foundations of everyday practical reasoning

J Philos Logic (2013) 42:831–862DOI 10.1007/s10992-013-9296-0

Foundations of Everyday Practical Reasoning

Hanti Lin

Received: 14 November 2012 / Accepted: 4 July 2013 / Published online: 13 October 2013© Springer Science+Business Media Dordrecht 2013

Abstract “Since today is Saturday, the grocery store is open today and will be closedtomorrow; so let’s go today”. That is an example of everyday practical reasoning—reasoning directly with the propositions that one believes but may not be fully certainof. Everyday practical reasoning is one of our most familiar kinds of decisions but,unfortunately, some foundational questions about it are largely ignored in the stan-dard decision theory: (Q1) What are the decision rules in everyday practical reasoningthat connect qualitative belief and desire to preference over acts? (Q2) What sort oflogic should govern qualitative beliefs in everyday practical reasoning, and to whatextent is that logic necessary for the purposes of qualitative decisions? (Q3) Whatkinds of qualitative decisions are always representable as results of everyday practi-cal reasoning? (Q4) Under what circumstances do the results of everyday practicalreasoning agree with the Bayesian ideal of expected utility maximization? This paperproposes a rigorous decision theory for answering all of those questions, which isdeveloped in parallel to Savage’s (1954) foundations of expected utility maximiza-tion. In light of a new representation result, everyday practical reasoning provides asound and complete method for a very wide class of qualitative decisions; and, tothat end, qualitative beliefs must be allowed to be closed under classical logic plus awell-known nonmonotonic logic—the so-called system P.

Keywords Practical reasoning · Qualitative decision theory · Qualitative belief ·Subjective probability · Defeasible reasoning · Nonmonotonic logic

H. Lin (�)School of Philosophy, Australian National University,Coombs Building, Canberra 0200, Australiae-mail: [email protected]

mailto:[email protected]

832 H. Lin

1 Introduction

Our everyday decision is sometimes based on reasoning like the following: “sincetoday is Saturday, the grocery store is open today and will be closed tomorrow; solet’s go today”. It seems natural to say that the agent believes or accepts that todayis Saturday, and that she reasons directly with the propositions that she believes butmay not be fully certain of. It is an example of everyday practical reasoning, oneof our most familiar ways for making decisions—but, unfortunately, it is largelyignored in the standard decision theory. According to the standard, Bayesian theory,rational choice is merely a matter of expected utility maximization, in which every-day practical reasoning appears to play no role. That seems not quite right. Perhapsexpected utility maximization is an ideal to strive for, but life is too short to apply thefull Bayesian apparatus to every decision that arises in a day and it often suffices todecide qualitatively. One may, for instance, base her decision upon everyday practicalreasoning as illustrated in the grocery example. The picture just sketched, however,raises some foundational questions:

(Q1) In everyday practical reasoning, what are the decision rules that connectqualitative belief and desire to preference over acts?

(Q2) What sort of logic should govern qualitative beliefs when it is used to guideeveryday practical reasoning? To what extent is that logic necessary for thepurposes of qualitative decisions?

(Q3) What kinds of qualitative decisions are always representable as results ofeveryday practical reasoning?

(Q4) Under what circumstances does everyday practical reasoning agree with theBayesian ideal of expected utility maximization?

This paper proposes a decision theory to answer all of those questions. In lightof a new representation result, everyday practical reasoning provides a sound andcomplete method for a very wide class of qualitative decisions; and, to that end, qual-itative beliefs must be allowed to be closed under classical logic plus a well-knownnonmonotonic logic—the so-called system P.1 That result provides a foundation foreveryday practical reasoning, in the sense that Savage’s [27] representation theoremprovides a foundation for expected utility maximization.

The proposed theory improves upon the earlier works concerning qualitativedecision on a number of fronts. Qualitative decision has been extensively studied inlogic-based artificial intelligence.2 But, in most of those works, a logic for defeasi-ble/nonmonotonic reasoning is simply assumed (e.g., [4]). Instead, the theory to bedeveloped below provides a representation theorem that determines the nonmono-tonic logic that is necessary for the purposes of everyday practical reasoning. Therole of qualitative beliefs in decision-making has been studied in philosophy and eco-nomics. Bratman [3] and Cohen [5], for example, expound the idea that qualitative

1System P was originally due to [1] for the logic of indicative conditionals. It was recognized in artificialintelligence to be the logic that underlies one of the main approaches to classical planning theory, calledcircumscription [16, 21, 29].2For reviews, see [6, 30].

Foundations for Everyday Practical Reasoning 833

beliefs can serve as premises in practical reasoning—I take that idea seriously anddevelop it into a rigorous decision theory. Morris [22] shows how the logic forbeliefs may depend on one’s preference relations, but he does so by defining quali-tative beliefs as fully certain propositions, which excludes daily beliefs such as “thegrocery store is open today”.3 No such restriction is imposed in the proposed theory.Some representation theorems for qualitative decisions have been provided in the lit-erature, but none of them concerns everyday practical reasoning as illustrated in thegrocery example. Brafman and Tennenholtz [2], for instance, provide a representa-tion theorem that completely characterizes the Maximin decision rule. Dubois et al.[7] prove that a certain class of preference relations is representable by a qualitativedecision rule they call the likely dominance rule, together with belief representationsbased on possibility measures. Although Dubois and colleagues do not intend to talkabout about everyday practical reasoning, it is their work that makes me realize thatit is possible to construct a Savage-like axiomatic foundation for everyday practicalreasoning. To that end, we have to weaken their axiomatization for preference andmodify some of their modeling assumptions (please see Appendix F for details).

The proposed decision theory can be employed to resolve an issue betweentwo approaches to epistemology: Bayesian epistemology and the more traditionalepistemology that concerns itself with justified qualitative beliefs and knowledge.The qualitative concept of “believing propositions” is essential to traditional epis-temology because, for example, believing is necessary for knowing. But Bayesianepistemologists have long complained that the concept of qualitative beliefs is tooambiguous to be taken seriously [15]. Bayesians seem to have the following reason:one of the most important roles of belief is to guide action; so, as a methodologicalprinciple, any concept that models doxastic states should be clarified by explaining itsrole in rational decision-making—provided that we are to take that concept seriouslyin philosophical or scientific discourses. Bayesian epistemologists have followed thatprinciple for clarifying the concept of probabilistic degrees of belief. And they chal-lenge traditional epistemologists to clarify the concept of qualitative beliefs in thesame fashion—that is known as the Bayesian challenge to traditional epistemology.The present paper itself will be neutral about whether the Bayesian challenge is soundor not. Instead, we can take the present paper to provide a case-by-case answer tothe Bayesian challenge: if the challenge is unsound, then traditional epistemologistsdo not need to respond; if the challenge is sound, then the challenge can be met byanswering questions (Q1)–(Q4) with the help of the decision theory to be developedbelow. The present paper aims to focus on the foundations of everyday practical rea-soning, and the philosophical issues regarding the Bayesian challenge have to bediscussed in detail in a future paper.

The core of the proposed decision theory—the new representation theorem—isprovided in Section 4. Before that, we will examine some examples of everydaypractical reasoning. Those examples reveal the decision rules basic to everyday prac-tical reasoning (Section 2), and illustrate the daily cases in which everyday practicalreasoning serves as a qualitative means for the Bayesian ends (Section 3). The

3To be more precise, Morris defines qualitative beliefs to be the negations of decision-theoretically nullpropositions.

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answers to the four questions (Q1)–(Q4) will be summarized in the concludingsection.

2 Decision Rules for Everday Practical Reasoning

In this section, we will examine some examples of everyday practical reasoning. Thegoal is to formulate the decision rules that lie behind the examples.

Recall the grocery store example mentioned in the introduction. It can beunderstood in terms of the following, more explicit reasoning:

(1) Today is Saturday.(2) So the grocery store is open today and will be closed tomorrow.(3) So going today would result in some food, and going tomorrow would result in

no food.(4) So going today would result in a more desirable outcome than going tomorrow.(5) Therefore, I prefer going today rather than tomorrow.

The agent believes (1), from which she defeasibly infers (2). Given (2), the agentreasons to (3), which concerns the outcomes that would be produced by the possibleacts on the table. Then, the agent’s desire for food leads to (4). The reasoning from(4) to (5) goes beyond logic; it rests on a decision rule so banal as to easily escapenotice:

(Cliche Rule) Prefer act a to act b if you believe that the outcome of a is moredesirable than the outcome of b.

The Cliche Rule can be taken as a kind of dominance argument. Assuming theHintikka-Kripke semantics for beliefs, one’s qualitative belief state can be modeledas a set B of possible worlds such that one believes a proposition if and only if thatproposition is true at every world in B. Then the Cliche Rule reads as follows. Preferact a to act b if the proposition “the outcome that a would produce is more desirablethan the outcome that b would produce” is true at every world in your belief state B.Namely, prefer a to b if, for each possible state s of the world in B, the outcome that awould produce at s is more desirable to you than the outcome that b would produce ats. So the Cliche Rule is like a strong dominance argument that requires quantificationover, not all possible states of the world, but only those contained in your beliefstate B. Note that the Hintikka-Kripke semantics has a feature: qualitative beliefs areclosed under conjunction. In the following, qualitative beliefs will be modeled in amore liberal way without presupposing conjunctive closure—instead, we will justifyconjunctive closure.

