INVERTED SPACE: MINIMAL VERIFICATIONISM,
PROPOSITIONAL ATTITUDES, AND COMPOSITIONALITY
Jon Cogburn1
Department of Philosophy and Linguistics Program
Louisiana State University
Roy Cook2
Department of Philosophy
The University of Saint Andrews
ABSTRACT: We use the duality theorem of projective geometry to describe an inverted
spectrum type thought experiment, and then show how this undermines the
verificationism of Michael Dummett. In closing we discuss varieties of compositionality
to suggest that a limited form of holism can preserve most of Dummett’s view.
e-mail: jcogbu1@lsu.edu
address: The Department of Philosophy
106 Coates Hall
Louisiana State University
Baton Rouge, LA 70803
USA
Telephone: (225) 578-2220
Fax: (225) 578-4897
2
e-mail: rtc1@st-andrews.ac.uk
address: Arché
School of Philosophical and Anthropological Studies
The University of St. Andrews
Fife KY16 9AL
SCOTLAND
Telephone: 44-1334-462428, 44-1334-462486
Fax: 44-1334-462485
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COGBURN & COOK
INVERTED SPACE: MINIMAL VERIFICATIONISM,
PROPOSITIONAL ATTITUDES, AND COMPOSITIONALITY
JON COGBURN & ROY COOK
While Dummett is most noted for his contributions to debates about the theory of
meaning1 and his arguments for intuitionistic logic,2 he has also provided a novel defense
of verificationism independent of these issues. In fact, he attempts to defend a minimal
form of verificationism upon which he and his non-Intuitionist, non-Anti-Realist,
opponents can agree. We do not seek to present a detailed exegesis of this defense.
Rather, we will explain the explanatory burden Dummett places upon verification
conditions.
After thus demonstrating the utility of Dummett’s position, we argue that it
cannot be right. The Duality Theorem of projective geometry straightforwardly entails
the existence of sentences distinct in meaning, yet identical in verification conditions.
This then raises the question of what else, other than inferential role, contributes to the
meaning of expressions.
We suggest that a limited form of holism, recapitulating appeals to relational
individuation in traditional metaphysics, renders (most of) Dummett’s view consistent
with the Duality Theorem. That is, the holistic verificationist can still say with Dummett
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that there is nothing to content over and above verification conditions, while admitting
that the content of some sentences depends upon the verification conditions of other
sentences as well.
1. THESIS: MINIMAL VERIFICATIONSM
Minimal Verificationism is the identification of the content of a sentence with its
canonical verification conditions.
In “Frege’s Distinction between Sense and
Reference,”3 and the appendix to “What is a Theory of Meaning? (I)”4 Dummett
motivates this view. Here, we will briefly reconstruct Frege’s puzzle in a manner that
illustrates how Dummett takes a weak form of verificationism to explain substitution
failures in propositional attitude contexts. This will serve to set in bold relief the problem
projective geometry poses for the Dummettian.
Frege argued that substitution failures must be explained in terms of some
component of meaning over and above mere reference. This argument can be presented
as depending on two premises, one of which is,
Fregean Compositionality
The logical properties of a sentence A are a function of the logical properties of the
words in A and the way those words are put together to form A.
This ought to entail,
Entailment of Fregean Compositionality
Whether or not B is a logical consequence of some set of sentences  depends
upon the logical properties of the parts of B and the parts of the sentences in  and
the way in which those parts are put together to form B and the sentences in .
Without an account of logicality these principles don’t really say anything. 5 The only
property of such an account which we use is the following:
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The Modal Principle
If it is possible for sentences in  to be true and  false, then  is not a logical
consequence of .
Nobody disputes this.
With these two fairly self-evident premises Frege’s puzzle can be presented in this
manner.6
1. Assume for reductio that the only logical property of a proper name like
“Augustus” is to whom the name refers.
2. Then, since two proper names like “Octavian” and “Augustus” name the same
person, those two names have the same logical properties.
3. Of the following two sentences,
(a) Livia believes that Augustus is happy,
(b) Livia believes that Octavian is happy,
it is clearly possible for (a) to be true and (b) to be false.
4. Thus, by the Modal Principle, (b) is not a logical consequence of (a), and, by the
Entailment of the Fregean Compositionality, this depends on the logical properties
of the parts of (b) and the parts of (a), and the way these words are put together to
form (b) and (a).
5. Some of the words in (a) and (b) must have different logical properties, since the
sentences are put together in exactly the same way.
6. But, given that the only different parts of (a) and (b) are the words “Augustus”
and “Octavian,” these words must have different logical properties.
7. Line 2 contradicts line 6.
8. Therefore, our original supposition that the only logical property of a proper
name like “Augustus” is to whom the name refers is false.
