
purely referential contexts) is not
sufficient
for a criterion of synonymy. The expectancy
is thus not going to be 1; though,
perhaps
it could be argued that it would be
high,
though this is not necessary, as will
become
clearer as the argument progresses.
1 > Pr(E | H&B) > 0
What, now, about not-H's likelihood?
It is
clear that it is contradictory to assert
that 'bachelor' is synonymous with
'unmarried
married male' given some bachelor is
married
and the noted auxiliary assumptions;
which
is to say that the set of statements
S: {not-H,
E, B} is inconsistent. Thus,
Pr(E | not-H&B) = 0.
To make this point clearer, consider
the
following proof which explicitly shows
that
S is an inconsistent set of statements,
and
therefore has a probability equal to
0 (as
asserted by (7)):
not-H
'Some bachelor is married' is true.
(from
1)
'bachelor' is synonymous with 'unmarried
male'. (E)
If two signs (or string of signs) are
synonymous,
then they are interchangeable salva
veritate
in all purely referential contexts.
(AA5,
which an element of B)
'Some...is---' is a purely referential
context.
(AA4, an element of B)
'Some unmarried male is married' is
true.
(from 2, 3, 4, and 5).
'Some unmarried male is married' is
a syntactical
falsity (and thus false). (AA2)
'Some unmarried male is married' is
true
and false. (from 6 and 7).
Since 8 is contradictory, and a result
of
deductions within the set of statements
S,
S is inconsistent, and therefore the
conjunction
of the statements has a probability
of 0:
Pr(not-H&E&B) = 0,
which is, given Bayes' theorem, logically
equivalent to (7).
Combining theorem (T1) (See Appendix)
with
(5), (6) and (7), it follows that:
Pr(H | E&B) = 1,
and, given theorem (T2):
Pr*(H | B) = 1,
thus showing that, given (A') and (S'),
H
is analytic. In general, I propose,
no prior
probability less than 1 and greater
than
0 can, through conditionalization,
become
either 1 or 0 unless the hypothesis
is sensitive
to evidence reporting how language
users
use the language, consisting of the
signs
(or string of signs) in question. Boyle's
law, for instance, which is on the
Bayesian
A-S distinction a synthetic proposition,
is not sensitive to evidence reporting
how
language users use the words 'pressure',
'volume' and 'temperature'. Rather,
it is
sensitive to evidence reporting relations
between barometer readings and volume
measurements
(assuming temperature is constant).
Thus,
given (A) and (S), Boyle's law (H1)
is synthetic
since for all times t, the prior probability
of H1 is less than 1 and greater than
0:
(t)[1 > Pr(at t)(H1 | AA1 &
... &
AAn) > 0].
The above explication is meant to apply
to
any statement whatsoever, whether it's
'F
= ma', 'God exists' or 'Any whole number
is a real number'. If a statement fails
to
fit the A-S distinction, i.e., if it
is neither
analytic nor synthetic, then I say
that the
putative statement is nonsciential;
that
is to say, it bears no significant
relation
to the theory of knowledge. Take, for
instance,
the sentence 'Objective moral values
exist'.
Is this a syntactical truth or falsity?
No.
Thus it is either analytic, in the
case that
it is true in virtue of its meanings,
partially
extrapolated from the linguistic behavior
of language users, or it is synthetic,
in
the case that it is sensitive to some
observed
datum (other than the linguistic behavior
of language users) to change its prior
probability.
If we are skeptical of either of these
options,
or have good reason to believe neither
option
is fulfilled for the sentence 'Objective
moral values exist', then there is
good reason
to suppose it is epistemologically
vacuous.
This is not to say that the sentence
is meaningless,
as the logical positivists insisted
upon.
Rather, this way is much like Sober's
(1999;
2008) criterion of testability--it
leaves
room for the prospect that the sentence
may,
sometime in the future, become relevant
to
the theory of knowledge. The sentence,
in
the case that it is neither analytic
nor
synthetic, is, so to speak, on the
"back
burner", until further notice.
For the
time being, however, it bears no significance
to epistemology, and thus, plays no
interesting
role in the edifice of our knowledge
of the
world such as Boyle's law or mathematical
propositions do. Nonsciential sentences,
therefore, play the same role of the
empiricists'
metaphysical sentences: they are dealt
with
in such a way as they are eliminated,
in
a sense, in order to solve (or dissolve)
certain longstanding philosophical
disputes,
such as the question of whether objective
moral values exist or not.
