1
COMPUTING, PHILOSOPHY, AND COGNITIVE
SCIENCE
an Eternal
Information
Braid
INTRODUCTION
Gordana Dodig Crnkovic and Susan Stuart Editors
COMPUTING, INFORMATION AND PHILOSOPHY
Chaitin
,
Alan Turing Lecture on Computing and Philosophy
:
Epistemology as Info
rmation Theory: From Leibniz to
Ω
Floridi
INVITED Information Logic
Allo:
Formalising Semantic Information. Lessons
f
rom Logical Pluralism
Johansson LG
INVITED
:
Causation

a synthesis of three approaches
Bynum
Georg Henrik von Wright Lecture on Ethics
:
A Copernican Revolution in Ethics?
LiuGang
: INVITED
:
An Oriental Approach to the Philosophy of
Information
Trojer
INVITED: Building Epistemological Infrastructures

interventions at
a technical university
BIOINFORMATION AND BIOSEMIOTICS
Marijuan

Moral
:
The Informational Architectures
o
f Biological Complexity
Århem
:
A Neurophysiological Approach to Consciousness: Integrating
Molecular, Cellular and System Level Information
Brattico
:
Complexity, cognition, and logical depth
Munn:
Functions and Prototypes
Brier
:
The Cybersemiotic framewor
k as a means to conceptualize the
difference between computing and semiosis
COGNITIVE SCIENCE AND PHILOSOPHY
Magnani
INVITED
:
Building Mimetic Minds. From the Prehistoric Brains
to the Universal Machines
Carsetti
:
Meaning and self

organisation in cognit
ive science
Lappi
:
On Facing Up To The Semantic Challenge
Milkowski
:
Is computationalism trivial?
Pylkkanen
:
Does dynamical modelling explain time consciousness?
2
Riegler
:
The paradox of autonomy: The interaction between humans and
autonomous cognitive arti
facts
ONTOLOGY
Smith

Ceusters
,
Carl Linnaeus
Lecture on
Ontology
,
: Ontology as the
Core Discipline of Biomedical Informatics. Legacies of the Past and
Recommendations for the Future Direction of Research
Hagengruber

Riss
: Knowledge in Action
Turner

Ede
n
: Towards A Programming Language Ontology
COMPUTATIONAL LINGUISTIC
S
Kn
uttila
INVITED
:
Language Technological Models as Epistemic Artefacts:
The Case of Constraint Grammar Parser
Hacken
Computational Linguistics as an Applied Science
Hirst
:
Views of Tex
t

Meaning in Computational
Linguistics: Past, present,
and future
3
COMPUTING, INFORMATION AND PHILOSOPHY
Epistemology as Information Theory: From Leibniz to
Ω
G. Chaitin,
IBM Research
Alan Turing Lecture on Computing and Philosophy, E

CAP'05, European
Computing and Philosophy Conference, Mälardalen University, Västerås,
Sweden, June 2005. For a digital video of this lecture, click here:
Abstract
In 1686 in his
Discours de métaphysique,
Leibniz points out that if an
arbitrarily complex theory is permi
tted then the notion of "theory" becomes
vacuous because there is always a theory. This idea is developed in the
modern theory of algorithmic information, which deals with the size of
computer programs and provides a new view of Gödel's work on
incompleten
ess and Turing's work on uncomputability. Of particular interest
is the halting probability
Ω
, whose bits are irreducible, i.e., maximally
unknowable mathematical facts. More generally, these ideas constitute a kind
of "digital philosophy" related to recen
t attempts of Edward Fredkin, Stephen
Wolfram and others to view the world as a giant computer. There are also
connections with recent "digital physics" speculations that the universe might
actually be discrete, not continuous. This
système du monde
is pre
sented as a
coherent whole in my book
Meta Math!,
which will be published this fall.
Introduction
I am happy to be here with you enjoying the delicate Scandinavian summer;
if we were a little farther north there wouldn't be any darkness at all. And I
am
especially delighted to be here delivering the Alan Turing Lecture.
Turing's famous 1936 paper is an intellectual milestone that seems larger and
more important with every passing year.
[For Turing's original paper, with commentary, see Copeland's
The Ess
ential
Turing.
]
People are not merely content to enjoy the beautiful summers in the far north,
they also want and need
to understand
, and so they create myths. In this part
of the world those myths involve Thor and Odin and the other Norse gods. In
this t
alk, I'm going to present another myth, what the French call a
système du
4
monde,
a system of the world, a speculative metaphysics based on
information and the computer.
[One reader's reaction (GDC): "Grand unified theories may be like myths, but
surely th
ere is a difference between scientific theory and any other
narrative?" I would argue that a scientific narrative is more successful than
the Norse myths because it explains what it explains more precisely and
without having to postulate new gods all the t
ime, i.e., it's a better
"compression" (which will be my main point in this lecture; that's how you
measure how successful a theory is).]
The previous century had logical positivism and all that emphasis on the
philosophy of language, and completely shunn
ed speculative metaphysics,
but a number of us think that it is time to start again. There is an emerging
digital philosophy and digital physics, a new metaphysics associated with
names like Edward Fredkin and Stephen Wolfram and a handful of like

minded i
ndividuals, among whom I include myself. As far as I know the
terms "digital philosophy" and "digital physics" were actually invented by
Fredkin, and he has a large website with his papers and a draft of a book
about this. Stephen Wolfram attracted a great
deal of attention to the
movement and stirred up quite a bit of controversy with his very large and
idiosyncratic book on
A New Kind of Science.
And I have my own book on the subject, in which I've attempted to wrap
everything I know and care about into
a single package. It's a small book, and
amazingly enough it's going to be published by a major New York publisher
a few months from now. This talk will be an overview of my book, which
presents my own personal version of "digital philosophy," since each o
f us
who works in this area has a different vision of this tentative, emerging world
view. My book is called
Meta Math!
, which may not seem like a serious title,
but it's actually a book intended for my professional colleagues as well as for
the general pu
blic, the high

level, intellectual, thinking public.
"Digital philosophy" is actually a neo

Pythagorean vision of the world, it's
just a new version of that. According to Pythagoras, all is number

and by
number he means the positive integers, 1, 2, 3,
...

and God is a
mathematician. "Digital philosophy" updates this as follows: Now everything
is made out of 0/1 bits, everything is digital software, and God is a computer
programmer, not a mathematician! It will be interesting to see how well this
vi
sion of the world succeeds, and just how much of our experience and
theorizing can be included or shoe

horned within this new viewpoint.
5
[Of course, a system of the world can only work by omitting everything that
doesn't fit within its vision. The questio
n is how much will fail to fit, and
conversely, how many things will this vision be able to help us to understand.
Remember, if one is wearing rose colored glasses, everything seems pink.
And as Picasso said, theories are lies that help us to see the truth
. No theory is
perfect, and it will be interesting to see how far this digital vision of the
world will be able to go.]
Let me return now to Turing's famous 1936 paper. This paper is usually
remembered for inventing the programmable digital computer via a
mathematical model, the Turing machine, and for discovering the extremely
fundamental halting problem. Actually Turing's paper is called "On
computable numbers, with an application to the
Entscheidungsproblem,
" and
by computable numbers Turing means "real
" numbers, numbers like
e
or π =
3.1415926... that are measured with infinite precision, and that can be
computed with arbitrarily high precision, digit by digit without ever stopping,
on a computer.
Why do I think that Turing's paper "On computable numbe
rs" is so
important? Well, in my opinion it's a paper on epistemology, because we only
understand something if we can program it, as I will explain in more detail
later. And it's a paper on physics, because what we can actually compute
depends on the laws
of physics in our particular universe and distinguishes it
from other possible universes. And it's a paper on ontology, because it shows
that some real numbers are
uncomputable
, which I shall argue calls into
question their very existence, their mathematic
al and physical existence.
[You might exclaim (GDC), "You can't be saying that before Turing and the
computer no one understood anything; that can't be right!" My response to
this is that before Turing (and my theory) people could understand things, but
t
hey couldn't measure how well
they understood them. Now you can
measure that, in terms of the degree of compression that is achieved. I will
explain this later at the beginning of the section on computer epistemology.
Furthermore, programming something for
ces you to understand it better, it
forces you to really understand it, since you are explaining it
to a machine
.
That's sort of what happens when a student or a small child asks you what at
first you take to be a stupid question, and then you realize that
this question
has in fact done you the favor of forcing you to formulate your ideas more
clearly and perhaps even question some of your tacit assumptions.]
To show how strange uncomputable real numbers can be, let me give a
particularly illuminating exam
ple of one, which actually preceded Turing's
6
1936 paper. It's a very strange number that was invented in a 1927 paper by
the French mathematician Emile Borel. Borel's number is sort of an
anticipation, a partial anticipation, of Turing's 1936 paper, but th
at's only
something that one can realize in retrospect. Borel presages Turing, which
does not in any way lessen Turing's important contribution that so
dramatically and sharply clarified all these vague ideas.
[I learnt of Borel's number by reading Tasic'
s
Mathematics and the Roots of
Postmodern Thought,
which also deals with many of the issues discussed
here.]
Borel was interested in "constructive" mathematics, in what you can actually
compute we would say nowadays. And he came up with an extremely stran
ge
non

