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on sentences, and it deserves very careful study. The sentence model is
so deeply ingrained both in our everyday and our philosophical
thought about thought that we will do well to understand it very ex-
plicitly. Otherwise it may mislead us in important ways. A way to begin
to explore this model is to examine it in its most naked form, namely,
that in which discursive thinking is analogized to the unfolding of a
formal system.


A formal system is usually laid out in the following way. First you say
what elementary symbols will be used: ps and qs, say, or As and Bs, and
xs and ys, wedges, horseshoes, parentheses, and so forth. Next you ex-
plain how to construct well-formed formulas (WFFs) from these ingre-
dients, typically with the aid of recursion.Then you may (but need not)
lay down axioms. And last, you lay down rules (this in the metalanguage,

of course) that will move you from WFFs already laid down or derived
to new WFFs. This laying down of symbols, well-formedness rules, ax-
ioms, and inference rules, is traditionally done by displaying tokens of
symbols, using these as examples of the types of symbols to be put down
and manipulated in accordance with the system™s rules.
But how are other tokens of these same symbol types to be recog-
nized? By what criterion will they be tokens of the same types? Typi-
cally, nothing is said about this. One supposes, traditionally, that some
understood but unmentioned parameter on sameness of shape is what
binds the tokens into types. For our purposes it will be important ex-
plicitly to recognize this implicitly designated part of a formal system.
Formal systems have, I will say, besides (1) basic WFFs and rules for con-
structing more WFFs, (2) axioms or postulates, and (3) rules of infer-
ence, also (4) “symbol-typing rules” or just “typing rules,” that is, im-
plicit rules telling what is to count as the same symbol or WFF again.
It is well known that in developing formal systems, rules can in gen-
eral be substituted for axioms or postulates and also, axioms or postu-
lates can be substituted for rules, though the latter (since the tortoise™s
historic conversation with Achilles “ Carroll 1894), it is supposed, not
without residue. Typing rules have generally been ignored. But in fact,
as I will illustrate, typing rules also can sometimes replace axioms or in-
ference rules.
A very simple interaction among axioms, inference rules, and typing
rules happens to be the interaction of the axiom, say, “B = B,” with the
two inference rules “replace . . . B . . . with . . . B . . . ” and “replace . . .
B . . . with . . . B . . . ,” and with the typing rule “B and B count as
symbols of the same type.” Now it is usually supposed that although
one can trade off rules and axioms to a degree in this way, still the dif-
ference between the systems that result from these swaps is objective.
Certainly, it is not so that there is really no difference between an ax-
iom and a rule or no difference between a rule of inference and a typ-
ing rule. It is just that in some cases one can substitute one for another
to determine the same set of theorems.What I will be urging, however,
is that distinctions among these three categories break down in crucial
ways when we hypothesize representation in the mind or brain. My
conclusion will be that there is no difference in this context between a
mental equals-sign marker, an identity rule and a duplicates marker.
To quell the suspicion that identity is somehow a special case here,
let me first illustrate the trade off between typing rules and axioms
and/or inference rules with a completely different kind of example.

