Von Wright's preference logic reconsidered - Semantic Scholar

1 downloads 0 Views 201KB Size Report
Mar 3, 2006 - and that von Wright's logic of preference yields a notion of better that comes very close to ...... King's College Publications, London, 2005. 23 ...
Von Wright’s preference logic reconsidered Patrick Girard Stanford University March 3, 2006 A logic of preference does not pertain to the formation of preferences, but rather to a pre-established hierarchy of alternatives, ordered by some underlying relation. A wide variety of relations, corresponding to reasons behind the ordering of alternatives, may be seen as preference relations. For example, alternatives may be ordered by the pleasure or utility they bring about, they may be ranked in terms of being better, nicer, and so on. The logics we will consider abstract from the origin of the preferential alternative hierarchies, and simply assumes that the alternatives are ordered by some preference relation. As it is the custom since the seminal work of von Wright, we assume the relation to be asymmetric and transitive. In section 1, we give a detailed analysis of von Wright’s seminal work [26], entitled The logic of preference. We will see that a key notion in this work is the notion of preferences ceteris paribus (all other things being equal ). Section 2 is devoted to tracking down ways to give appropriate semantics to this latter notion. In the last section, we look at the semantics for preference logic proposed in [23], which uses a basic modal language augmented with the existential modality. We will compare this version with von Wright’s original version, and see that the major difference between the two lies in the ceteris paribus clause. This is further investigated in a follow-up paper ([11]). In the appendix are presented alternative approaches to preference by Shoam, Lewis, Huang, and van Benthem and Liu.

1

1

Von Wright’s preference logic

Although the notion of preference circulated in many disciplines in the first half of the 20th century, especially in economics and social choice theory (cf. [28]), it is G. H. von Wright in [26] that initiated the study of the logic of preference per se. Von Wright acknowledges Halld´en [12] to be a precursor of his work, but only to the extent that Halld´en investigated a logic of betterness and that von Wright’s logic of preference yields a notion of better that comes very close to Halld´en’s notion. Nevertheless, we take von Wright to be the founder of preference logic, and present the main line of his approach.

1.1

What notion are we working with?

In [26], von Wright takes a notion of preference as primitive and investigates how it can be treated logically. His notion of preference is based on the three following principles: 1) preferences are intrinsic and subjective, 2) they are between states of affairs, and 3) preferences are evaluated all other things being equal. By intrinsic preferences, von Wright means preferences that are not based on other considerations. For example, I could prefer going to swim than going to run, not because I like swimming better, but because my doctor told me that running is bad for my back. In this case, my preference is derivative of external reasons, and is for that reason what von Wright calls an extrinsic preference. The account von Wright’s wants to give is an account in which preferences are primitive, and reduces to “what, in ordinary language, we mean by ‘to like better (more)’ ”. That preferences are taken to be subjective is incidental to the formal treatment, and is simply the claim that preferences are always someone’s preferences (p.12). As is common in logical investigations of epistemic notions, a logic of preference is not a logic that dictates how to build preferences, but rather how to reason about a given set of preferences, and how to infer new information from that set. Assuming that preferences are intrinstic and subjective plays exactly that role of providing a primitive notion of preference upon which a theory is built. The second principle is that preferences are between generic states of affairs, i.e., states of affairs that “may or may not obtain on any given occasion and that [...] can obtain on more that one occasion.”(p.18). This is meant as a limitation of the scope of preferences to first-order preferences about factual matters. Von Wright’s motivation to limit his account to this 2

type of preferences is that other types, such as for example preferences of instruments or preferences of actions, are derivative of preferences of states of affairs, although he does not attempt to provide such a derivation (p.12). The third principle, also know as the ceteris paribus principle, is probably the most contentious of the three principles (cf. p.31-32). It is meant to yield a notion of unconditional preferences, in the sense that a change in the world might influence the preference order between two states of affairs, but if all conditions stay constant in the world, than so does the preference order. Von Wright’ paradigmatic example is that one might prefer a raincoat to an umbrella, in the sense that one would rather loose the umbrella than the raincoat, unless one also looses her boots. That is, given that she does not loose her boots, she prefers the raincoat to the umbrella, but she prefers the umbrella if the loss of the raincoat is accompanied by the loss of her boots. If she looses her boots no matter what, than her preference is kept unchanged, and she likes the raincoat better. Hence, in either case where the loss of the boots is kept constant in the evaluation of two states of affairs, she prefers the raincoat, but if the world changes for one state and not for the other, than her preferences are influenced accordingly. We will study ceteris paribus preferences in more detail below.