constructive real number. You list all possible yes/no questions in
French in an immense, an infinite list of all possibilities. This will be what
mathematicians call a denumerable or a countable infinity of questions,
because it can be put into a o
ne

to

one correspondence with the list of
positive integers 1, 2, 3, ... In other words, there will be a first question, a
second question, a third question, and in general an
N
th question.
You can imagine all the possible questions to be ordered by size,
and within
questions of the same size, in alphabetical order. More precisely, you
consider all possible strings, all possible finite sequences of symbols in the
French alphabet, including the blank so that you get words, and the period so
that you have se
ntences. And you imagine filtering out all the garbage and
being left only with grammatical yes/no questions in French. Later I will tell
you in more detail how to actually do this. Anyway, for now
imagine
doing
this, and so there will be a first question,
a second question, an
N
th question.
And the
N
th digit or the
N
th bit after the decimal point of Borel's number
answers the
N
th question: It will be a 0 if the answer is no, and it'll be a 1 if
the answer is yes. So the binary expansion of Borel's number
contains the
answer to every possible yes/no question! It's like having an oracle, a Delphic
oracle that will answer every yes/no question!
How is this possible?! Well, according to Borel, it isn't really possible, this
can't be, it's totally unbelievable
. This number is only a mathematical fantasy,
it's not for real, it cannot claim a legitimate place in our ontology. Later I'll
show you a modern version of Borel's number, my halting probability
Ω. And
I'll tell you why some contemporary physicists, real physicists, not
mavericks, are moving in the direction of digital physics.
7
[Actually, to make Borel's number as real as possible, you have to avoid the
problem of filtering out all the yes/no qu
estions. And you have to use decimal
digits, you can't use binary digits. You number all the possible finite strings
of French symbols including blanks and periods, which is quite easy to do
using a computer. Then the
N
th digit of Borel's number is 0 if th
e
N
th string
of characters in French is ungrammatical and not proper French, it's 1 if it's
grammatical, but not a yes/no question, it's 2 if it's a yes/no question that
cannot be answered (e.g., "Is the answer to this question "no"?"), it's 3 if the
answe
r is no, and it's 4 if the answer is yes.]
Geometrically
a real number is the most straightforward thing in the world,
it's just a point on a line. That's quite natural and intuitive. But
arithmetically
,
that's another matter. The situation is quite diffe
rent. From an arithmetical
point of view reals are extremely problematical, they are fraught with
difficulties!
Before discussing my Ω number, I want to return to the fundamental question
of what does it mean to understand. How do we explain or comprehend
something? What is a theory? How can we tell whether or not it's a successful
theory? How can we measure how suc
cessful it is? Well, using the ideas of
information and computation, that's not difficult to do, and the central idea
can even be traced back to Leibniz's 1686
Discours de métaphysique.
Computer Epistemology: What is a mathematical or scientific theory? H
ow
can we judge whether it works or not?
In Sections V and VI of his
Discourse on Metaphysics,
Leibniz asserts that
God simultaneously maximizes the variety, diversity and richness of the
world, and minimizes the conceptual complexity of the set of ideas
that
determine the world. And he points out that for any finite set of points there
is always a mathematical equation that goes through them, in other words, a
law that determines their positions. But if the points are chosen at random,
that equation will
be extremely complex.
This theme is taken up again in 1932 by Hermann Weyl in his book
The
Open World
consisting of three lectures he gave at Yale University on the
metaphysics of modern science. Weyl formulates Leibniz's crucial idea in the
following ext
remely dramatic fashion: If one permits arbitrarily complex
laws, then the concept of law becomes vacuous, because there is always a
law! Then Weyl asks, how can we make more precise the distinction between
mathematical simplicity and mathematical complexi
ty? It seems to be very
hard to do that. How can we measure this important parameter, without
8
which it is impossible to distinguish between a successful theory and one that
is completely unsuccessful?
This problem is taken up and I think satisfactorily re
solved in the new
mathematical theory I call
algorithmic information theory.
The
epistemological model that is central to this theory is that a scientific or
mathematical theory is a computer program for calculating the facts, and the
smaller the program,
the better. The complexity of your theory, of your law,
is measured in bits of software:
program (bit string)

> Computer

>
output (bit string)
theory

> Computer

>
mathematical or scientific facts
Understanding is compression!
Now Leibniz's
crucial observation can be formulated much more precisely.
For any finite set of scientific or mathematical facts, there is always a theory
that is exactly as complicated, exactly the same size in bits, as the facts
themselves. (It just directly outputs th
em "as is," without doing any
computation.) But that doesn't count, that doesn't enable us to distinguish
between what can be comprehended and what cannot, because there is always
a theory that is as complicated as what it explains. A theory, an explanatio
n,
is only successful to the extent to which it compresses the number of bits in
the facts into a much smaller number of bits of theory. Understanding is
compression, comprehension is compression! That's how we can tell the
difference between real theories
and
ad hoc
theories.
[By the way, Leibniz also mentions complexity in Section 7 of his
Principles
of Nature and Grace,
where he asks the amazing question, "Why is there
something rather than nothing? For nothing is simpler and easier than
something."]
W
hat can we do with this idea that an explanation has to be simpler than
what it explains? Well, the most important application of these ideas that I
have been able to find is in metamathematics, it's in discussing what
mathematics can or cannot achieve. Yo
u simultaneously get an information

theoretic, computational perspective on Gödel's famous 1931 incompleteness
theorem, and on Turing's famous 1936 halting problem. How?
9
[For an insightful treatment of Gödel as a philosopher, see Rebecca
Goldstein's
Incom
pleteness.
]
Here's how! These are my two favorite information

theoretic incompleteness
results:
You need an
N

bit theory in order to be able to prove that a specific
N

bit program is "elegant."
You need an
N

bit theory in order to be able to determine
N
bits of
the numerical value, of the base

two binary expansion, of the halting
probability
Ω
.
Let me explain.
What is an elegant program? It's a program with the property that no program
written in the same programming language that produces the same output is
smaller than it is. In other words, an elegant program is the most concise, the
simp
lest, the best theory for its output. And there are infinitely many such
programs, they can be arbitrarily big, because for any computational task
there has to be at least one elegant program. (There may be several if there
are ties, if there are several p
rograms for the same output that have exactly
the minimum possible number of bits.)
And what is the halting probability Ω? Well, it's defined to be the probability
that a computer program generated at random, by choosing each of its bits
using an independent toss of a fair coin, will eventually halt. Turing is
interested in whether or not
individual programs halt. I am interested in
trying to prove what are the bits, what is the numerical value, of the halting
probability Ω. By the way, the value of Ω depends on your particular choice
of programming language, which I don't have time to dis
cuss now. Ω is also
equal to the result of summing 1/2 raised to powers which are the size in bits
of every program that halts. In other words, each
K