This kind of argument also reduces to absurdity the assumption that the only logical
properties of predicate expressions are the sets of objects to which they in fact refer.
Also note that this kind of argument reduces to absurdity the much stronger assumption
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that names and predicate expressions refer to entities and sets of entities at possible
worlds. That is, the existence of cointensional predicates, names, and sentences that
differ in content undermines the doctrine that the only relevant logical property of a name
or predicate is its extension at possible worlds. Clearly, we could have two predicates
coextensional at all possible worlds yet still not intersubstitutable in belief contexts.7
Verificationism’s great strength is the answer it provides to Frege’s puzzle. This
can best be shown if, for greater perspicuity, we represent “Livia believes that Augustus
is happy” and “Livia believes that Octavian is happy” as,
believe'(Livia', ^((happy')Augustus')),
believe'(Livia', ^((happy')Octavian')),
where (following Richard Montague8) the dash ( ' ) signifies that the word to which it is
attached is the formal language translate of the corresponding natural language word, and
the up-arrow sign (^) works to pick out the content of that which it binds. Unlike
Montague, Dummettians understand this intensional content to be the verification
conditions of the formulas bound by the up-arrow operator.
Dummettians are not, in fact, forced to eschew the standard possible worlds
framework for explaining other modal entailments! For example, one might hold that
believe'(Livia',^((happy')Augustus'))
is true at a possible world if and only if at that world the ordered pair consisting of Livia
and the canonical verification conditions for the proposition “Augustus is happy” is in the
extension of the set of ordered pairs for the denotation of “believe'.” For Dummettians,
this doesn’t entail
believe'(Livia', ^((happy')Octavian')),
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precisely because the canonical verification conditions for “(happy')Augustus'” are
different from those of “(happy')Octavian'.”
Before criticizing Minimal Verificationism we need to be clear about what it is
not. Minimal Verificationism differs in two respects from the full fledged Anti-Realism
of Michael Dummett and contemporary Dummettians such as Dag Prawitz, Neil Tennant,
and Crispin Wright. To be clear, Minimal Verificationism is an essential component of
Dummett’s philosophy.
However, it is logically independent of Dummett’s more
controversial beliefs. First, Minimal Verificationism does not entail that verification
conditions must be accessible to human beings. This issue of accessibility (and not
verificationism itself) is Dummett’s challenge to the metaphysical realist.
This is related to a second point. Dummett’s Minimal Verificationism is not
committed to an epistemic view of Fregean sense. “Livia believes that Augustus is
happy” does not entail “Livia believes that Octavian is happy” precisely because
“Augustus” and “Octavian” make different contributions to canonical verification
conditions of the sentences they occur in. But this does not mean that Livia has any
privileged epistemic access to the canonical verification conditions of “Augustus is
happy” and “Octavian is happy.”
Dummettian Anti-Realists attempt to explain grasp of meaning of a sentence in
terms of a person’s ability to recognize verifications of that sentence. 9 However, as
William Taschek points out, Kripke’s canonical criticisms of the Russellian description
theory of proper names and Putnam’s influential arguments against semantic internalism
can be marshalled with little effort against any such epistemic notion of sense.10
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Pace Dummett the Anti-Realist, Kripke and Putnam conclude that the meaning of
words transcends what any particular competent speaker needs to know in order to be
competent with those words.
Verificationism.
But this does not undermine Dummett’s Minimal
If Putnam and Kripke are right, Fregean Sense should not be
epistemically construed as something competent speakers must grasp.11 However, this
externalist position is clearly consistent with the position that canonical verification
conditions individuate meaning.12
2. ANTI-THESIS: MINIMAL VERIFICATIONISM
AND PROJECTIVE GEOMETRY
In the manner that we have shown, Dummett concludes from Frege’s puzzle that
verification conditions possess just the right amount of intensionality to individuate
content. Unfortunately for the Dummettian, projective geometry allows us to recapitulate
Frege’s argument, but this time against the idea that verification conditions are all there is
to meaning. That is, projective geometry clearly entails that two sentences can have
indiscernible verification conditions yet distinct content. Thus, it follows that Minimal
Verificationism faces the same problem that it was supposed to solve.
First, some background. Geometers of the early nineteenth century noticed that
their proofs were often complicated by the fact that they would have to consider more
than one case, depending on whether two lines involved in the construction intersected or
were parallel. This is because of a sort of lack of symmetry in Euclidean geometry: Two
distinct points always determine a unique line, yet two distinct lines determine a unique
point of intersection only if they are not parallel. The geometers realized that things
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would be much simpler if two lines always determined a unique point, and the simple
solution was to stipulate that such points do exist. Of course the next question concerns
where the point of intersection between two parallel lines is located, and the answer is “at
infinity.” Thus, we have a new geometrical plane, the projective plane, where every pair
of distinct points determines a unique line, and any two distinct lines intersect at exactly
one point.
