Popper’s Deductive Predictions, Ceteris Paribus Clause

and AGM Belief Revision

Frank Zenker

Calgary, October 2006

04/18/23 2

Two Parts The Thread example

Prediction as a form of Deductive Entail-ment, I & T -> P

Critique

Identifying relevant factors, I, produces a ceteris paribus clause, in testing T.

CP & I & T -> P

Belief revision (AGM-style) on consistent, deductively closed proposition-set Cn(K).

Epistemic Entrench-ment/Minimal Change

Revision of initial conditions in AGM.

Mercury Anomaly

Part IPopper‘s Thread Example

(1) I & T -> P

(2) ~P Ergo ~ (I & T) Falsification (m.p)

(2’) I & ~P Ergo ~T T-Falsification (m.p)

(2’) I & T & ~P Ergo interfering factor Immunization

04/18/23 4

Outline Part I T (Popper) A predicted event is deducible

from laws and initial conditions (alone).

Objection (Canfield and Lehrer): You need laws initial conditions and COMP. Thus, not T. Plus, you can’t have COMP.

Rebuttal (Stegmüller): COMP not needed still. COMP already implied in initial cond.

COMP as list of interfering factors, ceteris paribus clause in theory testing.

04/18/23 5

Popper 1959, § 12 “To give a causal explanation of an event

means to deduce a statement describing it, using as premises of the deduction one or more universal laws, together with certain singular statements, the initial conditions. (…) It is from universal statements in conjunction with initial conditions that we deduce the singular statement. (…) We call this statement a specific or singular prediction.“

04/18/23 6

Popper (1959)

We can say that we have

given a causal explanation of

the breaking of a certain

piece of thread if we have

found that the thread has a

tensile strength of 1 lb. and

that a weight of 2 lbs. was

put on it.

2 lbs.

1 lb.

Objection

Canfield & Lehrer (1961)

04/18/23 8

Enter the Magnet Canfield & Lehrer (1961)

1) L (Tx Wx, Bx) 1´) L (Tx Wx, Bx)

2) Tx Wx 2´) Tx Wx Mx

3) Bx 3´) ---

L = Law

T = Thread

W = Weight

M = Magnet

Mx is consistent with Tx Wx, but with an interfering factor, we would not predict Bx.

04/18/23 9

Reductio this “(…) shows that in prediction the event

E predicted is not deducible from the laws and initial conditions that are the premises of the prediction.” CL (1961:206)

Laws must contain completeness assumptions, COMP, consistent with the antecedent of the law, but inconsistent with the consequent.

Rebuttal

(Coffa 1968)

Lakatos 1978

Stegmüller 1983

04/18/23 11

Stegmüller (11969, 1983:189) “But it is not necessary to raise this

condition [COMP, FZ], to begin with.”

“If a DN-systematization is correct and that means complete, then the negations of those statements which describe ‘interfering factors’ must follow from the class of singular premises, possibly from the ‘global’ hypothesis that the system is closed.”

04/18/23 12

Well Just How

do these negations of statements which describe interfering factors follow?

‘Follow’ in the sense of ‘logically follow’

04/18/23 13

Setup I = Initial Conditions

T = Theory

P = Prediction

F = Interfering Factor

Fm = Magnet

(1) I & T -> P

(1’) I & T -> P & ~F ?

Since (2) I -> ~Fm?

How does

‘is a weight of 1 x-unit loaded onto a rope of 0,5 x-unit tensile strength’ imply the absence of, e.g. a magnet?

04/18/23 14

Back to Popper (1959)

We can say that we have

given a causal explanation of

the breaking of a certain

piece of thread if we have

found that the thread has a

tensile strength of 1 lb. and

that a weight of 2 lbs. was

put on it.

2 lbs.

1 lb.

04/18/23 15

What is Weight? Weight = Mass (of Object) x KG

Acceleration (via Gravity) m/s2

Where, at the earth’s surface, Gravity = 9.8 m/s2, unit for mass is the KILOGRAM

“The word ‘weight’ denotes a quantity of the same nature as a ‘force’ [N]”. (Comité International des Poids et Mesures, 15 October

1887)

04/18/23 16

What is Tensile Strength?