To formulate the Cliche Rule in decision-theoretic terms, it is most convenient toadopt Savage’s [27] framework. Let a decision problem be modeled by an orderedpair (S, O), where S is a non-empty set of states that are mutually exclusive andjointly exhaustive and O is a non-empty set of outcomes. An act is a function a fromS to O, where a(s) denotes the outcome that a would produce if s were the actual


state.4 An agent’s preference over acts is modeled by a binary relation � over acts,where a � b means that act a is at least as preferable as act b to the agent. Call� a preference relation. Let � induce the “equally preferable” relation ∼ and the“strictly preferable” relation � by the standard definition.5 Preference is supposedto be determined by desire and belief. Let one’s qualitative desire be modeled by abinary relation ≥ over outcomes, where o ≥ o′ means that outcome o is at least asdesirable as outcome o′ to the agent. Call ≥ a desirability order. Let ≥ induce the“equally desirable” relation ≡ and the “strictly more desirable” relation > by thestandard definition.6 A proposition is a subset of S, which is true at all and only thestates it contains. So, for example, the proposition expressed by ‘the outcome that awould produce is more desirable than the outcome that b would produce’ is identifiedwith the set {s ∈ S : a(s) > b(s)}, which is abbreviated as [[a > b]]. In general,let the notation [[a R b]] be defined as follows, where a, b are acts and R is a binaryrelation between outcomes:7

[[a R b]] = the proposition expressed by ‘the outcome that a would produce bearsrelation R to the outcome that b would produce’

= {s ∈ S : a(s) bears relation R to b(s)}.The agent’s qualitative belief is modeled by a set Bel, which contains exactly thepropositions that she believes. When proposition A belongs to Bel, write Bel(A)

to emphasize what it means: the agent believes A. Then the Cliche Rule can bereformulated as follows: a � b if Bel([[a > b]]).

Here is another example of everyday practical reasoning that illustrates other deci-sion rules. Suppose that Alice needs the projector for a presentation in class. Shebelieves that the projector is already in the seminar room. She also believes that, ifthe projector is not in the seminar room, it is in the department office. So she decidesin favor of the following act, which is quite reasonable:

a: search the seminar room first and, if the projector is not found, go to thedepartment office.

Alice prefers the above act to the following, lazy act:

b: search the seminar room and give up if the projector is not found.

4This Savage-style setup assumes the so-called constant acts, which many authors find very unrealistic(e.g., [28]). In fact, constant acts are not essential to the proposed decision theory, and they are assumedhere only for the sake of simplicity—see Appendix E for details.5Namely, a ∼ b iff a � b � a, and a � b iff a � b � a.6Namely, o ≡ o′ iff o ≥ o′ ≥ o, and o > o′ iff o ≥ o′ ≥ o.7That notation is borrowed from statistics. Think of a as a random variable: proposition/event [[a > 3]] isa shorthand for {s ∈ S : a(s) > 3}.

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We want to explain why Alice prefers a to b. Consider the following decision matrix,where the rows are acts and the columns are states concerning where the projector is:

in seminar room in department office in neither

act a projector found projector found energy wastedact b projector found projector not found energy saved

comparison a(s) b(s) a(s) > b(s) a(s) < b(s)

Recall that Alice believes that the projector is already in the seminar room, so shebelieves that the outcomes that a and b would produce are equally desirable. But merebelief of that does not suffice for taking a and b to be equally preferable, as shownby Alice’s preference of a to b. That would require full certainty:

(Sure Equivalence Rule) Take acts a and b to be equally preferable if you aresure that the outcomes that a and b would produce are equally desirable.

So, by the above rule, Alice is not sure that the outcomes that a and b would produceare equally desirable, although she does believe that they are. The key to explain-ing Alice’s preference seems to be what she believes conditionally, which can beunderstood in terms of the following practical reasoning:

(6) If the projector is not in the seminar room, it is in the department office.(7) So if a and b would not produce equally desirable outcomes, it is a that would

produce a better outcome.(8) Therefore, I prefer a to b.

So Alice’s preference can be explained by a conditional version of the Cliche Rule:

(Conditional Cliche Rule) Prefer act a to act b if: you are not sure that theoutcomes that a and b would produce are equally desirable, and yet you believethat, if those outcomes turn out to be not equally desirable, it is a that wouldproduce a better outcome.

What one believes conditionally is modeled as follows. A conditional belief setis a set Bel of ordered pairs of propositions.8 When (A, B) belongs to Bel, writeBel(A ⇒ B) to emphasize what it means: the agent believes that, if A is true, B istrue.9 For the sake of simplicity, let conditional belief reduce the other two doxastic

8Although the English statement of the Conditional Cliche Rule emplys the phrase ‘believes that, if A, thenB’, the present work is compatible with different interpretation of that phrase. It is compatible with theinterpreation that the belief of a conditional is the belief of the proposition expressed by the conditional. Itis also compatible with the alternative interpretation that the belief of a conditional is a conditional beliefthat is irreducible to the belief of any proposition. That is why I model the concept of ‘believes that, if A,then B’ by ordered pairs of propositions.9Remark on notation: For readers familiar with nonmonotonic logic, think of Bel(A ⇒ B) as A |∼ B.Although the latter notation is standard in nonmonotonic logic, I do not adopt it because it is usuallyunderstood to formalize “if A, normally B” while I wish to formalize “the agent believes that, if A, thenB”. An alternative notation is Bel(B|A); read: the agent believes B given A. Although that notation wouldfit the practice in probability theory better, it reverses the order of antecedent A and consequent B thatis commonly adopted in nonmonotonic logic and, hence, is awkward for stating the logical principlesstandardly called “left equivalence” and “right weakening”. Note that A ⇒ B is not intended to denote anelement of an algebra; think of Bel( · ⇒ · ) as a binary relation between propositions.


attitudes as follows. The agent believes that A is true just in case she believes that,if the tautology � is true, A is true. The agent is sure that A is true just in caseshe believes that, if A is false, anything (including the contradiction ⊥) is true. Insymbols:

Bel(A) iff Bel(� ⇒ A);Sure(A) iff Bel(¬A ⇒ ⊥);

where � is the tautology S and ⊥ is the contradiction ∅.10 So the three decision ruleswe have introduced can be reformulated as follows:

(Cliche Rule) a � b if Bel([[a > b]]).(Conditional Cliche Rule) a � b if ¬Sure([[a ≡ b]]) and Bel([[a ≡ b]] ⇒ [[a > b]]).(Sure Equivalence Rule) a ∼ b if Sure([[a ≡ b]]).

Those three rules provide sufficient conditions for strict or equal preferences interms of desirability order ≥, conditional belief set Bel, and no more. So they pro-vide a qualitative means for decision-making. Call them the basic rules for everydaydecision.

3 Qualitative Means for Bayesian Ends

Everyday practical reasoning is not meant to replace full Bayesian rationality. When,for example, the agent is considering stock investment, the situation is typically souncertain that the agent has no ground for judging whether one investment plan wouldyield a more desirable outcome than another investment plan on the table. In thatcase the Cliche Rule would be silent, saying nothing about which to prefer. Wheneveryday practical reasoning is silent, the agent can move to a more refined deci-sion procedure, such as the Bayesian ideal of expected utility maximization. What’simportant to our daily life is, rather, that when everyday practical reasoning yieldssome recommendation, it may serve as a qualitative means for the Bayesian ends.

Suppose, for example, that an agent is running out of food at home and believesthat the grocery store is open today:

s1: store open today s2: store closed today

a: going today satisfying dinner energy wasted the preferred actb: not going today no dinner energy saved

comparison a(s) > b(s) a(s) < b(s)

10It may appear surprising that the concept of “sure” can be reduced to conditional beliefs. But here is aheursitic argument that motivates the reduction. Argue as follows that Bel(¬A ⇒ ⊥) implies P(A) = 1.Suppose for reductio that Bel(¬A ⇒ ⊥) but P(A) < 1. So P(¬A) > 0 and, hence, P(⊥ | ¬A)

is defined and equal to 0. But we should never believe a conditional with zero conditional probability,so ¬Bel(¬A ⇒ ⊥), which contradicts our supposition. So if Bel(¬A ⇒ ⊥), then P(A) = 1 (and,furthermore, P(⊥ | ¬A) is not defined). I propose that we stipulate that the converse is also true, andobtain the biconditional: Bel(¬A ⇒ ⊥) iff P(A) = 1. I think one of the anonymous referees for his orher critical comment, which inspires this heuristic argument.

838 H. Lin

By applying the Cliche Rule, the agent prefers going today rather than not (i.e.,a � b). That particular application maximizes expected utility if and only if:

∑

i=1,2

P(si)U(a(si)) >∑

i=1,2

P(si)U(b(si)),

where P and U are the agent’s underlying probability measure and utility function,respectively.11 So, for maximization of expected utility, it suffices to require that:

P(store open today) >1

1 + U(satisfying dinner)−U(no dinner)U(energy saved)−U(energy wasted)

.

Hence, it suffices that probability P(store open today) is high enough; namely,the qualitative belief to which the Cliche Rule applies is sufficiently probable.Or, it suffices that utility difference U(satisfying dinner) − U(no dinner) is highenough; namely, the extra gain of utility for the preferred act a is sufficiently highgiven the truth of the qualitative belief (s1). Or, it suffices that utility differenceU(energy saved) − U(energy wasted) is low enough; namely, the extra loss in utilityfor the preferred act a is sufficiently low given the falsity of the qualitative belief (s2).

A more stringent condition has to be met if the possible applications of the basicrules to arbitrary acts in a decision problem are guaranteed to be consistent withBayesian preference. Let a decision scenario be defined by a quadruple (P, U,Bel, ≥), which characterizes aspects of one and the same agent’s mental state. (i) P is aprobability distribution defined on S, which is assumed to contain only finitely manystates for the sake of simplicity. (ii) U : O → R is a utility function. (iii) Bel is aconditional belief set over the propositions over S. (iv) ≥ is a desirability order overO. In the rest of this section, definitions, principles, and theorems are all relative toan arbitrary, fixed decision scenario (P, U,Bel, ≥).