bit program that halts
contributes 1/2
K
to Ω.
And what precisely do I mean by an
N

bit mathematical theo
ry? Well, I'm
thinking of formal axiomatic theories, which are formulated using symbolic
logic, not in any natural, human language. In such theories there are always a
finite number of axioms and there are explicit rules for mechanically
deducing consequen
ces of the axioms, which are called theorems. An
N

bit
theory is one for which there is an
N

bit program for systematically running
through the tree of all possible proofs deducing all the consequences of the
10
axioms, which are all the theorems in your form
al theory. This is slow work,
but in principle it can be done mechanically, that's what counts. David Hilbert
believed that there had to be a single formal axiomatic theory for all of
mathematics; that's just another way of stating that math is static and
perfect
and provides absolute truth.
Not only is this impossible, not only is Hilbert's dream impossible to achieve,
but there are in fact an infinity of irreducible mathematical truths,
mathematical truths for which essentially the only way to prove them
is to
add them as new axioms. My first example of such truths was determining
elegant programs, and an even better example is provided by the bits of Ω.
The bits of Ω are mathematical facts that are true for no reason (no reason
simpler than themselves),
and thus violate Leibniz's principle of sufficient
reason, which states that if anything is true it has to be true for a reason.
In math the reason that something is true is called its proof. Why are the bits
of Ω true for no reason, why can't you prove w
hat their values are? Because,
as Leibniz himself points out in Sections 33 to 35 of
The Monadology,
the
essence of the notion of proof is that you prove a complicated assertion by
analyzing it, by breaking it down until you reduce its truth to the truth o
f
assertions that are so simple that they no longer require any proof (self

evident axioms). But if you cannot deduce the truth of something from any
principle simpler than itself, then proofs become useless, because
anything
can be proven from principles
that are equally complicated, e.g., by directly
adding it as a new axiom without any proof. And this is exactly what happens
with the bits of Ω.
In other words, the normal, Hilbertian view of math is that all of
mathematical truth, an infinite number of t
ruths, can be compressed into
a finite number of axioms. But there are an infinity of mathematical
truths that cannot be compressed at all, not one bit!
This is an amazing result, and I think that it has to have profound
philosophical and practical implic
ations. Let me try to tell you why.
On the one hand, it suggests that pure math is more like biology than it is like
physics. In biology we deal with very complicated organisms and
mechanisms, but in physics it is normally assumed that there has to be a
t
heory of everything, a simple set of equations that would fit on a T

shirt and
in principle explains the world, at least the physical world. But we have seen
that the world of mathematical ideas has infinite complexity, it cannot be
explained with any theo
ry having a finite number of bits, which from a
11
sufficiently abstract point of view seems much more like biology, the domain
of the complex, than like physics, where simple equations reign supreme.
On the other hand, this amazing result suggests that even
though math and
physics are different, they may not be as different as most people think! I
mean this in the following sense: In math you organize your computational
experience, your lab is the computer, and in physics you organize physical
experience and
have real labs. But in both cases an explanation has to be
simpler than what it explains, and in both cases there are sets of facts that
cannot be explained, that are irreducible. Why? Well, in quantum physics it is
assumed that there are phenomena that w
hen measured are equally likely to
give either of two answers (e.g., spin up, spin down) and that are inherently
unpredictable and irreducible. And in pure math we have a similar example,
which is provided by the individual bits in the binary expansion of
the
numerical value of the halting probability Ω.
This suggests to me a quasi

empirical view of math, in which one is more
willing to add new axioms that are not at all self

evident but that are justified
pragmatically, i.e., by their fruitful consequence
s, just like a physicist would.
I have taken the term quasi

empirical from Lakatos. The collection of essays
New Directions in the Philosophy of Mathematics
edited by Tymoczko in my
opinion pushes strongly in the direction of a quasi

empirical view of math
,
and it contains an essay by Lakatos proposing the term "quasi

empirical," as
well as essays of my own and by a number of other people. Many of them
may disagree with me, and I'm sure do, but I repeat, in my opinion all of
these essays justify a quasi

emp
irical view of math, what I mean by quasi

empirical, which is somewhat different from what Lakatos originally meant,
but is in quite the same spirit, I think.
In a two

volume work full of important mathematical examples, Borwein,
Bailey and Girgensohn hav
e argued that experimental mathematics is an
extremely valuable research paradigm that should be openly acknowledged
and indeed vigorously embraced. They do not go so far as to suggest that one
should add new axioms whenever they are helpful, without bothe
ring with
proofs, but they are certainly going in that direction and nod approvingly at
my attempts to provide some theoretical justification for their entire
enterprise by arguing that math and physics are not that different.
In fact, since I began to es
pouse these heretical views in the early 1970's,
largely to deaf ears, there have actually been several examples of such new
pragmatically justified, non

self

evident axioms:
12
the P not equal to NP hypothesis regarding the time complexity of
computations,
the axiom of projective determinacy in set theory, and
increasing reliance on diverse unproved versions of the Riemann
hypothesis regarding the distribution of the primes.
So people don't need to have theoretical justification; they just do whatever is
needed to get the job done...
The only problem with this computational and information

theoretic
epistemology that I've just outlined to you is that it's based on the computer,
and there are uncomputable reals. So what do we do with contemporary
physics w
hich is full of partial differential equations and field theories, all of
which are formulated in terms of real numbers,
most of which are in fact
uncomputable,
as I'll now show. Well, it would be good to get rid of all that
and convert to a
digital physic
s.
Might this in fact be possible?! I'll discuss
that too.
Computer Ontology: How real are real numbers? What is the world made
of?
How did Turing prove that there are uncomputable reals in 1936? He did it
like this. Recall that the possible texts in Fre
nch are a countable or
denumerable infinity and can be placed in an infinite list in which there is a
first one, a second one, etc. Now let's do the same thing with all the possible
computer programs (first you have to choose your programming language).
So
there is a first program, a second program, etc. Every computable real can
be calculated digit by digit by some program in this list of all possible
programs. Write the numerical value of that real next to the programs that
calculate it, and cross off the
list all the programs that do not calculate an
individual computable real. We have converted a list of programs into a list
of computable reals, and no computable real is missing.
Next discard the integer parts of all these computable reals, and just kee
p the
decimal expansions. Then put together a new real number by changing every
digit on the diagonal of this list (this is called Cantor's diagonal method; it
comes from set theory). So your new number's first digit differs from the first
digit of the fir
st computable real, its second digit differs from the second digit
of the second computable real, its third digit differs from the third digit of the
third computable real, and so forth and so on. So it can't be in the list of all
computable reals and it h
as to be uncomputable. And that's Turing's
uncomputable real number!
13
[
Technical Note:
Because of
synonyms
like .345999... = .346000... you
should avoid having any 0 or 9 digits in Turing's number.]
Actually, there is a much easier way to see that there a
re uncomputable reals
by using ideas that go back to Emile Borel (again!). Technically, the
argument that I'll now present uses what mathematicians call
measure theory
,
which deals with probabilities. So let's just look at all the real numbers
between 0 an
d 1. These correspond to points on a line, a line exactly one unit
in length, whose leftmost point is the number 0 and whose rightmost point is
the number 1. The total length of this line segment is of course exactly one
unit. But I will now show you that
all the computable reals in this line
segment can be covered using intervals whose total length can be made as
small as desired. In technical terms, the computable reals in the interval from
0 to 1 are a set of measure zero, they have zero probability.
Ho
w do you cover all the computable reals? Well, remember that list of all
the computable reals that we just diagonalized over to get Turing's
uncomputable real? This time let's cover the first computable r
eal with an
interval of size ε/2, let's cover the second computable real with an interval of
size ε/4, and in general we'll cover the
N
th computable real with an interval
of size ε/2
N
. The total length of all these intervals (which can conceivably
overlap
or fall partially outside the unit interval from 0 to 1), is exactly equal
to ε, which can be made as small as we wish! In other words, there are
arbitrarily small coverings, and the computable reals are therefore a set of
measure zero, they have zero prob
ability, they constitute an infinitesimal
fraction of all the reals between 0 and 1. So if you pick a real at random
between 0 and 1, with a uniform distribution of probability, it is infinitely
unlikely, though possible, that you will get a computable rea
l!
What disturbing news! Uncomputable reals are not the exception, they are the
majority! How strange!
In fact, the situation is even worse than that. As Emile Borel points out on
page 21 of his final book,
Les nombres inaccessibles
(1952), without makin
g
any reference to Turing, most individual reals are not even uniquely
specifiable, they cannot even be named or pointed out, no matter how non

