Induction without

Probabilities

John D. Norton
Center for Philosophy of Science
Department of History and Philosophy of Science
University of Pittsburgh
This page at www.pitt.edu/~jdnorton/Goodies

Elsewhere, I have urged that there is no universal logic of

inductive inference. Rather the inductive logic appropriate to a
given domain is fixed by the particular facts that that prevail Elsewhere?
Where!? See "A
there. It is easy to give quick and simple examples of this
Material Theory of
sort of thinking. The trouble is that these examples are just Induction." "A
that: they are quick and simple, so the whole thing looks Little Survey of
frivolous. What's needed is a weightier example that shows Induction."
more clearly what I have in mind. That is what I'm going to
lay out here.

It is possible to have non-probabilistic,

indeterministic systems in physics. These are systems
whose past state fails to fix their future states and does
not even give probabilities for the different possible
futures. I don't assume that there really are such systems.
All I assume is that they are a perfectly respectable
possibility and that, if we lived in a world with them, we
would expect to be able to infer inductively about them.
The principal claim is that their odd physics determines a
correspondingly odd, non-probabilistic logic of induction.

The story will unfold using a particular example of an

This example is developed in indeterministic system, "the dome." That example is
Section 8.3 of "Probability used solely for concreteness and expository
Disassembled" and in
"Induction without convenience. Nothing in the general argument
Probabilities." depends on the details of the dome example.

1. Picking the Right Inductive

Logic

If one idea about inductive inference has taken root in

recent philosophy of science, it is that inductive inference
and probability theory are intimately connected. Indeed
many people seem to believe that probability theory
provides the One True Logic of induction. Probability
theory didn't start out that way. It was originally developed
as a physical theory, the theory of chances associated with
gambling. Very soon, it was noticed that probabilities
 behaved just like we'd want degrees of belief to behave.
The connection to inductive inference was made.

We now tend to use the word "probability" in two senses

reflecting these two uses. Sometimes it refers to a physical
property of a system. There is probability of a half that a
coin tossed fairly will come up heads. Sometimes it refers
to our degree of belief. I may entertain a probability of half
that it will rain today, so I take an umbrella.

There are occasions in which we will want to combine the

uses. What should my degree of belief be that a coin
tossed fairly comes up heads? The objective probability--
the chances--are a half. So it seems inescapable that I
should set my subjective, probabilistic degree of belief
equal to that same half. In his celebrated "A Subjectivist's
Guide to Objective Chance," David Lews laid out his
"PrincipalPrinciple." In effect it says just that. If there are
objective chances to be had, your subjective probabilities
should match them.

Look at what just happened. We want to know what our

degrees of belief about a coin toss should be. We notice
which physical facts goven the coin toss. Those physical
facts then determine what our degrees of belief should be.
Indeed the notion is strong enough to determine not just
what the degrees of belief should be, but what rules
govern them. The objective chance of getting a head or a
tail is just the sum of the objective chances of each
outcome individually. So our subjective degree of belief of
getting a head or a tail must be the sum of our subjective
belief in each individually.

The Moral

That seems to me to be just one illustration of a broader

idea. The physical facts that prevail in some domain fix the
inductive logic appropriate to that domain. That this
circumstance is universally so is the central idea of the
"material theory of induction." It says that inductive
inferences are not ultimately licensed by universally
applicable logical schemas, as is the case in deductive
inference. Rather, when we seek the grounding of an
inductive inference, our search ends in material facts
that hold only locally. In the case of the coin toss, it ends
in the stochastic properties of tossed coins. Our degrees of
belief ought to conform to the probability calculus just
because the physical chances of the coin tosses conform to
that same calculus.
Or another example: once we know the mass of one
(or a few) electrons, we know the mass of them all.
But once we know the mass of the star that is our sun,
we certainly don't know the mass of all stars. Why
does the inference from "one..." to "all..." work in
The "generally" has to be
one case but not the other? It is the same added. Another type of
inference form. Both infer from "one..." to "all..." It fundamental particle, the
works where it does because of the prevailing facts. neutrino, comes in varieties
Electrons are factually like that. They are fundamental that turn out, on recent
particles and fundamental particles of one type research, to have different
masses. The "generally" is
generally all have the same mass. Know the mass of
important. It makes the
one and, generally, you know the mass of them all.
inference inductive.
That fact licenses the inference. Stars are not Otherwise we'd just deduce
governed by the same facts, so the same inference the mass of all electrons
form cannot be used for their masses. from one.