The agent’s underlying Bayesian preference �Bayes is defined as follows: for allacts a, b,

a �Bayes b iff∑

s∈S

P (s)U(a(s)) ≥∑

s∈S

P (s)U(b(s)).

Let �basic be the preference relation that results from application of exactly the threebasic rules for everyday decision: for all acts a, b,

a �basic b iff

{either: Bel([[a > b]]),or: ¬Sure([[a ≡ b]]) and Bel([[a ≡ b]] ⇒ [[a > b]]); (1)

a ∼basic b iff Sure([[a ≡ b]]). (2)

When a �basic b and b �basic a, it means that the basic rules are silent about the com-parison between a and b. Say that the basic rules for everyday decision is consistentwith Bayesian preference if and only if:

11P(s) abbreviates the strictly correct notation P({s}).


(Consistency with Bayesian Preference) For all acts a, b : S → O,

a �basic b =⇒ a �Bayes b;a ∼basic b =⇒ a ∼Bayes b.

A necessary and sufficient condition is provided in the following result:

Theorem 1 Suppose that state space S and outcome space O are both finite, thatthere are outcomes o0, o1 ∈ O such that o0 < o1, that Sure(�), and that Bel(A ⇒A) for each proposition A ⊆ S. Then, (Consistency with Bayesian Preference) holdsif and only if all of the following three conditions hold:

1. ≥ is represented by U in the sense that, for all outcomes o, o′ ∈ O,

o > o′ =⇒ U(o) > U(o′),o ≡ o′ =⇒ U(o) = U(o′);

2. for each proposition A ⊆ S,

Sure(A) ⇐⇒ P(A) = 1;3. for all propositions A, B ⊆ S such that A ⊇ B and ¬Sure(¬A) (which is

equivalent to P(A) > 0 due to condition 2),12

Bel(A⇒B) =⇒ P(B|A) >�U

�U + δU,

where:

δU = min{ |U(o) − U(o′)| : o, o′ ∈ O and o > o′ },�U = max{ |U(o) − U(o′)| : o, o′ ∈ O }.

It has long been speculated that qualitative belief is related to high probability.In light of the third condition, each qualitative belief has to be highly probable ifeveryday practical reasoning is consistent with Bayesian preference. Note that it isnot required that each highly probable proposition be believed, which is one of thepremises that lead to the lottery paradox [17]. Furthermore, the threshold �U/(�U+δU) that defines high probability depends on contextual factors such as O (the setof outcomes that are relevant for the agent in the present context) and U (the agent’sutility function in the present context).

4 Representation Theorem

Everyday practical reasoning is a kind of qualitative decision. But what kinds ofqualitative decisions can be represented as results of everyday practical reasoning?Furthermore, what should be the logic for extending one’s qualitative beliefs ineveryday practical reasoning? To what extent is that logic necessary for the pur-

12Conditional probability P(B|A) is defined in the standard way as P(B ∩ A)/P (A), which exists just incase P(A) > 0.

840 H. Lin

poses of qualitative decisions? This section provides a new representation theoremfor answering those questions.

A preference relation � over acts is said to be simplified qualitatively if and onlyit satisfies the following four axioms.

Axiom 1 (Reflexivity) For each act a, a � a.

Axiom 2 (Transitivity) For all acts a, b, c, if a � b and b � c, then a � c.

It is allowed that two acts a, b are non-comparable in the sense that a � b and b � a

(for the reason mentioned before: qualitative decision may be too crude to result inany judgement and, hence, have to be silent in that case). Following the standardpractice, an outcome o will sometimes be abused to denote the (constant) act thatproduces the same outcome o at every state. So, since � is a relation between acts,o � o′ means that the constant act o bears relation � to the constant act o′.

Axiom 3 (Dominance) For all acts a, b, if a(s) � b(s) for every state s ∈ S, thena � b.

The above three axioms are standard, albeit not entirely free of controversy.13 Thelast axiom is substantial:

Axiom 4 (Qualitative Simplification) For all acts a, b, a′, b′, if the following holds:

for each states ∈ S,

a(s) � b(s) iff a′(s) � b′(s),a(s) ∼ b(s) iff a′(s) ∼ b′(s),a(s) ≺ b(s) iff a′(s) ≺ b′(s),

then we have:

a � b iff a′ � b′.

The (Qualitative Simplification) axiom is not meant to capture all possible qualitativedecisions; for example, it can be violated by the maximin decision rule.14 Instead,that axiom captures preferences that result from decision procedures in which theagent compares two acts by considering only the following questions: “Which actwould produce more desirable outcomes in which states”? “In which states wouldthey produce equally desirable outcomes”? Furthermore, (Qualitative Simplification)

13See, for example, [20] for discussion about the validity of transitivity. See Section 5 for getting rid ofconstant acts.14In the following example, the maximin rule favors a over b and favors b′ over a′, which violates(Qualitative Simplification):

s1 s2

a 1 46b 0 47

s1 s2

a 101 46b 100 47


is not intended to be a universal principle of rationality. As explained in Section 3,one may resort to Bayesian decision in cases in which qualitative decision is silentbecause it is too coarse to yield any recommendation.

A preference relation � is said to be represented by the basic rules for everydaydecision with respect to desirability order ≥ and conditional belief set Bel if and onlyif, for all acts a, b,

a � b iff

{either: Bel([[a > b]]),or: ¬Sure([[a ≡ b]]) and Bel([[a ≡ b]] ⇒ [[a > b]]);

a ∼ b iff Sure([[a ≡ b]]).In other words, � results from application of exactly the three basic rules for everydaydecision. A conditional belief set Bel is said to be consistent if and only if ¬Bel(⊥),i.e. ¬Bel(� ⇒ ⊥). Say that Bel is closed under system P if and only if it satisfies thefollowing closure properties:15 for all propositions A, B, C ⊆ S,

Bel(A ⇒ A) (Reflexivity)

Bel(A ⇒ B)

B ⊆ C (Right Weakening)

Bel(A ⇒ C)

Bel(A ⇒ B)

Bel(A ⇒ C) (Cautious Monotonicity)

Bel(A ∩ B ⇒ C)

Bel(A ⇒ B)

Bel(A ⇒ C) (And / Conjunctive Closure)Bel(A ⇒ B ∩ C)

Bel(A ⇒ C)

Bel(B ⇒ C) (Or / Case Reasoning)

Bel(A ∪ B ⇒ C)

By the reduction rule that Bel(A) iff Bel(� ⇒ A), it is routine to verify thatqualitative beliefs Bel(·) are closed under classical logic given (Reflexivity), (RightWeakening) and (And). By the reduction rule that Sure(A) iff Bel(¬A ⇒ ⊥),fully certain propositions Sure(·) are closed under classical logic given (Reflexivity),(Right Weakening), (Cautious Monotonicity), and (Or). Then we have:

15Strictly speaking, system P also contains:

Bel(A ⇒ B)

A is logically equivalent to C (Left Equivalence)Bel(C ⇒ B)

But given the non-syntactical treatment adopted here, two propositions are logically equivalent iff they areidentical. So (Left Equivalence) holds trivially.

842 H. Lin

Theorem 2 (Sound Representation) Suppose that desirability order ≥ over out-comes is reflexive and transitive, and that conditional belief set Bel is consistent andclosed under system P. Let preference relation � over acts be represented by thebasic rules for everyday decision with respect to Bel and ≥. Then � is simplifiedqualitatively.

Note that desirability order ≥ is only required to be reflexive and transitive, allow-ing for non-comparability: o ≥ o′ and o′ ≥ o. Non-comparable outcomes do occurin qualitative decision. One may take some outcomes to be genuinely incomparable.Or one may have to treat two outcomes as non-comparable in the sense of suspend-ing judgment about which outcome is at least as desirable as the other, because thetime is too short for thinking it through.

Complete representation requires a weak assumption about outcomes. Introducethe following notation for non-comparability: a � b iff a � b and b � a. Define thefollowing condition:

(Regularity)

1. There are three distinct outcomes o0, o1, o2 in O such that o0 ≺ o1 ≺ o2.2. If there are non-comparable outcomes in O with respect to �, then:

2.1 there is at least one pair of non-comparable outcomes in O thathave a common upper or lower bound in O with respect to �,and

2.2 the non-comparability relation � is not transitive for at leastone example: there are three outcomes o, o′, o′′ in O such thato � o′ � o′′ � o.

Unique representation requires a weak assumption about conditional belief sets.Say that Bel is closed under the minimal system for nonmonotonic logic if and onlyif it satisfies:

(Reflexivity), already listed in system P;

(Right Weakening), already listed, too;

Bel(A ⇒ B)

B ⊆ C (Hypothetico-Deductive Monotonicity)16

Bel(A ∩ C ⇒ B)

Bel(A ⇒ B) (Weak And)

Bel(A ⇒ A ∩ B)

16Philosophers of science speak of hypothetico-deductivism as the view that observing a logical conse-quence C of a theory B provides evidence in favor of the theory. Since it would be strange to retracttheory C in light of new, positive evidence B, I refer to the proposed principle as Hypothetico-DeductiveMonotonicity.


This system is so weak and uncontroversial that it is derivable in every systemever studied in non-monotonic logic—that is why I call it “minimal”.17 In partic-ular, the minimal system does not presuppose conjunctive closure, i.e. the (And)rule in system P, which is what we want to justify rather than presuppose. Thenwe have:

Theorem 3 (Complete Representation) Assume the (Regularity) condition. Sup-pose that preference relation � over acts is simplified qualitatively. Then � isrepresented by the basic rules for everyday decision with respect to, first, a uniquedesirability relation ≥ and, second, a unique conditional belief set Bel that is closedunder the minimal system. Furthermore:

1. the uniquely determined ≥ is reflexive and transitive;2. the uniquely determined Bel is consistent and closed under system P.