1.2

The logic

The language von Wright works with is a propositional language whose propositions range over states of affairs, augmented with a binary preference relation P such that ‘pP q’ expresses that the state of affairs p is preferred to the state of affairs q. There is a restriction in the inductive definition of the language, namely that in ‘ϕP ψ’, ‘ϕ’ and ‘ϕ’ can be either propositional or Boolean, but cannot be preference expressions. We will follow von Wright and write pP q instead of ϕP ψ, but remind the reader that p and q in pP q stand for Boolean combinations of the propositional variables. This reflects the second principles we outlined above, viz. that preferences are first-order. We will assume that the rule of substitution of equivalents holds throughout the exposition of von Wright’s logic. Making this assumption explicit will make our life easier, especially with principles relating preferences involving disjunctive components. Von Wright’s formal notion of preference relies on 5 basic principles, divided in two groups (p.21-32):

3

1. The first group gives two formal properties of the relation of preference: (a) asymmetry, and (b) transitivity. The asymmetry of the relation is obvious; if one prefers p to q, then it is not the case that one prefers q to p as well. Transitivity has a strong intuitive appeal, although it leads to paradoxical situations. For a discussion of the paradoxes of transitive preference, see Hansson [13]. There are technical ways to deal with the paradoxes, but these come out more as patch-work than advances in the logic of preference itself. For this reason, and for the sake of generality, we leave the discussion of the paradoxes aside. 2. The second group provides three principles of transformation of any preference expression into a so-called normal form. (a) Given two generic states p and q, to say that p is preferred to q is to say that a state of affairs with p∧¬q as components is preferred to a state of affairs with ¬p ∧ q. This principle relates a formal understanding of preferences with changes in world. It says that an agent who prefers p to q and supposing, for example, that both p and q obtains in the actual world, would prefer loosing q than loosing p. Similarly, if both ¬p and ¬q obtain in the world, then the agent would favor an acquisition of p over q, and so on. In von Wright’s words: “to say that a subject prefers p to q is tantamount to saying that he prefers p ∧ ¬q to ¬p ∧ q as end-states of contemplated changes in his present situation (whatever that be).” This latter principle was called “conjunctive expansion” by Jennings in [15], who provides an extended philosophical criticism of it. The trust of his argument is that conjunctive expansion should be taken as a principle of choice rather than preference. Conjunctive expansion predates von Wright’s manuscript, and was introduced in the field of deontic logic by Halld´en in [12], p.27 ff. For an interesting (but short) discussion of Halld´en’s principle, see [6], in which Castaneda, after providing a counterexample, still provides the following in its defense, at least for its intuitive appeal: “When St. Paul said “better to marry than to burn” he 4

meant “it is better to marry and not burn than not to marry and burn.”” For further discussion of conjunctive expansion, see Hansson [13]. (b) Disjunctive preferences are conjunctively distributive, i.e., (p ∨ q)P r ≡ pP r ∧ qP r

(1)

pP (q ∨ r) ≡ pP q ∧ pP r

(2)

and

We call this principle ’distribution’. It allows to analyze disjunctions as conjunctions in preference expressions, and is quite appealing. For instance, if I prefer flying to taking either a bus or a train, then I prefer flying to taking a bus, and I prefer flying to taking a train. Von Wright investigates the consequences of distribution when combined with conjunctive expansion. The discussion may summarized in by the following principles: (p ∨ q)P r ≡

(p ∧ q ∧ ¬r)P (¬p ∧ ¬q ∧ r) ∧ (p ∧ ¬q ∧ ¬r)P (¬p ∧ ¬q ∧ r) ∧ (¬p ∧ q ∧ ¬r)P (¬p ∧ ¬q ∧ r)

(3)

(p ∧ ¬q ∧ ¬r)P (¬p ∧ q ∧ r) ∧ (p ∧ ¬q ∧ ¬r)P (¬p ∧ q ∧ ¬r) ∧ (p ∧ ¬q ∧ ¬r)P (¬p ∧ ¬q ∧ r).