The limited capacity

to retain structural

integrity, expressible

as a directed force.

Second axiom

F = ma

2 kg

1 kg

04/18/23 17

Force A force of one

NEWTON will accelerate a mass of one KILOGRAM at the rate of one meter/second per second.

2 kg

1 kg

1 NEWTON

2 NEWTONS

Sum: 1 NEWTON, downwards

04/18/23 18

F = ma The force that the thread-suspended

object is subject to in the downward direction depends on its acceleration, i.e., the strength of the gravitational field assumed in this application.

With a magnet on top, the value for mass remains constant, but thereis an additional force. 2 kg

04/18/23 19

Identifying relevant factors If I identify the object below the string as

being of a given weight (rather than a mass), I assume the object is subject only to the force of the gravitational field.

In this sense, being an object of a given weight (vs. mass) precludes the presence of a magnet, just as the force exerted by a now-present magnet would falsify the net-force exerted onto the object.

04/18/23 20

vs. ├ One can always add

forces and, thus, change the net sum of the system.

In a deductive recon-struction, however, adding a hypothesis is useless.

A ├ B A C ├ B2 kg

1 kg

1 N

2 NEWTONS

1,2 N

04/18/23 21

Hence, It is not the initial conditions that exclude

the presence of interfering factors.

Rather, exclusion of interfering factors is a function of having formed the sum, i.e., of having assumed a particular finite number of relevant factors (CP).

Assumption is relevant in theory testing, but has no logical form in I & T -> P.

04/18/23 22

Calculating P vs. Testing C The schema

‘I & T -> P’ reads:

‘If I and T are true, then P’.

This is calculating P.

If one adds factors, I and T remain true.

The schema ‘CP & I & T -> P’ reads

‘If I & T is all that is true, then P’.

This is testing T.

Hypotheses, H, make CP false and imply ~P.

‘CP & I & T -> P’, ‘~ P’ Ergo ‘~CP & H’

Part II

04/18/23 24

Outline Epistemic Entrenchment -- AGM axioms

Rationality postulate -- Minimal Change

Applied to the Test of one Empirical Theory

Law: x [Fx -> Gx], Instance: Fa -> Ga

Initial values: Fa, Ceteris paribus clause: CP

Falsific./ Immuniz.: Aux-Hypotheses

Case: Mercury‘s perihelion shift.

AGMAlchorrón, C., Gärdenfors, P. & Makinson, D. (1985). On the

logic of theory change. Partial meet contraction functions and their associated revision functions. Journal of Symbolic Logic 50, 510-530.

Peter Gärdenfors (1988). Knowledge in Flux. Cambridge: CUP.

--- (2000). Conceptual Spaces. Cambridge: MIT Press.

04/18/23 26

AGM (late 1980s) State of Knowledge/Belief, K, as deduct-

ively closed set of propositions, Cn (K)…

…with an EE-ordering that determines comparative retractability (Rott) of items.

Prioritized belief revision. The new and inconsistent belief/ propositions, ~p, must be integrated in a conservative way.

04/18/23 27

AGM Revision Functions K + p = Cn(K {p}) [Expansion]

Whatever is consistent can be integrated.

K p = Cn(K {p}) [Contraction]If p is inconsistent with Cn(K), the least entrenched item in K is deleted „to make room“ (minimal change principle).

K * p = Cn(K {p}) + p [Revision]Contraction before revision.

04/18/23 28

EE-Postulates (EE1) Transitivity

For any A, B and C, if A ≤ B and B ≤ C, then A ≤ C.

(EE2) Dominance

For any A and B, if A |- B, then A ≤ B

(EE3) Conjunctiveness

For all A and B in K, A ≤ A B or B ≤ A B.

(EE4) Minimality

When K = K, A K iff A ≤ B for all B

(EE5) Maximality

If B < A for all B, then |- A.

04/18/23 29

Applying the EE-Postulates Yields an ordering exclusively determined

by deductive relations in K.

If one were to map law, x [Fx -> Gx], instance, Fa -> Ga, and initial values, Fa, accordingly, what ordering do we get?

