ARTICLE 02 - BELIEFS (B.E.R.D.)

SECTION DESCRIPTION

01 Conceal Evidence.
02 Truthful.
03 Contrary Evidence.
04 Orders of Belief.
05 Bayesian Confirm/Evidential Support.
06 Base Rate Fallacy.
07 Comparative Incremental Evidence.
08 Equivalent Assertions.
09 Effective Incremental Evidence.
10 Differential Incremental Evidence.
11 Degrees of Evidence.
12 Weak Evidence Principle.
13 Comparative Degree of Evidence.
14-16 Likelihood Ratios.
17 Measuring Evidence Relations.
18 Law of Likelihood.
19 Subjective Probability.
20 Scientific Concept of Evidence.
21 Hypothesis.
22 Objective Evidence.
23 Objective Likelihood.
24 Weak Likelihood Principle.
25 Confirmed Hypotheses.
26-27 Belief & Experience.

SECTION DESCRIPTION

28 Simple Conditioning.
29 Jeffery Conditioning.
30 Belief Conditioning.
31 Weak Evidence Principle.
32 Consequence Simple/Jeffery Conditioning.
33-34 Probability & Ordinal Similarity.
35 Consequence of Jeffery Conditioning.
36 Likelihood Simple/Jeffery Condition.
37 Reasoner Categories.
38 Reasoner Types.
39 Godel Incomplete/Doxastic Undecidability.
40 Inaccuracy & peculiarity.
41 Self-fulfilling Beliefs.
42 Inconsistent Belief in One's Stability.
43-44 Credibility.
45-46 Claims.
47 Degrees of Credibility.
48 Claims on Observation.
49 Observation & Memory Reliability.
50 Doubt of Credibility.
51-52 Experts.
53 Unexamined Life.
54-55 Deception.
56 Proving Claims.

§02.01

Do not think it worthwhile to hold a belief by concealing evidence, for it is sure to be discovered.

§02.02

Be truthful even if the truth is inconvenient, for it is more inconvenient when you try to conceal it.

§02.03

An action can be assumed or implied not to change a given property of a situation unless there is evidence to the contrary.

§02.04

§02.04.1

§02.04.1.1

§02.04.1.2

§02.04.2

Beliefs come in varying gradations of strength and an ideally Rational individual's graded beliefs can be represented by subjective probability function P.

P(H) measures level of confidence or degree of conditional belief in H's truth:

"H" = df "firm opinion of belief ."

"P" = df "Subjective Probability of ."

PE(H) measures confidence of H on the supposition that E is a fact.

§02.05

§02.05.1

§02.05.1.1

§02.05.1.2

§02.05.2

§02.05.2.1

§02.05.2.2

§02.05.3

§02.05.3.1

§02.05.3.2

§02.05.3.3

§02.05.3.3.1

Bayesian Confirmation and Evidential Support.

Conformational Relativity.

Evidential relations must be relative to individuals and their degrees of belief.

Evidence relative to the information encoded in the subjective probability P.

Evidence Proportionism.

Rational believer will proportion their confidence in a hypothesis H to their total evidence for H, so that subjective probability for H reflects the overall balance of their reasons for or against its truth.

Level of confidence should vary directly with the strength of the evidence in favor of H's truth.

Incremental Confirmation.

The body of data provides incremental evidence for H to the extent that conditioning of the data raises H's probability.

Receiving the data increases or decreases individuals total evidence for the truth.

When probabilities measure total evidence, the increment of evidence that E provides for H is a matter of disparity between PE(H) and P(H).

With "odds" = df "OE(H) and O(H)."

§02.06

§02.06.1

§02.06.2

§02.06.3

Base Rate Fallacy; Total Evidence versus Incremental Evidence.

Known as Base Rate Neglect or Base Rate Bias.

If presented with related base rate information (i.e. generic, general information) and specific information (information only pertaining to a certain case), the mind tends to ignore the former and focus on the latter.

Base rate neglect is a specific form of the more general Extension neglect.

§02.07

§02.07.1

§02.07.1.1

§02.07.1.2

§02.07.1.3

Comparative Incremental Evidence Account.

