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1
THE PARADOX OF IDENTITY*
William J. Greenberg
ABSTRACT:
Call a semantics for singular terms extensionalist if it embraces (1) and classical if it embraces
(2).
1. The meaning of a singular term is exhausted by its reference.
2. The reference of a singular term is an entity that is logically simple.
Call a semantics adequate if it distinguishes material identity (the identity of a and b) and
formal identity (the identity of a and a).
Frege reacts to the inadequacy of classical extensionalist semantics by rejecting (1). This he
does without a sideways glance at (2), whose background ontology he implicitly accepts.
In contrast, my account of the difference between material and formal identity replaces that
background ontology, the so-called ontology of individuals (van Heijenoort's term), with an
ontology whose ground-level objects are ontologically differentiated and logically complex. The
semantics I urge for singular terms, while extensionalist* in the sense of (1), is thus a non-
classical semantics in which singular terms take structured individuals, or complexes (as I will
say), as their referents. For, unlike the logically simple units of the ontology of individuals,
complexes keep (true) 'a = b' and 'a = a' apart.
*Epistemologia XIX, 1996, pp. 207-226
2
1.0 Introduction
1
'A Theory of Complexes'
2
presented a formal theory of the identity-relation. The foundation-
stone of this theory is the particular-cum-complex, an entity which is ontologically differ-
entiated and logically complex. The deductive apparatus of the theory is that of the formal
system P.
3
Viewed as an abstract calculus, P says nothing about the world. Like other theoretical models,
however, P can be endowed with empirical content by identifying C-complexes with particu-
lars. The effect of this identification is to replace the standard ontology of quantification theory
by an ontology of complexes. I will warrant this identification by showing that important logico-
semantic features of particulars which the standard ontology cannot explain, can be explained
in terms of the principles governing the relations of complexes and their constituents. An
ontology of complexes is capable of such explanation because complexes incorporate essential
features of the particularity of individual things from which the standard ontology prescinds.
In Appearance and Reality, F. H. Bradley cites the two aspects of particularity I have in mind:
If we take up anything considered real, no matter what it is, we find in it two aspects.
There are always two things we can say about it; and if we cannot say both we have not
got reality. There is a 'what' and a 'that', an existence and a content, and the two are
inseparable. That anything should be, and should yet be nothing in particular, or that a
quality should not qualify and give a character to anything is obviously impossible.
(Appearance and Reality, p. 162, cited in Murphy's 'Substance and Substantive,' U.C.
Publications in Philosophy, Vol. 9, 1927, pp. 63-87)
Complexes incorporate these aspects of an individual thing's particularity, for every complex
encapsulates a haecceity and an individual.
In contrast, the ground-level objects of the "ontology of individuals", as Van Heijenoort calls the
standard ontology of quantification theory, are qualitatively indefinite--and intrinsically so. The
translation of everyday assertions about ordinary objects into the language of quantification
theory thus involves replacement of the qualitatively definite individual things and events such
assertions are about by ground-level objects of the standard ontology, "bare individuals" with
1
I am grateful to Paul Schachter, John Olney, and an anonymous referee for comments on earlier
versions of this paper--and to the College of Humanities of the University of Puerto Rico at Rio
Piedras for reductions in load which helped make completion of this paper possible.
2
This journal
3
P is presented in Appendix Two.
3
no "inner structure, . . . mere pegs"
4
. Logico-semantic analyses which proceed by translating
such assertions into the language of quantification theory
5
thus obliterate the nexus of that and
what constitutive of the particularity of the objects these assertions are about. As a result, such
analyses fail in two related respects. First, they fail to convey the complex states of affairs to
which truthful singular identity and existence statements refer; and second, they fail to uncover
a basis for the "referential opacity" of singular terms in propositional attitude and modal
contexts. Nowhere is the insufficiency of classical analysis more apparent than in its inability to
resolve the so-called "problem of identity".
1.1 The Problem of Identity
No instruction in semantics is required to grasp that (1) and (2) do not mean the same thing.
(1) Hesperus = Hesperus.