Consider, first, laws of commutativity. Standard renderings of the propo-
sitional calculus require that the equivalence of A&B to B&A and of
A∨B to B∨A be either introduced as axioms, derived as theorems, or
(aberrant but possible) given as special rules of inference. Suppose, how-
ever, that one were to construct a system in which the difference be-
tween left and right on the paper is ignored when grouping symbol to-
kens into types. No distinction is drawn between “p” and “q,” or
between “b” and “d,” and so forth. Similarly, “A⊃H” is the same string
as “H‚A.” More interesting, “A∨H” is the same string as “H∨A,” and, if
we use the traditional dot instead of “&,” “A·H” is the same string as
“H·A.” Here a symbol typing rule does duty for one or a couple of ax-
ioms, theorems, or rules.
For another example, suppose we read right-left distinctions as usual
and play instead with up and down. We read “p” as a symbol of the
same type as “b” except that it has been turned upside down. Then we
use turning upside down for the negation transformation. We negate
propositional constants and variables by turning them upside down; we
negate strings by turning the whole string upside down. Double nega-
tion elimination now no longer appears as an axiom, theorem, or rule.
It can™t be stated, or can™t be differentiated from p implies p. Of course,
we have to be terribly careful. We must not use any symbols that are
symmetrical top to bottom, or we won™t be able to tell whether they
have undergone the negation transformation or not. On the other
hand, suppose we use the traditional symbol “§” for conjunction. The
effect is that De Morgan™s laws need not be stated, are indeed unstatable,
being mere fallout from the symbol-typing rules. For example, suppose
that you turn “p” and “q” each upside down, put a wedge between
them, and then turn the whole string over, thus saying that it is neither
the case that not-p nor that not-q. The result, “p§q”, is a string that is
more naturally read straight off as saying, simply, that p and q. The pos-
sibility of swapping symbol-typing rules for axioms or rules of inference
is not then an artifact resulting merely from the peculiarities of identity.
Next, I would like to argue that when we turn from a representa-
tional system written on paper in a public language to a representational
system in the mind or brain, both the distinction between axioms and
inference rules and the distinction between typing rules and inference
rules tend to break down.
Consider first the apparently nearly self-evident truth that although
it is possible to build a logical system that has no axioms but only rules,
it is not possible to build a system with no rules but only axioms. The

rules of a system cannot all be represented explicitly (Carroll 1894). In
a traditional formal system, each axiom or hypothesis is written down
on a separate line of paper.The system unfolds as new sentences or for-
mulae are derived from these by rule and written down below. In such
a system, adding axioms “ writing down more sentences at the top of
the page “ won™t by itself determine how these axioms will be used in
order to guide derivations within the system. Rules telling how the ax-
ioms, as well as the other strings, are to be manipulated must be given
in a metalanguage which the system builder reads and, if appropriately
inspired by them, then performs the appropriate transformations.
But if the system is unfolding not on paper but in the head, the
“user” is just another part of the head. Since representations in the head
are just head-structures designed to vary according to how the world
varies, this user part may itself constitute a representation. Taking a
childish example, imagine an inference machine designed to perform
inferences using universal categorical sentences as major premises and
constructed in the following manner. Premises representing that All As
are Bs are entered by constructing a sliding board between two ports,
the top port being an A-shaped hole, the bottom a B-shaped hole. Sen-
tences ascribing predicates to individuals consist of pieces of putty the
colors of which name individuals and the shapes of which (“A,” “B,”
“C”) ascribe properties to them. These pieces of putty are gently
pushed across the tops of the constructed slides, where they enter the
ports through which they fit, proceeding to the bottom where they are
pressed down through the lower ports and change their shapes accord-
ingly. They thus become conclusions. All As are Bs and a is an A thus
yields a is a B.
Surely this image could be improved on, but the principle should be
clear.There is no reason why one premise has to lie passively beside the
next in a representational system in the mind or head. There is no rea-
son why the structure, the mechanism, that operates upon a representa-
tion during inference may not itself be a representation that has been
molded or tuned to perform its appointed tasks, reflecting something in
the dynamics, uniformities, or logic of some aspect of the environment,
for which aspect it stands.Ways that various individuals™ inferencing sys-
tems are put together can themselves be representations, so long as they
are determined by learning under the influence of individuals™ environ-
ments, such that variations in ways of being put together correspond,
systematically, to variations in environments, according to the design of
the learning systems.