A startling result was soon discovered. Consider the following meta-theorem, true
of the projective plane.
Duality Theorem
Given any statement true of the projective plane, replace every occurrence of “line”
by “point,” “point” by “line,” “intersect at” by “lie on,” and “lie on” by “intersect.”
The resulting sentence, called the dual of the original, is also true of the projective
plane.
An example may be helpful. Consider the following classic theorem of Pappus of
Alexandria (circa 320 A.D.).
Pappus’ Theorem
Given points A, B, and C on a line L, and A', B', and C' on a line L', let P be the
intersection of A'B and AB', Q the intersection of AC' and A'C, and R the
intersection of BC' and B'C. Then P, Q, and R are collinear.
The dual of Pappus’ Theorem is the following.
Pappus’ Dual
Given lines A, B, and C passing through a point L, and lines A', B', and C' passing
through a point L', let P be the line determined by the point of intersection of A and
B' and the point of intersection of A' and B, Q the line determined by the
intersection point of A and C' and the intersection of A' and C, and R the line
determined by the intersection of B and C' and the intersection of B' and C. Then
P, Q, and R intersect at a single point.
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The Duality Theorem tells us that this is true of the projective plane, merely in virtue of
the truth of Pappus’ theorem.
Projective geometry demonstrates that a verificationist account of meaning is
fundamentally flawed; Pappus’ Theorem and its dual have completely indiscernible
verification conditions, and thus should be equivalent in terms of meaning on the
verificationist view. This conclusion, however, seems absurd, given the obvious intuitive
differences between the two theorems.
A few objections need to be dealt with. The first objection is that the proof of
Pappus’ Theorem and the proof of its dual are not in any way indiscernible, since the
proof of Pappus’ Theorem uses classical Euclidean geometrical reasoning, yet the proof
of its dual consists of this proof plus an application of the Duality Theorem cited above.
Thus, the proof of the dual theorem contains an extra step.
This objection misses the point entirely; however, in refuting it we are able to
highlight the aspect of projective geometry that should worry the verificationist. The
metatheorem regarding sentences and their duals not only allows us to transform one true
statement into another by means of the translation rules. It also allows us to transform
any valid proof of a theorem into a valid proof of its dual. Thus, if we restrict ourselves
to the resources of projective geometry, any sentence and its dual end up having
indiscernible proof-conditions.
The verificationist might retort that a sentence and its dual do not have
indiscernible proof conditions. Of course, she will admit that for any proof of a sentence
about the projective plane, there is a structurally similar proof of its dual, but the proofs
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are not indiscernible. One need only notice that every time the word “line” occurs in the
first proof, the word “point” occurs in the second, and vice versa.13
This objection is untenable, however, unless the verificationist can defend the
claim that “point” and “line” have different meanings. If the meaning of a linguistic item
is given wholly by its role in verifications, however, then the fact that “projective line”
and “projective point” have different meanings cannot be defended.
For the
verificationist, two words (or the same word in a different speaker’s idiolect) have the
same meaning if and only if they contribute to sentences’ verification conditions in the
same ways (that is, where P[a/x] denotes the result of substituting occurrences of the
word a for all occurrences of the word x in the sentence P, a and b have the same
meaning if and only if for all sentences P, P[a/x] and P[b/x] have the same verification
conditions).
Thus, for the verificationist the only way for “projective line” and
“projective point” to diverge in meaning is for there to be some divergence in the roles
they play in derivations. There must be some proof in which one of the words plays a
certain role where the other cannot.
Assume the word “point” plays a particular role in a proof  of a sentence S.
Then there is a proof ', structurally isomorphic to , where “line” plays the role “point”
played in , and where the conclusion is the dual of S. Since the very issue at stake is the
difference in meaning between S and the dual of S, it seems impossible to defend the
claim that “line” is not playing the very same role in ' as “point” was playing in , or
even to find any grounds for claiming that  and ' are distinct proofs in the first place.
We are left with two choices. We can jettison some of the most basic truths of
one of the most fruitful areas of contemporary mathematical research, or abandon the
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view that verification conditions individuate content. We take it as obvious that the first
option is unacceptable.14 But then we must conclude that the central plank of Dummett’s
program collapses.
3. SYNTHESIS? VARIETIES OF COMPOSITIONALITY
The extent to which Dummett’s program collapses depends in large part on the extent to
which it can be reconfigured with minimal mutilation. How might one accommodate the
Duality Theorem and still hold that verification conditions are centrally involved in an
explanation of content? The most direct way would be to view the Duality Theorem as
forcing the admission that content is individuated by verification conditions and
something else. But then we must ask: What else, other than inferential role, contributes
to the meaning of expressions?