(2) Hesperus = Phosphorus.
Of course, it is one thing to note that sentences such as "Hesperus = Hesperus" and "Hesperus =
Phosphorus" differ in what they assert--and quite another thing to adduce why this should be
so.
I will venture an explanation of this from the standpoint of a referential view of meaning. Such
a view holds that "Hesperus" and "Phosphorus" are names, and that the meaning of a name is
what it stands for. In coming to terms with what sets (1) and (2)--and such pairs of sentences
generally--apart, I will adopt a referential view of meaning--not only, as is sometimes done, for
proper names like "Hesperus" and "Phosphorus", but for other singular terms
6
as well. On the
view of meaning I have in mind, "Hesperus" will mean Hesperus, and "Phosphorus"
4
Van Heijenoort, Jean (1976). 'Set-Theoretic Semantics,' in Selected Papers, p. 48.
5
The logical technique for obtaining such objects is well-known. First, an object, say an apple, is
separated from its color, so becoming an x such that red(x). Divested in turn of its other
attributes--its shape, feel, taste, kind,..., etc.--the apple emerges as an x such that red(x) and
round(x) and soft(x) and sweet(x) and (fruit)x ... etc. (Francis Pelletier, Review of E.J. Lowe's
'Kinds of Being,' History and Philosophy of Logic 13, 1992, pp. 125-128.) The entity which
stands in for "x" is thus no longer an apple, but--like Aristotelian matter or Thomas's materia
insigna--an attributeless bearer-of-attributes.
6
By singular term, I mean any nominal referring expression which purports to pick out a
particular--the category "particular" being broadly construed so as to include things, processes
and events. Singular terms include proper names ("Scott", "Vulcan", "the Palmdale Bulge"),
definite descriptions ("the author of Waverley", "the largest prime number", "the round square"),
and singular possessive phrases ("Euclid's fifth axiom", "Smith's hangover", "Yugoslavia's
tragedy").
4
Phosphorus. As a result, (1) will assert--as intuitively and pre-theoretically it appears to assert--
that Hesperus is identical with Hesperus; and (2) will assert--as intuitively and pre-theoretically
it appears to assert--that Hesperus is identical with Phosphorus. What sets apart (1) and (2) will
thus reflect some difference between the identity of Hesperus and Hesperus and the identity of
Hesperus and Phosphorus--and so, ultimately, some difference between Hesperus and
Phosphorus, which--notwithstanding the historic principle of the Indiscernibility of Identicals
7
--
are, as (2) suggests, one and the same entity.
1.2 Extensions as Meanings: Chronicle of an Exile Foretold
A referential theory of meaning for singular terms does two things. First, it equates the
meaning of a singular term with its reference. In this way, such a theory links naming and
meaning. Second, it equates the reference of a singular term with a particular, concrete or
abstract
8
. In this way, a referential theory avoids making singular terms the vehicles of a strictly
conceptual content.
9
A referential theory thus assigns meanings in such a way as to satisfy EX1.
EX1: The meaning of a singular term is a particular.
An extensional semantics encapsulates a referential theory of meaning for singular terms. Call
a semantics which satisfies EX1, EX1-extensional.
The second criterion for an extensional semantics is of a different sort. Whereas EX1 equates
the meaning of a singular term with its reference, EX2 exacts a tribute from sentence meanings.
EX2: When in a sentence, a singular term is substituted for another with the same
meaning, the meaning of the sentence remains constant.
7
According to the Indiscernibility of Identicals, if x and y are identical, every property of x is also
a property of y--and conversely. So if, as I maintain, Hesperus and Phosphorus are numerically
identical yet differ in some respect, then the Indiscernibility of Identicals is false.
8
"Hesperus" and "Phosphorus" refer to concrete particulars, while "nine" and "the number of
planets" refer to abstract ones.
9
There is no general agreement concerning what the intension of a singular term might be. The
distinction between intension and extension, however, is often taken to reflect Frege's distinction
between the particular a singular term stands for and the conceptual content it expresses.