constructively, because of the limitations of human languages, which permit
only a countable infinity of possibl
e texts. The individually accessible or
nameable reals are also a set of measure zero. Most reals are un

nameable,
with probability one! I rediscovered this result of Borel's on my own in a
slightly different context, in which things can be done a little m
ore
rigorously, which is when one is dealing with a formal axiomatic theory or an
14
artificial
formal language instead of a natural human language. That's how I
present this idea in
Meta Math!
.
So if most individual reals will forever escape us, why should
we believe in
them?! Well, you will say, because they have a pretty structure and are a nice
theory, a nice game to play, with which I certainly agree, and also because
they have important practical applications, they are needed in physics. Well,
perhaps n
ot! Perhaps physics can give up infinite precision reals! How? Why
should physicists want to do that?
Because it turns out that there are actually many reasons for being skeptical
about the reals, in classical physics, in quantum physics, and particularly
in
more speculative contemporary efforts to cobble together a theory of black
holes and quantum gravity.
First of all, as my late colleague the physicist Rolf Landauer used to remind
me, no physical measurement has ever achieved more than a small number
of
digits of precision, not more than, say, 15 or 20 digits at most, and such high

precision experiments are rare masterpieces of the experimenter's art and not
at all easy to achieve.
This is only a practical limitation in classical physics. But in quant
um physics
it is a consequence of the Heisenberg uncertainty principle and wave

particle
duality (de Broglie). According to quantum theory, the more accurately you
try to measure something, the smaller the length scales you are trying to
explore, the highe
r the energy you need (the formula describing this involves
Planck's constant). That's why it is getting more and more expensive to build
particle accelerators like the one at CERN and at Fermilab, and governments
are running out of money to fund high

ener
gy physics, leading to a paucity of
new experimental data to inspire theoreticians.
Hopefully new physics will eventually emerge from astronomical
observations of bizarre new astrophysical phenomena, since we have run out
of money here on earth! In fact,
currently some of the most interesting
physical speculations involve the thermodynamics of black holes, massive
concentrations of matter that seem to be lurking at the hearts of most
galaxies. Work by Stephen Hawking and Jacob Bekenstein on the
thermodynam
ics of black holes suggests that any physical system can contain
only a finite amount of information, a finite number of bits whose possible
maximum is determined by what is called the Bekenstein bound. Strangely
enough, this bound on the number of bits gr
ows as the surface area of the
physical system, not as its volume, leading to the so

called "holographic"
15
principle asserting that in some sense space is actually two

dimensional even
though it appears to have three dimensions!
So perhaps continuity is an
illusion, perhaps everything is really discrete.
There is another argument against the continuum if you go down to what is
called the Planck scale. At distances that extremely short our current physics
breaks down because spontaneous fluctuations in the q
uantum vacuum
should produce mini

black holes that completely tear spacetime apart. And
that is not at all what we see happening around us. So perhaps distances that
small
do not exist
.
Inspired by ideas like this, in addition to
a priori
metaphysical bia
ses in favor
of discreteness, a number of contemporary physicists have proposed building
the world out of discrete information, out of bits. Some names that come to
mind in this connection are John Wheeler, Anton Zeilinger, Gerard 't Hooft,
Lee Smolin, Set
h Lloyd, Paola Zizzi, Jarmo Mäkelä and Ted Jacobson, who
are real physicists. There is also more speculative work by a small cadre of
cellular automata and computer enthusiasts including Edward Fredkin and
Stephen Wolfram, whom I already mentioned, as well
as Tommaso Toffoli,
Norman Margolus, and others.
And there is also an increasing body of highly successful work on quantum
computation and quantum information that is not at all speculative, it is just a
fundamental reworking of standard 1920's quantum m
echanics. Whether or
not quantum computers ever become practical, the workers in this highly
popular field have clearly established that it is illuminating to study sub

atomic quantum systems in terms of how they process qubits of quantum
information and h
ow they perform computation with these qubits. These
notions have shed completely new light on the behavior of quantum
mechanical systems.
Furthermore, when dealing with complex systems such as those that occur in
biology, thinking about information proce
ssing is also crucial. As I believe
Seth Lloyd said, the most important thing in understanding a complex system
is to determine how it represents information and how it processes that
information, i.e., what kinds of computations are performed.
And how ab
out the entire universe, can it be considered to be a computer?
Yes, it certainly can, it is constantly computing its future state from its
current state, it's constantly computing its own time

evolution! And as I
believe Tom Toffoli pointed out, actual co
mputers like your PC just hitch a
ride on this universal computation!
16
So perhaps we are not doing violence to Nature by attempting to force her
into a digital, computational framework. Perhaps she has been flirting with
us, giving us hints all along, that
she is really discrete, not continuous, hints
that we choose not to hear, because we are so much in love and don't want
her to change!
For more on this kind of new physics, see the books by Smolin and von
Baeyer in the bibliography. Several more technica
l papers on this subject are
also included there.
Conclusion
Let me now wrap this up and try to give you a present to take home, more
precisely, a piece of homework. In extremely abstract terms, I would say that
the problem is, as was emphasized by Ernst
Mayr in his book
This is Biology
,
that the current philosophy of science deals more with physics and
mathematics than it does with biology. But let me try to put this in more
concrete terms and connect it with the spine, with the central thread, of the
id
eas in this talk.
To put it bluntly, a closed, static, eternal fixed view of math can no longer be
sustained. As I try to illustrate with examples in my
Meta Math!
book, math
actually advances by inventing new concepts, by completely changing the
viewpoin
t. Here I emphasized new axioms, increased complexity, more
information, but what really counts are new ideas, new concepts, new
viewpoints. And that leads me to the crucial question, crucial for a proper
open, dynamic, time

dependent view of mathematics,
"Where do new mathematical ideas come from?"
I repeat, math does not advance by mindlessly and mechanically grinding
away deducing all the consequences of a fixed set of concepts and axioms,
not at all! It advances with new concepts, new definitions, new
perspectives,
through revolutionary change, paradigm shifts, not just by hard work.
In fact, I believe that this is actually the central question in biology as well as
in mathematics, it's the mystery of creation, of creativity:
"Where do new mathematic
al and biological ideas come from?"
"How do they emerge?"
17
Normally one equates a new biological idea with a new species, but in fact
every time a child is born, that's actually a new idea incarnating; it's
reinventing the notion of "human being," which c
hanges constantly.
I have no idea how to answer this extremely important question; I wish I
could. Maybe
you
will be able to do it. Just try! You might have to keep it
cooking on a back burner while concentrating on other things, but don't give
up! All it
takes is a new idea! Somebody has to come up with it. Why not
you?
[I'm not denying the importance of Darwin's theory of evolution. But I want
much more than that, I want a profound, extremely general mathematical
theory that captures the essence of what
life is and why it evolves. I want a
theory that gets to the heart of the matter. And I suspect that any such theory
will necessarily have to shed new light on mathematical creativity as well.
Conversely, a deep theory of mathematical creation might also
cover
biological creativity.
A reaction from Gordana Dodig