The Bayesian A-S distinction, aside
from
dealing with so-called metaphysics,
also
does justice to the apparent epistemological
difference between logic and mathematical
propositions, on the one hand, and
ordinary
and scientific propositions, on the
other.
Consider the sentences '2+2=4' (H2)
and 'The
universe is 13.4 billion years old'
(H3).
The apparent epistemological difference
is
difficult to accurately describe without
assuming the A-S distinction itself,
but
it is at least clear that the difference
is one of tenacity: we are more willing
to
forgo the universe's age than we are
that
two plus two equals four. The Bayesian
A-S
distinction accounts for this difference
in the following way, where the asterisk
"*" is again taken to mean
some
new probability evaluation (after considering
three old values, viz. the hypothesis'
prior
probability, its likelihood, and its
negation's
likelihood), and B is a set of suitable
auxiliary
assumptions:
Pr*(H2 | B) = 1
1 > Pr*(H3 | B) > 0
Another positive feature of the Bayesian
A-S distinction is that it accounts
for the
Prior-Post distinction. All truths
that are
known a priori are analytic, and all
synthetic
truths are known a posteriori. Notice,
however,
that this does not imply that all analytic
truths are a priori known or that all
truths
known a posteriori are synthetic. In
fact,
the Bayesian A-S distinction's application
to sentences such as 'No bachelor is
married'
implies there is analytic a posteriori
knowledge;
on the other hand, it implies that
synthetic
a priori knowledge is impossible. These
corollaries
are due to the general principle that
if
a hypothesis is sensitive to any type
of
observational evidence at all, and
it can
be known at all to be either probably
true
or probably false, then it is a posteriori
known; otherwise, the hypothesis is
known,
if known at all, a priori. Since this
distinction
may be applied to any statement whatsoever,
and not every statement falls into
either
category, the distinction appears tenable.
It may still be objected, however,
that,
as Quine tacitly suggests, that admitting
the possibility of analytic a posteriori
knowledge hurts any A-S distinction.
Nevertheless,
this type of objection is misguided.
For
Quine's objection to using behavioral
data
to support our assertion that a statement
is analytic relied upon the fact that
if
a statement is at any time sensitive
to any
type of data, whatsoever, it is no
different
from other types of statements ordinarily
thought to be sensitive to observational
evidence such as scientific laws. With
the
Bayesian A-S distinction, however,
we've
shown how there is still an epistemologically
substantial difference between scientific
laws that utilize an array of observational
evidence and statements that utilize
only
linguistic behavior as observational
evidence.
There is, moreover, a likeness of the
Bayesian
A-S distinction to Kant's A-S distinction,
in that Kant explicated the A-S distinction
and the Prior-Post distinction such
that
all analytic truths were a priori,
but not
all synthetic truths were a posteriori.
The
Bayesian A-S distinction merely switches
things up. To give the Bayesian Prior-Post
distinction more explicitly, where
B is some
set of auxiliary assumptions:
H is known (to be true or false) a
priori
if and only if (t)[Pr(at t)(H | B)
= 1 or
0].
H is known (to be true or false) a
posteriori
if and only if ($t)[1 > Pr(at t)(H
| B)
> 0].
Thus, it is clear that all propositions
known
(to be true or false) a priori are
also analytic;
but for analytic statements, it is
possible
that for some there exists a time such
that
its probability is less than 1 and
greater
than 0, yet, some time afterwards,
its probability
sticks to either 1 or 0; i.e. analytic
a
posteriori knowledge is possible, though
synthetic a priori knowledge is not.
Though these pragmatic advantages highlighted
above benefit philosophy, particularly
in
the branch of epistemology, there is
no reason
to believe that the Bayesian A-S distinction
is capable of solving all the problems
of
philosophy. For one, Bayesian epistemology
itself has not been fully developed
into
an acceptable framework for the theory
of
knowledge (or, more radically, as a
replacement
for traditional epistemology). I don't
think
that this issue is due to the fact
that some
people are just unwilling to change
their
views; rather, I think further research
is
required in order to flesh out genuine
issues
with Bayesianism (See Howson and Urbach
1993;
Jeffrey 2007; Sober 2008 for discussion).