Relative to subjective probability function P:

E incrementally confirms, dis-confirms or is irrelevant to H if and only if PE(H) <>= P(H) ;

H receives a greater increment or lesser decrement of evidential support from E than from E* if and only if PE(H) exceeds PE*(H) ;

Be it odds or probabilities.

§02.08

§02.08.1

§02.08.2

Each of the following is equivalent to the assertion that E provides incremental evidence in favor of H:

PR(H, E) > 1 | OR(H, E) > 1;

PD(H, E) > 1 | OD(H, E) > 1.

§02.09

"Effective increment of evidence" = df "amount where the incremental evidence that E provides for H exceeds the incremental evidence that ~E provides for H."

§02.10

"Differential incremental evidence" = df "amount where the incremental evidence that E provides for H exceeds the incremental evidence that E* provides for H."

§02.11

§02.11.1

§02.11.2

Effective evidence is a matter of the degree to which a person's total evidence for H depends on their opinion about E.

When PE(H) | P~E(H) or OE(H) | O~E(H) are far apart, the person's belief about E has a greater effect on their belief about H.

A great deal hangs on E's truth-value when it comes to questions about H's truth-value.

§02.12

§02.12.1

§02.12.2

§02.12.3

For any H, E* and E with positive probability, the following are equivalent:

E provides more incremental evidence than E* does for H;

PR(H,E) > PR(H,E*) | OR(H,E) > OR(H,E*) ;

PD(H,E) > PD(H,E*) | OD(H,E) > OD(H,E*).

§02.13

§02.13.1

§02.13.1.1

§02.13.1.2

There can be disagreement over the comparative degree to which a single item of data incrementally confirms two distinct hypotheses.

All differences between the measure involve:

If the total evidence in favor of a hypothesis should be measured in terms of probability or odds;

If disparities in total evidence are best captured as ratios or differences:

P = Total

O = Total

Ratio

PR(H,E) = PE(H) / P(H)

OR(H,E) = OE(H) / O(H)

Difference

PD(H,E) = PE(H) - P(H)

OD(H,E) = OE(H) - O(H)

§02.14

Likelihood ratios of §2.13; Differences between each multiplying measure and adding counterpart:

P = Total

O = Total

Ratio

PR(H, E,) = LR(H, T; E)

OR(H,E) = LR(H,~H;E)

Difference

PD(H,E) = P(H) [LR(H,T;E) - 1]

OD(H,E) = O(H) [LR(H,~H;E) - 1]

§02.14.01

§02.14.02

The likelihood term that appears in a given multiplying measure is diminished by 1 in its associated addition measure.

In each addition measure the diminished likelihood term is multiplied by an expression for H's probability – P(H) or O(H).

§02.15

§02.15.01

Does a given piece of data provide a greater increment of evidential support for a more probable hypothesis than it does for a less probable hypothesis when both hypotheses predict the data equally well?

"Difference" measures says yes, "Ratio" measures say no.

§02.16

§02.16.01

§02.16.01.01

§02.16.01.02

§02.16.01.03

§02.16.02

§02.16.02.01

§02.16.02.02

§02.16.02.03

§02.16.03

§02.16.03.01

§02.16.03.02

§02.16.03.03

The status of the likelihood ratio:

Probability as total evidence reading:

"PR(H,E)" = df “incremental change in total evidence.”

“LR(H,E)” = df “incremental change in net evidence.”

“LR(H,H*,E)” = df “incremental change in balance of evidence that E provides for H over H*.”

Odds as total evidence reading:

“OR(H,E)” = “incremental change in total evidence.”

“LR(H,E)2 “ = “incremental change in net evidence.”

“LR(H,H*;E)/LR(~H,~H*; E)” = “incremental change in balance of evidence that E provides for H over H*.”

"Likeli-hoodist" reading:

Neither P nor O measures total evidence because evidential relations are essentially comparative; they always involve balance of evidence.

“LR(H,E)” = df “balance of evidence that E provides H over H*.”

“LR(H,H*;E)” = df “balance of evidence that E provides H over H*.”