We suggest that a moderate form of holism provides the most verificationist
position for which the Dummettian can hope. That is, the holistic verificationist can still
say that there is nothing to content over and above verification conditions, while
admitting that the content of some sentences depends upon the verification conditions of
other sentences as well.
In the case of our projective geometry example, this would involve concluding
that the difference in meaning between “projective point” and “projective line” is not to
be found in how we use these words, but rather in how we use other words. If this is
right, then “projective point” means something different from “projective line” merely
because “Euclidean point” means something different from “Euclidean line.” A theorem
in projective geometry and its dual have different meanings merely because “Euclidean
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point” and “Euclidean line” have different meanings. In good holistic fashion, this is to
affirm that the meaning of projective theorems is parasitic on the meaning of theorems
outside of projective geometry.15
One might wonder what such holistic verificationism has to recommend for itself,
other than being the position involving the minimal amount of retreat for the
Dummettian. Rather than address this question directly, we will conclude by clarifying
the position as a means to making such evaluation possible.
First, it should be noted that the revised Dummettian position is similar to that of
William Taschek’s, as presented in “On Ascribing Beliefs: Context in Context.”16
Taschek argues that compositionality breaks down with ascription of propositional
attitudes.
As we will show, the envisioned revised Dummettian view agrees with
Taschek’s in this respect. However, while Taschek gives up what we have earlier called
“Fregean Compositionality,” the revised Dummettian view is consistent with Fregean
Compositionality and inconsistent with a position we call “Logician’s Compositionality.”
In addition, the revised view is also consistent with the form of compositionality most
relevant to empirical semantics. As a way of further sketching out the holistic solution to
the problem with the Duality Theorem, we will explain these other two kinds of
compositionality, showing how the revised Dummmettian view can accommodate one of
them and not the other.17
To illustrate how the revised Dummettian view can accommodate two forms of
compositionality and not a third we shall have to be slightly more rigorous about how we
envision a compositional verificationist semantics for the attitudes to interact with a
standard truth-conditional accounts. Contemporary verificationists utilize Heyting, or
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proof theoretic, semantics to recursively correlate verification conditions with sentences
of formal languages. For first order logic the correlations can be given in this manner.
Heyting Semantics Definition of Truth18
 is true if and only if there exists a verification k of ,
where one inductively defines what it is for k to be a verification of  as follows:
1. If  is atomic,  = (1 1, . . . , n) where 1 is an n-ary predicate, and each i
is either an individual variable or constant, then k verifies  if and only if k yields a
general method that determines for which n-tuples of objects, (<1, . . . , n>), (1
1, . . . , n) holds.
2. If  is a conjunction,  = (1 & 2), then k verifies  precisely when k yields a
general method that enables us to find a verification k1 of 1 and k2 of 2.
3. If  is an implication,  = (1  2), then k verifies  precisely when k yields
a general method that, from every construction l1 verifying 1,enables us to find a
construction l2 verifying 2.
4. If  is a negation,  = (1), then k verifies  precisely when k verifies
(1  ), where the constant  has no construction verifying it.
5. If  is a disjunction,  = (1  2), then k verifies  precisely when one can
extract from k information about which of the terms i of the disjunction is true
and a construction ki verifying that term i.
6. If  is an existential,  = (x 1(x)), then k verifies 1 precisely when k enables
us to determine for which object a, 1(a) holds.19
7. If  is a universal,  = (x 1 (x)), then k verifies 1 precisely when k yields a
general method that, for every object a, enables us to find a verification ka of the
proposition 1 (a).
Thus, the above clauses associate informal verification conditions with any formula of
first order logic. For example, for a formula of the form x y  (x,y), we have
x y  (x,y) is true if and only if there is a verification k such that:
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1. By 7, k yields a general method that, for every object a, enables us to find a
verification ka of the proposition y  (a,y).
2. By 6, k yields a general method that, for every object a, enables us to find a
verification ka such that ka enables us to determine for which object b,  (a,b)
holds.
Of course the above definition is incomplete pending clarification of notions such as “one
can extract information,” and “we give a general method.”20
We see no reason why the kind of mixed framework mentioned earlier in our
discussion of Livia’s beliefs, one which utilizes both verificationist and possible world
semantics, cannot account for standard de dicto/de re amibiguities such as the
following.21
John believes that a fish walks.
believe'(John',^x[fish'(x) & walk'(x)])
x[fish'(x) & believe’(John', ^walk'(x)])]
Every man believes that a fish walks.
y[man'(y)  believe'(y,^x[fish'(x) & walk'(x)])
x[fish'(x) & y[man'(y)  believe’(y, ^[walk' (x)])]]
Within the mixed framework, any clause embedded in the up-arrow operator gets
interpreted by the Heyting Semantic definition of truth, while the rest of the formula gets
interpreted in some form of standard possible worlds semantics.
