Lecture 1: The Humean Thesis on Belief Hannes Leitgeb LMU Munich October 2014

What do a perfectly rational agent’s beliefs and degrees of belief have to be like in order for them to cohere with each other?

Plan: Give an answer in terms of a joint theory of belief and degrees of belief. The Humean Thesis on Belief 1 Consequence 1: The Logic of Belief 2 Consequence 2: The Lockean Thesis 3 Consequence 3: Decision Theory 4 Conclusions 5 In all of this I will focus on (inferentially) perfectly rational agents only. (There is related literature in computer science: Benferhat et al. 1997, Snow 1998.)

The Humean Thesis on Belief ... an opinion or belief is nothing but an idea, that is different from a fiction, not in the nature or the order of its parts, but in the manner of its being conceived ... An idea assented to feels different from a fictitious idea, that the fancy alone presents to us: And this different feeling I endeavour to explain by calling it a superior force, or vivacity

The Humean Thesis on Belief ... an opinion or belief is nothing but an idea, that is different from a fiction, not in the nature or the order of its parts, but in the manner of its being conceived ... An idea assented to feels different from a fictitious idea, that the fancy alone presents to us: And this different feeling I endeavour to explain by calling it a superior force, or vivacity, or solidity, or firmness, or steadiness. [ ... ] its true and proper name is belief, which is a term that every one sufficiently understands in common life. [ ... ] It gives them [the ideas of the judgment] more force and influence; makes them appear of greater importance; infixes them in the mind; and renders them the governing principles of all our actions. (Treatise, Section VII, Part III, Book I) ... the mind has a firmer hold, or more steady conception of what it takes to be matter of fact, than of fictions. (Treatise, Appendix)

Tradition in Hume interpretation has it that beliefs are lively ideas. In my interpretation, beliefs are steady dispositions. (Loeb 2002) A disposition to vivacity is a disposition to experience vivacious ideas, ideas that possess the degree of vivacity required for occurrent belief. Some dispositions to vivacity are unstable in that they have a tendency to change abruptly ... Such dispositions, in Hume’s terminology, lack fixity. Hume in effect stipulates that a dispositional belief is an infixed disposition to vivacity ... (Loeb 2010) there must be a property that plays a twofold role. The presence of the property must constitute a necessary condition for belief. In addition, establishing that the beliefs produced by a psychological mechanism have that property must constitute a sufficient condition for establishing justification, other things being equal. My claim is that stability is the property that plays this dual role, one within Hume’s theory of belief, the other within Hume’s theory of justification. (Loeb 2002)

In a nutshell: Beliefs are stable dispositions to have ideas with high “degree of vivacity” on which acting, reasoning, and asserting is then based. This is plausible also independently of (Loeb on) Hume: our epistemic lives depend on stability under perception, supposition, communication, ... .

Now we are going to explicate this: (Rational) Degree of vivacity ≈ subjective probability P (cf. Maher 1981) Let us call then the following the Humean thesis on rational belief: It is rational to believe a proposition just in case it is rational to have a stably high degree of belief in it (“infixed disposition to vivacity”).

Now we are going to explicate this: (Rational) Degree of vivacity ≈ subjective probability P (cf. Maher 1981) Let us call then the following the Humean thesis on rational belief: It is rational to believe a proposition just in case it is rational to have a stably high degree of belief in it (“infixed disposition to vivacity”). But what is a stably high degree of belief? ֒ → Skyrms (1977, 1980) on resiliency with respect to conditionalization!

Now we are going to explicate this: (Rational) Degree of vivacity ≈ subjective probability P (cf. Maher 1981) Let us call then the following the Humean thesis on rational belief: It is rational to believe a proposition just in case it is rational to have a stably high degree of belief in it (“infixed disposition to vivacity”). Let W be a finite non-empty set of possible worlds. A proposition is a subset of W , and P is a probability measure over propositions.

Now we are going to explicate this: (Rational) Degree of vivacity ≈ subjective probability P (cf. Maher 1981) Let us call then the following the Humean thesis on rational belief: It is rational to believe a proposition just in case it is rational to have a stably high degree of belief in it (“infixed disposition to vivacity”). Let W be a finite non-empty set of possible worlds. A proposition is a subset of W , and P is a probability measure over propositions. The Humean Thesis Explicated (First Approximation) If Bel is a perfectly rational agent’s class of believed propositions at a time, and if P is the same agent’s subjective probability measure at the same time, then HT r Y : Bel ( X ) iff for all Y, if Y ∈ Y and P ( Y ) > 0 , then P ( X | Y ) > r. But stably high probability with respect to what class Y of propositions?

