What Achilles Did and the Tortoise
Wouldn’t
Catherine Legg
Agenda
Background: Semantics  Metaphysics in Contemporary
Analytic Philosophy
1.


Truth: Metaphysical Realism vs. Conventionalism
Validity - ?
Carroll’s Fable of Achilles and the Tortoise
2.
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

What does this fable teach us?
How might the puzzle be solved?
Logical action: the “Hardness of the Iconic Must”
Peirce’s Sign Theory
3.


Icon, Index, Symbol
Icons share properties with their objects: partial identity
What Peirce’s Sign Theory Has to Say re. Carroll’s Fable
4.

Partial identity between sign and required inference act
Semantics-Driven Metaphysics: A Reprise
5.

Beyond Metaphysical Realism and Conventionalism
1. Background: Semantics  Metaphysics in
Contemporary Analytic Philosophy
Representationalist model of language: the primary purpose of language
is to state “facts” in mind-independent reality.
This naturally leads to a certain metaphysical realism:
Account of real objects
No account of interpretation
Term in
language, e.g.
‘”cat”
DENOTES,
REFERS TO
Thing in world
Dyadic structure
Proposition in
language, e.g.
“The cat is on the
mat”
STATES
Fact in world
 But what makes it the case that a proposition is ‘fact-stating’?
 Early (strict) logical positivism: statements have “literal
significance” by virtue of offering empirical hypotheses:
 Thus “the cat is on the mat” is literally significant because an
appropriately cat-on-mat type experience might be had in relevant
situations.
 However crisp criteria for genuinely empirical hypotheses were
rather harder to find than logical positivists initially supposed.
 Quine (more purely semantic): a proposition is fact-stating
if its bound variables have values:
 Thus “the cat is on the mat” is literally significant because in x(Cx
& Oxm), suitably interpreted, x binds to George.
George
 Many philosophers find this view of language unsatisfying.
 This relation of ‘denoting’ or ‘stating’ seems too abstract,
‘supernatural’. (Putnam: ‘noetic rays’ between words and world)
 It is tempting to deny the reference, producing a conventionalism,
according to which terms do not denote, but have other socially
sanctioned and taught functions:
“John Key is the Prime
Minister of New
Zealand.”
“That is not a very
funny joke”
“Your RAE score is…..”
Account of interpretation
No account of real objects
 Debates between metaphysical realism





and conventionalism are usually fought
out with respect to truth.
But we don’t just use language to state
facts. We also use language to reason.
The study of valid reasoning is called
logic.
What (if any) metaphysics underlies this
logic?
Representationalists often talk of truthmakers for our truths: entities whose very
existence somehow makes truths true.
Are there ‘validity-makers’? If so, what
could they be?
2. Lewis Carroll (Charles Lutwidge Dodgson)
 27 January 1832 – 14 January 1898
 English author, mathematician,
Anglican deacon and
photographer.
 His most famous writings are Alice's
Adventures inWonderland and its sequel
Through the Looking-Glass, as well as
the poems: “The Hunting of the
Snark” and “Jabberwocky”.
 But he was also, of course, an
extremely talented logician.
Carroll’s Puzzle of Achilles and the Tortoise
 The great warrior and his friend with a shell together
examine a clearly valid argument. Something like this:
ARG) Socrates is a human being.
PREMISE: P1
All human beings can be killed. PREMISE: P2
Therefore Socrates can be killed. CONCLUSION: C
 Most of us* see a pattern in ARG) which makes us inclined




to do something.
What?
If we believe P1 and P2 are true, infer that C is true too.
But how does this work? What makes it happen?
And what do we mean by make?
* English speakers who are rational
ARG) Socrates is a human being.
PREMISE: P1
All human beings can be killed. PREMISE: P2
Therefore Socrates can be killed. CONCLUSION: C
This looks good on the page.
I do believe P1 and P2.
But I don’t see why I have to believe C.
Please explain
?! You’re kidding me.
What do you mean,
you can’t see why? It’s
totally obvious!
No actually, it’s really not obvious to me.
I’m a tortoise so I’m a little slow.
Please explain it to me.
Good grief!
Well it happens to be objectively true that:
If P1 and P2 are both true, then C is
true also.
So we can add this to the argument.
ARG) Socrates is a human being.
All human beings can be killed.
If P1 and P2 are true then C is also.
Therefore Socrates can be killed.
Um, well, no actually.
I accept P1 and P2, and P3.
But I still don’t see why all of that means I have to
accept C.
I’m a tortoise so I’m a little slow. Please explain it
to me.
PREMISE: P1
PREMISE: P2
PREMISE: P3
CONCLUSION: C
Can you see it now?
You’re kidding me!!!
Tortoise
No I’m not, honestly
It looks as though something else needs to be
added to the argument…
Achilles
ARG) Socrates is a human being.
PREMISE: P1
All human beings can be killed. PREMISE: P2
If P1 and P2 are true then C is also. PREMISE: P3
If P1, P2 and P3 are true then C is also. PREMISE: P4
Therefore Socrates can be killed. CONCLUSION: C
Can you see it now?
Um…..
Tortoise
…to be continued.
(What should Achilles have
said to the Tortoise?)
Achilles
What does this fable teach us?
 It adroitly highlights both the existence and the puzzlement
of a certain feature of our necessary reasoning which
representationalism encourages us to miss: its link to action.
 The fable exposes a bindingness on the actions of rational
agents (specifically their inferencing) which appears a-causal,
yet nonetheless intriguingly compelling.
 Here one glimpses an internalism about logic, whose analogy
to metaethics is arguably no accident. If one is not motivated
to act by logical norms, it seems that one does not fully
understand them.
 Via his recalcitrance the Tortoise usefully makes the
bindingness of valid argument on action  normally invisible
due to its ubiquity  visible.
How might the puzzle be solved?
 The disparity in understanding between Achilles and the