5
It is sometimes supposed that a semantics which is EX1-extensional must also be EX2-exten-
sional:
If [singular terms] have no other semantic role but to refer, then it appears that if two
[singular terms] refer to the same individual, then a principle ... is warranted ...that says
that substitution of one [term] for the other will...preserve...the proposition expressed.
(Donnellan, 1990: 202)
10
But this is not so. Whether a singular term's reference is its meaning, and whether the meaning
of a sentence remains constant when one singular term is replaced by another with the same
meaning, are separate questions--as are whether "Hesperus" and "Phosphorus" mean Hesperus
and Phosphorus, and whether the mean- ing of "Hesperus = Hesperus" is the same as that of
"Hesperus = Phosphorus". Whether a theory of meaning is EX1-extensional is independent of
whether it is EX2-extensional. In my semantics, the concrete particulars Hesperus and
Phosphorus exhaust the meaning of "Hesperus" and "Phosphorus". The EX2-semanticist and I
thus both commence by treating the referents of singular terms as their meanings. But here we
part ways. For the tribute exacted by the EX2-semanticist's commitment to Hesperus and
Phosphorus is one that "Hesperus = Phosphorus" and "Hesperus = Hesperus" cannot pay. And
so the EX2-semanticist casts out Hesperus and Phosphorus from the Garden of Meaning. I will
not recount the ensuing fall from semantic grace of extensions generally.
1.3 Sense and Reference
Thomas garners points toward promotion by working the semantic circuit in the Valley of the
Shadow of Doubt--for all it's worth (and then some). A highly regarded younger scholar,
Thomas does the right thing in philosophical semantics.
At desk. Reflective. When in a sentence, a singular term is substituted for another with the
same meaning, the meaning of the sentence remains constant. Brow Darkens. Some people still
say meaning is naming. They say "Hesperus" means Hesperus and "Phosphorus" means
Phosphorus. Leaps to his feet. But Hesperus IS Phosphorus. So if "Hesperus" meant Hesperus
and "Phosphorus" meant Phosphorus, "Hesperus = Hesperus" and "Hesperus = Phosphorus"
would mean the same thing. But they don't! Sniffs. Meaning as naming. Extensional semantics.
What a crock!
Mephisto. From behind desk. Wait a minute! Not so durn fast! Maybe it's not meaning
as naming that's a crock. Maybe it's something else. Like "The meaning of a sentence remains
10
For my "singular terms"/"term", read Donnellan's "proper names"/"name".
6
constant when a singular term is substituted for another with the same meaning". Maybe
there's your crock. Who's to say?
Thomas. Stamps foot. Mine is NOT a crock!
Mephisto. "This is a crock and that is not." O lover of wisdom, `give me an ARGUMENT.
Thomas. Resolute. All right. Meaning as naming is a crock because it is inconsistent
with a Self-Evident Principle of Philosophical Semantics...
Mephisto. Evident, schmevident...
Thomas. ...and Touchstone of Self-Consistent Thought, mere reflection on--it's Frege's
Principle, you know--suffices--(Mephisto smirks) Meaning as naming? I'll show you! Advances
on Mephisto, brandishing reprint.
Voice. Unswerving devotion alone to Transcendent Truth would never have provoked
Thomas's willy nilly flight from extensions. Had "Hesperus = Phosphorus" and "Hesperus =
Hesperus" meant the same thing, as it was an article of faith that extensionally proper such sen-
tences should--or had it been possible to predict such perturbations on the basis of the regnant
ontology, Thomas would never have cast his lot in with those creatures of darkness that EX2-
semanticists call intensions. Nor would Thomas have been likely to prostrate himself before
whatever avatar of Fregean semantics was currently making the rounds of his culture circle.
Unfortunately, "Hesperus = Phosphorus" and "Hesperus = Hesperus" didn't mean the same
thing; this was something on which Thomas--and members of his culture circle
11
--fervently
agreed. As for the ground-level objects of the standard ontology, no amount of prodding could
force them to cough up the difference between "a = a" and "a = b".