If the difference between rules of inference and premises of inference
in a cognitive system is not clearly marked, the difference between typ-
ing rules and identity axioms or rules for a cognitive system is altogether
chimerical. To see exactly why this is so, two more failures of parallel
between formal systems and the way representations are used by cogni-
tive systems need to be recognized. One concerns the conventional and
public nature of symbol typing in formal systems. The other concerns
the use of duplicates as the sameness markers in formal systems.
Earlier I remarked that, typically, nothing is explicitly said about the
symbol typing rules when a formal system is laid down, but that it is as-
sumed that these unspoken rules concern parameters or limits on vari-
ation in physical form. Better, we pretend that these unspoken rules con-
cern sameness, that is, concern duplication, of physical form. The rules
we actually employ are derived, at the start, from irregular rather dis-
junctive conventions for determining what is the same symbol type.
There are, for example, numerous conventional styles of writing the
same letter by hand, and numerous type fonts, across which the shape of
a particular letter may vary in a rather irregular way. Compare this, for
example, with the typing of words, where practices can be quite disor-
derly. The rules may be quite disjunctive and may include many excep-
tions. For example, in English, contrasting pronunciations of “schedule”
(s-k-e-dule versus sh-e-dule) count as tokens of the same word type
while exactly the same contrast between the pronunciations of “mask”
and “mash” or “skin” and “shin” produces different word types. More-
over, the methods we use in practice to determine what counts as an-
other token of the same symbol type concern not just shape but how
the person who wrote the symbol intended it to be taken. Recall that
most actual formal systems are originally developed in somebody™s
handwriting on paper and that a typical way of passing them on is by
writing on a blackboard.What counts as an “a” or as an “˜” then is what
was intended to be an “a” or a “˜,” having been purposefully copied,
carefully or carelessly, competently or incompetently, with or without
consciously added style, on the model of earlier “a”s or “˜”s.2 Why do
we keep up this pretense that formal symbols are typed by nondisjunc-
tive exceptionless rules on shape?
There is one innocent reason and, I believe, one lingering guilty rea-
son why we do this. The innocent reason is that proofs of consistency

2 For fuller discussion of this etiological principle in grouping words and other symbols into
types, see (Millikan 1984, Chapter 4, in press c; Kaplan 1990).

and completeness in formal systems work, exactly, by treating systems as
if their symbols and WFFs corresponded to well-defined simple physi-
cal shapes and well-defined configurations of these. This assumption
makes the proofs easy, indeed, possible, and nothing is distorted thereby.
The guilty reason is that it is implicitly assumed that the only true
marker of sameness in content, the only way sameness of content can
be directly represented, is by duplicating representations. That is, the pas-
sive picture theory of the act of identifying hovers over. Or it may be
assumed that identifying is reacting the same way to what is identified,
so that the same physical form will be needed to produce the same re-
action again.
But ease in proving consistency and completeness is clearly irrelevant
to how cognitive systems work. And the passive picture theory, and the
repetition theory of identifying, I have argued, are mistaken. If we keep
clear on these issues it becomes evident that there is nothing different in
principle going on in a mind that uses “sameness of form” to mark iden-
tity if that mind happens to define equivalence classes for these forms dis-
junctively, or with numerous exceptions, dividing up its space of forms in
as gerrymandered a way as you please. All that is needed is that the typ-
ing rules used remain consistent with one another. Should the brain
mark sameness by equivalence classes of physical type, there is no distinc-
tion in principle between systems that mark neatly by perfect duplication
of some aspect of form, and systems that mark messily with lists and dis-
junctions and exceptions.There is nothing magical about simple nondis-
junctive typing rules where classes of physical forms mark samenesses.
Now in a public symbol system, which similarities determine that two
symbol tokens are of the same type depends on the conventional prac-
tices of the language community.That “defence” is read as the same word
as “defense” and also as the word “DEFENSE,” for example, is a matter of
public convention. That “Cicero” is not read as the same word as
“Tully” is equally a matter of public convention. Certain physical forms
and not others are grouped into the same representational type because
someone, in this case the general public, reads them that way. And if an
individual user of the public language comes along who happens to, or
learns to, read “Cicero” and “Tully” as equivalent in type “ these pro-
duce thoughts of exactly the same type “ that doesn™t change the con-
ventional typing rules for words used by the community. It doesn™t
change the typing rules for the public language.
But for mental representations, there is no distinction like that be-
tween public convention and private response. Whatever the individual