Crnkovic: "Regarding Darwin and Neo

Darwinism I agree with you

it is a very good idea to go beyond. In my
view there is nothing more beautiful and convincing than a good
mathematical theory. A
nd I do believe that it must be possible to express
those thoughts in a much more general way... I believe that it is a very crucial
thing to try to formulate life in terms of computation. Not to say life is
nothing more than a computation. But just to exp
lore how far one can go with
that idea. Computation seems to me a very powerful tool to illuminate many
things about the material world and the material ground for mental
phenomena (including creativity)... Or would you suggest that creativity is
given by
God's will? That it is the very basic axiom? Isn’t it possible to relate
to pure chance? Chance and selection? Wouldn’t it be a good idea to assume
two principles: law and chance, where both are needed to reconstruct the
universe in computational terms? (l
ike chaos and cosmos?)"]
Appendix: Leibniz and the Law
I am indebted to Professor Ugo Pagallo for explaining to me that Leibniz,
whose ideas and their elaboration were the subject of my talk, is regarded as
just as important in the field of law as he is
in the fields of mathematics and
philosophy.
The theme of my lecture was that if a law is arbitrarily complicated, then it is
not a law; this idea was traced via Hermann Weyl back to Leibniz. In
18
mathemati
cs it leads to my Ω number and the surprising discovery of
completely lawless regions of mathematics, areas in which there is absolutely
no structure or pattern or way to understand what is happening.
The principle that an arbitrarily complicated law is n
ot a law can also be
interpreted with reference to the legal system. It is not a coincidence that the
words "law" and "proof" and "evidence" are used in jurisprudence as well as
in science and mathematics. In other words, the rule of law is equivalent to
t
he rule of reason, but if a law is sufficiently complicated, then it can in fact
be completely arbitrary and incomprehensible.
Acknowledgements
I wish to thank Gordana Dodig

Crnkovic for organizing E

CAP'05 and for
inviting me to present the Turing lectu
re at E

CAP'05; also for stimulating
discussions reflected in those footnotes that are marked with GDC. The
remarks on biology are the product of a week spent in residence at
Rockefeller University in Manhattan, June 2005; I thank Albert Libchaber for
invi
ting me to give a series of lectures there to physicists and biologists. The
appendix is the result of lectures to philosophy of law students April 2005 at
the Universities of Padua, Bologna and Turin; I thank Ugo Pagallo for
arranging this. Thanks too to
Paola Zizzi for help with the physics references.
References
Edward Fredkin,
http://www.digitalphilosophy.org/
.
Stephen Wolfram,
A New Kind of Science,
Wolfram Media, 2002.
Gregory Chaitin,
Meta Math!,
Pantheon, 2005.
G. W. Leibniz,
Discourse on Metaphysics, Principles of Nature and
Grace, The Monadology,
1686, 1714, 1714.
Hermann Weyl,
The Open World,
Yale University Press, 1932.
Thomas Tymoczko,
New Directions in the Philosophy of
Mathematics,
Princ
eton University Press, 1998.
Jonathan Borwein, David Bailey, Roland Girgensohn,
Mathematics
by Experiment,
Experimentation in Mathematics,
A. K. Peters, 2003,
2004.
Rebecca Goldstein,
Incompleteness,
Norton, 2005.
B. Jack Copeland,
The Essential Turing,
Oxford University Press,
2004.
Vladimir Tasic,
Mathematics and the Roots of Postmodern Thought,
Oxford University Press, 2001.
Emile Borel,
Les nombres inaccessibles,
Gauthier

Villars, 1952.
19
Lee Smolin,
Three Roads to Quantum Gravity,
Basic Books, 2001
.
Hans Christian von Baeyer,
Information,
Harvard University Press,
2004.
Ernst Mayr,
This is Biology,
Harvard University Press, 1998.
J. Wheeler, (the "It from bit" proposal),
Sakharov Memorial
Lectures on Physics,
vol. 2, Nova Science, 1992.
A. Zeili
nger, (the principle of quantization of information),
Found.
Phys.
29
:631

643 (1999).
G. 't Hooft, "The holographic principle,"
http://arxiv.org/hep

th/0003004
.
S. Lloyd, "The computational universe,"
http://arxiv.org/quant

ph/0501135
.
P. Zizzi, "A minimal model for quantum gravity,"
http://arxiv.org/gr

qc/0409069
.
J. Mäkelä, "Accelerating observers, area and e
ntropy,"
http://arxiv.org/gr

qc/0506087
.
T. Jacobson, "Thermodynamics of spacetime,"
http://arxiv.org/gr

qc/9504004
.
20
Information Logic
Luciano Floridi
Dipart
imento di Scienze Filosofiche, Università degli Studi di Bari; Faculty
of Philosophy and IEG, Computing Laboratory, Oxford University.
Address for correspondence: Wolfson College, OX2 6UD, Oxford, UK;
luciano.floridi@philosophy.oxford.ac.uk
Abstract
One
of the open problems in the philosophy of information is whether there
is an
information logic
(
IL
), different from
epistemic
(
EL
) and
doxastic logic
(
DL
), which formalises the relation “
a
is informed that
p
” (
I
a
p
) satisfactorily.
In this paper, the probl
em is solved by arguing that the axiom schemata of the
normal modal logic (
NML
)
KTB
(also known as
B
or
Br
or Brouwer’s
system) are well suited to model the relation of “being informed”.
Keywords
Brouwer’s system; Doxastic logic; Entailment property; Epi
stemic logic;
Gettier problem; Information logic; Normal modal logic
KTB
; Veridicality
of information.
21
Introduction
As anyone acquainted with modal logic (
ML
) knows,
epistemic logic
(
EL
)
formalises the relation “
a
knows that
p
” (
K
a
p
), whereas
doxastic lo
gic
(
DL
)
formalises the relation “
a
believes that
p
” (
B
a
p
). One of the open problems in
the philosophy of information (
Floridi [2004b]
) is whether there is also an
information logic
(
IL
), differ
ent from
EL
and from
DL
, that formalises the
relation “
a
is informed that
p
” (
I
a
p
) equally well.
The keyword here is “equally” not “well”. One may contend that
EL
and
DL
do not capture the relevant relations very well or even not well at all.
Hocutt [1972]
, for example, provides an early criticism. Yet this is not the
point here, since all I wish to argue in this paper is that
IL
can do for “being
informed” what
EL
does for “kno
wing” and
DL
does for “believing”. If one
objects to the last two, one may object to the first as well, yet one should not
object to it more.
The proposal developed in the following pages is that the normal
modal logic (
NML
)
KTB
(also known as
B
,
Br
or Bro
uwer’s system
1
) is well
suited to model the relation of “being informed”, and hence that
IL
can be
constructed as an informational reading of
KTB
. The proposal is in three
sections.
In section one, several meanings of “information” are recalled, in
order t
o focus only on the “cognitive” sense. Three main ways in which one
may speak of a “logic of (cognitive) information” are then distinguished.
Only one of them is immediately relevant here, namely, “
a
is informed that
p
” as meaning “
a
holds the information
that
p
”. These clarifications are
1
The name was assigned by
Becker [1930]
. As
Goldblatt [2003]
remarks:
“The connection with Brouwer is remote: if ‘not’ is translated to ‘impossible’
(¬
), and ‘implie
s’ to its strict version, then the intuitionistically acceptable
principle p
¬¬p becomes the Brouwersche axiom”. For a description of
KTB
see
Hughes and Cresswell [1996]
.
22
finally used to make precise the specific question addressed in the rest of the
paper.
In section two, the analysis of the informational relation of “being
informed” provides the specifications to be satisfied by its accur
ate
formalization. It is then shown that
KTB
successfully captures the relation of
“being informed”.
In section three, the conclusion, I sketch some of the work that lies
ahead, especially as far as the application of information logic is concerned to
dea
l with some key issues in epistemology.
Throughout the paper the ordinary language of classical,
propositional calculus (
PC
) and of normal, propositional modal logic (see for
example
Girle [2000]
) will be presupposed. Implication (
→) is used in its
“material” sense; the semantics is Kripkean; Greek letters are metalinguistic,
propositional variables ranging over well