A large part of Richard Montague’s lasting heritage is due to his discovery of how
to derive formulas such as these in a typed lambda calculus, step-by-step with the
syntactic derivation of the natural language sentence.22 Since the possible success of
something like Montague’s endeavor is an essential part of the claim that a natural
(rather than formal) language is compositional, we shall state this requirement explicitly.
Where + is the function under which the set of expressions of the logical language is
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closed (intuitively, the closure of the formation rules for the language), * is the function
under which the set of syntactic structures is closed (the closure of the grammatical rules
for some limited fragment of the natural language in question), and t is the translation
function from syntactic components (of the natural language) to expressions of the logical
language,23 Montague’s fragment satisfied the
Homomorphism Requirement
t(*(x1, . . . , xn)) = +(t(x1), . . . , t(xn)).
Again, this intuitively requires that when units are conjoined by the syntax, their formal
language translates are also conjoined. It should be clear why this is the linguistically
interesting sense of compositionality. Grammars satisfying Montague’s Homomorphism
Requirement really do show how the meanings of whole sentences are determined by the
meanings of their parts and how those parts are put together. Every application of * puts
together syntactical parts, and the Homomorphism Requirement requires for each such
application an application of + to put together the semantic parts.
The kind of holism envisioned for the Dummettian does not prohibit the form of
compositionality encoded by the Homomorphism Requirement. Indeed the Dummettian
verificationist is free to utilize the Montagovian step-by-step method of deriving the
above formulas (alongside the syntactic structures of the natural language sentences),
while jettisoning the use of intensional model theory to explain the formulas embedded in
the up-arrow operator, instead using some form of constructivist semantics for those
formulas.
However, this strategy carries a steep price. As noted, the Heyting style clauses
are essentially informal. Rigorously defining a consequence relation requires tightening
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up these clauses.24 Unfortunately, it may be a consequence of the Duality Theorem that
hope of such further rigorization is chimerical.
While the Homomorphism Requirement is a property of the syntax and semantics
interface, Logician’s Compositionality is a property of the logical language to which the
syntactic algebra is mapped (by the homomorphism in question). Where LM is equal to
the smallest set containing the morphology (or nonlogical vocabulary) M and closed
under +, where LM' is equal to the smallest set containing the morphology M ' and closed
under +, +(x1, . . . , xn) is a well formed formula of both LM and LM', i is an interpretation
of LM, and i' is an interpretation of LM' we can give this other requirement as
Logician’s Compositionality
If i(x1) = i' (x1), . . . , i(xn) = i' (xn),
then i(+(x1, . . . , xn)) = i'(+(x1, . . . , xn)).
This property basically says that the interpretation of a sentence (and hence the logical
properties of that sentence) depend on the parts of that sentence and those parts alone.
Note that the Homomorphism Constraint and Logician’s Compositionality together place
extremely minimal restrictions on what the syntax, logical language, and interpretation
functions are like.
By showing how semi-formal constructive semantics can be brought into a
Montague style architecture we have shown that by going verificationist you can hold
onto the Homomorphism Requirement. However, the holistic solution to the problem
with the Duality Theorem entails that Logician’s Compositionality is violated. By the
solution, the proper Heyting style interpretation of the subsentential units in a given
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projective theorem crucially depends on whether or not the subsentential units of
sentences in Euclidean Geometry are also interpreted.
Indeed, one could argue that without an interpretation of the Euclidean
vocabulary, projective geometry is inconsistent. The notion that predicates that have the
same meaning ought to apply to the same objects is incontrovertible. Yet if we accept
this then we find that a Heyting style interpretation of projective geometry alone yields
inconsistency. The following are trivial truths regarding the projective plane: “There is a
point,” “There is a line,” “No point is a line,” and “No line is a point.” But these facts,
combined with the synonymy of “point” and “line” (which we argued follows from the
Duality Theorem and verificationism) and the fact that they should therefore apply to the
same objects, produce an inconsistency. Given this, it is clear that we must admit that
there is more to the meanings of expressions than their roles in proofs, even in
mathematics. This is elegantly accomplished by forfeiting Logician’s Compositionality,
as accepting the holistic conclusion that the logical properties of the parts of a sentence
are often dependent upon the logical properties of other sentences in which those words
don’t occur.25
Here we have merely suggested a possible Dummettian solution to the problem
we presented, and not attempted to rigorously model this solution. Indeed, there is an
interesting research program lurking here. Very promising work concerning formal
languages with two logics is suggested to those who accept: (1) something like the
Montagovian homomorphism between the syntactic algebra and the algebra of the logical
language, (2) something like Montague’s intensional semantics for non-attitude contexts,
(3) some rigorization of Heyting semantics for attitude contexts. It is a consequence of
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our discussion that a test case for this involves formally modeling the monotonicity
failure involved in the holistic solution to the verificationist’s problem with projective
geometry. That is, if one could define a formal consequence relation such that a set of
projective theorems are inconsistent on their own, yet consistent when conjoined with the
Euclidean theorems, then one would go a long way towards formalizing the mixed view
in a way that formally models the revised (holistic) Dummettian position. We would be
very excited if there were any linguistic evidence concerning how a non-constructive and
constructive logic could be put together in a single language in this context.26
On the other hand, necessity may be a virtue; we may here be at the limit of
formal semantics. The pathological resistance of the attitudes to a satisfactory formal
treatment strikes us as much evidence for the view that we have reached such a limit.