mind/brain treats as the same mental word again IS the same word
again. For mind-language there are no conventions “ there is only the
private user. Nor is there any reason why mental typing should not
evolve in an individual mind or brain over time. If the private user
changes her habits, then the typing rules for her mental representations
will change. This is because the typing rules ARE nothing but her dis-
positions to coidentify.
For the mind, there also is no distinction like that between an iden-
tity axiom or postulate, A = B, written at the top of the page, and a
typing rule. For there is no distinction like that between what is writ-
ten on the paper and what is written in the structure of the reader “ in
the structure responsible for conforming the reader™s reactions to a cer-
tain typing rule. One structure responsible for brain coidentifying pat-
terns is on a par with any other; all are equally “written” in the brain.
Write an identity sentence, that is, a structure responsible for producing
certain coidentifications, in neuronal patterns instead of in graphite,
and the distinction between identity sentence and interpreting mecha-
nism vanishes. Whether the mechanism in the mind effects only that
the mental Ciceros get coidentified with the mental Ciceros, or also that
the mental Ciceros get coidentified with the mental Tullys, this mecha-
nism is no more or less of an extra postulate one way than the other.
What is the alternative? That to be an identity postulate it would have,
literally, to be physically shaped like this: “Tully = Cicero”? Marking
sameness, however that™s done, and fixing identity beliefs is exactly the
same thing.
Thus if we think carefully about the effects of an equals marker on
the system that understands it, the distinction between it and a Straw-
son marker collapses. What effect are we to imagine mental Cicero =
Tully to have if not, precisely, that it changes the mind™s dispositions to
mental typing? Henceforth, mental Cicero and mental Tully will behave
as representational equals. They become the same mental word, that is,
they are ready to be coidentified. But if this is so, the mental equals
marker behaves exactly like a Strawson marker. It merges two thought
types into one, threatening equivocation in thought, and doling out to
each thinker just one mode of presentation per object.
We must conclude, I think, that the peculiar effect of the Strawson
markers was on us, on our understanding, not on the operation of the
cognitive systems modeled. Systems that use Strawson markers grasp
identities by explicitly changing their mental vocabularies, replacing two
representations with one. Systems that use equals markers do exactly the

same thing but implicitly, changing merely the typing rules for their
mental vocabularies, that is, merely the functions of the symbol forms.


One possibility concerning negative identity judgments and the undo-
ing of identity markings hasn™t been dealt with yet. Imagine a system
that keeps a log of the various changes made in the representational sys-
tem as identities are marked, and keeps a log of mediate inferences that
pivot on these marked representations. Compare the way modern word
processing programs can keep track of the last two or three hundred
commands carried out. Then if a mistake is discovered, the “undo” but-
ton can be pressed until the system is returned to the point where the
original false coidentification was made. Different pieces of information
that were attached to the same Strawson dot at different times then have
different statuses in case of emergency. Indeed, might we say that they
represent predicates attached to the same subject but under different
modes of presentation? How many of the various purposes of Frege™s
modes of presentation could differences of this sort serve?
I have not explored these questions because I think such a model is
completely lacking in psychological plausibility. Imagine keeping such a
log on all the times you have ever reidentified or made inferences about
your husband or mother! Of course it is true that were I seriously to sup-
pose, say, that Mark Twain was not Samuel Clemens, I might have some
idea how to guess which of my beliefs about this double person should be
attached to which name. Certain facts would cohere with Twain™s role as
an author, others perhaps with his role as public speaker or builder of the
Twain house in Hartford. But this untangling would certainly not be done
on the basis of a memory of when and in what order I had discovered or
inferred what about Twain. It would be done using a theory about how I
had got two men so mixed in my mind, and by speculating about which
items of information are most likely to have come from which source.