§02.17

§02.17.01

§02.17.02

§02.17.03

Measurable relationships of evidence (are different):

Odds measure total evidence – neither PR(H, E) nor LR(H, H*; E) plays a fundamental role in the theory of evidence.

Always need to account for, explicitly or implicitly, ~H or ~H*.

While each of the 2 functions (probability, likelihood) can figure as one component in a meaningful measure of confirmation, neither tells us anything about Incremental Evidence taken by itself.

§02.18

§02.18.01

§02.18.01.01

§02.18.01.02

Law of Likelihood

If H implies that the probability of E is x, while H* implies that the probability of E is x*, then E is evidence supporting H over H*.

If and only if, x exceeds x*; And

The likelihood ratio, x/x*, measures the strength of this support.

§02.19

§02.19.01

A person's subjective probability for a hypothesis merely reflects their degree of uncertainty about its truth.

Subjective probability does not need to be tied in any way to the amount of evidence they have in its favor.

§02.20

§02.20.01

Any scientifically respectable concept of evidence must analyze the evidential impact of E on H solely in terms of likelihoods; not unconditional or marginal probabilities.

Likelihoods are better known and more objective.

§02.21

§02.21.01

Hypotheses deductively entail a definite probability for the data.

A hypothesis is supported by any body of data it renders probable.

§02.22

§02.22.01

§02.22.02

Neither probability ratio nor probability difference will capture the sort of objective evidence required by science because their values depend on "subjective" terms.

The a posteriori probability distribution that is generally unknown or personal.

Should we tolerate subjective probabilities in one's account of evidential relations?

§02.23

Determine objective likelihoods in situations where the predictive connection from hypothesis to data is itself the result of inductive inferences.

§02.24

§02.24.01

§02.24.02

§02.24.03

Weak Likelihood Principle

If LR(H, H*; E) ≥ 1 and LR(~H, ~H*; ~E) ≥ 1, with one inequality strict, then E provides more incremental evidence for H than for H* and ~E provides more incremental evidence for ~H than for ~H*.

Relations depend only on inverse probabilities.

Conditioning on E increases H's probability and its odds strictly more than those of H*.

§02.25

Hypotheses are confirmed by data they predict.

§02.26

§02.26.01

§02.26.02

§02.26.03

Direct experience may depend causally on a believer's prior probability.

Immediate belief changes impose constraints using the form, "the a posteriori probability Q has such-and-such properties."

Experience limits or constraints to a certain degree.

Can prior opinions be used to justify the choice of a posteriori probability from among the many that might satisfy a given restraint.

§02.27

§02.27.01

A person is justified in adopting eligible post-beliefs if those beliefs depart minimally from current opinions.

Rational learners should proportion their beliefs to the strength of the evidence they acquire.

§02.28

§02.28.01

§02.28.02

§02.28.03

§02.28.04

Simple Conditioning: If a person with a "priori" such that 0 < P(E) < 1 has a learning experience whose sole immediate effect is to raise their subjective probability for E to 1, then their post-learning "posteriori" for any proposition H should be Q(H) = PE(H).

"Simplest learning" = df "learner becomes certain of the truth of some proposition of E where they were previously uncertain."

The constraint is that all hypotheses inconsistent with E must be assigned probability zero.

Process where the prior probability of each proposition H is replaced by a posterior proposition that coincides with the prior probability of H conditional on E.

A rational believer who learns for certain that E is true should factor this into their doxastic system by conditioning on it.

§02.29

§02.29.01

§02.29.02

Jeffrey Conditioning (Probability Kinematics): If a person with a prior such that 0 < P(E) < 1 has a learning experience whose sole immediate effect is to change their subjective probability for E to q, then their post-learning posterior for any H should be:

Q(H) = qPE(H) + (1 – q)P~E(H)

The direct effect of a learning experience will be to alter the subjective probability of some proposition without raising it to 1 or lowering to 0.

§02.30

Conditioning is the only belief revision rule that allows learners to correctly proportion their posterior beliefs to the new evidence received.

§02.31

§02.31.01

Weak Evidence Principle: If, relative to a prior P, E provides at least as much incremental evidence of H as for H*, and if H is antecedently more probable than H*, then H should remain more probable than H* after any learning experience whose sole immediate effect is to increase the probability of E.

