computational coherence daniel schoch chiang mai university department of economics

Computational Coherence

Daniel SchochChiang Mai UniversityDepartment of Economics

Computational Coherence Daniel Schoch

Content

1. Terminology

2. Coherentist Epistemology

3. Inferential Coherence

4. Thagard’s Computational Model

5. Qualitative Decision Theory

6. COHEN


1. Terminology – Belief System

A Belief System is a consistent set S of propositions.

It is normally considered to be deductively closed, i.e. any sentence p, which can be derived from a subset S’S, is contained in S.

Propositions p not in S are called disbelieved (not to be mixed up with ¬p being believed).


Terminology – Evidences

In epistemology, two kinds of propositions are distinguished:

Evidences: A singular statement describing a basic belief, e.g. an observation or an utterance of some person.

Non-evidential beliefs: A general statement having the power to explain evidences, e.g. a theory or a hypothesis.


Terminology - Inference

In Coherence Theory we consider a general inference relation between propositions,

p1,…,pn -> p Logical deduction “” is a special case of

inference, others might be induction, explanation, etc. Note that “deduction” in the sense of Sherlock Holmes is abduction.

Propositions can be isolated or inferentially connected: Some propositions p1,…,pn are inferentially connected, if p1,…,pk-1,pk+1,…,pn-> pk holds for some k.


2. Coherentist Epistemology

Two epistemological paradigms compete:Fundamentalism: Justification based on

evidences: Given some evidences, how justified is the belief that p?

Coherentism: Taking all formerly beliefs and new incoming information, select some propositions to form the most plausible belief system.


Example

Consider the following two propositions: John is in Rome on Nov. 4th 2007, 14:30. John is in Bangkok on Nov. 4th 2007, 14:31.

The two propositions do not logically contradict, but together they are implausible (nobody can travel that fast), they are incoherent.

Even if both information came from reliable sources, we could not believe them simultaneously.


Basic Beliefs

The two epistemological paradigms differ in their treatment of evidences:

Fundamentalism: Evidences have a distinguished epistemological status from non-evidential beliefs. In particular, they are justified through the process of observation.

Coherentism: Evidential and non-evidential beliefs have the same epistemological status. They are mutually justified through their interferential connections.


Justification and Inference

Fundamentalism:

External Justification

Coherentism:

Internal Justification

p Perception that p

Inferencethat q

p q

Reliable process Pure Inference


Treatment of Evidences

Fundamentalism: It is decisive for evidences through which process they emerge. They are justified through the reliability of the process.

Coherentism: Evidences are regarded as if they spontaneously emerge in the mind of the subject. Reliability is purely subjective and only plays a subordinate role in justification.


Computational Justification

Fundamentalism: External Justification Needs Meta-beliefs

on reliability Probabilistic Bayesian NetsBovens, Luc, and Stephan

Hartmann, Bayesian Epistemology (Oxford:

Clarendon Press, 2003).

Coherentism: Internal Justification No meta-beliefs

required Gradational Inference algorithmPaul Thagard, Computational

Philosophy of Science (MIT Press, 1988, Bardford Book, 1993).


Revision of Evidential Beliefs

Fundamentalism: Only conditional

inference Bayesian Nets have

fixed input and output propositions

Evidential beliefs are given, no revision

Coherentism: Both conditional and

unconditional inference

Inference algorithm treats propositions equal

Evidential beliefs can be revised


3. Inferential Coherence

We assume that there is an abstract inference relation p1,…,pn -> p between propositions.

Inferential Coherence is a measure of plausibility of belief systems S, such that:

1. For every p1,…,pn,pS with p1,…,pn -> p, the degree of coherence rises.

2. For every p1,…,pn,pS with p1,…,pn -> ¬p, the degree of coherence sinks dramatically.

3. Inferential relations p1,…,pn -> p, p1,…,pn -> q from common premises p1,…,pn are better (more coherent) than from different.

4. Inferences from fewer premises are better (more coherent) than from more.


Explanatory Coherence

Thagard interpretes the inference relation p1,…,pn -> p as an explanation of p (explanandum) by the explanans p1,…,pn. The conditions then read:

1. A successful explanation increases coherence.2. An explanatory anomaly decreases coherence.3. A unified explanation by a single explanans is

better than two unrelated explanations.4. A simple explanation with fewer premises is

stronger and therefore has higher coherence.For condition 3 see Bartelborth, Thomas, Explanatory Unification, in:

Synthese 130 (2002), 91-207; for condition 4 see Schoch, Daniel, Explanatory Coherence, Synthese 122/3 (2000), 291-311.