What determines the identity of the least entrenched belief, i.e., of the deletion-candidate?

04/18/23 30

Handout

04/18/23 31

Handout

04/18/23 32

Thus Both an immunization and a falsification

strategy can be described as obeying the minimal change principle.

The belief to be given up depends only on the pre-ordering. Thus, minimal change is general…and empty.

AGM epistemic entrenchment does not model tenacity of a law (Feyerabend).

Ceteris Paribus

1. Fa -> Ga, Fa therefore Ga

2. CP (Fa -> Ga), Fa & CP therefore Ga

04/18/23 34

Pre-Conceptions We read ‚ceteris paribus‘ propositionally,

i.e., as all other things being equal (absent, negligible).

CP as infinite non-existence statement.

It is never the case that all other things are equal. Moreover, equal to what?

However, we cannot test a theory without taking a verdict on the CP clause.

04/18/23 35

CP as „hypotheses-storage“ If a prediction fails, one can reject CP and

„draw auxilliary hypotheses from it“.

The CP clause is usually not well-corroborated. Read: It is least entrenched!

The deductive schema of theory testing is a schema created by criticism. It eliminates the auxilliary hypotheses and relegates them to background knowledge.

Lakatos (1978)

Mercury Anomaly

Roughly 1857 through 1915

04/18/23 37

Completeness Clause CP & Fa & (Fa -> Ga) therefore Ga.

Every application of an empirical theory T must assume (the truth of) a provisoe that, in a given case, no factors other than those specified in the antecedent(-statement) are present which could affect the event described in the consequent(-statement).

Hempel (1988)

04/18/23 38

Example

Deterministic law of succession

Stellar-System at t1

System at t2

predicted

„observed“

unaccounted factor

04/18/23 39

A very small deviance Mercury orbits the sun once in three months.

Mercury‘s perihelion takes one Platonic year (260 000 years) and describes a „rosette“.

Mercury

PerihelionSun

04/18/23 40

Partial Explanation: 39’’ missingGravitational influence on Mercury

Venus 280’’.6 Jupiter 152’’.6

Earth 83’’.6 Saturn 7’’.2

Mars 2’’.6 Uranus 0’’.1

Total 526’’.7 Observed: + 39”

(Le Verrier 1859:11, after Roseveare 1982:11).

04/18/23 41

Stick to the law!

There is a rule in philosophy… that admits

of no dispute… we are never entitled to

challenge the universality of laws that,

within our experience, have nowhere failed

– until every other mode of overcoming the

difficulty has proved of no avail.

Nichol (1848:63f.), after Musgrave (1976)

04/18/23 42

The classic Immunization Example

Obeying normal-science conservativity, matter hypothesis exploiting pertubation were forwarded.

(i) Vulkan, (ii) belt of asteroids, (iii) matter distribution that causes zodiac-light.

Each is less testable than its predecessor.

As late as the 1950s, „matter was sought“, though Einstein gave the correct orbit.

04/18/23 43

Matter-Hypotheses

Vulkan?

AsteroidsZodiac-Light

CP as least entrenched belief -- AGM-style

Testing a Theory vs.

Calculating a Prediction.

04/18/23 45

CP as least entrenched beliefK Cn(K)

n p v ~p

+1 x Fx->Gx Fa -> Ga

+2 Fa

+3 CP CP & Fa & (CP & Fa ->Ga) Ga

04/18/23 46

Step by Step Prediction

CP & Fa & (CP & Fa -> Ga) |- Ga. Observation

Fa & Ga‘ (where Ga‘ |- ~Ga) Question

CP? (where ~CP |- ~Ga) Immunization

~CP (where H |- ~CP) New Prediction

CP2 & H & Fa & (CP2 & Fa & H -> Ga‘) |- Ga‘

04/18/23 47

Merits The CP clause‘s role as a necessary

completeness assumption in theory-testing finds a logical representation.

New hypotheses „drawn from“ CP, raise its index, indicating severity of an anomaly.

With CP appearing at the level of the applied law-instance, tenacity of law vs. applied instance can be respected (Klee).

Thank you.

mail@frankzenker.de