If these problems happened to be solved
without
adversely affecting the Bayesian A-S
distinction
outlined above, then perhaps it could
claimed
that it achieves the seemingly impossible
task of having solved all the problems
of
philosophy. For all problems of knowledge
would be directed to logic, mathematics
and
science, and hence have at least solvability
within some particular field of knowledge--the
issue of whether there are unsolvable
problems
to which philosophy is helplessly committed
to go to work on or not in thus answered
negatively; there are no such legitimate
issues; all substantive problems are
solvable,
in principle, within the domain of
science,
in general, including logic and mathematics.
The pragmatic value of the Bayesian
A-S distinction
may, thus, seems clear and does not
stray
too far from its explicandum. Extensionally,
the favorable cases of the explicandum
are
equivalent to favorable cases of the
explicatum.
Quine's alternative to usage of the
A-S distinction,
his so-called "holism", achieves
its putative merit for receiving the
same
pragmatic support as the Bayesian A-S
distinction
does. My overall contention, however,
is
that such a radical pragmatism as Quine's
holism is avoidable. We can achieve
the same
success within the current best framework
for understanding science (in general),
paradigmatic
of modeling genuine knowledge, viz.
probability
theory, whilst preserving the time-honored
A-S distinction.
Notes
[1] I am deeply indebted to Robert Greg Cavin
and Michael Sechman for discussion and Martin
Young and Richard Swinburne for helpful comments.
[2] I am following Carnap (1947; 1950) in
introducing the notion of explication, i.e.,
clarification of a term or idea that sharpens
its use in a wider variety of contexts. The
explicandum is that which is explicated,
and the explicatum is that which does the
explicating. This idea is similar (though
not precisely parallel) to definition, which
involves a definiendum, that which is defined,
and the definiens, that which does the defining.
[3] Though, Ayer attempted to defend such
a phenomenalistic interpretation of "truth
based on empirical investigation" briefly
in Language, Truth and Logic. A more rigorous
trial came later (1940); but, Ayer eventually
disbanded with the view altogether for so-called
"sophisticated realism" (1973).
[4] I would modify this quote slightly to
say that when drawing a distinction the permissible
solutions to the problems it reveals are
equally important for establishing the distinction’s
worth.
[5] So-called "everyday" propositions
are those which report much of what is the
topic of ordinary, nonphilosophical discourse,
e.g. "I told you that last night",
"Grandma's house is fun", "The
freeway is on the left-hand side", etc.
[6] Some might object that, following Quine's
(1968) general advice, this criticism hinges
on the dubious notion of a "proposition".
However, as Sober (2000) points out, "It
does not depend on any particular 'theory
of propositions' -- i.e., on any particular
view about how propositions are individuated.
What is required is some distinction between
a sentence and 'what the sentence says' (Boghossian
1996, p. 380)". That is, we need only
make a sentence/what-a-sentence-says distinction,
which is indispensable insofar as we wish
to retain semantics as a genuine field of
logic.
[7] It is a short step from the sentence/what-a-sentence-says
distinction that Carnap draws in regards
to synthetic sentences to the idea that both
analytic and synthetic sentences express
propositions which are true or false regardless
of what object-language they are uttered
in (i.e., they'd be true by either logical
laws alone or empirical facts – both of which
would be sentences of the metalanguage).
[8] This supposition seems plausible in light
of the fact that Quine's holism, sketched
at the end of "Two Dogmas", does
away with the Prior-Post distinction altogether.
[9] These names are those I've given to Quine's
otherwise unnamed criteria of synonymy. C-S1
is discussed at length in section 2, and
C-S2 and C-S3 in section 3 of "Two Dogmas".
[10] This is roughly the route taken by Bates
(1950). Specifically, Bates argues that interchangeability
salva veritate would only suffice as a criterion
of synonymy in nonextensional languages that
include modal contexts. Quine, as we'll see,
objects to this sort of maneuver, viz. C-S3,
as helplessly question-begging.
References
Ayer, Alfred Jules. Language, Truth and Logic.
1936. 2nd ed. New York, NY: Dover, 1952.
- - -. The Foundations of Empirical Knowledge.
London: Macmillan, 1940.
- - -. The Central Questions of Philosophy.