Thus, insuperable difficulties involved in attempting a rigorization of the revised
Dummettian view might paradoxically be indirect evidence in its favor.27
LOUISIANA STATE UNIVERSITY
BATON ROUGE, LA 70803
USA
jcogbu1@lsu.edu
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NOTES
1
For discussions, see Appiah, A., For Truth in Semantics, Cambridge: Blackwell, 1986,
most of the articles in Wright, C., Realism, Meaning, and Truth, 2nd ed., Oxford:
Blackwell, 1993, Tennant, N., Anti-Realism and Logic: Truth as Eternal, Oxford:
Clarendon Press, 1987, and Tennant, N., The Taming of the True, Oxford: Clarendon
Press, 1997.
2
For discussions, see chapter 6 of (Tennant, ibid.).
3
Dummett, M., “Frege’s Distinction between Sense and Reference”, in Truth and Other
Enigmas, Cambridge: Harvard University Press, 1978, pp. 116-144
4
Dummett, M., “What is a Theory of Meaning? (I)”, in Dummett, The Seas of Language,
Oxford: Clarendon Press, 1976a, pp. 1-33
5
It may be the case that our modern conceptions of logicality and consequence came
about because of commitment to Fregean Compositionality. We have no problem with
this as far as it goes. However, for an important critique, see Hintikka, J., The Principles
of Mathematics Revisited, Cambridge: Cambridge University Press, 1996. For criticisms
see Cook, R. & Shapiro, S., “Hintikka’s Revolution, The Principles of Mathematics
Revisited”, in The British Journal of the Philosophy of Science Vol. 49, 1998, pp. 302316, and Tennant, N., “Review essay on J. Hintikka’s The Principles of Mathematics
Revisited”, in Philosophia Mathematica, Vol. 6, 1998, pp. 90-115. For another critique
of the standard modern concept of logical consequence, see Etchemendy, J., The Concept
of Logical Consequence, Cambridge: Harvard University Press, 1990. For interesting
critiques of that critique, see Ray, G., “Logical Consequence: A Defense of Tarski”, in
Journal of Philosophical Logic, Vol. 25, 1996, pp. 617-677, Sher, G., “Did Tarski
Commit “Tarski’s Fallacy”?”, in Journal of Symbolic Logic, Vol. 61, 1996, pp. 653-686,
and Shapiro, S., “Logical Consequence: Models and Modality”, in Schirn, M., ed.,
Philosophy of Mathematics Today, Oxford: Oxford University Press, 1998, pp. 131-156.
6
We are not trying to make an exegetical claim about what Frege actually intended,
though see Taschek, W., “On Sinn and Bedeutung: A Critical Reception.”, in Ricketts,
T., ed., Cambridge Companion to Frege, Cambridge: Cambridge University Press,
forthcoming for an interpretation of Frege along the lines of the argument presented here.
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COGBURN & COOK
7
In talking about just names, predicates, and sentences, we pretend that natural language
a formal language. However, it should be clear that Frege’s argument does not depend
on this for its validity.
8
Montague, M., Formal Philosophy, New Haven: Yale University Press, 1974.
9
See (Dummett, 1976a) and Dummett, M., “What is a Theory of Meaning? (I)”, in
Dummett, The Seas of Language, Oxford: Clarendon Press, 1976b, pp. 34-93.
10
See (Taschek, forthcoming). For Kripke’s discussion, see Kripke, S., Naming and
Necessity, Cambridge, MA: Harvard University Press, 1980. For Putnam’s discussion,
see Putnam, H., “The Meaning of ‘Meaning’”, in Gunderson, K. (Ed.), Language, Mind,
& Logic, Oxford: Oxford University Press, 1975.
11
Both Kripke and Putnam (1975) can be interpreted as arguing that explanatory
demands upon the theory of reference motivate Minimal Verificationism. For example,
Kripke appeals to verificationist considerations when he criticizes other theories of
reference as being circular. What Kripke takes to be wrong about circularity is that it
renders a proposed definition uninformative.