Examples Explanation

1. A successful explanation of an evidence is performed by a theory and other evidences, e.g. a falling ball can be described by the law of gravitation plus air resistance.

2. An explanatory anomaly occurs, when an evidential belief contradicts (or incoheres with) some proposition, which is explained by other beliefs. E.g. Bohr’s atomic model contradicts Maxwell’s electrodynamics.

3. A unified theory brings together formerly separated parts of science, e.g. unification of sublunar (Galilei) and celestial (Kepler) physics by Newton.

4. A simpler theory supersedes a complicated one, e.g. Kepler’s theory needs less parameters than Kopernikus’.


Coherence and Belief

Two Perspectives of Coherence:1. Global (Belief Choice): Consider

competing total belief systems S1,…,Sn, choose the most coherent one.

2. Local (Belief Change): Given a belief system S and a proposition p, investigate whether a.) p can be accepted or not, andb.) S has to be changed or not.


The Dual Pathway Approach

The local perspective is exemplified in Thagard’s dual pathway model:


4. Thagard’s Computational Model

For simplicity, we take a set of (atomic) propositions p1,…,pN and consider only belief systems S which can be formed by the pi and their negation ¬pi, that is, we take S to be the deductive closure of a consistent subset of P:={p1,…,pN, ¬p1,…, ¬pN}.

Then each such belief system S can be described by assigning to every pk a truth value vk with

vk = 1, if pkS,vk = 0, if ¬pkS,

vk = ½, else.Thagard uses an artificial neural network model to

implement his theory of coherence.


Biological Neurons

A biological neuron has two states. If the sum of the electrical inputs from the dendrites exceeds a threshold, it transmits a signal to the output axon. Otherwise it remains inactive. Dendrites can be excitatory or inhibitory.


Computational Neurons

An artificial neuron is represented by a mathematical function, f(x1,…,xn), which is 1, if Σkwk·xk>S, and 0 else. Here, w1,...,wn are the weights and S is the threshold. Positive weights represent excitatory, negative weights represent inhibitory connections.


The Neural Network Model

Thagard uses an Artificial Neural Network Model to implement a measure of coherence:

Each proposition corresponds to one neuron. Coherence between two propositions correspond to

excitatory relations between the corresponding neurons.

Incoherence between two propositions correspond to inhibitory relations between the corresponding neurons.

In contrast to nature, Thagard’s Neuronal Nets only have symmetric connections (Hopfield model)!


Hopfield Contra Nature

Biological Neural Net

Asymmetric recurrent network, very difficult to understand.

Hopfield model

Symmetric, thus theoretically well understood.


Explanatory Relations

Thagard reduces each explanatory relation to binary relations between neurons.

If ‘p1, . . . , pm explain p’, then:

a.) For each i, pi and p cohere.

b.) For each i, j, ij, pi and pj cohere.

c.) The degree of coherence is inversely proportional to m.


Critique

Explanatory and competitive relations can not be reduced to relations between two propositions only.

For example, if three propositions {p,q,r} are inconsistent, nevertheless each pair {p,q}, {p,r}, and {q,r} can nevertheless be consistent.

If the rule ‘p,q explains r’ is reduced to coherence of the three pairs, it is undistinguishable from ‘p,r explains q’.

Thus the Hopfield model can not adequately represent explanatory coherence.


5. Qualitative Decision Theory

Qualitative Decision Theory (QDT) deals with preferences over certain propositions or goals described by propositions.

For example, conditional constraints can be goals in this sense.

A coherentist constraint could be that if p and q incohere, they should not both be true.

Coherence Theory can be considered a QDT- maximization problem over constraints: Find the belief system which satisfies the most constraints.


Quantitative Representation

Although QDT deals with qualitative aspects, it could nevertheless be represented quantitatively, just as preferences over qualitative alternatives can have numerical utility representation.

This makes the coherence problem easily tractable: one has to find the valuation v1,…,vN for the propositions p1,…,pN which maximize the coherence measure.


Additive Representation

The most straightforward representation for a multi-factor utility is additive.

We assume that the total coherence is the sum of all coherence measures for the individual constraints of our coherence theory.

Additive representation guaranties some nice invariance properties.


Quantitative Coherence

Consider, for example, the inference relation p1,…,pn -> p: If p1,…,pn are true, then p must also be true.