Requires an agent to retain their views about the relative probability of 2 hypotheses when acquiring evidence that supports the more probable hypothesis more strongly.

§02.32

§02.32.01

§02.32.01.1

§02.32.01.2

Consequence: If a person's prior is such that LR(H, H*; E) ≥ 1, LR(~H, ~H*; ~E) ≥ 1 and P(H) > P(H*), then any learning experience whose sole immediate effect is to raise their subjective probability for E should result in a posterior such that Q(H) > Q(H*).

Assuming Q is defined on the same set of propositions as P is defined:

Simple conditioning is the unique correct method of belief revision for learning experiences that make E certain.

Jeffrey conditioning is a unique correct method when learning merely alters one's subjective probability for E.

§02.33

§02.33.01

§02.33.01.1

§02.33.01.2

Of probabilities: If H and H* both entail E when P(H) > P(H*), then LR(H, H*; E) = 1 and LR(~H, ~H*; ~E) > 1.

Simple conditioning on E is the only rule for revising subjective probabilities that yields posterior with the following properties for any prior such that P(E) > 0:

Q(E) = 1

Ordinal Similarity: If H and H* both entail E, then P(H) ≥ P(H*) IFF Q(H) ≥ Q(H*).

§02.34

§02.34.01
§02.34.02

Given §2.33, Corollary:

If H and H* entail E, then P(H) > P(H*) IFF Q(H) > Q(H*).
If H and H* entail ~E, then P(H) > P(H*) IFF Q(H) > Q(H*).

§02.35

§02.35.01

§02.35.02

Consequence

For every proposition H, QE(H) = PE(H) and Q~E(H) = P~E(H).

Necessary and sufficient for Q to arise from P by Jeffrey conditioning
on E.

§02.36

The weak likelihood principle entails that simple and Jeffrey conditioning on E are the only rational ways to revise beliefs in response to a learning experience whose sole immediate effect is to alter E's probability.

§02.37

§02.37.01

§02.37.01.1

§02.37.02

§02.37.02.1

§02.37.03

§02.37.03.1

§02.37.03.2

§02.37.04

§02.37.04.1

§02.37.05

§02.37.05.1

§02.37.06

§02.37.06.1

§02.37.06.2
§02.37.06.3

§02.37.07

§02.37.07.1

§02.37.08

§02.37.08.1

§02.37.08.2

§02.37.09

§02.37.09.1

§02.37.09.2

§02.37.10

§02.37.10.1

§02.37.10.2

§02.37.10.3

§02.37.10.3.1

§02.37.11

§02.37.11.1

§02.37.11.2

§02.37.11.3

§02.37.11.3.1

§02.37.12

§02.37.12.1

§02.37.13

Reasoner Categories:

Accurate reasoner: Never believe any false proposition.

(modal axiom T) Ɐp: βp → p

Inaccurate reasoner: Believes at least one false proposition.

ⱻp: ¬p ^ βp

Conceited reasoner: Believes her/is beliefs are never inaccurate.

β[¬ⱻp(¬p^βp)] OR β[Ɐp(βp → p)]

A conceited reasoner with a rationality of at least §2.38.01 (Type 1) will necessarily lapse into inaccuracy.

Consistent reasoner: Never simultaneously believes a proposition and its negation.

(modal axiom D) ¬ⱻp: βp ^ β¬p OR Ɐp: βp → ¬β¬p

Normal reasoner: One who, while believing P, also believes s/he believes P.
(modal axiom 4) Ɐp: βp → ββp

Peculiar reasoner: Believes proposition P while also believing s/he does not believe P.

Although a peculiar reasoner may seem like a strange psychological phenomenon, a peculiar reasoner is necessarily inaccurate but not necessarily inconsistent.

EX: Moore’s Paradox

ⱻp: βp ^ β¬βp

Regular reasoner: One who, while believing p → q, also believes Bp → Bq.
ⱯpⱯq: β(p → q) → β(βp → βq)

Reflexive reasoner: One for whom every proposition P has some proposition Q such that the reasoner believes q ≡ (βq → p).
Ɐp: ⱻqβ( q ≡ ( βq → p))

If a reflexive reasoner of §2.38.05 (Type 4) believes βp → p, s/he will believe P. A parallelism to Lob's theorem for reasoners.