2. The Dome: Indeterminism

Without Probabilities

Let's try for a more interesting example. The dome--as

described elsewhere in these pages--is an indeterministic,
Newtonian system. A mass sits at the apex of a dome. In
full conformity with Newtonian theory, it may remain there
for ever; or it may spontaneously move away from the
apex in any direction at any time T. It is a true case of
indeterminism. There are no hidden processes that I'm
ignoring that might fix the time T of spontaneous motion.
The mass is not bumped off the dome apex by unnotice
vibrations in the dome; or by enough random collisions
with air molecules to dislodge it. The example is
sufficiently idealized so that no such processes are at
work. It is manifestation of a less recognized fact about
Newtonian physics. Some of its systems are just
indeterministic.

For present purposes, however, the important fact is this:

Newtonian theory provides no probabilities for the time of
spontaneous motion T or the direction of the motion. It
merely says spontaneous motion at this or that time T is
 possible. This is such an essential point that it is worth
repeating. Newtonian theory provides no probabilities for
the time of spontaneous motion T or the direction of the
motion.

So the question is: given that we know the mass is at rest

at time t=0, what should our beliefs be for spontaneous
motion at any given later time?
For experts: nothing essential in the claim about induction depends on
the details of the dome. We could use just about any indeterministic
system. For example there are many Newtonian "supertask" systems that
manifest spontaneous motions. Or, there is a simple recipe for making an
indeterministic system. Take any theory with an interesting gauge
freedom, i.e. one whose gauge degrees of freedom can change in time.
Now declare that there is a factually true gauge. Since the signature
of a gauge freedom is that the theory's equations fail to fix the gauge
quantities, we now have an indeterministic theory in which future facts
are not fixed by all past facts.

3. Inductive Inference
about Radioactive
Decay

Before we tackle the case of the dome, let's look at a more

familiar case in which everything goes pretty much as
you'd expect. Indeed it looks very much like the dome,
so a default supposition might well be that both deserve
the same analysis. Consider a radioactive atom, which has
a fairly short half life. The atom will just sit there doing
nothing and then, without any immediate trigger,
spontanously decay, just as the mass on the dome
spontaneously moves.

The time at which the atom decays is governed by the law

of radioactive decay. For present purposes, the one detail
in it that we care about is that the law depends upon a
time constant τ that is a distinctive property of each
radioactive element. The time constant τ sets the overall
time scale for the decay. A big τ means that we are likely
 to wait a long time before the decay happens. A small τ
means that we are likely to wait a short time before the
decay happens.

The precise formula that governs these times is this

expression for P(t), the probability of decay sometime
over time t:

P(t) = (1 - exp(-t/τ))

Plotting the probability P(t) against t gives this curve for

the case of a time constant τ=1. When t is very small--say
t = 0.1 = one tenth of τ--then the probability that decay
has happened is small. Do the sums and it comes out at
about 0.1. When t is large--say t = 5 = five times τ--the
probability that decay has happened is large. Do the sums
and it comes out at about 0.99.

The time constant is closely related to the half-life of the

atom.
The half life t1/2=τ ln 2.

An Inductive Inference
Problem

So now let's pose an inductive inference problem

concerning this atom. We'll have an hypothesis H and
evidence E:

E: At t=0, the atom has not decayed

H: The atom decays sometime in the ensuring time
interval t.

The quantity we want is

[H|E] = the degree of support evidence E lends to
hypothesis H

What should [H|E] be? This is a case covered by Lewis'

"Principal Principle." We have physical chances, P(t). So
our subjective propbabilities ought to match. That is, we
would set

[H|E] = P(H|E) = P(t)

where the notation P(H|E) is a sort of hybrid notation that

says that our degrees of support are probabilities.
 Under this identification, our beliefs move in concert
with the probabilities. For the case of τ=1 shown in the
graph, we have little belief that the atom will decay in the
first 0.1 units of time; but 0.99 belief that it will once 5
units have elapsed.

4. How NOT to Infer Inductively

Over the Dome

Consider the analogous inductive inference problem

for the dome. We'll have an hypothesis H and evidence E:

E: At t=0, the mass is motionless at the apex.

H: The mass begins to move sometime in the ensuring
time interval t.

What is the degree of support [H|E] that E lends to H?