12
Thomas. To Mephisto. And so it was that Gottlob Frege, with his doctrine of sense and
reference, cast extensions out from the Garden of Meaning. In the words of the Master:
It is natural...to think of there being connected with a sign (name, combination of
words, letter), besides that to which the sign refers, which may be called the reference
of the sign, also what I should like to call the sense of the sign, wherein the mode of
presentation is contained. (Frege, 1892, 1992: 24)
11
With the exception of Nathan Salmon.
12
Indeed, when questioned about identity or difference, they would but give forth with a surly "Yea, Yea"
or "Nay, Nay".
7
It is to Frege, the Founder of Intensional Semantics, that we owe the insight that the senses of
"Hesperus" and "Phosphorus" are ingredients in the meaning of "Hesperus = Hesperus" and
"Hesperus = Phosphorus". Not their reference, their sense. Substitute reference for sense and
there is nothing to set sentences like these apart. Suppose the reference of "a" and "b"deter-
mines what "a = a" and "a = b" mean. In the Master's words:
...the cognitive value of a = a becomes essentially that of a = b, provided a = b is true.
(ibid.)
On the other hand, if the sense of "a" and sense of "b" determine what "a = a" and "a = b"
mean, the cognitive value of "a = a" is different from that of "a = b", even when "a = b" is true.
For although "a" and "b" are, then, identical in reference, they are different in sense.
What IS a sense, you ask? Well, some say a sense is an object's mode of presentation.
Others say it's an individual concept. Some say it's a criterion for a word's application. Others
say it's an incomplete state of affairs. Some say it's a context in which a reference is found in
the world.
13
Others say (brightening)...it's a function! A mathematical mapping! Take a sheet
of paper. Draw a vertical line. On one side, put--put--Falters. I don't KNOW what a sense is.
Takes a deep breath. Alright, alright. A sense--a sense--is whatever it is about "a" and "b" that
makes "a = a" and "a = b" differ in cognitive value. Mephisto smiles.
Voice. Mephisto's smile means Mephisto knows something that Thomas doesn't.
Indeed, Mephisto is about to give Thomas the Real Low Down about Hesperus and Phosphorus-
-with oodles of fire and brimstone, but minus the fonts and curlicues with which Thomas and
his symbol-thumping cohorts religiously anoint the ponderous, mulled over dullness of their
journal submissions; and with put-downs galore of "Frege fairy-tale semantics" that will make
Thomas, his editor friends, their sustaining subscribers--and all those fervid little investors in
Frege Futures over at the Oxford and Cambridge presses do a not-so-slow burn. Reflects. If
Mephisto opens his trap again, this thing will never get published. To Mephisto. I'm sorry, Bub,
but I'm TIRED of goosing electrons! Yells. Pull the plug on Mephisto! Mephisto squeals.
1.4 The Word Well-Lost
Mephisto called our attention to the fact that it was Thomas's uncritical adherence to a
presupposition common to every variety of Fregean semantics--that the meaning of a sentence
remains constant when a singular term is substituted for another with the same meaning--
which led him to conclude, from the identity of Hesperus and Phosphorus, and cognitive dis-
13
See Panayot Butchvarov, "Identity", in Peter A. French et. al. (eds.), Contemporary Studies in the
Philosophy of Language, U. of Minnesota Press, Minneapolis, 1979, p. 163 f.
8
tinctness of "Hesperus = Phosphorus" and "Hesperus = Hesperus", that the customary
reference of "Hesperus" and "Phosphorus" plays no role in the meaning of "Hesperus = Phos-
phorus" and "Hesperus = Hesperus". What Mephisto was unable to do--thanks to a less than
timely intervention on the part of the Voice, was explain what it is about the customary
reference of "Hesperus" and "Phosphorus" that makes "Hesperus = Hesperus" and "Hesperus =
Phosphorus" cognitively distinct.
What is it, then, about the customary reference of singular terms which causes assertions of
formal and material identity so to misbehave? Contra countless journal articles which take
their cue from turn-of-the-century writings by Gottlob Frege and Bertrand Russell, my account
of what sets such assertions apart will not evoke the language in which these are cast. Instead
the point of departure for my account of the fall of Frege's Principle will be a less-than-modern
view of identity and the entities it relates.