(4)

and pP (q ∨ r) ≡

We show how to derive (3) using conjunctive expansion and (1). By conjunctive expansion, (p ∨ q)P r ≡ ((p ∨ q) ∧ ¬r)P (¬(p ∨ q) ∧ r). Now, by propositional logic, (p ∨ q) ∧ ¬r is equivalent to: (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ ¬r) ∨ (¬p ∧ q ∧ r) and ¬p ∧ ¬q ∧ r ≡ ¬(p ∨ q) ∧ r

5

(5)

Hence, by substitution of equivalents in (5), we get: ((p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ ¬r) ∨ (¬p ∧ q ∧ ¬r))P (¬p ∧ ¬q ∧ r) (6) Finally, (3) follows from two applications of (1). (c) Ceteribus paribus preferences (holistic nature of preference). As we discussed above, the ceteris paribus nature of preference is the strongest assumption von Wright makes about preferences. As we expressed, it says that a preference of p over q can be analyzed so long as the world stays constant in any other respect. As von Wright expresses it, “a subject favors a change to p ∧ ¬q over a change to ¬p ∧ q, irrespective of what the state of the world is, but assuming that it does not change in other features beside p and q”(p.31). We postpone again the discussion of ceteris paribus until section 2. From the ceteris paribus notion of preference, one can define notions of ‘good’ (pP ¬p) and ‘bad’(¬pP p). These notions are not absolute notions of ‘Good’ and ‘Bad’, but rather subjective notions relative to an agent, irrespective of consequentialist consequences or of the moral status of states of affairs. Hence, one may see preference logic as grounding deontic logic, and this has been a motivation for further investigations (see e.g., [7]).

1.3

Von Wright’s decision procedure

A striking feature of von Wright’s approach to preference logic, a feature that is pervasive in modal logic investigations from the rise of modal logic in the work of C. I. Lewis up to the 70’s, is the syntactic character of the approach. Instead of looking for a semantics of preference, from which a set of axioms is discovered and proven to be complete, von Wright lays down a set of postulates that are meant to capture the notion of preference. These postulates, as we will now see, are principles that transform an arbitrary wellformed formula into a so-called ‘normal form’ that can be analyzed either in a truth-table or in a graph reminiscent of Kripke models. Both procedures are suggestive of an underlying semantics, but we will see that abstracting a semantics is not an easy task. The procedure to transform a well-formed formula ϕ into its normal form is based on the 5 principles outlined above. First, one uses the principles of 6

the second group to transform the given formula into a formula whose constituents are state descriptions, i.e., conjunctions of literals from the universe of discourse of ϕ (cf. definition 1 below). Second, following the principles of the first group, one assigns truth-values to these state descriptions and computes the resulting truth-tables. We need the following definition: Definition 1 (Universe of discourse). Let ϕ be a formula. The set of propositions that occur in ϕ is called the universe of discourse of ϕ and is denoted UD(ϕ). Let γ be a formula. We apply the transformation principles to all preference subformulas of γ of the form ϕP ψ. The transformation principles corresponding to the principles of the second group are given by: 1. Conjunctive expansion: given ϕP ψ, to get (ϕ ∧ ¬ψ)P (¬ϕ ∧ ψ). • For example, transform (p ∧ q)P r into (p ∧ q ∧ ¬r)P (¬(p ∧ q) ∧ r) ≡ α . Multiple applications of conjunctive expansion yield equivalent formulas. For example, applying it twice to pP q yield: (p ∧ ¬q) ∧ (¬¬p ∨ ¬q)P (¬p ∨ ¬¬q) ∧ (¬p ∧ q)

(7)

but (p∧q)∧(p∨q) ≡ p∧q, hence (7) is equivalent to (p∧¬q)P (¬p∧¬q), which results from a single application of conjunctive expansion. 2. (a) Replace the constituents of the P -expression obtained from the previous step by their disjunctive normal form. • For example, transform α into (p ∧ q ∧ ¬r)P [(¬p ∧ q ∧ r) ∨ (p ∧ ¬q ∧ r) ∨ (¬p ∧ ¬q ∧ r)] ≡ α′ . (b) Replace atomic P -expressions with conjunctions of atomic P -expressions using distribution. • For example, letting β ≡ p ∧ q ∧ ¬r, α′ becomes [βP ¬p ∧ q ∧ r] ∧ [βP p ∧ ¬q ∧ r] ∧ [β¬p ∧ ¬q ∧ r] 7