formed formulae of the object
language of the corresponding
NML
; and until section 2.6 attention is focused
only on t
he axiom schemata of the
NMLs
in question.
1. Three logics of information
“Information” may be understood in many ways, e.g. as signals, natural
patterns or nomic regularities, as instructions, as content, as news, as
synonymous with data, as power or as
an economic resource and so forth. It
is notoriously controversial whether even most of these senses of
“information” might be reduced to a fundamental concept.
2
However, the sort
of “information” that interests us here is arguably the most important. It i
s
“information” as
semantic content
that, on one side, concerns some state of a
system, and that, on the other side, allows the elaboration of an agent’s
2
For an overview see
Floridi [2004a]
and
Floridi [2005b]
. Personally, I am
very sceptical about attempts to find a unified theory of information and
hence a unique logic that would capture all its interesting
features.
23
propositional knowledge of that state of the system. It is the sense in which
Matthew is informed that
p
, e.g. that “the train to London leaves at 10.30
am”, or about the state of affairs
f
expressed by
p
, e.g. the railway timetable.
In the rest of the paper, “information” will be discussed only in this intuitive
sense of
declarative
,
objective
and
semanti
c content
that
p
or about
f
(
Floridi
[2005a]
). This sense may loosely be qualified as “cognitive”, a neutral label
useful to refer here to a whole family of relations expressing p
ropositional
attitudes, including “knowing”, “believing”, “remembering”, “perceiving”
and “experiencing”. Any “non

cognitive” sense of “semantic information”
will be disregarded.
3
The scope of our inquiry can now be narrowed by considering the
logical anal
ysis of the
cognitive
relation “
a
is informed that
p
”. Three related
yet separate features of interest need to be further distinguished, namely
a) how
p
may be informative for
a
.
For example, the information that ¬
p
may or may not be informative
dependin
g on whether
a
is already informed that (
p
q
). This aspect of
information
–
the
informativeness
of a message
–
raises issues of e.g. novelty,
reliability of the source and background information. It is a crucial aspect
related to the quantitative theory
of semantic information (
Bar

Hillel and
Carnap [1953]
, see
Bar

Hillel [1964]
;
Floridi [2004c]
), to the logic of
3
There are many plausible contexts in which a stipulation (“let the value of x
= 3” or “suppose we discover the bones of a unicorn”), an invitation (“you
are cordially invited to the college party”), an order (“close the window!”), an
instructio
n (“to open the box turn the key”), a game move (“1.e2

e4 c7

c5” at
the beginning of a chess game) may be correctly qualified as kinds of
information understood as semantic content. These and other similar, non

cognitive meanings of “information” (e.g. to
refer to a music file or to a
digital painting) are not discussed in this paper, where semantic information
is taken to have a declarative or factual value i.e. it is suppose to be correctly
qualifiable alethically.
24
transition states in dynamic system, that is, how change in a system may be
informative for an observer (
Barwise and Seligman [1997]
) and to the theory
of levels of abstraction at which a system is being considered (
Floridi and
Sanders [2004]
;
Floridi and Sanders [forthcoming]
);
b) the process through which
a
becomes informed that
p
.
The informativeness of
p
makes possible the pro
cess that leads from
a
’s
uninformed (or less informed) state
A
to
a
’s (more) informed state
B
.
Upgrading
a
’s state
A
to a state
B
usually involves receiving the information
that
p
from some external source
S
and processing it. It implies that
a
cannot
be i
nformed that
p
unless
a
was previously uninformed that
p
. And the
logical relation that underlies this state transition raises important issues of
timeliness and cost of acquisition, for example, and of adequate procedures of
information processing, includ
ing introspection and metainformation, as we
shall see. It is related to communication theory (
Shannon and Weaver [1949
rep. 1998]
), temporal logic, updating procedures (
Gärdenfors [1988]
), and
recent trends in dynamic epistemic logic (
Baltag and Moss [2004]
);
c) the state of the epistemic agent
a
, insofar as
a
holds
the information that
p
.
This is the
statal
condition into which
a
enters, once
a
has acquired the
information (actional state of being informed) that
p
. It is the sense in which
a witness, for example, is informed (holds the information) that the suspect
was with her at the time when the crime was committed. The distinction is
standard among grammarian
s, who speak of passive verbal forms or states as
“statal” (e.g. “the door
was shut
(state) when I last checked it”) or “actional”
25
(e.g. “but I don't know when the door
was shut
(act)”).
4
Here, we are
interested only in the
statal
sense of “is informed”. T
his sense (c) is related to
cognitive issues and to the logical analysis of an agent’s “possession” of a
belief or a piece of knowledge.
Point (a) requires the development of a logic of “being informative”; (b)
requires the development of a logic of “beco
ming informed”; and (c) requires
the development of a logic of “being informed (i.e. holding the information)”.
Work on (a) and (b) is already in progress.
Allo [2005]
and
Sanders
[forthcoming]
, respectively, develop two lines of research complementary to
this paper. In this paper, I shall be concerned with (c) and seek to show that
there is a logic of information comparable, for adequacy, fle
xibility and
usefulness, to
EL
and
DL
.
Our problem can now be formulated more precisely. Let us
concentrate our attention on the most popular and traditional
NML
,
obtainable through the analysis of some of the well

known characteristics of
the relation of
accessibility (reflexivity, transitivity etc.). These fifteen
5
NMLs
range from the weakest
K
to the strongest
S5
(see below Figure 1). They are
also obtainable through the combination of the usual axiom schemata of
PC
with the fundamental modal axiom sche
mata (see below Figure 2). Both
EL
and
DL
comprise a number of cognitively interpretable
NML
, depending on
the sets of axioms that qualify the corresponding
NML
used to capture the
relevant “cognitive” notions. If we restrict our attention to the six most
popular
EL
and
DL
–
those based on systems
KT
,
S4
,
S5
and on systems
KD
,
4
I owe to Christopher Kirwan this very
useful clarification; in a previous
version of this paper I had tried to reinvent it, but the wheel was already
there.
5
The number of
NMLs
available is infinite. I am grateful to Timothy
Williamson and John Halleck who kindly warned me against a misleadin
g
wording in a previous version of this paper.
26
KD4
,
KD45
respectively
–
the question about the availability of an
information logic can be rephrased thus: among the popular
NMLs
taken into
consideration, is there one, not belongin
g to {
KT
,
S4
,
S5
,
KD
,
KD4
,
KD45
},
which, if cognitively interpreted, can successfully capture and formalise our
intuitions regarding “
a
is informed that
p
” in the (c) sense specified above?
A potential confusion may be immediately dispelled. Of course, th
e
logical analysis of the cognitive relation of “being informed” can sometimes
be provided in terms of “knowing” or “believing”, and hence of
EL
or
DL
.
This is not in question, for it is trivially achievable, insofar as “being
informed” can sometimes be co
rrectly treated as synonymous with
“knowing” or “believing”.
IL
may sometime overlap with
EL
. The interesting
problem is whether “being informed” may show properties that typically (i.e.,
whenever the overlapping would be unjustified) require a logic diffe
rent from
EL
and
DL
, in order to be modelled accurately. The hypothesis defended in
the following pages is that it does and, moreover, that this has some
interesting consequences for our understanding of the nature of the relation
between “knowing” and “be
lieving”.
27
S4
S5
K
KT
KD
K4
KD4
K5
K45
KD5
KD45
A7
A7
A7
A7
A4
A8
A8
A8
A7
A6
A6
A8
A8
A6
A6
A6
A9
A9
A6
A6
A8
A7
A8
A7
A6
KTB
KDB
KB5
KDB5
KB
A4
A9
A7
A7
A4
A9
A4
A4
A4
A4
A9
A9
A4
A4
A4
A4
A7
Figure 1 Fifteen Normal Modal Logics
Note that
K
DB5
is a “dummy” system: it is equivalent to
S5
and it is added to the
diagram just for the sake of elegance.
Synonymous
T = M = KT
B = Br = KTB
D = KD
Equivalent
axiomatic systems
B = TB
KB5 = KB4, KB45
S5 = T5, T45, TB4, TB5, TB45, DB4, DB5, DB45
Platonic axis
Symmetric R
A7
Socratic axis
Euclidean R
A8