Many find in the analysis of the radioactive decay of an

atom a template that they cannot resist applying to the
dome. The law of radioactive decay has an important
property. It is the unique decay law that has a "no
memory" property. If the atom has not decayed after 1
time unit, or 5 time units, or 10 time units, or whatever,
then the probability of decay in the next unit of time
always comes out to be the same. The atom does not
remember how long it has been sitting, when the next
time unit comes along.

That looks very promising. The distinctive thing about the

physics of the dome is that it also has this "no memory"
property. Whether the spontaneous motion happens at
some moment is quite independent of how long the mass
has been sitting at the apex. So why not set our degrees
of support [H|E] equal to probabilities governed by the
same formula as in the law of radioactive decay?

Why not? The reason is that time constant τ. Any

instance of the law of radioactive decay has a time
constant τ in it. And that time constant exercises a
powerful influence on chance of the spontaneous event. To
see this, here are graphs of P(t)=P(H|E) for values of
τ=0.1, τ=1 and τ=10:
 When τ is small, then we become near certain of the
spontaneous event very soon, say within t=0 to t=1. If τ is
large, then we expect to wait a long time before the
spontaneous event occurs.

So what's the problem? Nothing in the physics of the

dome fixes a time constant or any sort of time scale for
the spontaneous motion of the mass. The physics is
completely silent on how soon the motion may happen. It
just says "it's possible." So if we are to use the
probabilistic formula, we must add a time scale. The
contradicts the basic spirit of the "Principal Principle" and
the more general idea that the physics present should
dictate the inductive logic. If we want to use the
probabilistic law, we have to add more physical properties
to the system than the system has. And that just seems
wrong.

All we want to do is reason inductively about a system; we

want to be inductive logicians. But somehow we have
ended up as physicists, proposing new physical
properties that the system--by construction--does not
have.

Something has gone very wrong and, in my view, what has

gone wrong is quite simple. We are trying to force the
wrong inductive logic onto the dome.

An Improper Proposal

There's a loophole I need to address. The probability

formula of the law of radioactive decay is the unique
 probability law with the "no memory" property that the
dome also has. So if any probability formula would work
for the dome, that one would have to be the one.

Statisticians sometimes stretch the rules and use

probability distributions that aren't really probability
distibutions. One that could be used here is a uniform,
improper distribution. It would assign the same small
amount of probability ε to every unit time interval as
shown in the graph below. It is "improper" since the
probability assigned to all the unit time intervals taken
together is not one, as the probability calculus
demands, but it is infinity. While this seems a fatal problem, it
turns out that if you are careful about how you use them, these improper
distributions need not cause serious harm. But that is a topic for another
place.

Tempting as this improper distribution may be, it suffers

the same problem as the proper distribution. It still adds
physical properties to the dome. For a consequence of it is
that spontaneous motion in time t=1 to t=2 has probability
ε and spontaneous motion in the time interval t=2 to
t=4 is 2ε. Motion in the one interval is twice as
probable--no more no less--than motion in the other. But
nothing in the physics licenses this precise judgment. All
the physics says is that motion in each interval is
"possible"--and that is all.

Once again, we have passed from being inductive logicians

to being physicists, adding more physical properties to the
system than Newton's theory has already given it.

5. How to Infer Inductively Over

the Dome

The probability calculus is the wrong calculus to use as an

inductive logic for the dome. So what is the right one?
How could we know? We've already seen how it can go in
another case. Radioactive decay is governed by chances
and we can let those chances pick out our inductive logic.

We can do exactly the same thing with the dome. However

the physics of the dome is more impoverished than that of
radioactive decay. So we are going to get a more
impoverished logic. It is a somewhat mechanical
exercise to read the relevant inductive logic from the
physics. To do this, let's define

E: the mass is at rest at the apex of the dome at t=0.

H(t1,t2): The spontaneous motion happens in the time
interval t=t1 to t=t2,
where we write "(t1,t2)" and shorthand for that interval of time. So
H(10,20) just says that the spontaneous motion happens sometime in
t=10 to t=20.

Then we just take what the physics says and translate it

into the logic using the same sort of transformations as we
did in the case of radioactive decay. The chance of decay in
t=5τ is 0.99; so our degree of belief in that decay is 0.99.
However the indeterministic physics of the dome doesn't
give us real valued degrees. It actually says rather little. It
just says that a spontaneous motion in this or that time is
possible. That's it. No degrees of possibility: not 50%
possible, not 95% possible; and no comparative measures:
not more possible, less possible, twice as possible. Just
possible.

