Computational Coherence

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Computational Coherence

• Daniel Schoch
• Chiang Mai University
• Department of Economics

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Content

• Terminology
• Coherentist Epistemology
• Inferential Coherence
• Thagards Computational Model
• Qualitative Decision Theory
• COHEN

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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).

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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.

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Terminology - Inference

• In Coherence Theory we consider a general
  inference relation between propositions,
  p1,,pn -gt 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,pk1,,pn-gt
  pk holds for some k.

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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.

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Example

• Consider the following two propositions
• John is in Rome on Nov. 4th 2007, 1430.
• John is in Bangkok on Nov. 4th 2007, 1431.
• 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.

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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.

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Justification and Inference

• Fundamentalism
• External Justification

• Coherentism
• Internal Justification

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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.

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Computational Justification

• Fundamentalism
• External Justification
• Needs Meta-beliefs on reliability
• Probabilistic
• Bayesian Nets
• Bovens, Luc, and Stephan Hartmann, Bayesian
  Epistemology (Oxford Clarendon Press, 2003).

• Coherentism
• Internal Justification
• No meta-beliefs required
• Gradational
• Inference algorithm
• Paul Thagard, Computational Philosophy of Science
  (MIT Press, 1988, Bardford Book, 1993).

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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

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3. Inferential Coherence

• We assume that there is an abstract inference
  relation p1,,pn -gt p between propositions.
• Inferential Coherence is a measure of
  plausibility of belief systems S, such that
• For every p1,,pn,p?S with p1,,pn -gt p, the
  degree of coherence rises.
• For every p1,,pn,p?S with p1,,pn -gt p, the
  degree of coherence sinks dramatically.
• Inferential relations p1,,pn -gt p, p1,,pn -gt q
  from common premises p1,,pn are better (more
  coherent) than from different.
• Inferences from fewer premises are better (more
  coherent) than from more.

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Explanatory Coherence

• Thagard interpretes the inference relation
  p1,,pn -gt p as an explanation of p (explanandum)
  by the explanans p1,,pn. The conditions then
  read
• A successful explanation increases coherence.
• An explanatory anomaly decreases coherence.
• A unified explanation by a single explanans is
  better than two unrelated explanations.
• 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.

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Examples Explanation

• 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.
• An explanatory anomaly occurs, when an evidential
  belief contradicts (or incoheres with) some
  proposition, which is explained by other beliefs.
  E.g. Bohrs atomic model contradicts Maxwells
  electrodynamics.
• A unified theory brings together formerly
  separated parts of science, e.g. unification of
  sublunar (Galilei) and celestial (Kepler) physics
  by Newton.
• A simpler theory supersedes a complicated one,
  e.g. Keplers theory needs less parameters than
  Kopernikus.

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Coherence and Belief

• Two Perspectives of Coherence
• Global (Belief Choice) Consider competing total
  belief systems S1,,Sn, choose the most coherent
  one.
• 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.

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The Dual Pathway Approach

• The local perspective is exemplified in Thagards
  dual pathway model

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4. Thagards 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
  Pp1,,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 pk?S, vk 0, if pk?S,
• vk ½, else.
• Thagard uses an artificial neural network model
  to implement his theory of coherence.

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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.

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Computational Neurons

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

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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, Thagards Neuronal Nets
  only have symmetric connections (Hopfield model)!

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Hopfield Contra Nature

• Biological Neural Net
• Asymmetric recurrent network, very difficult to
  understand.

• Hopfield model
• Symmetric, thus theoretically well understood.

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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, i?j, pi and pj cohere.
• c.) The degree of coherence is inversely
  proportional to m.

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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.

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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.

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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.

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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.

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Quantitative Coherence

• Consider, for example, the inference relation
  p1,,pn -gt 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.

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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 vp 1 - vp is
  the value of p.
• If vp is the value of p, and vq is the value of
  q, then vpq vp vq is the value of pq.
• This is a valid system of fuzzy logic according
  to Gottwald.
• Gottwald, Siegfried, Fuzzy Sets and Fuzzy Logic,
  Vieweg, 1993.

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Fuzzy Coherence

• Now we can specify the value function for the
  inference rule p1,,pn -gt p
• For the case of p1,,pn incohering, we obtain
• We see that the latter is a special case of the
  former for vp0!

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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 -gt p
• Here, w is an optional number giving the weight
  of the rule.

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COHEN Syntax

• Each rule p1,,pn -gt pis regarded as an
  explanatory relation.
• Negation is designated by an ! prefix.
• An incoherence/competitive relation between
  p1,,pn is written as p1,,pn -gtwhich is
  interpreted as p1,,pn explaining a
  contradictory statement.
• Evidential support is written as an explanatory
  relation without premises, -gt p

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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.

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Lavoisiers 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.

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Lavoisiers Oxygen Hypothesis (2)

• 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.

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Lavoisiers Oxygen Hypothesis (3)

• Oxygen Hypotheses
• OH1 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 Hypotheses
• PH1 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.

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Lavoisiers Oxygen Hypothesis (4)
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.

• Oxygen Explanations
• OH1 OH2 OH3 -gt E1
• OH1 OH3 -gt E3
• OH1 OH3 OH4 -gt E4
• OH1 OH5 -gt E5
• OH1 OH4 OH5 -gt E6
• OH1 OH5 -gt E7
• OH1 OH6 -gt E8

• Phlogiston Explanation
• PH1 PH2 PH3 -gt E1
• PH1 PH3 PH4 -gt E2
• PH5 PH6 -gt E5
• Competetive Relations
• 20 PH3 OH3 -gt
• 20 PH6 OH5 -gt

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Lavoisiers Oxygen Hypothesis (5)

• 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.
• Thagards 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.