A6

A7
Cartesian axis
Transitive R
A6
Aristotelian axis
Reflexive R
A9

A4
28
2. Modelling “being informed”
Let us interpret the modal operator
as “is informed that”. We may then
replace the symbol
with
I
for “being informed”, include an explicit
reference to the informed agent
a
, and w
rite
□
p
=
I
a
p
to mean
a
is informed (holds the
information) that
p
.
6
As customary, the subscript will be omitted whenever we shall be dealing
with a single, stand

alone agent
a
. It will be reintroduced in § 2.4, when
dealing with multiagent
IL
. Next, w
e can then define
in the standard way,
thus
U
a
p
=
def
¬
I
a
¬
p
to mean
a
is uninformed (is not informed, does not
hold the
information) that ¬
p
; or
for all
a
’s information (given
a
’s
information base), it is possible
that
p
.
Simplifying,
a
’s infor
mation base can be modelled by representing it as a
dynamic
set
D
a
of sentences of a language
L
.
7
The intended intepretation is
6
A
de re
interpretation is obtainable by interpreting
I
a
p
as “there is the
information that
p
”.
29
that
D
a
consists of all the sentences, i.e. all the information, that
a
holds at
time
t
. We then have that
I
a
p
means that
p
D
a
, and
U
a
p
means that
p
can be
uploaded in
D
a
while maintaining the consistency of
D
a
, that is,
U
a
p
means
(
p
D
a
) “salva cohaerentiae”.
8
Note that
a
need not be committed, either
doxastically (e.g. in terms of strengths of belief,
Lenzen [1978]
) or
epistemically (e.g. in terms of degrees of certainty) in favour of any element
in
D
a
.
Given that
IL
might actually overlap and hence be confused with
EL
or
DL
, the most plausible conjecture is that an
IL
that can capture our
intuitions, and hence sati
sfy our requirements regarding the proper
formalization of
Ip
, will probably bear some strong resemblance to
EL
and
DL
. If there is any difference between these three families of cognitive logics
it is likely to be identifiable more easily in terms of sati
sfaction (or lack
thereof) of one or more axioms qualifying the corresponding
NML
. The
heuristic assumption here is that, by restricting our attention to the fifteen
NMLs
in question, we may be able to identify the one which best captures our
requirements.
It is a bit like finding where, on a continuous map, the logic of
7
Dynamic sets are an important class of data structures in which sets of
items, indexed by keys, are maintained
. It is assumed that the elements of the
dynamic set contain a field (called the key) by whose value they can be
ordered. The phone directory of a company is a simple example of a dynamic
set (it changes over time), whose key might be “last name”. Dynamic
sets can
change over the execution of a process by gaining or losing elements. Of the
variety of operations usually supported by a dynamic set, three are
fundamental and will be assumed in this paper:
Search(S,k)
= given a set
S
and a key value
k
, a query
operation that returns a
pointer
x
to an element in
S
such that
key
[
x
] =
k
, or nil if no such element
belongs to
S
.
Insert(S,x)
= an operation that augments the set
S
with the element
x
.
Delete(S,x)
= an operation that removes an element pointed to by
x
fr
om
S
(if
it is there).
8
As Patrick Allo has noted in a personal communication, this can also be
expressed in terms of safety of inclusion of
p
in
D
a
.
30
information may be placed: even if we succeed in showing that
KTB
is the
right
NML
for our task, there is still an infinite number of neighbouring
NMLs
extending
KTB
.
9
For ease of referenc
e, the axiom schemata in question are
summarised and numbered progressively in Figure 2, where
φ
,
χ
and
ψ
are
propositional variables referring to any wff of
PC
.
Following Hintikka’s standard approach (
Hintikka [1962]
), a
systematic way to justify the choice of some axiom schemata is by trying to
i
dentify a plausible interpretation of a semantics for the corresponding
NML
.
We shall now consider the 12 axiom schemata and show that
IL
shares only
some of them with
EL
and
DL
.
Label
Definitions of Axiom Schemata
Name of the axiom or the
corresponding
N
ML
Frame
A
1
φ
(χ
φ)
=
N
st
axiom of
PC
A
2
(φ
(χ
ψ))
((φ
χ)
(φ
ψ))
=
O
nd
axiom of
PC
A
3
(¬ φ
¬ χ)
(χ
φ)
=
P
rd
axiom of
PC
A
4
□φ
φ
=
KT
or
M
, K2, veridicality
Reflexive
A
5
□(φ
χ)
(□φ
□χ)
=
K
, distribution, deductive cogency
A
6
□φ
□□φ
=
QI=
S4
, K3, KK, reflective thes is
or pos itive intros pection
Trans itive
A
7
φ
□
φ
=
KTB, B
,
Br
, Brouwer’s axiom
=
or=mlatonic=thesis
=
pymmetric
=
A
8
φ
□
φ
=
S5
, reflective, Socratic thes is or
negative intros pection
Euclidean
A
9
□φ
φ
=
KD
,
D
, consistency
Serial
A
10
□(φ
χ)
(□(χ
ψ)
□(φ
ψ))
=
pingle=agent=transmission
=
=
A
11
□
x
□
y
φ
□
x
φ
=
K4
, multiagent transmission,
or Hintikka’s axiom
=
=
cigure=O=Axiom= schemata=of=the=éroéositional=
NMLs
discussed in the paper
9
Many thanks to John Halleck for calling my attention to this point and to
Miyazaki [2005]
.
31
2.1
IL
satisfies A
1
, A
2
, A
3
, A
5
Trivia
lly, we may assume that
IL
satisfies the axioms A
1

A
3
. As for A
5
, this
specifies that
IL
is distributive, as it should be. If an agent
a
is informed that
p
q
, then, if
a
is informed that
p
,
a
is also informed that
q
. Note that,
although this is entirely
uncontroversial, it is less trivial. Not all “cognitive”
relations are distributive. “Knowing”, “believing” and “being informed” are,
as well as “remembering” and “recalling”. This is why Plato is able to argue
that a “mnemonic logic”, which he seems to ba
se on
K4
, may replace
DL
as a
foundation for
EL
.
10
However, “seeing” and other experiential relations, for
example, are not: if an agent
a
sees (in a non metaphorical sense) or hears or
experiences or perceives that
p
q
, it may still be false that, if
a
s
ees (hears
etc.)
p
,
a
then also sees (hears etc.)
q
.
The inclusion or exclusion of the remaining seven axioms is more
contentious. Although logically independent, the reasons leading to their
inclusion or exclusion are not, and they suggest the following c
lustering. In §
2.2,
IL
is shown to satisfy not only A
9
(consistency) but also A
4
(veridicality).
In § 2.3, it is argued that
IL
does not have to satisfy the two “reflective”
axioms, that is A
6
and A
8
. And in § 2.4, it is argued that
IL
should satisfy the
“transmissibility” axioms A
10
and A
11
. This will leave us with A
7
, to be
discussed in § 2.5.
2.2
Consistency and Truth: IL
satisfies A
9
and A
4
In
DL
, A
9
replaces the stronger A
4
, which characterizes
EL
: whereas
p
must
be true for the epistemic agent
a
to
know that
p
, the doxastic agent
a
only
10
On Plato’s interpretation of knowledge as recollection see especially
Phaedo
72e