Tortoise cannot be remedied by further explanatory signs. For as
Achilles’ extra conditionals tend to infinity, no progress is
made.
But if the Tortoise is not lacking further signs in his failure to
grasp the relationship between P1 and P2 and C, what is he
lacking? He is surely lacking something!
He is failing to see a structural isomorphism shared by the original
sign (ARG)) and an act (inferring C from P1 and P2).
A structural isomorphism between a written sign and an act
might seem to be a curious idea.
And how can we explain Achilles’ grasp of it?
Arguably, by recourse to a kind of sign which current
representationalist semantics overlooks.
Charles Sanders Peirce
 Sep. 10, 1839 – April 19, 1914
 American philosopher, logician,
mathematician, scientist…
 In 1934, the philosopher Paul Weiss
called Peirce: “the most original and
versatile of American philosophers
and America's greatest logician”.
 Peirce placed logic within the
broader context of a theory of
signs, or semiotics.
1ns
3ns
2ns
Three Peircean categories…
…based on 3 valencies:
monad
triad
dyad
1
unit
(in itself)
REPRESENTAMEN
e.g. “cat”
3
medium
(bringing into
INTERPRETANT
relation)
Uses of “cat” to
correlate
2 (to another)
OBJECT
refer to cats
Account of real objects
AND account of interpretation
The structure of the sign
1(icon)
Picks out its object
by resembling it.
E.g. map of NZ
3
2
index
Picks out its object by brute
denotation (direct reference)
E.g. “Gordon Brown”
What does this kind of
signification best suit?
symbol
Picks out its object by
arbitrary convention
E.g. “train”
This kind of signification best suits general
terms. Conventionalism arguably treats all
signification on this model
This kind of signification best suits discrete
particular objects. Metaphysical realism
arguably treats all signification on this model
Three kinds of sign (Mixed forms are most usual)
An icon and its object share a property (thus
partaking in partial identity)
 Since indices serve as pure pointers, whatever internal
properties they have are irrelevant to their signification.
 Symbols have nothing in common with their objects by definition
since their establishing convention is arbitrary.
 But as icons represent their objects by resembling them, the basis
of that resemblance must be some shared property.
 Thus Peirce wrote:
“…a pure icon does not draw any distinction between itself and its object…
whatever it is like, it in so far is.”
 This claim is made within a broad framework of scholastic realism
(realism about universals)
ARG) Socrates is a human being.
PREMISE: P1
All human beings can be killed. PREMISE: P2
Therefore Socrates can be killed. CONCLUSION: C
 How do we apprehend the validity of this argument?
 Plausibly, by drawing a kind of diagram in our heads,
something like this:
So he can’t
be out here
All things that can
be killed
All humans
Socrates is in
here
 This diagram is an iconic sign insofar as the parts of an
icon, in Peirce’s words “…bear the same relationship to
one another as the parts of the object they represent”.
 The object in this case is logical form.
4. What Peirce’s Sign Theory has to say about
Carroll’s Puzzle
 Written signs such as ARG) represent logical form to the human
mind so that it can be understood and acted upon. And what more
could we ask to say that a system of signs has genuine content?
 Yet ARG) does not state logical form in any way understood by
contemporary mainstream philosophy.
 ARG) does not denote logical form in a metaphysical realist
sense. There are no further objects (validity-makers) picked out,
over and above the objects picked out by ARG)’s individual
sentences (i.e. Socrates, humans, dead people).
 ARG) does not fit the conventionalist picture either, though,
because logical form does not rest on, and cannot be altered by,
arbitrary convention. Its necessity is directly seen. (I call this “the
hardness of the iconic must”.)
 And insofar as the implications for action are isomorphic with the
written sign, these are directly apprehended also.
 Thus there is an important sense in which the Tortoise fails to read
the argument at all. He reads and accepts P1, P2 and C, but he fails
to see the structure which upon acceptance of P1 and P2 compels
recognition that C must follow. And it is this (iconic) structure that
makes the sign ARG) what it is.
 Thus, further, what explains the intriguing bindingness between sign
and inference  which Achilles grasps but cannot explain to the
Tortoise  is identity.
 What else could explain it?

D
5. Semantics-Driven Metaphysics: A Reprise
 We may understand Quine’s ‘bound variable’ criterion of
ontological commitment in Peircean terms as an attempt to
place the full burden of representation on indexical signs.
 This leads realist philosophers to ask questions such as: “Does
term X denote an existent object?”, and it often seems hard to
answer “yes” for key terms in manifestly important human
discourses [e.g. “the Good”, “God”….]
 On the other hand, we have seen that those unsatisfied with
metaphysical realism’s problematization of such terms often
counter with a conventionalism arguing that the term does
not denote but has some other socially sanctioned and taught
function.
 We may see such conventionalism in Peircean terms as trying to
understand all representation as symbolic.
 Metaphysical realism and conventionalism are too often

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taken for polar opposites in philosophy: an argument against
metaphysical realism is frequently assumed without question to
be an argument for conventionalism, & vice versa.
This dichotomy is false.
A third kind of representation exists, which does not consist in
brute denotation or arbitrary convention, but presents structure
directly to the mind’s eye. It is iconic
It is barely glimpsed in formal semantics today. Yet it is this kind
of sign that represents logical form – hardly a trivial part of our
conceptual scheme.
If we could only recognize that the symbol, index and icon
all have a unique, irreducible semantic role, and reality
correspondingly comprises real habits, real particulars and
real structures, we could take an unanticipated leap towards
understanding this most contested term.
Thank you!
clegg@waikato.ac.nz