1.5 Identity As Oneness in Substance
Identity, Aristotle informs us in Metaphysics, is oneness, in matter or substance.
En sentido esencial, las cosas son idénticas del mismo modo en que son unidad, ya que
son idénticas cuando es una sola su materia (en especie o en número) o cuando su
sustancia es una. (Met., V, 9, 1018 a 7. Cited in Nicola Abbagano, Diccionario de
Filosofía, Fondo de Cultura Económica, México, 1986, p. 640).
Concerning Aristotle's view of identity as a kind of oneness, Nicholas White observes:
...it is clear that in saying that X and Y are one, Aris-totle does not simply mean that [X
and Y] are parts of the same compound entity; he also means that [X and Y] are in some
sense the same, as each other. (1971: 187)
Aristotle is thus not "keeping separate the use of 'X and Y are one' to mean that they are in
some way identical from its use to say that they make up a unitary entity". (ibid.)
Now, if identity is oneness in substance, "Hesperus = Phosphorus" and "Hesperus = Hesperus"
assert different things. Thus, suppose that X = Y iff X and Y are one in substance. Suppose also
that X and Y are one in substance iff for some Z, X and Y are parts of Z. Then X = Y iff for some Z,
X and Y are parts of Z. Therefore, if identity is oneness in substance, material identity is a
relation that relates one part of a compound entity to another such part.
Not so formal identity. Consider. Since X = Y iff X and Y are one in substance, Y = Y iff Y and Y
are one in substance. Similarly, since X and Y are one in substance iff for some Z, X and Y are
9
parts of Z, Y and Y are one in substance iff for some Z, Y is a part of Z.
14
Hence Y = Y iff for some
Z, Y is a part of Z. Thus, unlike material identity, which relates one part of a compound entity to
another such part, formal identity relates each part of such an entity to itself.
To sum up so far: If identity is oneness in substance, the ground of the cognitive difference
between "Hesperus = Phosphorus" and "Hesperus = Hesperus" is not a mode of presentation,
or an individual concept, or a criterion for a word's application, or an incomplete state of
affairs, or a context in which a reference is found in the world, or a function or mathematical
mapping of any kind. The ground of this difference is simply the circumstance in the world
which makes each assertion true: in the case of "Hesperus = Phosphorus", that there is some
compound entity of which Hesperus and Phosphorus are parts; and in the case of "Hesperus =
Hesperus", that there is some compound entity of which Hesperus is a part.
An ontological foundation is now taking shape for Aristotle's statement at 1018a79:
...la identidad de cualquier modo es una unidad, ya sea que la unidad se refiere a
pluralidad de cosas, ya sea que se refiera a una única cosa, considerada como dos, como
resulta cuando se dice que la cosa es idéntica a sí misma. (Ibid.)
As we will see, the road--from a foundation for Aristotle's statement at Metaphysics 1018a79,
to the shipwreck of Sense, to a solution for the problem of identity--the road, from the return
of extensions to the Garden of Meaning, to the twilight of Fregean semantics, to a recon-
ciliation of the theory of meaning and theory of reference--this road passes by way of the
particular-cum-complex.
1.6 New Foundations From Old
White's remarks about the "state of Aristotle's thinking in Metaphysics V" are meant to show
why Aristotle "is neglecting to give a clear account of the notion of identity":
14
Render "X and Y are one in substance just in case for some Z, X and Y are parts of Z" as:
(i).
xy(one(x,y)
z(part(x,z) & part(y,z))
Now (ii) follows from (i) by Universal Instantiation:
(ii)
y(one(y,y)
z(part(y,z) & part(y,z))
But (ii) is logically equivalent to (iii).
(iii)
y(one(y,y)
z(part(y,z))
Therefore, if X and Y are one in substance just in case for some Z, X and Y are parts of Z, then Y
and Y are one in substance just in case for some Z, Y is a part of Z.