75 and
Meno
82b

85.
32
needs to be consistent in her beliefs. There are at least four reasons why
IL
should be characterized as satisfying A
9
:
1) A
9
specifies that, in
IL
, the informational agent
a
is consistent, but so can
be our ordinary
informed agent in everyday life:
Ip
Up
. If
a
holds the
information that the train leaves at 10.30 am then, for all
a
’s information, it is
possible that the train leaves at 10.30 am, in other words,
p
can be uploaded
in
a
’s information base
D
a
while main
taining the consistency of
D
a
;
2) even if (1) were unconvincing,
IL
should qualify
a
as consistent at least
normatively, if not factually, in the same way as
DL
does. If
a
holds the
information that the train leaves at 10.30 am, then
a
should not hold the
information that the train does not leave at 10.30 am. The point is not that
doxastic or informational agents cannot be inconsistent,
11
but that A
9
provides an information integrity constraint: inconsistent agents should be
disregarded. Again, to appreciate
the non

trivial nature of a normative
approach to A
9
, consider the case of a “mnemonic logic”: it might be
factually implausible and only normatively desirable to formalise “
a
remembers that
p
” as implying that, if this is the case, then
a
does not
rememb
er that ¬
p
. Matthew may remember something that actually never
happened, or he might remember both
p
(that he left the keys in the car) and
¬
p
(that he left the keys on his desk) and be undecided about which memory
is reliable. Likewise, if a database co
ntains the information that
p
it might,
unfortunately, still contain also the information that ¬
p
, even if, in principle,
it should not, because this would seriously undermine the informative nature
of the database itself (see next point 3), and although
it is arguable (because
of A
4
, see below) that in such case either
p
or ¬
p
fail to count as information;
11
It might be possible to develop a modal approach to QC (quasi

classical)
logic in order to weaken t
he integrity constraint, see
Grant and Hunter
[forthcoming]
.
33
3) objections against
IL
satisfying A
9
appear to be motivated by a confusion
between “becoming informed” and “being informed”, a distinction
emphasis
ed in § 2.1. In the former case, it is unquestionable that
a
may
receive and hence hold two contradictory messages (e.g.,
a
may read in a
printed timetable that the train leaves at 10.30 am, as it does, but
a
may also
be told by
b
that the train does not l
eave at 10.30 am). However, from this it
only follows that
a
has the information that the train leaves at 10.30 am, but
since
p
and ¬
p
erase each other’s value as pieces of information
for a
,
a
may
be unable, subjectively, to identify which information
a
holds. It does not
follow that
a
is actually informed both that the train leaves at 10.30 am and
that it does not;
4) if
IL
satisfies the stronger A
4
then,
a fortiori
,
IL
satisfies A
9
. Accepting
that
IL
satisfies A
9
on the basis of (1)

(3) is obviously not
an argument in
favour of the inclusion of A
4
. At most, it only defuses any argument against it
based on the reasoning that, if
IL
did not satisfy A
9
, it would fail to satisfy A
4
as well. The inclusion of A
4
requires some positive support of its own, to
wh
ich we now turn.
According to A
4
, if
a
is informed that
p
then
p
is true. Can
this be right? Couldn’t it be the case that one might be qualified as being
informed that
p
even if
p
is false? The answer is in the negative, for the
following reason. Includin
g A
4
as one of
IL
axioms depends on whether
p
counts as information only if
p
is true. Now, some critics (
Colburn [2000]
,
Fox [1983]
,
Dodig

Crnkovic [2005]
and, among situation theorists,
Devlin
[1991]
) may still be unconvinced about the necessarily veridical nature
of
information, witness the debate between
Floridi [2004c]
and
Fetzer [2004]
.
However, more recently, it has
been shown in
Floridi [2005a]
that the
Dretske

Grice approach to the so

called standard definition of information as
34
meaningful data
12
remains by far the mos t plaus ible. In s hort,
p
counts as
information only if
p
is true because:
“[…]
false
information and
mis

information are not kinds of
information
–
any more than decoy ducks and rubber ducks are kinds
of ducks” (
Dretske [1981]
, 45).
“False information is not an inferior kind of information; it just is not
information” (
Grice [1989]
, 371).
As in the case of knowledge, truth is a necessary condition for
p
to qualify as
information. In
Floridi [2005a]
this is established by proving that none of the
reasons us
ually offered in support of the alethic neutrality of information is
convincing, and then that there are several good reasons to treat information
as encapsulating truth and hence to disqualify misinformation (that is, “false
information”) as pseudo

inform
ation, that is, as not (a type of) information at
all. The arguments presented there will not be rehearsed here, since it is
sufficient to accept the conclusion that either one agrees that information
encapsulates truth or (at least) the burden of proof is
on her side.
Once the veridical approach to the analysis of semantic information
is endorsed as the most plausible, it follows that, strictly speaking, to hold
(exchange, receive, sell, buy, etc.) some “false information”, e.g. that the
train leaves at 1
1.30 am when in fact it leaves at 10.30 am, is to hold
(exchange, receive, sell, buy, etc.) no information at all, only some semantic
content (meaningful data). But then,
a
cannot hold the information (be
informed) that
p
unless
p
is true, which is precise
ly what A
4
states. Mathew is
not informed but misinformed that Brazil lost the world cup in 2002 because
Brazil won it. And most English readers will gladly acknowledge that
12
Other philosophers who accept a truth

based definition of information are
Barwise and Seligman [1997]
and
Graham [1999]
.
35
Matthew is informed about who won the world cup in 1966 only if he holds
that Engl
and did.
The mistake
–
arguing that
a
may be informed that
p
even if
p
is
false, and hence that
IL
should not satisfy A
4
–
might arise if one confuses
“holding the information that
p
”, which we have seen must satisfy A
4
, with
“holding
p
as information”, w
hich of course need not, since an agent is free
to believe that
p
qualifies as information even when
p
is actually false, and
hence counts as mere misinformation.
As far as A
4
is concerned, “knowing that
p
” and “being informed
that
p
” work in the same way
. This conclusion may still be resisted in view of
a final objection, which may be phrased as dilemma: either the veridical
approach to information is incorrect, and therefore
IL
should not satisfy A
4
,
or it is correct, and therefore
IL
should satisfy A
4
,
yet only because there is no
substantial difference between
IL
and
EL
(information logic becomes only
another name for epistemic logic). In short, the inclusion of A
4
among the
axiom schemata qualifying
IL
is either wrong or trivial.
The objection is inter
esting but mistaken. So far,
IL
shares all its
axiom schemata with
EL
, but information logic allows truth

encapsulation
without epistemic collapse because there are two other axiom schemata that
are epistemic but not informational. This is what we are goin
g to see in the
next section.
2.3
No reflectivity
:
IL
does not satisfy A
6
, A
8
Let us begin from the most “infamous” of
EL
axiom schemata, namely A
6
.
One way of putting the argument in favour of A
4
and against A
6
, is by
36
specifying that the relation of “inf
ormational accessibility”
13
H
in the system
that best formalises “being informed/holding the information that
p
” is
reflexive
without being
reflective
, reflectivity being here the outcome of a
transitive relation in a single agent context, that is, “introsp
ection”, a rather
more common label that should be used with some caution given its
psychologistic overtones.
If
H
were reflective (if the informational agent were introspective),
IL
should support the equivalent of the
KK
or
BB
thesis, i.e.,
Ip
IIp
.
How
ever, the
II
thesis is not merely problematic, it is unjustified, for it is
perfectly acceptable for
a
to be informed that
p
while being (even in
principle) incapable of being informed that
a
is informed that
p
, without
adopting a second, meta

informationa
l approach to
Ip
. The distinction
requires some unpacking.
On the one hand, “believing” and “knowing” (the latter here
understood as reducible to some doxastic relation) are mental states that,
arguably, in the most favourable circumstances, could impleme
nt a
“privileged access” relation, and hence be fully transparent to the agents
enjoying them, at least in principle and even if, perhaps, only for a Cartesian
agents. Yet
KK
or
BB
remain controversial (see
Williamson [1999]
,
Williamson [2000]
for arguments against them). The point here is that
defenders of the inevitability of the
BB
or
KK
th
esis may maintain that, in
principle, whatever makes it possible for
a
to believe (or to know) that
p
, is
also what makes it possible for
a
to believe (or to know) that
a
believes (or
knows) that
p
.
B
and
BB
(or
K
and
KK
) are two sides of the same coin. Mo
re
precisely, if
a
believes (or knows) that
p
, this is an internal mental fact that
could also be mentally accessible, at least in principle, to a Cartesian
a
, who
13
The choice of the letter
H
is arbitrary, but it may graphically remind one of
the
H
in Shannon’s famous equation and in the expression “holdi
ng the
information that
p
”.
37
can be presumed to be also capable of acquiring the relevant, reflective
mental state of bel
ieving (knowing) that
a
believes (or knows) that
p
.
Translating this into information theory, we are saying that either there is no
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