Someone uses the name ‘Socrates.’ How are we supposed to know to whom he
refers? By using the description which gives sense of it. According to Kneale, the
description is ‘the man called “Socrates.” ’ And here, (presumably, since this is
supposed to be so trifling!) it tells us nothing at all. Taking it in this way it seems
to be no theory of reference at all. (Kripke, (1980, p. 70))
Kripke is requiring the theory to be such that knowing the conditions for successful
reference of a name as stipulated by the theory is sufficient for being able to locate the
referent. Thus, for Kripke, if the theory is informative, then it provides canonical criteria
by which one can determine the referent of the words in question. But such canonical
criteria are exactly what is required by Minimal Verificationism. Thus, there is a sense in
which any informative theory of reference is a verificationist one.
12
For a discussion of these issues that takes seriously the arguments for bifurcating sense,
while still defending a unary externalist theory, see Bilgrami, A., Belief and Meaning,
Cambridge: Blackwell, 1992.
13
In all that follows we simplify a bit, as the argument will actually only run if similar
things are said concurrently about “intersect” and “lie on.” In other words, strictly
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speaking we need to argue for the claim that there is nothing in the verificationist view
preventing us from drawing the more complex conclusion both that “line” and “point”
mean the same thing and that “intersect” and “lie on” are also synonymous.
14
Some orthodox intuitionists, already eschewing the law of excluded middle, might be
perfectly satisfied to view our demonstration as a refutation of projective geometry. We
have no sympathy for this position. Constructive approaches to mathematics must, in a
sense, “save the phenomena” by either making explicit the constructive content of
classical mathematics, or reformulating classical theories constructively. Following the
articles in Shapiro, S., ed., Intensional Mathematics: Studies in Logic and the
Foundations of Mathematics 113, Amsterdam: North Holland, 1985, we view
constructive approaches to mathematics as mathematics in their own right that can (pace
most constructivists), and (pace most mainstream mathemeticians) ought to, coexist
peacefully with classical mathematics.
15
One might note a similarity to certain other cases involving the violation of Leibniz’s
Law, such as the classical metaphysical discussions in Hochberg, H., “Elementarism,
Independence, Ontology”, Philosophical Studies, Vol. 12, 1961, 36-42 , Allaire, E.,
“Bare Particulars”, Philosophical Studies, Vol. 14, 1963, pp. 1-7, Allaire, E., “Another
Look at Bare Particulars”, Philosophical Studies , Vol. 16, 1965, pp. 16-20, Hochberg,
H., “Universals, Particulars, and Predication”, Review of Metaphysics, Vol. 19, 1965, pp.
87-103, Hochberg, H., “Ontology and Acquaintance”, in Philosophical Studies, Vol. 17,
1966, pp. 49-54, as well as the debate about quantum physics in Barnette, R.L,
“Discussion: Does Quantum Mechanics Disprove the Principle of the Identity of
Indiscernibles”, in Philosophy of Science, Vol. 45, 1978, 466-470, Cortes, A., “Leibniz’s
Principle of the Identity of Indiscernibles: A False Principle”, in Philosophy of Science,
Vol. 43, 1976, pp. 491-505, Teller, P., “Discussion: Quantum Physics, The Identity of
Indiscernables, and Some Unanswered Questions”, in Philosophy of Science, Vol. 50,
1983, pp. 309-319, and Redhead, M. & Teller, P., “Particle Labels and the Theory of
Indistinguishable Particles in Quantum Mechanics” in The British Journal of the
Philosophy of Science, Vol. 43, 1992, 201-218. Indeed, if the correct response to those
cases is that relational properties of objects (either subatomic or citizens of the kind of
possible worlds that constitute philosophers’ Gedanken-experiments) are essential
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COGBURN & COOK
properties involved in the individuation of objects from one another, then the analogy is
clear and persuasive. Moreover, one finds much of this early metaphysics recapitulated
in the philosophy of language. For example, “bare particulars” (a modern version of
Lockean substance) are extremely similar to Fodorian meaning atoms (See Fodor, J.,
Concepts, Where Cognitive Science Went Wrong, Oxford: Clarendon Press, 1998). Here
philology recapitulates ontology.
16
Taschek, W., “On Ascribing Beliefs: Content in Context”, in The Journal of
Philosophy, Vol. 95, 1998, pp.323-353.
17
Which aspects of the revised Dummettian view should be thought of as notational
variants of Taschek’s positions, and which are in fundamental disagreement, is a subtle
issue. If we understand Taschek’s dialectic right, he seeks to block the argument we have
called “Frege’s Problem” by giving up Fregean Compositionality, and then goes on to use
holistic considerations to explain the failure of substitution in propositional attitude
contexts. In this sense Taschek’s theory might be thought of as dispensing with Fregean
sense. Pace Taschek, the holist verificationist keeps Fregean Compositionality and hangs
onto Fregean senses, albeit characterizing them holistically. Thus, it seems that the
revised view is fundamentally different from Taschek’s. However, given that Taschek
doesn’t consider the other two kinds of compositionality we go on to consider, the issue
is somewhat underdetermined.