Unstable reasoner: One who believes that s/he believes some proposition, but in fact does not believe it.

ⱻp: ββp ^ ¬βp
An unstable reasoner is not necessarily inconsistent.

Stable reasoner: Not unstable. That is, for every P, if s/he believes βp then s/he believes P.

Ɐp: ββp → βp

Note that stability is the converse of normality.

A reasoner believes s/he is stable if for every proposition P, s/he believes ββp → βp.

Example: Believing, "If I should ever believe that I believe P, then I really will believe P."

Modest reasoner: One for whom every believed proposition P, βp → p only if s/he believes P.

Never believes βp → p unless s/he believes P.

Ɐp: β(βp → p) → βp

Any reflexive reasoner of §2.38.05 (Type 4) is modest.

Example: Lob’s Theorem.

Queer reasoner: §2.38.06 (Type G) reasoner

Believes s/he is inconsistent – but is wrong in this belief.

Timid reasoner: Does not believe P, is "afraid to" believe P, if s/he believes βp → βꞱ.

§02.38

§02.38.01
§02.38.01.1
§02.38.01.2

§02.38.01.3

§02.38.01.3.1

§02.38.01.3.2

§02.38.01.4

§02.38.01.4.1

§02.38.02

§02.38.02.1

§02.38.02.2

§02.38.02.3

§02.38.03

§02.38.03.1

§02.38.03.2
§02.38.03.3
§02.38.03.4

§02.38.03.4.1

§02.38.04

§02.38.04.1

§02.38.04.1.1

§02.38.05
§02.38.05.1
§02.38.05.1.1

§02.38.06
§02.38.06.1
§02.38.06.1.1

Reasoner Types:

Type 1 Reasoner: Has a complete knowledge of propositional logic – sooner or later believes every tautology.

Any proposition proven by truth tables.

The set of beliefs (past, present, future) is logically closed under modus ponens.

If s/he ever believes P and believes p → q (P implies Q) then s/he will sooner or later believe Q.

⊢PCp ⇒ ⊢βp

ⱯpⱯq: ((βp ^ β(p→ q)) → βq)

Belief distributes over implication, as it is logically equivalent to:

ⱯpⱯq: β(p→ q) → (βp→βq)

Type 1* Reasoner:

Everything covered in §2.38.01.

Type 1* reasoner has "a shade more" self awareness than a §2.38.01 (type 1) reasoner.

ⱯpⱯq: β(p→ q) → β(βp→βq)

Type 2 Reasoner:

S\he follows §2.38.01.

If for every P and Q s\he (correctly) believes: "If I should ever believe both P and p→q, then I will believe Q."

Believes the logically equivalent proposition: β(p→ q) → (βp→βq)

Knows her\is beliefs are closed under modus ponens.

ⱯpⱯq: β((βp ^ β((p → q)) → βq)

Type 3 Reasoner:

S\he is §2.37.05 (normal) and §2.38.03 (type 2).

Ɐp: βp → ββp

Type 4 Reasoner:
S\he is §2.38.04 (type 3) and believes s/he is §2.37.05 (normal).
β[Ɐp(βp → ββp)]

Type G Reasoner:
Believes s\he is §2.37.11 (modest).
β[Ɐp(β(βp → p) → βp)]

§02.39
§02.39.01

§02.39.02

§02.39.02.1

"Godel Incompleteness and doxastic undecidability"
There exists a statement which the reasoner must either remain forever undecided about or lose his\her accuracy.

For any formal system F, there exists a mathematical statement which can be interpreted as "This statement is not provable in formal system F."

If the system F is consistent, neither the statement nor its opposite will be provable in it.

§02.40

§02.40.01

§02.40.02

§02.40.02.1

§02.40.02.2

§02.40.02.2.1

§02.40.03

"Inaccuracy and peculiarity"

Statement: " I will never believe this sentence."