In light of the existence result in the above theorem, the basic rules for everydaydecision plus nonmonotonic logic P provide a complete approach to decision-making if the situation makes it reasonable for the agent to base her choiceupon qualitatively simplified preferences (assuming the (Regularity) condition). Inlight of the uniqueness result, if the agent wishes to generate qualitatively simpli-fied preferences by employing exactly the three basic rules for everyday decision,then the nonmonotonic logic in use must be system P (assuming the minimalsystem).

5 Decision Rules in an Elegant Form

The three basic rules for everyday decision take an elegant form when we assumenonmonotonic logical system P. The elegant form facilitates proof of the representa-tion theorem.

As a first step, we can prove that, under appropriate conditions, the Cliche Rule isredundant because it can be subsumed under its conditional version:

Proposition 1 Suppose that ≥ is reflexive and transitive and Bel is consistentand closed under system P. Then, the triggering condition of the Cliche Rule (i.e.Bel(S ⇒ [[a > b]]) implies the triggering condition of the Conditional Cliche Rulei.e. Bel([[a ≡ b]] ⇒ [[a > b]]) and ¬Bel([[a ≡ b]] ⇒ ∅)).

17The weakest nonmonotonic logics ever studied are perhaps those validated by the probability-thresholdsemantics; see [11], [12], and [24]. In the present terminology, that semantics can be defined as follows. Foreach probability-threshold t and each probability measure P over the propositions, define the followingconditional belief set: BelP,t (A ⇒ B) iff either P(B|A) ≥ t or P(A) = 0. The class of such conditionalbelief sets validates the minimal system, while it invalidates system P (e.g., (Cautious Monotonicity),(And), and (Or)).

844 H. Lin

Recall that a preference relation � over acts is said to be represented by the basicrules for everyday decision with respect to ≥ and Bel just in case:

a � b ⇐⇒{

either: Bel(S ⇒ [[a > b]]),or: ¬Bel( [[a ≡ b]] ⇒ ∅) and Bel( [[a ≡ b]] ⇒ [[a > b]]). (3)

a ∼ b ⇐⇒ Bel( [[a ≡ b]] ⇒ ∅); (4)

So, given what Proposition 1 supposes, (3) is equivalent to the following:

a � b ⇐⇒ ¬Bel( [[a ≡ b]] ⇒ ∅) and Bel( [[a ≡ b]] ⇒ [[a > b]]). (5)

We can even combine everything into a single formula:

Proposition 2 (Equivalent Form) Suppose that ≥ is reflexive and transitive and Belis consistent and closed under system P. Then, (3)+(4) is equivalent to the following,elegant form:

a � b ⇐⇒ Bel( [[a ≡ b]] ⇒ [[a > b]] ). (6)

6 Comparison with Savage’s Representation Theorem

The representation theorem assumes little about the cardinality of states, outcomes,and acts: S is non-empty by definition, O contains at least three outcomes by the(Regularity) condition and, hence, there are at least three acts in OS . In contrast,Savage’s [27] representation theorem for expected utility maximization requires infi-nite many states. Indeed, unique representation for maximization of expected utilityrequires infinitely many objects—states, outcomes, or acts—in order to construct afine-grained yardstick for measuring (sets of) probabilities.18 In contrast, the repre-sentation result that we just obtained for everyday practical reasoning is applicableeven when the states in S correspond to very coarse-grained possibilities, such as “itwill rain today” vs. “it will not”. Those two states may be all the possibilities that theagent wishes to distinguish (due to, for example, lack of time). In the coarse-grainedstate “it will rain today”, the act “bring the umbrella” would produce an outcome thatcorresponds to an unspecific description, such as “you are kept dry or at least nottoo wet”. It is quite unspecific, but may suffice for the agent’s purposes in context.Such coarse-grained states and unspecific outcomes are common in everyday practi-cal reasoning, so it is important that the proposed decision theory does apply to thosecases.

18For another example, Gilboa and Schmeidler [10] assume infinitely many acts (for any two acts f, g andreal number r ∈ (0, 1), the mixture rf + (1 − r)g is a distinct act).


The (Qualitative Simplification) axiom implies Savage’s (Independence) axiom:the preference relation between any two acts remains the same if, in a state inwhich the two acts produce equally desirable outcomes, the original outcomes arereplaced by other outcomes that are still equally desirable. However, the (Indepen-dence) axiom, if taken as a universal principle of rationality, is controversial: it hasan influential counterexample known as the Ellsberg paradox [8]. Suppose that wehave an urn. We know for sure that it contains 30 red balls and 60 other balls thatare either black or yellow. We have no idea how many black or how many yellowballs there are. We also know for sure that the balls are well mixed so that eachball is as likely to be drawn as any other. You are now given a choice betweenthe following two gambles, whose payoffs depend on the color of the ball youdraw:

red black yellow

gamble a $100 $0 $0gamble b $0 $100 $0

You are also given the choice between the following two gambles (with a differentdraw from the same urn):

red black yellow

gamble c $100 $0 $100gamble d $0 $100 $100

Most people prefer a to b and prefer d to c, which violates (Independence) and, hence,(Qualitative Simplification).

Then, what is the implication of the Ellsberg paradox for the proposed deci-sion theory? Recall that (Qualitative Simplification) is not meant to be a universalprinciple of rationality, but only intended to characterize the generality of every-day practical reasoning as an approach to qualitative decision. So people’s reactionsin the Ellsberg case implies only the following: when facing the Ellsberg case, weshould not base the decision upon everyday practical reasoning. Indeed, an ordi-nary person would believe only that the color of the ball to be drawn is eitherred, black, or yellow—so the Cliche Rule would say nothing about which gam-bles to prefer. The other two basic rules for everyday decision would be silent,too. In that case, the agent may adopt a more refined decision procedure, possiblya generalization of expected utility maximization that accommodates the Ellsbergcase (e.g., [10]).

846 H. Lin

7 Concluding Remarks

The answers to the four foundational questions (Q1)–(Q4) for everyday practicalreasoning are summarized below, respectively:

(Q1) In everyday practical reasoning, what are the decision rules that connectqualitative belief and desire to preference over acts?

Answer: They include, at least, what I call the basic decision rules foreveryday decision: the Cliche Rule, its conditional version, and the SureEquivalence Rule (Section 2).

(Q2) What sort of logic should govern qualitative beliefs when it is used to guideeveryday practical reasoning? To what extent is that logic necessary for thepurposes of qualitative decisions?

Answer: It should be nonmonotonic logical system P. That logic isnecessary whenever an agent wishes to generate qualitatively simplifiedpreferences by employing exactly the three basic rules for everyday decision(Section 4).

(Q3) What kinds of qualitative decisions are always representable as results ofeveryday practical reasoning?

Answer: Given certain weak assumptions, decisions based upon qualita-tively simplified preferences are always representable as results of everydaypractical reasoning that only makes use of the three basic decision rules(Section 4).

(Q4) Under what circumstances does everyday practical reasoning agree with theBayesian ideal of expected utility maximization?

Answer: It suffices that the following three conditions hold. First, one’sdesirability order agrees with the her utility function. Second, a proposi-tion is fully certain if and only if it has probability 1. Third, each of one’s(conditional) beliefs has a high (conditional) probability, where the thresh-old that defines high probability is given in terms of the utilities involved inthe decision problem in context. It is not required that every highly probableproposition be believed (Section 3).

The proposed decision theory may have the following extensions or applications.The proposed theory concerns what an agent believes and prefers at a frozen time.So the next step is to relax that restriction and address the questions that emergefrom a diachronic setting. For example, how should an agent’s preference changein light of new information, and how should her qualitative desire and belief berevised? Does the answer imply any relation between Bayesian conditioning and


qualitative belief revision? If it does, is it compatible with the relations betweenBayesian conditioning and qualitative belief revision that are derived from epistemicconsiderations (e.g., [18, 19])?

The proposed decision theory justifies the use of nonmonotonic logical systemP from a pragmatic, non-epistemic perspective. It would be interesting to see howthe famous, strictly stronger system R corresponds to a more restricted class ofqualitative decisions (which can be obtained by adding more axioms to constrain thepreference relations over acts).

The three basic rules for everyday decision provide complete representationsfor qualitatively simplified preferences. It would be interesting to investigate otherclasses of preference relations in relation to other types of practical reasoning (say,practical reasoning with the notion of “being more likely than”).

The proposed theory explains how everyday practical reasoning can serve as aqualitative means for the Bayesian ideal. So, if everyday practical reasoning hascomputational advantage over Bayesian decision, the present work will have appli-cation to bounded rationality as studied in psychology and economics.19 In fact,the present work characterizes what everyday practical reasoning is and what it cando, which has to be known before we can proceed to determine its computationaladvantage.

Last but not the least, the present work employs an important methodology inBayesian epistemology (i.e., representation of preference) to study concepts in tra-ditional epistemology: qualitative beliefs and nonmonotonic inference. It wouldbe interesting to see whether Bayesian methodologies can contribute more toissues in traditional epistemology. For example, consider the issue called pragmaticencroachment in traditional epistemology: how do pragmatic concerns influencewhat counts as knowledge or justified beliefs?20 The present work may be ableto make a Bayesian contribution to that issue as follows. First, assume that anagent is justified in believing a set of propositions only if she can employ thosepropositions as premises for everyday practical reasoning without contradictingBayesian practical rationality. But freedom from such contradiction requires, byTheorem 1, that each of those qualitative beliefs has high subjective probability,exceeding the threshold �U/(�U + δU ) that depends on pragmatic features ofthe context (as explained at the end of Section 3). That may explain why justi-fied beliefs require a higher standard when stakes are high: high stakes raise themaximal difference �U of utilities and, hence, raise the probability lower bound�U/(�U + δU ) for a qualitative belief to be counted as justified. That is noth-ing more than a sketch of ideas, and I have to leave further development to a futurepaper.