How then does the Fregean avoid the Strawson image with its threat of
equivocation in thought and its frugal offer of just one mode of pre-
sentation per object? The trick is to imagine that how a thought func-
tions has no effect on its content. One assumes that how the mind un-
derstands its thoughts is irrelevant to their significance. Throughout I
have assumed, on the contrary, that use does affect representational

value. I have assumed that what marks content sameness in thought is
whatever the cognitive systems read as marking sameness, or what they
are designed to read as marking sameness. I assumed that if thought to-
kens are marked to function as representing the same, this will affect
their representational value. In particular, if this marking conflicts with
other factors relevant to representational value, say, with the information
content of the tokens so marked, or with other ways their associated
referents may be determined, then there will be equivocation in con-
tent. Sameness is represented yet different things are represented.Visag-
ings of conceptual contents need not be consistent, nor is inconsistency
in conceptual content discovered by a priori inspection.
The Fregean view assumes, on the contrary, that insertion of a same-
ness marker (an identity judgment), hence change in the employment
of the marked terms, has no bearing on content. Placing a mental equals
sign between mental Cicero and mental Tully has no effect on the rep-
resentational value of either, even if Cicero is not in fact Tully. Similarly,
if duplicated thoughts are in fact thoughts of the same, each token of
Cicero referring again always to the same (rather than acting, say, like the
English word “he”), this depends in no way on the fact that duplication
is what is read by the mind as marking identity. Thought typing is de-
termined independently of thought use.
This Fregean assumption implies, I believe, that thoughts are not men-
tal representations. For we cannot suppose that a representation could be
a mental representation, a representation for mind, yet that its representa-
tional value was independent of its effect upon mind, independent of how
the mind reads it.And, of course, Frege himself did not hold that thoughts
are mental representations. Fregean senses are abstract entities that bear
their contents quite independently of whether or how a mind “grasps”
them.The conclusion that classical Fregean modes of presentation are not
compatible with a representational theory of mind is not then a criticism
of Frege. But if we propose to defend any sort of representational theory
of mind, we cannot also keep Fregean modes of presentation.


According to Frege there are informative and also uninformative iden-
tity claims. Uninformative identities are so called because they do not
inform us of anything not already immediately known a priori. Presum-
ably these claims also cannot be false. Frege is not supposing that there
might be false identities that we cannot help but affirm. The Fregean

senses that figure in uninformative identities function psychologically as
would thoughts marked with duplicates markers. But on Frege™s view
this way of functioning is not, of course, what determines that duplicate
graspings of duplicate Fregean senses always have the same content.
Function has no effect on content. On a representationalist view, same-
ness markings do force both marked thoughts to refer to the same
thing, hence if the markings are wrong, forces both to refer equivocally,
but on a Fregean view, the referents of duplicate thoughts are deter-
mined independently of the mind™s way of being governed by them.
That the thinker identifies the referents as one and the same is in no
way responsible for them being the same. What is the guarantee, then,
that the referents of duplicate thoughts actually ARE the same? (Or if
what you mean by “duplicate thoughts” includes that they have the same
referent, what is to guarantee that the mind that grasps two thoughts
can tell whether these thoughts are indeed duplicates?) How can there
be uninformative identities that are at the same time certain to be real
identities and not merely false appearances of identity?
This line of questioning highlights the internalist assumption built
into the Fregean position. What is duplicated when “the very same
thought” is repeated must be something that is simultaneously (1) com-
pelled always to bring with it the same referent and (2) capable of be-
ing unmistakably known by the mind, when the mind duplicates its en-
tertainment, as being the very same thought. That, I take it, is one role
of a Fregean sense: it always determines the same referent regardless of
the context, the grasper (understander), or the use, and its identity is
transparent to mind.3 More generally, that which completely determines
the referent must be exactly the selfsame as that which, when dupli-
cated, constitutes a grasp of sameness. Otherwise the appearance of
sameness might not be veridical.
It follows that whatever determines the reference must be entirely
internal to mind. Reference cannot be affected at all by, say, the exter-
nal causes of thoughts, or their natural informational content, for there
can be no certain internal or a priori mark proving the external causes

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