If §2.37.3 reasoner (conceited) believes they are always accurate, then they will become accurate.

Reasoner believes the statement, which makes it false.

Reasoner is inaccurate in believing that the statement is true.

If the reasoner had not assumed their own accuracy, they would never have lapsed into an inaccuracy.

S ≡ ¬BS

(¬S → S) → S

(BS → S) → S

B((BS → S) → S)

B(BS → S) → BS

B(BS → S)

¬S

Reasoner is §2.37.6 (peculiar): Believes that s\he does not believe the statement – B(¬BS), which follows from BS because S ≡ ¬BS - even though s\he actually believes it.

§02.41

§02.41.01

Self-fulfilling Beliefs

For systems, we define reflexivity to mean that for any p (in language of that system) there is some q such that q ≡ (Bq → p) is provable in the system.

For any reflexive system of §2.38.5 (type 4), if Bp → p is provable in the system, so is p.

§02.42

§02.42.1

Inconsistent Belief in One's Stability

If a §2.37.4 (consistent) and §2.37.8 (reflexive) reasoner of §2.38.5 (type 4) believes that s\he is §2.37.10 (stable), then s\he will become §2.37.9 (unstable) – inconsistent.

§02.43

§02.43.1

§02.43.2

Two grounds for suspicions of credibility:

The claim itself;

The source of the claim.

§02.44

§02.44.1

§02.44.2

Always ask on credibility:

When does a claim’s content lack credibility?

When does a source lack credibility?

§02.45

§02.45.1

§02.45.2

A claim itself can fail when it conflicts with:

Observation(s);

Background Information or Knowledge.

§02.46

§02.46.1

§02.46.2

Suspicions of a claim or its source:

Lack of initial plausibility can make some claims lack credibility.

There is a connexion between our belief of a claim and the source of a claim.

§02.47

§02.47.1

§02.47.2

§02.47.3

§02.47.4

There are degrees of credibility (credible information):

Interested Parties = “Person who stands to gain from our belief in a claim.”

Disinterested Parties = “No benefit/influence in what we belief either way.”

Irrelevant Considerations = “Physical characteristics, attributes, gender, height, race, etc.”

Persons Occupation = “Adds to their credible knowledge or ability; takes away from moral character or truthfulness.”

§02.48

§02.48.1

§02.48.1.1

§02.48.1.2

§02.48.1.3

§02.48.1.4

§02.48.2

§02.48.2.1

§02.48.2.2

On claims by observations.

Observation and short term memory can be fallible and influenced by, but not limited to:

emotions;

psyche;

physical;

measuring instruments or processes.

Observations differ among people by, but not limited to:

Better seeing/hearing/remembering;
Influential beliefs, hopes, fears and expectations.

§02.49

§02.49.1

Observation and memory have the same level of reliability.

Both can be deceptive; there can be a time where no one, single person is credible.

§02.50

§02.50.1

§02.50.2

Kinds of doubt about the credibility of a source:

Doubt real knowledge about the issue in question;

Doubt truthfulness, objectivity, accuracy or integrity.

§02.51

§02.51.1

§02.51.2

When judging expertise:

Take into account education and expertise;

Take into account accomplishments, reputation or position.

§02.52

§02.52.1

§02.52.2

If 'experts' are seen as disagreeing, suspend judgement until more information is shown.

The ability to become an expert about a Subject is not equivalent to actually being an expert.

Claims from experts in the field is not equivalent to claims put forth by non-experts.

§02.53

§02.53.1

§02.53.2

Do not feel envious or happiness of those who live an unexamined life.

Only the unexamined life will think it has happiness.

The unexamined life is not worth living.

§02.54

§02.54.1

Be skeptical of advertising and promotion, where intelligence is suspended for a specific gain.

Is there a gain or improvement as the intent or concern.

§02.55

Euphemisms and Dysphemisms are both deceptive depending on its purpose and use.

§02.56

§02.56.1

§02.56.2

There are times we cannot prove a claim, but we can hint that there exists some proof (evidence or authority) without saying what it is.

No real proof or evidence suggests poor research or propaganda.

Critical Thinking begins by offering evidence of a claims truth or falsity.

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