19See, e.g., [25] for a review on the literatue about bounded rationality.20E.g., [9, 13, 14, 23, 26, 31].

848 H. Lin

Acknowledgments The author is indebted to Dubois and his colleagues’ pioneering work, with-out which I would not see the possibility of a foundation for everyday practical reasoning. I amalso indebted to David Etlin, Kevin Kelly, Hannes Leitgeb, Olivier Roy, Vincenzo Crupi, for exten-sive discussions. I am also indebted to the participants of the 10th Conference on Logic and theFoundations of Game and Decision Theory (Sevilla, June 2012) and the 5th Workshop on Fron-tiers of Rationality and Decision (Groningen, August 2012), especially Branden Fitelson, GiacomoBonanno, Michael Trost, Daniel Eckert, and Jan-Willem Romeijn. I am also indebted to the three anony-mous referees of LOFT and the two anonymous referees of the Journal of Philosophical Logic fordetailed comments. I am also indebted to Clark Glymour, Horacio Arlo-Costa, Kevin Zollman, ArthurPaul Pedersen, and Teddy Seidenfeld for comments on my earlier thoughts that lead to the presentpaper.

Appendix A: Proof of Theorem 1

Proof of the “If” Side Suppose that conditions 1-3 holds. Without danger ofconfusion, write a � b to abbreviate a �basic b. We will use the following results:

P([[a > b]]) · δU > P([[a ≥ b]]) · �U =⇒ EUP (a) > EUP (b); (7)

P([[a ≡ b]]) = 1 =⇒ EUP (a) = EUP (b). (8)

The proof is given below, which relies on the representability of ≥ by U:

EUP (a) − EUP (b) =∑

s ∈ S

P (s) [U(a(s)) − U(b(s))]

=∑

s ∈ [[a>b]]P(s) [U(a(s)) − U(b(s))]

+∑

s ∈ [[a≡b]]P(s) [U(a(s)) − U(b(s))]

+∑

s ∈ [[a ≥b]]P(s) [U(a(s)) − U(b(s))]

≥∑

s ∈ [[a>b]]P(s) δU + 0 −

∑

s ∈ [[a ≥b]]P(s)�U

= P([[a > b]]) δU − P([[a ≥ b]])�U.

P ([[a ≡ b]]) = 1 ⇒ P(X) = 1, where X = {s ∈ S : a(s) ≡ b(s)}⇒ P(Y ) = 1, where Y = {s ∈ S : U(a(s)) = U(b(s))} ⊆ X

⇒ EUP (a) = EUP (b).


Prove (Consistency with Bayesian Preference) as follows:

a � b ⇒{

either: Bel([[a > b]]),or: ¬Sure([[a ≡ b]]) and Bel([[a ≡ b]] ⇒ [[a > b]]);

⇒{

either: Bel(� ⇒ [[a > b]]),or: Bel([[a ≡ b]] ⇒ [[a > b]]);

⇒ P([[a > b]] | [[a ≡ b]]) >�U

�U + δU(by condition 3)

⇒ P([[a > b]] ∩ [[a ≡ b]])P ([[a ≡ b]]) >

�U

�U + δU

⇒ P([[a > b]])P ([[a > b]] ∪ [[a ≥ b]]) >

�U

�U + δU

⇒ P([[a > b]])P ([[a > b]]) + P([[a ≥ b]]) >

�U

�U + δU

⇒ P([[a > b]]) δU > P([[a ≥ b]])�U

⇒ EUP (a) > EUP (b) (by (8)).

a ∼ b ⇒ Sure([[a ≡ b]])⇒ P([[a ≡ b]]) = 1 (by condition 2)

⇒ EUP (a) = EUP (b) (by (8)).

Proof of the “Only If” Side Suppose that (Consistency with Bayesian Preference)holds. Argue as follows that ≥ is represented by U. Suppose that o1 ≡ o2. So[[o1 ≡ o2]] = �. Since Sure(�), it follows from the Sure Equivalence Rule thato1 ∼ o2. So, by (Consistency with Bayesian Preference), EUP (o1) = EUP (o2).It follows that U(o1) = U(o2). By the same argument, o1 > o2 implies U(o1) >

U(o2).Argue as follows that Sure(A) iff P(A) = 1. By hypothesis, o0 < o1. So, by rep-

resentability, U(o0) < U(o1). Let f be the act that maps each state s ∈ A to the worseoutcome o0, and maps the other states to the more desirable outcome o1. Compare f

to the constant act o0. If Sure(A), then, by the Sure Equivalence Rule, f ∼ o0, andhence, by (Consistency with Bayesian Preference), EUP (f ) = EUP (o0), which isthe case only if P(A) = 1. Suppose that ¬Sure(A). By hypothesis, Bel(¬A ⇒ ¬A)

(Reflexivity). It follows from the Conditional Cliche Rule that f � o0. Then, by(Consistency with Bayesian Preference), EUP (f ) > EUP (o0), which is the caseonly if P(A) = 1.

Suppose, for reductio, that condition 3 is violated; namely, there exist proposi-tions A, B ⊆ S such that A ⊇ B, Bel(A ⇒ B), ¬Sure(¬A) (so P(A) > 0), but

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P(B|A) ≤ �U�U+δU

. Since some outcome is better than some other outcome, δU > 0.

So P(B|A) ≤ �U�U+δU

< 1. It follows that:

P(A ∩ B)

P (A ∩ B)≤ �U

δU.

Then, the definition of �U and δU guarantees that there exist outcomes o0 ≤ o1 <

o2 ≤ o3 ∈ O such that:

P(A ∩ B)

P (A ∩ B)≤ U(o3) − U(o0)

U(o2) − U(o1)= �U

δU. (9)

Let a, b be the acts defined by the following conditions:

(i) for all s ∈ A ∩ B = B, a(s) = o2 > o1 = b(s);(ii) for all s ∈ A ∩ B, a(s) = o0 < o3 = b(s);

(iii) for all s ∈ A, a(s) = b(s) = o0.

Then, the upper bound in (9) guarantees that EUP (a) ≤ EUP (b). Argue as followsthat a � b. By the reductio hypothesis, Bel(A ⇒ B) and ¬Sure(¬A). Note thatA = [[a ≡ b]] and B = [[a > b]]. So, by the Conditional Cliche Rule, a � b. Torecap, we have that a � b but EUP (a) ≤ EUP (b), which contradicts (Consistencywith Bayesian Preference).

Appendix B: Proof of Propositions 1 and 2: Decision Rules in the Elegant Form

Proof of Proposition 1 Suppose that Bel(S ⇒ [[a > b]]). Since ≥ is reflexive andtransitive, [[a > b]] ⊆ [[a ≡ b]]. So, by (Right Weakening), Bel(S ⇒ [[a ≡ b]]).Since Bel(S ⇒ [[a > b]]) and Bel(S ⇒ [[a ≡ b]]), we have Bel([[a ≡ b]] ⇒ [[a >

b]]) by (Cautious Monotonicity). Suppose for reductio that Bel(S ⇒ [[a > b]]) andBel([[a ≡ b]] ⇒ ∅). Note that Bel([[a ≡ b]] ⇒ ∅) implies that Bel([[a ≡ b]] ⇒[[a ≡ b]]) by (Right Weakening), and we already have that Bel([[a ≡ b]] ⇒ [[a ≡ b]])by (Reflexivity). So, by (Or), Bel([[a ≡ b]] ∪ [[a ≡ b]] ⇒ [[a ≡ b]]). Namely,Bel(S ⇒ [[a ≡ b]]). But Bel(S ⇒ [[a > b]]) (by hypothesis). So, by (And), Bel(S ⇒[[a ≡ b]] ∩ [[a > b]]). But ≥ is a preorder, so Bel(S ⇒ ∅), which contradicts theassumption that Bel is consistent.

Proof of Proposition 2 Argue that (3)+(4) implies (6) as follows. Assume the basicrules (3)+(4). Suppose that a � b (and we want to show that Bel([[a ≡ b]] ⇒ [[a >

b]])). Case 1: a ∼ b. Then, by (4), Bel([[a ≡ b]] ⇒ ∅). So, by (Right Weakening),Bel([[a ≡ b]] ⇒ [[a > b]]). Case 2: a � b. Then, by (3), either Bel(S ⇒ [[a > b]]),or Bel([[a ≡ b]] ⇒ [[a > b]]). But the first disjunct implies the second disjunct, byProposition 1. Hence, the second disjunct is true; namely, Bel([[a ≡ b]] ⇒ [[a > b]]).To prove the converse, suppose that Bel([[a ≡ b]] ⇒ [[a > b]]) (and we want to showthat a � b). Case 1: Bel([[a ≡ b]] ⇒ ∅). Then apply (4) to derive a ∼ b and, thus,a � b. Case 2: ¬Bel([[a ≡ b]] ⇒ ∅). Then apply (3) to derive a � b and, thus,a � b.


Argue that (6) implies (3)+(4) as follows. Assume the elegant form of the basicrules (6). For the left-to-right side of (4), suppose a ∼ b. So a � b and b � a.Then, by (6), Bel([[a ≡ b]] ⇒ [[a > b]]) and Bel([[b ≡ a]] ⇒ [[a > b]]). The latteris equivalent to Bel([[a ≡ b]] ⇒ [[a < b]]). So, by (And), Bel([[a ≡ b]] ⇒ [[a >

b]] ∩ [[a < b]]). But the conjunction [[a > b]] ∩ [[a < b]] equals ∅, for ≥ is reflexiveand transitive. So we have that Bel([[a ≡ b]] ⇒ ∅), which is the right-hand side of(4), as required. For the right-to-left side of (4), suppose that Bel([[a ≡ b]] ⇒ ∅).Then, by (Right Weakening), we have both Bel([[a ≡ b]] ⇒ [[a > b]]) andBel([[a ≡ b]] ⇒ [[a < b]]). The latter is equivalent to Bel([[b ≡ a]] ⇒ [[a > b]]).So, by (6), a � b and b � a. Hence, a ∼ b, as required. To prove (3), note that,by definition, a � b iff (i) a � b and (ii) a ∼ b. By (6), the first conjunct (a � b)is equivalent to Bel([[a ≡ b]] ⇒ [[a > b]]). By (4), the second conjunct (a ∼ b) isequivalent to ¬Bel([[a ≡ b]] ⇒ ∅). So a � b iff Bel([[a ≡ b]] ⇒ [[a > b]]) and¬Bel([[a ≡ b]] ⇒ ∅), which implies (3) by Proposition 1.