10
Rather than thinking about what it is for X and Y to be identical, he has his mind fixed on
what it is for an entity to be unitary. . . This discussion drags the treatment of sameness
along on its coattails. (ibid., 188)
Nevertheless, an account of the notion of identity which specifies what it is for X and Y to be
identical need not, as White suggests, dispense with an Aristotelian notion of sameness as
oneness in substance. If particular and complex are one and the same entity, not only are the
usual properties of identity engendered by the mutual relations of complexes and their
constituents, but these relations also provide a foundation--as evidenced by the theorems of
Theory P which follow--for the Aristotelian notion of sameness as oneness in substance.
T1a,b and T2a,b specify identity-conditions for the constituents of complexes. T1a spells out
when individuals are materially identical; and T1b, when individuals are self-identical.
T1a
xy(x = y
wz(w.Z cont x & w.Z cont y))
(Individuals are materially identical just in case there is some complex of which they are
parts.)
T1b
x(x = x
wz(w.Z cont x))
(An individual is self-identical just in case there is some complex of which it is a part.)
T2a,b do the same for haecceities, T2a spelling out when haecceities are materially identical;
and T2b, when haecceities are self-identical.
T2a
xy(X = Y
wz(w.Z emb X & w.Z emb Y))
(Haecceities are materially identical just in case there is some complex of which they are
parts.)
T2b
x(X = X
wz(w.Z emb X))
(A haecceity is self-identical just in case there is some complex of which it is a part.)
The "b" member of each theorem-pair is a corollary of its "a" counterpart. The material
identity of the individuals x and y is thus sufficient for the self-identity of x and y, as is the
material identity of the haecceities X and Y for the self-identity of X and Y.
T3a,b,c,d specify identity conditions for complexes (a complex being, arguably, an Aristotelian
tode ti in modern garb). T3a spells out when complexes are materially identical; T3b, when a
11
complex is self-identical; T3c, when C-complexes are materially identical--a C-complex being a
complex whose constituents correspond; and T3d, when a C-complex is self-identical.
T3a
uvwx(u.V = w.X
yz(y.Z cont u & y.Z cont w & y.Z emb V & y.Z emb X))
(Complexes are materially identical just in case there is some complex of which their con-
stituents are parts.)
T3b
xy(x.Y = x.Y
wz(w.Z cont x & w.Z emb Y))
(A complex is self-identical just in case there is some complex of which its constituents
are parts.)
T3c
xy(x.X = y.Y
wz(w.Z cont x & w.Z cont y & w.z emb X & w.Z emb Y))
(C-complexes are materially identical just in case there is some complex of which their
constituents are parts.)
T3d
x(x.X = x.X
wz(w.Z cont x & w.Z emb X))
(A C-complex is self-identical just in case there is some complex of which its constituents
are parts.)
T3b is a corollary of T3a, as is T3d a corollary of T3c. For complexes as well as their
constituents, material identity is thus a sufficient condition for self-identity.
1.7 Bringing It All Home
It's time to give Mephisto his due. He demands to be told what all those symbols, and the
particular-cum-complex, have to do with Frege's Principle and the problem of identity.
Now, Frege's Principle says that the meaning of a sentence remains constant when a singular
term is substituted for another with the same meaning. If Frege were right, "Hesperus" couldn't
possibly mean Hesperus and "Phosphorus" couldn't possibly mean Phosphorus. Granted that
Hesperus = Phosphorus, if "Hesperus" meant Hesperus and "Phosphorus" meant Phosphorus,
"Hesperus = Phosphorus" and "Hesperus = Hesperus" would have to mean the same thing. But
they don't. Therefore, either "Hesperus" doesn't mean Hesperus and "Phosphorus" doesn't
mean Phosphorus--or Frege got it wrong, and the meaning of a sentence does not remain
constant when a singular term is substituted for another with the same meaning.
Frege got it wrong. Let me say why.