18
These clauses have been taken, with modifications, from Dragalin, A., Mathematical
Intuitionism, Introduction to Proof Theory, Providence: American Mathematical Society,
1980.
19
Utilizing these clauses will, of course, require standard use of alphabetic variants.
20
There is little agreement among contemporary verificationists about how this is to be
done. For example, Neil Tennant and Dag Prawitz think that the introduction and
elimination rules of natural deduction style proof systems can be thought of as a very
good way to make the above precise. See Prawitz, D., Natural Deduction: A ProofTheoretical Study, Stockholm: Almqvst and Wiksell, 1965, Prawitz, D., “On the Idea of a
General Proof Theory”, in Synthese, Vol. 27, 1974, pp. 63-77, and (Tennant, 1997).
Michael Dummett disagrees and urges the use of Beth Frames to do this. See Dummett,
M., The Logical Basis of Metaphysics, Cambridge: Harvard University Press, 1991. In
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mathematics proper there are many different proposals for ways to firm up the
constructive notions in the above definitions (e.g. (Dragalin, 1980).
21
Three of these are taken from Dowty, D., Wall, R., & Peters, S., Introduction to
Montague Semantics, Dordrecht: Kluwer, 1981.
22
This is the major point of miscommunication between philosophers of language and
empirical semanticists. “The question of compositionality” in empirical semantics and
computational linguistics almost always involves the extent to which a Montague style
syntax-semantics interface is viable for the fragment of language in question. On the
other hand, philosophical discussions of compositionality almost never involve this
question. For a particularly clear and accessible discussion of Montague’s historical
contribution in this regard see Partee, B., “The Development of Formal Semantics in
Linguistic Theory,” in Lappin, S., ed., The Handbook of Contemporary Semantic Theory,
Oxford: Blackwell, 1996, pp. 11-38.
Note that, like many philosophers,
transformationally minded syntacticians often also misinterpret Montague’s achievement
merely in terms of the development of intensional logic. Also note that computational
linguists do not misinterpret Montague’s contribution, as their grammars are forced to
possess the rigor that is now only a promissory note in transformational approaches.
23
We realize that the purist will balk at how we’ve put this, as an essential part of
compositionality debates in linguistics concern the extent to which the logical language
can be dismissed, so that a function directly from the syntactic algebra to model theoretic
interpretations can be given (see the relevant discussion (Dowty, Wall, & Peters, 1981)).
However, separating out the Homomorphism Requirement from what we go on to call
“Logician’s Compositionality” is, as far as we can tell, the only way to make discussions
of compositionality in philosophy (for example, the discussion in (Fodor, 1998) relevant
to actual empirical semantics and computational linguistics. This, and the fact that our
division captures the actual heuristic followed by working semanticists, justifies the
ineliminable role of the logical language in our characterization.
24
See (Dragalin, 1980) for many examples.
25
Note that we have not given up the view that the verification conditions of a sentence
are a function of the meaning of the parts of that sentence and the way they are put
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together! One might think that Taschek only forfeits Fregean Compositionality because
he states it in too strong a form. He writes,
An unrestricted principle of compositionality will require that of any complex
expression E containing an expression n, if the semantic content of some expresion
n’ is the same as (different from) that of n, then the expression E’ that results from
substituting n’ for n in E will possess the same (a different) semantically content as
(from) E. (Taschek, (1998, p. 329))
The parenthetical marks here might be mistaken. Let E be equal to the contradiction (P
& ~P) and replace “P” in this formula by “~P” to get the formula E’’ which is (~P &
~~P). Arguably, the two contradictions have the same content, though clearly the two
substituents do not. However, if one reads his dialectic as we have in footnote 13, then
nothing in his argument hangs on the mistake.
26
The philosophical and logical challenge is interesting enough independent of our
discussion of the duality theorem. For putting together constructive and non-constructive
logics in a different context, see the relevant work in (Shapiro, 1985).
27
We would like to dedicate this to our metaphysics teacher, Herbert Hochberg. A much
earlier incarnation of this paper was presented at the 1998 Third Annual Ohio State
University Colloquium for Crispin Wright. We thank Professor Hochberg, Professor
Wright, Robert Batterman, Emily Beck Cogburn, Ed Henderson, Robert Kraut, Hussein
Sarkar, Stewart Shapiro, Mary Sirridge, William Taschek, Neil Tennant, and John
Whittaker for very helpful conversations.
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