Appendix C: Proof of Theorem 2: Sound Representation

The proof of (Reflexivity), (Dominance), and (Qualitative Simplification) is routineverification given the lemma that, for all outcomes o, o′ ∈ O, o ≥ o′ iff o � o′,which we prove as follows. Argue as follows that if o > o′ then o � o′. Suppose thato > o′. So [[o > o′]] = �. By (Reflexivity), Bel(� ⇒ �), namely Bel(�), namelyBel([[o > o′]]). So, by the Cliche Rule, o � o′. Argue as follows that if o ≡ o′ theno ∼ o′. Suppose that o ≡ o′. So [[o ≡ o′]] = �. By (Reflexivity), Bel(⊥ ⇒ ⊥),namely Bel(¬� ⇒ ⊥), namely Sure(�), namely Sure(o ≡ o′). So, by the SureEquivalence Rule, o ∼ o′. For the last case, argue as follows that if o ≥ o′ theno � o′. Suppose that o ≥ o′. So [[o > o′]] = ⊥, [[o ≡ o′]] = ⊥, and [[o ≡ o′]] = �.Then, o � o′ follows from the hypotheses that � is represented by the basic rules foreveryday decision, and that Bel is consistent (i.e., ¬Bel(� ⇒ ⊥)).

The proof of (Transitivity) is provided below, in which we will make extensive useof system P with two of its derived rules:

Bel(A ⇒ B)

Bel(A ∪ X ⇒ B ∪ X)(Left−Right Or)

Bel(A ⇒ B)

B ⊆ C

Bel(A ∩ C ⇒ B)

(Hypothetico−Deductive Monotonicity)

Proof of Transitivity Abbreviate Bel(X ⇒ Y ) as X |∼ Y (which is the standard nota-tion in nonmonotonic logic). Suppose that a � b and b � c. With the help ofProposition 2, apply the elegant form of the basic rules to a � b:

[[a ≡ b]] |∼ [[a > b]] (10)

Apply (Left-Right Or) to (10); then we have:

[[a ≡ b]] ∪ ([[a ≡ b]] ∩ [[b ≡ c]]) |∼ [[a > b]] ∪ ([[a ≡ b]] ∩ [[b ≡ c]]) (11)

852 H. Lin

For the antecedent of (11), distribute the first disjunct into the conjunction, and weobtain:

[[a ≡ b]] ∪ [[b ≡ c]] |∼ [[a > b]] ∪ ([[a ≡ b]] ∩ [[b ≡ c]]) (12)

Apply the elegant form of the basic rules to b � c, and repeat the derivation from(10) to (12)—but with [[aRb]] and [[bRc]] exchanged. Then we have:

[[b ≡ c]] ∪ [[a ≡ b]] |∼ [[b > c]] ∪ ([[b ≡ c]] ∩ [[a ≡ b]]) (13)

Apply (And) to (12) and (13), we have:

[[a ≡ b]] ∪ [[b ≡ c]] |∼ ([[a > b]] ∪ ([[a ≡ b]] ∩ [[b ≡ c]])) (14)

∩([[b > c]] ∪ ([[b ≡ c]] ∩ [[a ≡ b]])) (15)

Verify that the consequent of the above formula entails [[a > c]] as follows: rewritethe consequent of the above formula by the identity: (A∪A′)∩ (B ∪B ′) = (A∩B)∪(A ∩ B ′) ∪ (A′ ∩ B) ∪ (A′ ∩ B ′), so that we obtain the disjunction of four disjuncts;then it is routine to verify that each of the four disjuncts entails [[a > c]]. Then, apply(Right Weakening) to (15); we have:

[[a ≡ b]] ∪ [[b ≡ c]] |∼ [[a > c]] (16)

Apply the De Morgan rule to the antecedent of (16); then we have:

¬([[a ≡ b]] ∩ [[b ≡ c]]) |∼ [[a > c]] (17)

Note that the consequent of (17) entails [[a ≡ c]]. So, apply (Hypothetico-DeductiveMonotonicity) to (17); then we have:

¬([[a ≡ b]] ∩ [[b ≡ c]]) ∩ [[a ≡ c]] |∼ [[a > c]] (18)

In the antecedent of (18), the second conjunct [[a ≡ c]] entails the first conjunct, sothe entire antecedent is equal to the second conjunct. Hence:

[[a ≡ c]] |∼ [[a > c]] (19)

Then, apply the elegant form of the basic rules to (19), we have that a � c, asrequired.

Appendix D: Proof of Theorem 3: Complete Representation

Let preference relation � be simplified qualitatively, i.e., satisfy axioms (Reflexiv-ity), (Transitivity), (Dominance), and (Qualitative Simplification). The full strengthof the (Regularity) condition is needless until Lemma 5. For now we only need toassume a strictly weaker condition: there exist outcomes 0, 1 ∈ O such that 0 ≺ 1.For each proposition A ⊆ S, let fA be the act defined by:

fA(s) ={

1, if s ∈ A;0, if s ∈ A.

(20)

Define conditional belief set Bel as follows:

Bel(A ⇒ B) iff fA∩B � fA∩B , (21)


for all propositions A, B ⊆ S. Define a desirability relation ≥ over outcomes asfollows:

o ≥ o′ iff o � o′ ,

for all outcomes o, o′ ∈ O.

Lemma 1 ≥ is reflexive and transitive.

Proof ≥ is, by definition, isomorphic to the restriction of � to constant acts, whichsatisfies (Reflexivity) and (Transitivity).

Lemma 2 Bel is consistent.

Proof Suppose for reductio that Bel(S ⇒ ∅). Then, by definition, fS∩∅ � fS∩∅.In other words, f∅ � fS . So, by definition, 0 � 1, which contradicts that 1 � 0.

Lemma 3 Bel satisfies (Reflexivity).

Proof Let A be an arbitrary proposition. By (Dominance), fA � f∅. In other words,fA∩A � fA∩A. So, by definition, Bel(A ⇒ A).

Lemma 4 Bel satisfies (Right Weakening).

Proof Suppose that Bel(X ⇒ A) and A ⊆ A′. Since Bel(X ⇒ A), fX∩A � fX∩A,by definition. Then, since A ⊆ A′, it follows from (Dominance) both that fX∩A′ �fX∩A and that fX∩A � f

X∩A′ . So we have established that fX∩A′ � fX∩A �fX∩A � f

X∩A′ . Then, by (Transitivity), fX∩A′ � fX∩A′ . Hence, by definition,

Bel(X ⇒ A′).

From now on, assume the full strength of (Regularity). So we have outcomes0, 1, 2 ∈ O such that 0 ≺ 1 ≺ 2. When the antecedent of the (Qualitative Simpli-fication) axiom holds, say that the two act-pairs in question, 〈a, b〉 and

⟨a′, b′⟩, are

ordinally similar.

Lemma 5 � is represented by the elegant form of the basic rules with respect to ≥and Bel; namely, for all acts a, b:

a � b ⇐⇒ Bel( [[a ≡ b]] ⇒ [[a > b]] ).

Proof Let a, b be arbitrary acts. Define propositions N = [[a ≡ b]] (‘N’ for ‘notequally good’), B = [[a > b]] (‘B’ for ‘better’). By the definition of Bel, it sufficesto prove the following equivalence:

a � b ⇐⇒ fN∩B � fN∩B. (22)

854 H. Lin

Here are the preliminaries. Let s be an arbitrary state in S. Then:

a(s) � b(s) ⇐⇒ a(s) > b(s)

⇐⇒ s ∈ B

⇐⇒ s ∈ N ∩ B

⇐⇒ fN∩B(s) = 1 � 0 = fN∩B(s)

⇐⇒ fN∩B(s) � fN∩B(s).

Similarly,

a(s) ≺ b(s) ⇐⇒ fN∩B(s) ≺ fN∩B(s).

Then we prove the equivalence (22) by the following cases.

Case 1: all outcomes in O are comparable with respect to �. Then we have:

a(s) ∼ b(s) ⇐⇒ a(s) ≡ b(s)

⇐⇒ s ∈ N

⇐⇒ fN∩B(s) = 0 ∼ 0 = fN∩B(s)

⇐⇒ fN∩B(s) ∼ fN∩B(s).

So 〈a, b〉 is ordinally similar to⟨fN∩B, fN∩B

⟩and, hence, (Qualita-

tive Simplification) applies: a � b iff fN∩B � fN∩B . (Whenever weapply (Qualitative Simplification) in the following, verification of ordinalsimilarity will be omitted because it is routine).

Case 2: some outcomes in O are non-comparable with respect to � and, hence, thesecond part of (Regularity) condition applies. Argue as follows that:

a � b =⇒ fN∩B � fN∩B.