Assume p1,…,pn are true:If p is indeed true, a term of positive

coherence is added.But if p is false, a negative

punishment term occurs.


Fuzzy Logic

To be more precise, we choose a system of fuzzy logic with values between 0 (false) and 1 (true): Negation is represented by the function 1-x, conjunction is represented by multiplication:

If vp is the value of p, then v¬p := 1 - vp is the value of ¬p.

If vp is the value of p, and vq is the value of q, then vp&q := vp · vq is the value of p&q.

This is a valid system of fuzzy logic according to Gottwald.

Gottwald, Siegfried, Fuzzy Sets and Fuzzy Logic, Vieweg, 1993.


Fuzzy Coherence

Now we can specify the value function for the inference rule p1,…,pn -> p:

For the case of p1,…,pn incohering, we obtain

We see that the latter is a special case of the former for vp=0!

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6. COHEN

Coherence Optimization of Hypotheses Explanatory Nets

The program COHEN accepts as an input a list of weighted inference rules of the form

[w:] p1,…,pn -> p

Here, w is an optional number giving the weight of the rule.


COHEN Syntax

Each rule p1,…,pn -> p

is regarded as an explanatory relation. Negation is designated by an ! prefix. An incoherence/competitive relation between p1,…,pn

is written as p1,…,pn ->

which is interpreted as p1,…,pn explaining a contradictory statement.

Evidential support is written as an explanatory relation without premises,

-> p


Termination Conditions

The program stops when one of the following events occur:

The net is exactly stable (within standard numerical accuracy).

The net is approximately stable.There is no further progress in

coherence.


Lavoisier’s Oxygen Hypothesis (1)

Around 1785, two competing theories explained chemical combustion and calcination processes:

Phlogiston Theory (Becher 1667, Stahl): Phlogiston was considered an element contained within combustible bodies, and released during combustion and calcination.

Oxygen Theory (Lavoisier 1775-77):Lavoisier formulated the principle that every reaction preserves mass, and observed weight increase during calcination, which he explained by absorbtion of a substance he called oxygen.

Thagard, Paul, Explanatory Coherence, in: Behavioral and Brain Sciences 12 (1989), 435-502.



The following evidences have to be explained:E1: During combustion, heat and light are given off.E2: Inflammability is transmittable from one body to

another.E3: Combustion only occurs in the presence of pure air.E4: The increase of weight in an incinerated body is

exactly equal to the weight of air absorbed.E5: Metals undergo calcination.E6: During calcination, bodies increase in weight.E7: During calcination, volume of air diminishes.E8: During reduction, effervescence appears.



Oxygen HypothesesOH1: Pure air contains oxygen

principle.OH2: Pure air contains matter of

fire and heat.OH3: During combustion,

oxygen from the air combines with the burning body.

OH4: Oxygen has weight.OH5: During calcination, metals

add oxygen to become calxes.

OH6: During reduction, oxygen is given off.

Phlogiston HypothesesPH1: Combustible bodies

contain phlogiston.PH2: Combustible bodies

contain matter of heat.PH3: During combustion,

phlogiston is given off.PH4: Phlogiston can pass from

one body to another.PH5: Metals contain phlogiston.PH6: During calcination,

phlogiston is given off.



Oxygen Explanations

OH1 OH2 OH3 -> E1OH1 OH3 -> E3OH1 OH3 OH4 -> E4 OH1 OH5 -> E5OH1 OH4 OH5 -> E6OH1 OH5 -> E7OH1 OH6 -> E8

Phlogiston Explanation

PH1 PH2 PH3 -> E1PH1 PH3 PH4 -> E2PH5 PH6 -> E5

Competetive Relations

20: PH3 OH3 ->20: PH6 OH5 ->

The interesting point about this example is that there are only two analytical contradictions. There is no need to implement the main hypotheses OH1 and PH1 as competing.



The program COHEN stops with an exactly stable net. All evidence and all oxygen hypotheses are exactly accepted with value one, PH3 and PH6 are exactly rejected with value zero. The other phlogiston hypotheses PH1, PH2, PH4 and PH5 exactly receive the indifferent value ½. This is plausible, since according to the reconstruction they conflict with some part of the oxygen theory system.

Thagard’s program ECHO tends towards the same result (relative to his scale)! Some minor deviations, e.g. in OH2 and OH6, which remain below the value of full acceptance, seem to be numerical artifacts - possibly caused by the connectionist algorithm.

computational coherence daniel schoch chiang mai university department of economics

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