Smaller Y s yields braver Bel s, larger Y s yields more cautious Bel s: The Humean Thesis Explicated HT r Y : Bel ( X ) iff for all Y, if Y ∈ Y and P ( Y ) > 0 , then P ( X | Y ) > r. Y ∈ Y iff P ( Y ) = 1: The Lockean Thesis. Reduces to: Bel ( X ) iff P ( X ) > r . (In general, not stable enough.) Y ∈ Y iff Bel ( Y ) (and P ( Y ) > 0): A coherence theory of belief. Y ∈ Y iff P ( Y ) > s : A modestly cautious proposal. Y ∈ Y iff Poss ( Y ) (and P ( Y ) > 0): Another cautious proposal. � ������ �� ������ � not Bel ( ¬ Y ) Y ∈ Y iff P ( Y ) > 0: The probability 1 proposal. Reduces to: Bel ( X ) iff P ( X ) = 1. (In general, too stable.)

Smaller Y s yields braver Bel s, larger Y s yields more cautious Bel s: The Humean Thesis Explicated HT r Y : Bel ( X ) iff for all Y, if Y ∈ Y and P ( Y ) > 0 , then P ( X | Y ) > r. Y ∈ Y iff P ( Y ) = 1: The Lockean Thesis. Reduces to: Bel ( X ) iff P ( X ) > r . Y ∈ Y iff Bel ( Y ) (and P ( Y ) > 0): A coherence theory of belief. Y ∈ Y iff P ( Y ) > s : A modestly cautious proposal. Y ∈ Y iff Poss ( Y ) (and P ( Y ) > 0): Another cautious proposal. � ������ �� ������ � not Bel ( ¬ Y )

Smaller Y s yields braver Bel s, larger Y s yields more cautious Bel s: The Humean Thesis Explicated HT r Y : Bel ( X ) iff for all Y, if Y ∈ Y and P ( Y ) > 0 , then P ( X | Y ) > r. Y ∈ Y iff P ( Y ) = 1: The Lockean Thesis. Reduces to: Bel ( X ) iff P ( X ) > r . Y ∈ Y iff Bel ( Y ) (and P ( Y ) > 0): A coherence theory of belief. Y ∈ Y iff P ( Y ) > s : A modestly cautious proposal. Y ∈ Y iff Poss ( Y ) (and P ( Y ) > 0): Another cautious proposal. � ������ �� ������ � not Bel ( ¬ Y ) Theorem The Humean thesis HT r Poss entails the Humean thesis for all other conditions above (assuming not Bel ( ∅ ) , and with appropriate thresholds).

The Humean Thesis Explicated If Bel is a perfectly rational agent’s class of believed propositions at a time, and if P is the same agent’s subjective probability measure at the same time, then (HT r Poss ) Bel ( X ) iff for all Y, if Poss ( Y ) and P ( Y ) > 0 , then P ( X | Y ) > r (and 1 where Poss ( Y ) iff not Bel ( ¬ Y ) 2 ≤ r < 1 ).

The Humean Thesis Explicated If Bel is a perfectly rational agent’s class of believed propositions at a time, and if P is the same agent’s subjective probability measure at the same time, then (HT r Poss ) Bel ( X ) iff for all Y, if Poss ( Y ) and P ( Y ) > 0 , then P ( X | Y ) > r (and 1 where Poss ( Y ) iff not Bel ( ¬ Y ) 2 ≤ r < 1 ). This will be our official explication of belief! ‘ Poss ( Y ) ’ matches the probabilistic condition ‘ P ( Y ) > 0’. The resulting constraint on Bel is not as strong as it might seem: ‘ Poss ’ expresses serious possibility (Levi 1980), not logical possibility. It is not a reduction of Bel to P . The explication is fruitful, as we are going to see.

Consequence 1: The Logic of Belief The logic of belief follows from the Humean thesis (and probability theory). Theorem If P is a probability measure, if Bel satisfies the Humean thesis HT r Poss , and if not Bel ( ∅ ) , then the following principles of doxastic logic hold: Bel ( W ) . If Bel ( X ) and X ⊆ Y, then Bel ( Y ) . If Bel ( X ) then Poss ( X ) . If Bel ( X ) and Bel ( Y ) , then Bel ( X ∩ Y ) . So HT r Poss entails that there is a least believed proposition B W that is non-empty and which generates the agent’s belief system Bel .

B W! X ! Bel(X) ! Y ! Poss(Y) !

Moral: Once belief is postulated to be “sufficiently” stable, the logical closure of belief becomes derivable.

An example from Bayesian networks : J. Pearl (1988), D. Barber (2012) Krombholz (2013) It)rained) The)sprinkler)was)left)on) R) S) S=1$ R=1$ 0.1) 0.2) T=1$ R$ S$ J=1$ R$ 1) 1) 1) 1) 1) J) T) 1) 1) 0) 0.2) 0) 0.9) 0) 1) 0) 0) 0) Tracey’s)lawn)is)wet) Jack’s)lawn)is)wet) “belief arises only from causation ... ” ( Treatise , Section IX, Part III, Book I)

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