Lecture 5 - Confirmation

Question 1

Qualitative confirmation

Answer 1

empirical evidence confirms hypothesis with respect to background knowledge

Question 2

Quantitative confirmation

Answer 2

empirical evidence confirms hypothesis to with respect to background knowledge

Question 3

Principles of Bayesianism:

Answer 3

• Measure principle: every subject has belief-measure, quantifying strength of blief of S that p  B(S, p, g); where 0 g 1 (g=0 utter disbelief, g=0,5 is maximum uncertainty, g=1 is complete conviction)
• Principle of synchronic coherence: belief measure is handled by probability theory, degree of belief is equal to probability that subject assesses to proposition
• Principle diachronic coherence: connects change of degree of belief over time

Question 4

Kolgomorov axioms of probability theory:

Answer 4

• Axiom of unity: For every proposition p in Γ: 0 ⩽ Pr(p) ⩽ 1, and Pr(⊤) = 1 and Pr(⊥) = 0
• Axiom of additivity: for incompatible propositions, Pr(p ∨ q) = Pr(p)+Pr(q)

Question 5

Theorem by Gaifman & Snir:

Answer 5

: in the long run, all the priors wash away by Bayesian updating, initial degrees of belief become irrelevant

Question 6

Qualitative confirmation and infirmation:

Answer 6

• Confirmation: if degree of belief increases with empirical evidence
• Infirmation: if degree of belief decreases with empirical evidence
• Irrelevance: if degree of belief stays the same with empirical evidence

Question 7

Verification & falsification

Answer 7

Verification: iff confirmation of h and condition probability of h is one (Pr(h/e)=1) and prior <1
Falsification: iff confirmation of hh and condition probability of not h is one (Pr(-h/e)=1) and prior of not h < 1

Question 8

Quantitative confirmation + degree

Answer 8

• Some evidence confirms hypothesis better than other evidence/hypotheses
• Therefore, degree of confirmation is warranted (µ(e,h))
• Degree of confirmation says something about: evidence, model, and time
• Verification is strongest possible confirmation, quantified with 10
• Falsification is strongest possible information, quantified with -10

Question 9

Hempels Raven paradox:

Answer 9

• If all ravens are black is true
• And if everything that is not black is not a raven is true
• These things are logically equivalent, and therefore have the same probability
• This would mean that seeing colored things that are not ravens would confirm that all ravens are black  this is weird

Question 10

Solution to Raven paradox

Answer 10

• Distinguishing evidence more accurately: black raven, black object that is not raven, colored or white raven, colored or white object that is not raven
• Factoring in background knowledge: more colored birds than black, more other birds than ravens

Question 11

Goodman’s New riddle of induction:

Answer 11

• All emeralds are green
• New predicates: grue (green before 2100) and bleen (green after 2100)
• New hypothesis: all emeralds are grue (now: inspected emeralds confirm both hypotheses because 2024)
• Real hypothesis (all emeralds are green) can be true as well as magical hypothesis (all emeralds are grue) even when all the evidence is there, therefore we cannot rule out magical hypotheses on the basis of evidence / induction
• Is Bayesian updating solution: for magical hypotheses prior is very low