What it induces in the

What the physics says:
inductive logic:

The present state does not fix the future The inductive logic for the
(indeterminism). The physics just tells us that a support [A|B] of A from B has
future state is necessary, possible or impossible. three values: nec, poss, imp.

If the motion happens in (10,20), then it [ H(0,100) | H(10,20) ] =

necessarily happens in (0,100). nec

Motion in any later non-zero interval is possible, [ H(0,10) | E ] = [ H(0,100) |

given E: the mass is at rest at the apex of the E]=
dome at t=0. [ H(10,20) | E ] = … = poss

If the motion happened in (0,10), it is impossible

[ H(20,30) | H(0,10) ] = imp
in (20,30).

The table gives a few obvious illustrations for a more

general system. With a little reflection, you'll see that it is
fully generated by a few simple rules. They are:

The complete inductive logic of the dome

[ A|B ]
= nec if B entails A
= imp if B entails not A
= poss otherwise

6. Bayesian Response I: The

 Simulation Trick

There is a common Bayesian rejoinder. While the strengths

[A|B] of the inductive logic above are not conditional
probability measures, we are just a few lines of
mathematics away from conditional probability measures.
Consider any probability measure at all that is adapted to
the behaviors of the indeterministic systems through

P(A|B)
= 1, if B entails A
= 0, if B entails not A

and

0 < P(A|B) < 1 otherwise.

Every so adapted probability measure induces the logic of

Section 5 by the rule

[ A|B ]
= nec, if P(A|B) = 1
= imp, if P(A|B) = 0
= poss, if 0 < P(A|B) < 1

Obviously there are other ways to define the logic of

Section 5 in terms of probability measures; finding them is
simply a challenge to our ingenuity.

Has this sort of possibility shown us that the probability

calculus is the One True and Universal Inductive Logic after
all? It has certainly not. The inductive logic of
indeterministic systems are inherently non-additive.
The degree of belief assigned to each of two mutually
exclusive, contingent propositions is the same as the
degree of belief assigned to their disjunction. If probability
measures are to have any meaning as a logic of induction
at all, their additivity is their essence. We add the
numerical probabilities of two mutually exclusive outcomes
to find the probabilities of their disjunction.

What the exercise above shows is that we can take one

sort of inductive logic, one of additive measures, and use it
to simulate another, with the non-additive degrees.
Since the additive measures of probability theory are now
offered as devices for generating all other logics, they have
ceased to be used a logic in their own right. They have
been reduced to a useful adjunct tool of computation and
are no more the Universal Inductive Logic than is the
differential calculus.

Of course we can readily simulate the additive

measures of probability theory by other adjunct tools. A
trivial example is provided by complex valued,
multiplicative measures M, for which we replace the
additivity axiom of probability by a multiplication axiom:
for mutually exclusive outcomes A and B, M(AvB) =
M(A).M(B). These multiplicative measures can simulate
additive measures through the formula P(A) = log
Re(M(A)), so these multiplicative measures can replicate
any result achievable with additive measures. However
that fact in no way makes these new measures the
Universal Logic of Induction.

7. Bayesian Response II:

Subjectivism

The discussion so far has sought to present an intractable

problem for Bayesians of all varieties. Perhaps subjective
Bayesians specifically have an escape. They hold that
probabilities may be assigned subjectively, initially, but
that as we conditionalize on new evidence, the whim of our
individual opinions will be overwhelmed by the weight of
evidence. Why cannot a subjective Bayesian assign a
specific probability measure to the time of excitation? It
merely represents that Bayesian’s opinion and makes
no pretense of being grounded in the facts. Why does
that fail?

First there is a general problem with subjective

Bayesianism that is independent of this example. It
changes the problem. Our original problem was to discern
the bearing of evidence. That has been replaced by a
different problem: to express one’s opinion, in such a way
that eventually the bearing of evidence will overwhelm
opinion.

Second, in the context of the present example, the

subjective project fails by its own standards. The
subjective approach can only be relevant to an analysis of
the bearing of evidence if there is no way to separate out
mere opinion from the objective bearing of evidence in the
probability measures; and if we have some reason to think
that mere opinion will eventually be overwhelmed by the
weight of evidence. Neither obtain.