12
Symbolize Hesperus by "H" and Phosphorus by "P". The identity of Hesperus and Hesperus
then comes to:
(3) H = H
and the identity of Hesperus and Phosphorus to:
(4) H = P
By hypothesis, Hesperus and Phosphorus are complexes--C-com-plexes, to be exact. As a
result, (3) and (4) go over into assertions about C-complexes, thus:
(5) h.H = h.H
(6) h.H = p.P
Now, T3c says that C-complexes are identical just in case there is some complex of which their
constituents are parts--so that by being parts of that compound entity these are one in
substance. By hypothesis, Hesperus and Phosphorus are C-complexes. Consequently, the
identity of Hesperus and Phosphorus involves the oneness in substance of the constituents of
both Hesperus and Phosphorus. Not so the self-identity of Hesperus--for, as T3d suggests,
Hesperus's self-identity depends, not upon the oneness in substance of the constituents of both
Hesperus and Phosphorus, but upon the oneness in substance of the constituents of Hesperus
alone. Therefore, since "Hesperus = Phosphorus" and "Hesperus = Hesperus" differ in what
they assert, "Hesperus = Phosphorus" and "Hesperus = Hesperus" differ in meaning, even
though Hesperus and Phosphorus are one and the same.
15
1.8 Dead Meat
Voice. Thomas's parting comment about Sense reflected the sorry state of official
wisdom-seeking in that domain. Although a tumble with Sense racked up more points for an
academic fast-tracker than a pit stop with Phlogiston or "Fido"-Fido, the contribution to a
solution of the enigma of Sense of a century's worth of tumbling had been vanishingly small.
Indeed, the sum total of what any wisdom-seeker might stand and knowingly deliver about
Sense, despite a bloat of sclerotic monographs, journal articles and lecture notes on that and
related topics, was epitomized in Thomas's parting words to Mephisto: "A sense is whatever it
is about 'a' and 'b' that makes 'a = a' and 'a = b' differ in cognitive value."
15
The solution to the so-called "Paradox of Analysis" proceeds along similar lines.
13
Why the sorry state of Sense? Was there no more to Sense than the self-indulgent and
pretentious set-theoretic ersatzism with which Thomas and his Frege-worshipping colleagues
sought to stave off the downfall of the regnant ontology? Was there no more to Sense than the
accumulatory zeal of investors in Frege Futures over at the Oxford and Cambridge presses? For
somebody with no interest in the sociology of knowledge, was Sense anything more than dead
meat?
Thomas. DEAD MEAT?
Voice. Philosophically speaking.
Thomas. Petulant. How would YOU know?
Voice. Here. Take a whiff. Thomas holds nose. Now look. Retching sound.
Thomas, muffled. THAT kind of dead is a matter for philosophical reflection. Louder.
Worms propose. Wisdom-lovers dispose. Out of control. Pull the plug on the Voice! Shudders.
Ferchrissake, did you see those fucking WORMS?
16
16
"Desideria: The Voice explained to me that the barbarians, being pagans or else forming part
of some heretical sect, did not hesitate to desecrate churches or other places dedicated to
religious observance. According to the Voice, this way of acting on the part of the barbarians
could be described as desecratory precisely because the places that they desecrated were sacred.
But what did it really mean--desecratory? It meant that the barbarians with their devastations did
not so much destroy churches as despoil them, once and for all, of their sacred character. Before
the desecration, the church was a place which one entered bareheaded, in a state of reverence,
walking slowly and speaking in a low voice; after the desecration it was nothing more than a
warehouse, a big shed, in fact a structure possibly intact but devoid of any sacred character."
Alberto Moravia, Time of Desecration (Playboy Paperbacks, 1981), p. 110.
14
APPENDIX ONE
Modus Ponens, Modus Tollens
17
His advanced logic class does not share Thomas's conviction that the notions intuitively valid
18
and valid in all set-theoretic structures are extensionally equivalent. After today's lecture they
will. Or else.
Addresses class. An argument due to Georg Kreisel (19 69, 1971) shows that the notions
intuitively valid and valid in all set-theoretic structures are extensionally equivalent. Goes to
board. Kreisel notes that
(I) D(A)
19
Val(A)
20
(II) Val(A)
V(A)
21
(III) V(A)
D(A)
jointly establish
(*) (V(A)
D(A)) & (D(A)
Val(A)).