Suppose that a � b. By (Regularity), there are outcomes o � o′ � o′′ � o.Define acts c, d, e as follows (the outcomes of the acts are specified withrespect to the states in the following four propositions, which are mutuallyexclusive and jointly exhaustive):

[[a > b ]] [[a < b ]] [[a b]] [[a b]]

c 2 0 o 0d 1 1 o 0e 0 2 o 0

Then, by (Qualitative Simplification) and by the same argument as in case1, we have: a �b iff c�d iff d �e. Since a �b (by hypothesis), we havethat c�d and d�e. Then, by (Transitivity), c�e. Recall that N =[[a ≡ b]]and B = [[a > b]]. So fN∩B and fN∩B can be expressed by:

[[a > b ]] [[a < b ]] [[a b]] [[a b]]

f N B 1 0 0 0f N B 0 1 1 0


Then, since o′′ � o, it follows from (Qualitative Simplification) thatc � e iff fN∩B � fN∩B . Since c � e, we conclude that fN∩B � fN∩B ,as required. Argue as follows that:

fN∩B � fN∩B =⇒ a � b.

Suppose that fN∩B � fN∩B . By (Regularity), there are non-comparableoutcomes o, o′ that have a common upper or lower bound. Case 2.1: o ando′ have a common upper bound o∗; namely: o∗ � o � o′ ≺ o∗. Defineacts g, h∗, i as follows:

[[a > b ]] [[a < b ]] [[a b]] [[a b]]

g 1 0 o 0h 0 1 o 0i 0 1 o 0

Then, by (Qualitative Simplification), fN∩B � fN∩B iff g � h∗. So,since fA∩B � fA∩B (by hypothesis), we have that g � h∗. But h∗ �i, by (Dominance). So, by (Transitivity), we have that g � i. Then, by(Qualitative Simplification), g � i iff a � b. Hence, a � b. Case 2.2: oand o′ have a common lower bound o∗. So we have that o∗ ≺ o � o′ � o∗.Define act h∗, to be compared to the acts g, i that we have defined:

[[a > b ]] [[a < b ]] [[a b]] [[a b]]

j 1 0 o 0k 1 0 o 0l 0 1 o 0

By the same argument, we have: j � k∗, by (Dominance); fN∩B � fN∩B

iff k∗ � l, by (Qualitative Simplification); k∗ � l (since fN∩B � fN∩B

by hypothesis); hence, j � l, by (Transitivity). But j � l iff a �, by(Qualitative Simplification). So a � b, as required.

By Case 1 and 2, we have established that a � b iff fN∩B � fN∩B , as required.

Lemma 6 For all propositions A, B:

Bel(A ⇒ B) ⇐⇒ Bel(A ⇒ A ∩ B).

Proof Bel(A ⇒ B) is equivalent to fA∩B � fA∩B (by definition), which isequivalent to fA∩(A∩B) � fA∩(A∩B) (by the Boolean algebra of sets), which isequivalent to Bel(A ⇒ A ∩ B) (by definition).

Lemma 7 Bel satisfies (And).

Proof Let A, B, X be arbitrary propositions. Suppose that Bel(X ⇒ A) andBel(X ⇒ B). Define acts i, j, k as follows (note that the five propositions in the

856 H. Lin

first row are mutually exclusive and jointly exhaustive, where concatenation meansconjunction):21

ABX ABX ABX A BX X

i 2 1 0 0 0j 1 0 2 1 0k 0 2 1 2 0

Since Bel(X ⇒ A), we have that Bel(X ⇒ A ∩ X) (by Lemma 6). According tothe table, X = [[i ≡ j ]] and A ∩ X = [[i > j ]]. So Bel([[i ≡ j ]] ⇒ [[i > j ]]). Then,by Lemma 5, i � j . By the same argument (with A replaced by B and ij replaced byjk), we have that j � k. Then, by (Transitivity), we have that i � k. So, by Lemma 5,Bel([[i ≡ k]] ⇒ [[i > k]]). But [[i ≡ k]] = X and [[i > k]] = ABX (by the table). Sosubstitution yields that Bel(X ⇒ ABX). Hence, by Lemma 6, Bel(X ⇒ A∩B).

To prove that Bel satisfies (Cautious Monotonicity), we will employ some earlierresults plus:

Lemma 8 Bel satisfies:

Bel(A ⇒ B)

B ⊆ X

Bel(A ∩ X ⇒ B)

(Hypothetico-Deductive Monotonicity)

Proof Suppose that Bel(A ⇒ B) and B ⊆ X. Then, since Bel(A ⇒ B), fA∩B �fA∩B . Also, by (Dominance), fA∩B � fA∩X∩B . So, to recap, we have: fA∩X∩B =fA∩B � fA∩B � fA∩X∩B . Then, by (Transitivity), fA∩X∩B � fA∩X∩B . So, bydefinition, Bel(A ∩ X ⇒ B).

Lemma 9 Bel satisfies (Cautious Monotonicity).

Proof It is routine to prove that (Cautious Monotonicity) follows from (Right Weak-ening), (And), and (Hypothetico-Deductive Monotonicity). So the present lemma isan immediate consequence of Lemmas 4, 7, and 8.

To prove that Bel satisfies (Or), we will employ some earlier results plus:

Lemma 10 Bel satisfies:

Bel(A ⇒ B)

A ∩ X = ∅

Bel(A ∪ X ⇒ B ∪ X)

(Weak Or)

Proof Suppose that Bel(A ⇒ B) and A ∩ X = ∅. So, by definition, fA∩B �fA∩B . By Boolean algebra, A ∩ B ⊆ (A ∪ X) ∩ (B ∪ X). Then, by (Dominance),

21This strategy of proving (And) is due to [7].


f(A∪X)∩(B∪X) � fA∩B . By A ∩ X = ∅ and Boolean algebra, A ∩ B = (A ∪ X) ∩(B ∪ X). So, to recap, f(A∪X)∩(B∪X) � fA∩B � fA∩B = f(A∪X)∩(B∪X). Hence, by(Transitivity), we have that f(A∪X)∩(B∪X) � f(A∪X)∩(B∪X). Then, by definition, wehave that Bel(A ∪ X ⇒ B ∪ X), as required.

Lemma 11 Bel satisfies (Or).

Proof It is routine to prove that (Or) follows from (And) plus (Weak Or). So thepresent lemma is an immediate consequence of Lemmas 7 and 10.

Lemma 12 Bel is consistent and closed under system P.

Proof Immediate from Lemmas 2, 3, 4, 7, 9, and 11.

Lemma 13 (Existence) Preference relation � is represented by the basic rules foreveryday decision with respect to 〈≥,Bel〉, where Bel is consistent and closed undersystem P and ≥ is reflexive and transitive.

Proof Immediate from Proposition 2 and Lemmas 1, 5, and 12.

Lemma 14 (Uniqueness) Suppose that � is represented by the basic rules for every-day decision with respect to

⟨≥′,Bel′⟩, where ≥′ is a desirability order over outcomes

and Bel′ is a conditional belief set closed under the minimal system: (Reflexivity),(Weak And), (Right Weakening), and (Hypothetico-Deductive Monotonicity). Then:≥′ = ≥ and Bel′ = Bel. And hence: ≥′ is reflexive and transitive, and Bel′ isconsistent and closed under system P.

Proof First, argue as follows that Bel′ is consistent. Suppose for reductio that it isnot consistent. Then, Bel′(S ⇒ ∅). Let ¬A be the negation of an arbitrary propo-sition. Since ∅ ⊆ ¬A, it follows from (Hypothetico-Deductive Monotonicity) thatBel′(¬A ∩ S ⇒ ∅), namely Bel′(¬A ⇒ ∅), namely Sure′(A). So Sure′(A), foreach proposition A ⊆ S. Then, by representability with respect to

⟨≥′,Bel′⟩, we have

that o1 ∼ o2 for all outcomes o1, o2 ∈ O. But that contradicts the (Regularity)condition.

Argue as follows that ≥′ = ≥. Let o1, o2 be arbitrary outcomes in O. Then it suf-fices to show that o1 ≥′ o2 iff o1 ≥ o2. Suppose that o1 ≥′ o2 (and we wantto show that o1 ≥ o2). Case 1: o1 ≡′ o2. Then [[o1 ≡′ o2]] = ∅, and, hence,Bel′([[o1 ≡′ o2]] ⇒ ∅) by (Reflexivity). Then, by representability with respect to⟨≥′,Bel′

⟩, we have that o1 ∼ o2. Then, by Lemma 13 and the basic rules with respect

to 〈≥,Bel〉, we have that Bel([[o1 ≡ o2]] ⇒ ∅). Note that [[o1 ≡ o2]] = S or ∅,because o1 and o2 are constant acts. Since Bel is consistent, [[o1 ≡ o2]] cannot beS and, hence, has to be ∅. So o1 ≡ o2. Hence, o1 ≥ o2. Case 2: o1 >′ o2. Then[[o1 >′ o2]] = S. So, by (Reflexivity), we have that Bel′(S ⇒ [[o1 >′ o2]]). Then, byrepresentability with respect to

⟨≥′,Bel′⟩, we have that o1 � o2. Then, by Lemma 13

and the basic rules with respect to 〈≥,Bel〉, we have that: ether Bel(S ⇒ [[o1 > o2]]),

858 H. Lin

or Bel([[o1 ≡ o2]] ⇒ [[o1 > o2]]). Then, by the consistency of Bel and by the sameargument as the preceding case, we have that [[o1 > o2]] = S. Namely, o1 > o2. Soo1 ≥ o2. We have established that o1 ≥′ o2 implies o1 ≥ o2, and the converse can bederived by the same argument with Bel and Bel′ exchanged, and with ≥ and ≥′. (Notethat we earlier employed the reflexivity and consistent of Bel; for Bel′, reflexivity isa premise and consistency has been established).