1973. New York, NY: Holt, Rinehart and Winston,
1974.
Boghossian, P. "Analyticity Reconsidered."
Nous 30 (1996): 360-91.
Carnap, Rudolf. "Elimination of Metaphysics
Through Logical Analysis of Language."
Logical Positivism. Ed. Alfred Jules Ayer.
Glencoe, Ill: The Free Press, 1959. 60-81.
Rpt. of “Überwindung der Metaphysik durch
logische Analyse der Sprache.” Erkenntnis
2 (1931): 220-241. Translation of Rudolf
Carnap
- - -. "Foundations of Logic and Mathematics."
1938. International Encyclopedia of Unified
Science. By Neils Bohr, et al. Ed. Rudolf
Carnap, C. W. Morris, and Otto Neurath. Vol.
1. Chicago, Ill: Chicago University Press,
1955. 143-213.
- - -. Introduction to Semantics and Formalization
of Logic: Two Volumes in One. 1942. N.p.:
Harvard University Press, 1943.
- - -. "Meaning Postulates." Philosophical
Studies: An International Journal for Philosophy
in the Analytic Tradition 3.5 (1952): 65-73.
- - -. Introduction to Symbolic Logic and
Its Applications. 1954. Trans. W. H. Meyer
and J. Wilkinson. New York, NY: Dover, 1958.
Howson, Colin, and Peter Urbach. Scientific
Reasoning: The Bayesian Approach. LaSalle,
Ill: Open Court, 1993.
Jeffrey, Richard. Subjective Probability:
The Real Thing. 2004. Cambridge: Cambridge
University Press, 2007.
Mates, Benson. "Synonymity." University
of California Publications in Philosophy
25 (1950). Rpt. in Semantics and the Philosophy
of Language. By Rudolf
Carnap, et al. Ed. Leonard Linsky. Urbana:
University of Chicago Press,
1952. 111-38.
Proust, Joëlle. Introduction. Questions of
Form: Logic and the Analytic Proposition
from Kant to Carnap. By Proust. Minneapolis:
University of Minnesota Press, 1989.
Quine, Williard Van Orman. "Two Dogmas
of Empiricism." The Philosophical Review
60 (1951): 20-43. Rpt. in From a Logical
Point of View. 2nd ed. N.p.: Harvard Universtiy
Press, 1953.
- - -. "Logic and the Reification of
Universals." 1953. From a Logical Point
of View. 2nd ed. N.p.: Harvard University
Press, 1980.
- - -. "Propositional Objects."
Critica 2.5 (1968). Rpt. in Ontological Relativity
and Other Essays. Columbia, NY: Columbia
University Press, 1969.
Quine, Willard Van Orman, and J. S. Ullian.
The Web of Belief. 1970. 2nd ed. New York,
NY: Random House, 1978.
Salmon, Wesley C. "Logical Empiricism."
A Companion to the Philosophy of Science.
Ed. W. H. Newton-Smith. Malden, MA: Blackwell
Publishing, 2001. 233-42.
Skyrms, Brian. "Coherence, Probability
and Induction." Philosophical Issues
2 (Nov. 1996): 227-31.
Sober, Elliott. "Testability."
Proceedings and Addresses of the American
Philosophical Association 74 (1999): 237-80.
- - -. "Quine's Two Dogmas." Proceedings
of the Aristotlean Society 73 (2000): 47-76.
- - -. "What is Wrong with Intelligent
Design?" Quarterly Review of Biology
82 (2007): 3-8.
- - -. Evidence and Evolution: The Logic
Behind the Science. Cambridge: Cambridge
University Press, 2008.
Appendix
The following theorem is derivable from the
probability calculus, p, q are r are any
propositions:
(T1) Pr(p | q&r) = 1 or 0 if and only
if [(Pr(p | r) = 1 or 0) or ((Pr(q | p&r)
> 0 & Pr(q | not-p&r) = 0) or
(Pr(q | p&r) = 0))].
The following theorem is a special case of
Jeffrey conditionalization, where the asterisk
"*" signifies an updated or "new"
probability.
(T2) Pr*(p | r) = Pr(p | q&r) =
Pr(p
| q&r) Pr(q | p&r) / [Pr(p
| r) Pr(q
| p&r) + Pr(not-p | r) Pr(q | not-p&r)]
|