A corollary of Kreisel's argument is that intuitively valid and valid in all set-theoretic structures
have the same extension.
(**) V(A)
Val(A))
(Mephisto emerges from behind podium.)
Mephisto. Wait a minute. Not so durn fast!
Thomas. YOU again!
Mephisto. You claim intuitively valid and valid in all set-theoretic structures have the
same extension.
17
"one philosopher's modus ponens is another philosopher's modus tollens." (Hilary Putnam,
'Realism Without Absolutes', International Journal for Philosophical Studies, Vol 1(2), p. 180).
18
That is, valid in every conceivable circumstance--in every possible state of things.
19
"A" ranges over formulas of first-order predicate logic with identity.
20
"D(A)" means "A is formally derivable", and "Val(A)" means "A is intuitively valid".
21
"V(A)" means "A is valid in all set-theoretic structures".
15
Thomas. Kreisel's argument shows that.
Mephisto. No, it doesn't. All it shows is what follows from what. Goes to board. Take
the negation of (**) as a premise, and you'll see what I mean.
(**)' ¬(V(A)
Val(A))
From the negation of (**), it follows that if (II) holds, either (I) or (III) does not:
(**)' _ (Val(A)
V(A))
(¬(D(A)
(Val(A)) v ¬(V(A)
D(A)))
But (II) is vouchsafed by the meaning of its terms. So if intuitively valid and valid in all set-
theoretic structures do not have the same extension, either (I) or (III) is false.
(**)' _ ¬(D(A)
Val(A)) v ¬(V(A)
D(A))
But (III) "is precisely the mathematical content of Gödel's completeness theorem".
22
Hence if
intuitively valid and valid in all set-theoretic structures are not extensionally equivalent, first-
order predicate logic with identity is either unsound or incomplete. Kreisel's argument thus
does not establish the extensional equivalence of intuitively valid and valid in all set-theoretic
structures. What Kreisel's argument does show is that if Gödel's completeness theorem holds,
the extensional equivalence of intuitively valid and valid in all set-theoretic structures is a sine
qua non for the soundness of first-order predicate logic with identity.
V(A)
D(A) _ (D(A)
Val(A))
(V(A)
Val(A))
Thomas snorts.
Mephisto. Thomas?
Thomas. Yes.
Mephisto. Is The Law of Contraposition a crock?
22
Kreisel (1971), p. 254
REFERENCES
A.C. Anderson and J Owens (eds.) [1990], Propositional Attitudes, CSLI, Stanford, 1990.
K.S. Donnellan [1989, 1990], "Belief and the Identity of Reference," First published
in Midwest Studies in Philosophy. Reprinted in Anderson et. al. [1990], pp. 201-214.
J. Etchemendy [1990], The Concept of Logical Consequence, Harvard University Press,
Cambridge, 1990.
G. Frege [1892, 1992], "On Sense and Reference," From Translations From the
Philosophical Writings of Gottlob Frege, P. Geach and M. Black (eds.), Reprinted in
Moore [1992], pp. 23-42.
W. J. Greenberg [1982], Aspects of a Theory of Singular Reference, UCLA Ph.D.
Dissertation (published in the Garland Series Outstanding Dissertations in Linguistics,
1985).
G. Kreisel [1969], "Informal Rigour and Completeness Proofs," in The Philosophy of
Mathematics, J. Hintikka (ed.), Oxford University Press, London, 78-94.
G. Kreisel [1971], "Mathematical Logic: what has it done for the philosophy of
mathematics?" R. Schoenman (ed.), Bertrand Russell: Philosopher of the Century, Allen
& Unwin, London, 1971.
A.W. Moore (ed) [1992], Meaning and Reference, Oxford University Press, New York,
1992.
J. van Heijenoort [1985], Collected Papers, Bibliopolis, Napoli, 1985.
N. White [1971], "Aristotle on Sameness and Oneness," Philosophical Review 80 (1971),
2 pp. 177-197.