Argue as follows that Bel′ = Bel. Let A, B be arbitrary propositions. It suffices toshow that Bel′(A ⇒ B) iff Bel(A ⇒ B). Suppose that Bel′(A ⇒ B) (and we want toshow that Bel(A ⇒ B)). Consider acts fA∩B and fA∩B . Also note that, since ≥′=≥,we have that 1 >′ 0. Case I: Bel′(¬A ⇒ ∅). Then, by representability with respect to⟨≥′,Bel′

⟩, fA∩B ∼ fA∩B . So fA∩B � fA∩B . Then, by the definition of Bel, we have

that Bel(A ⇒ B), as required. Case II: ¬Bel′(¬A ⇒ ∅). Also, since Bel′(A ⇒ B),we have that Bel′(A ⇒ A∩B) by (Weak And). Then, by representability with respectto

⟨≥′,Bel′⟩, fA∩B � fA∩B . So fA∩B � fA∩B . Then, by the definition of Bel, we

have that Bel(A ⇒ B), as required. For the converse, suppose that Bel(A ⇒ B) (andwe want to show that Bel′(A ⇒ B)). Then, by the definition of Bel, we have thatfA∩B � fA∩B . Case α: fA∩B ∼ fA∩B . Note that fA∩B(s) ≡′ fA∩B(s) iff s ∈ A.Then, by representability with respect to

⟨≥′,Bel′⟩, we have that Bel′(A ⇒ ∅). So, by

(Right Weakening), Bel′(A ⇒ B). Case β: fA∩B � fA∩B . Then, by representabilitywith respect to

⟨≥′,Bel′⟩, either Bel′(S ⇒ A ∩ B), or Bel′(A ⇒ A ∩ B). The first

disjunct implies the second disjunct. For, if Bel′(S ⇒ A∩B), then, since A∩B ⊆ A,we have, by (Hypothetico-Deductive Monotonicity), that Bel′(A ⇒ A∩B). It followsthat Bel′(A ⇒ A ∩ B). Then, by (Right Weakening), Bel′(A ⇒ B).

Theorem 3 follows immediately from the existence Lemma 13 and the uniquenessLemma 14.

Appendix E: Freedom from Constant Acts

Constant acts are not essential to the representation Theorem 2 plus 3. Whenever wemake use of o � o′, the only property we need for the proof is that, for each states, the outcome of act o in s is preferred to the outcome of act o′ in s—namely, acto is preferred to act o′ given each state s ∈ S. In the present paper, we never needto compare an outcome at a state with another outcome at a distinct state. That isrequired by the basic rules for everyday decision, and also required by the (Qualita-tive Simplification) axiom. The same holds not only for ‘�’ but also for ‘∼’ and ‘�’.Accordingly, modify the definitions in the paper as follows. A decision problem isspecified by a set S of states such that the states s in S are associated with disjointsets Os of outcomes, where o ∈ Os means that outcome o is possible in state s.22 Anact is a function a that maps each s ∈ S to an outcome a(s) ∈ Os (i.e., a is a choicefunction over the product space

∏s∈S Os). Desirability orders ≥ are binary relations

22Why disjoint? Getting an umbrella in a rainy state is an outcome distinct from getting an umbrella in asunny state.


over the set of all outcomes,⋃

s∈S Os . Preference relations � are understood as con-taining two kinds of data: for acts a and b, a � b means that a is at least as preferableas b; for outcomes o and o′ that are possible in the same state s, o � o′ means thato is at least preferable as o′ given that s is the actual state. So a(s) � b(s) is to beunderstood as conditional preference given s in axioms (Dominance) and (QualitativeSimplification). The (Regularity) condition in its original form is about the outcomeset O; now replace it by the requirement that, for each s ∈ S, (Regularity) holds forOs . Then the representation Theorem 2 and 3 are still true—except that, in the theo-rem for complete representation, the existence of the required desirability order ≥ isunique only up to restriction to Os for each state s ∈ S (that is, the relative desirabilitybetween two outcomes that are possible in different states may not be unique).

Constant acts are essential only to the “only if” side of Theorem 1. When each states is associated with its own set Os of possible outcomes, the “if” side of Theorem 1is still true; namely, we still have the same sufficiency condition for (Consistencywith Bayesian Preference).

Appendix F: Comparison with the Pioneering Work of Dubois et al. [7]

Dubois et al. [7] does not intend their pioneer work to be about everyday practicalreasoning, but it is their work that makes me realize that it is possible to providea Savage-like foundation for everyday practical reasoning. But, in order to captureimportant features of everyday practical reasoning, I have to weaken their axiomati-zation and modify their modeling assumptions in a number of ways, as explained thefollowing.

1. Axiomatization. First, the set of my axioms theirs (what they call: WS1, WSTP,S3, S5, and OI). I have four axioms. The first three axioms (Reflexivity),(Transitivity), and (Dominance) are so basic that they are entailed by Dubois etal.’s axiomatization (and probably entailed by almost all axiomatizations in theliterature). My last axiom is (Qualitative Simplification), which is an immediateconsequence of Dubois et al.’s Ordinal Invariance (OI) axiom. So my axioma-tization is at least as weak as Dubois. In fact, mine is strictly weaker, becausetheir axiomatization has a consequence that mine does not have, namely thatthe strict desirability order > is rankable, or equivalently, that the non-strictdesirability order ≥ is a weak order (i.e., reflexive, transitive, and complete).That is stated in their proposition 8, whose proof makes essential use of theirOrdinal Invariance (OI) axiom.

2. Desirability. There is a subtle difference in the ways we model desirabilityover outcomes. Dubois et al. take the concept of “being more desirable than”(>) as primitive, which is unable to distinguish “equally desirable” from “non-comparable”. Instead, I take the concept of “being at least as desirable as” (≥)as primitive in order to distinguish “equally desirable” from “non-comparable”,which is crucial for the project I pursue. In qualitative decision-making, onemay have to treat two outcomes as non-comparable (at least for the time being)

860 H. Lin

because, for example, the time available may be too short for the agent tothink it through. And that is very different from the judgment that two out-comes are equally desirable. As mentioned in the preceding paragraph, Duboiset al. require that a strict desirability order > over outcomes be rankable, inthe sense that the relation R defined below is transitive: o1R o2 iff o1 > o2and o2 > o1. In their formal framework, the statement “o1R o2” does not saywhether the two outcomes are equally desirable or non-comparable, and eachof the two interpretations would make the rankability requirement too strong.Case 1: if R is understood to express equal desirability for all instances, then therequirement is equivalent to saying that we take ≥ as primitive and require it tobe reflexive, transitive, and complete and, hence, there are no non-comparableoutcomes. Case 2: if R is understood to express non-comparable desirabilityfor all instances, then the requirement is equivalent to saying that we take ≥as primitive, that no two distinct outcomes are equally desirable, and that thenon-comparability relation of desirability is transitive. 23

3. Preference. There is a corresponding difference in the ways we model pref-erence over acts. Dubois et al. employ the concept of “being strictly morepreferable than” as primitive. But I employ the concept of “being at least aspreferable as” as primitive. I do so because I need to distinguish equal pref-erence from non-comparable preference, which is crucial for explaining howeveryday practical reasoning may work with Bayesian decision-making. Sup-pose that an agent is choosing between two acts. If those two acts are judgedto be equally preferable by a kind of qualitative decision-making, the agent isadvised to choose either. But things are very different if the two acts are leftnon-comparable because, say, everyday practical reasoning is too coarse to makethe two acts comparable. In that case, the agent is provided with no advice atall from everyday practical reasoning, and she may wish to consider resortingto a more refined style of decision-making (such as maximization of expectedutility), or just choosing one arbitrarily (in order to save some deliberation cost).

4. Belief. The kind of belief representation that Dubois et al. adopt is based on theidea that one proposition is “more likely than” another proposition, which theycall a subjective likelihood relation >L.24 They also explain how each subjectivelikelihood relation can be used to define a nonmonotonic consequence relation,or what I call a conditional belief set:

Bel(A ⇒ B) iff A ∩ B >L A ∩ B. (23)

So there is no essential difference in belief representations. But the non-monotonic logics derived are slightly different. The present work derives systemP, while Dubois and his colleagues derive system P minus (Reflexivity). The

23I thank an anonyous referee for asking me a stimulating question to help me think it through.24In fact, they ultimately represent >L by another kind of belief representation, called possibilitymeasures. I frame the discussion in terms of >L simply because it facilitates the comparison.


difference comes down to the following: Let A be a decision-theoretically nullproposition (i.e., the preference relation is independent of what outcomes areproduced at the states in A). I still have Bel(A ⇒ A) in conformity to (Reflex-ivity) and, furthermore, I have Bel(A ⇒ X) for each proposition X ⊆ S. ButDubois et al. have ¬Bel(A ⇒ A), which violates (Reflexivity) and the reason isrooted in their decision rule (see their discussion in Section 5.3).

5. Decision Rule. The difference in decision rules is the following. For Dubois etal., each strict desirability order > and each subjective likelihood relation >L

jointly determine a strict preference relation � as follows:

a � b iff [[a > b]] >L [[a < b]], (24)

which implies, by definition (23), that:

(Dubois et al.) a � b iff Bel([[a > b]] ∪ [[a < b]] ⇒ [[a > b]]). (25)

That is a restatement of Dubois et al.’s decision rule in my terminology. Nowwe can compare it to my decision rules. An elegant form for strict preferencein my theory is formula (5), which is reproduced in the following for ease ofcomparison:

(My Proposal) a � b iff ¬Bel([[a ≡ b]] ⇒ ⊥) andBel([[a ≡ b]] ⇒ [[a > b]]).(26)

Assuming that Bel satisfies (Left Weakening) and (Cautious Monotonicity) andthat ≥ is reflexive and transitive, the left-hand side of (25) is implied by thesecond conjunct in the left-hand side of (26).25 So Dubois et al’s criterion forstrict preference is weaker than my criterion, which is ultimately motivatedfrom examples of everyday practical reasoning. Furthermore, I have a decisionrule for equal preference (which is distinguished from non-comparability), butDubois et al. do not.

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