Here, the separation of whim and warrant can be

effected. The three-valued inductive logic (3) expresses
precisely what the evidence warrants. In so far as the
probability assigned goes beyond, it expresses mere
opinion. The translation from probability measures to the
three valued inductive logic (5) enables us to read
precisely how it goes beyond. Any probability that lies
strictly between 0 and 1 merely encodes the value poss.
Anything more, including the specific numerical value, is
opinion.

There will also be the familiar probabilistic dynamics

as we conditionalize on new evidence. However these
changing probabilities do not represent shifts in inductive
warrant. The new evidence will be the observation of
whether the excitation happens in successive time
intervals. In so far as the resulting shifts in probability
 assignments leave the probabilities strictly between 0 and
1, they amounts to pure shifts of opinion. If they force a
probability assignment of 0 or 1, the shift is deductively
generated, arising when the evidence deductively refutes
or verifies the hypothesis.

8. Some Concluding "Yes, But's..."

That's it. That's the system whose inductive logic is non-

probabilistic. Is it really that easy? I think so. The
inductive logic appropriate to any system is determined
by the facts prevailing in that system. The facts
governing the dome call for a much simpler inductive logic
than the probability calculus.

But you may have some nagging "yes, but..." worries.

Let's look at two.

Weird Physics

"That is all well and good, you say. The dome and other
indeterministic systems require a different sort of inductive
logic. Why should we care? Our world doesn't seem to
have these sorts of indeterministic systems."

That may well be the right thing to say when it

comes to inferring inductively about systems in Compare with deductive logic,
our world. However it does concede my main which has universal applicability.
We quite happily apply deductive
point. That--indeed, if--our world does not harbor
logic to weird systems like the
such indeterministic systems is a factual matter. dome. We don't say, "Oh that is a
So what is being accepted is that the applicability weird system, so modus ponens
of ordinary inductive logic depends upon certain does not apply." Should not a truly
facts about our world. That is what the material universal inductive logic apply to
theory of induction claims; there is no universal weird systems as well?
inductive logic.

The sort of indeterminism without probabilities might well

factually be the case for at least some processes in our
world. Perhaps the simplest arises in standard big bang
cosmology. Our cosmos emerged some 1010 and more
years ago from a singularity in spacetime. We remain
uncertain about whether that newborn cosmos came with
enough matter to halt cosmic expansion and lead to a big
crunch; or whether the density of matter was too small to
halt the expansion, which will go on forever.

Standard cosmology tells us that many initial densities are

possible, but it provides no probabilities for the different
possibilities. It is a natural temptation to try to assign
probabilities to the different values of matter density.
However to do so would add to what the physics tells us.
 Now that we know that our inductive logic depends upon
at least some facts, are we so sure that the only pertinent
facts are exotic ones like the absence of indeterministic
systems? Might not other less exciting facts also be
pertinent to our selection of the right inductive logic so
that multiple inductive logics might well be appropriate to
different domains in our ordinary experience?

No Physics

Here's another "yes, but..." "All this is fine for

inductive inference on physical systems whose properties
are fully specified," you may say, "but why should that sort
of thinking work in real inductive inference, that is,
inductive inferences in which we have to deal with the
uncertainties of the real world?"

I'm not moved by this worry. I do admit that the example

is a little contrived. That was the price paid for an example
in which we have complete control of all relevant facts
that govern the ways that uncertainty enters into our
analysis. For a complete control of those facts then makes
it quite easy to see just which inductive logic is
appropriate. In real world cases of inductive inference,
those facts that govern the uncertainties are less clear to
us. But that sort of muddiness seems no reason to me to
think that things are any different. There still are facts
governing our uncertainties and they will still dictate the
appropriate logic.

It will, however, be harder for us to see precisely what that

logic is because we are uncertain of the the pertinent
facts. Indeed it all seems to be working out just as it
should. In real world cases, we do struggle to see just
which is the right inductive logic to be applied. That is just
what you'd expect from the material theory of induction
when we are unsure of the facts that govern the
uncertainties. We routinely misdiagnose this problem,
however, as our continuing failure to have lighted upon
just the right universal logic of induction. The assured
failure of our quest for this One True Logic of Induction in
turn, it seems to me, go a long way in explaining why
induction has perennially been such a murky topic.
Copyright John D. Norton. September 18, 2006; March 27, 2007; November 11, 2008.