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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.
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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)]
|