3. Suppose s belongs to the universe of discourse of γ, but does not occur in the formula β resulting from the applications of the two previous steps, then replace a P -expression ϕP ψ by the conjunction [(ϕ ∧ s)P (ψ ∧ s)] ∧ [(ϕ ∧ ¬s)P (ψ ∧ ¬s)]. This is called ‘amplification’, and is applied for every s in the universe of discourse of γ that do not occur in β. This step does not apply in the example considered so far. We note for now that amplification is what evaluates preferences “all other things being equal”. Indeed, we make sure that every s in the universe of discourse of a formula that are not directly relevant to the evaluation of a preference subformula of γ are kept constant. This would not be the case, for example, if we could have a resulting sentence of the form ϕ ∧ rP ψ ∧ ¬r. This would express something of the form “I prefer loosing my umbrella and keeping my boots than loosing my raincoat and loosing my boots”, which would overturn my preference for my raincoat over my umbrella. It is quite clear how these transformation principles correspond to the principle of the second group given above, and may be summarized by the following set of postulates: pP q → ¬(qP p) (pP q) ∧ (qP r) → pP r pP q ≡ (p ∧ ¬q)P (¬p ∧ q) (a) pP q ∨ r ≡ pP q ∧ pP r (b) p ∨ qP r ≡ pP r ∧ qP r 5. pP q ≡ [(p ∧ r)P (q ∧ r)] ∧ [(p ∧ ¬r)P (q ∧ ¬r)]

1. 2. 3. 4.

Instead of postulate 4, von Wright gives the following: (p ∨ q)P (r ∨ s) ≡

(p ∧ ¬r ∧ ¬s)P (¬p ∧ ¬q ∧ r) ∧ (p ∧ ¬r ∧ ¬s)P (¬p ∧ ¬q ∧ s) ∧ (q ∧ ¬r ∧ ¬s)P (¬p ∧ ¬q ∧ r) ∧ (q ∧ ¬r ∧ ¬s)P (¬p ∧ ¬q ∧ s)

(8)

and as we saw above, this is a consequence of distribution combined with conjunctive expansion. We leave the derivation of (8) from postulate 4 as an exercise to the reader.

8

The first and second postulates simply reiterate that preferences are transitive and asymmetric, and the third gives the conjunction expansion principle of section 1.3. The fourth postulate stands for distribution. Finally, the fifth postulates should be applied for every proposition of the universe of discourse. It captures the ceteris paribus nature of preferences. 1.3.1

Truth tables

We now show how to use these principles to evaluate the consistency of an arbitrary formula. Let γ ′ be the formula resulting after performing the above transformations to all preference constituents of γ. To decide on the consistency of γ, we compute a truth-table assigning truth-values to the atomic constituents (taking constituents of the form ϕP ψ to be atomic) of γ ′ respecting the principles of the first group (asymmetry and transitivity) in the following way: 1. If a constituent of the form ϕP ψ is assigned the value T , then a constituent of the form ψP ϕ cannot be assigned the truth-value T in the same row. This makes sure that any row of the table respect asymmetry. 2. If both ϕP ψ and ψP δ are assigned the truth-value T , the constituent ϕP δ is also assigned the truth-value T . This makes sure that transitivity is respected in each row of the table. 3. Finally, ones computes the truth-table in a standard way. We call these three principles the truth-assignment principles. If the resulting computation displays at least one T in the column under the main connective, then the formula γ is said to be consistent. If all entries under the main connective are T ’s, then von Wright calls it a P-tautology. We give some examples. Example 1. On p.41-42, von Wright shows that the formula (pP ¬p) ∧ (¬qP q) → pP q is a P -tautology. Remembering the definition of good and bad in terms of pP ¬p and ¬qP q respectively, this formula expresses that if p is good and q is bad, then p is preferred to q. We refer the reader to the text for the detail of the proof. Example 2. γ ≡ (p ∨ ¬p)P q is not a P -tautology.

9

(p ∨ ¬p)P q ≡ p ∨ ¬p ∧ ¬qP ¬p ∧ p ∧ q (Conjunctive expansion) ≡ (p ∧ ¬q) ∨ (¬p ∧ ¬q)P ¬p ∧ p ∧ q (Substitution of equivalents) By distribution, we get [(p ∧ ¬q)P (¬p ∧ p ∧ q)] ∧ [(¬p ∧ ¬q)P (¬p ∧ p ∧ q)] We have no more transformation to apply, and we see that we end up with two atomic constituents, and that the two first truth-valuation principles need not apply on the resulting constituents, since assymetry and transitivity are vacuously respected. Hence, the original formula can be falsified and is not a tautology of the calculus. Example 2 expresses that tautologies are not always preferred to arbitrary formulas. This is not surprising if we remember that the relation von Wright describes is a strict notion of preference. If the relation was taken to be reflexive, then the preference that would obtain would have p ∨ ¬pP q as a thesis. The rejection of the thesis here only means that tautologies are not strictly preferred to anything else. 1.3.2

Using graphics and state descriptions

Von Wright gives an alternative way, this time graphical, for evaluating the consistency of formulas. It is quite reminiscent of Kripke semantics for modal logic in terms of possible worlds, and indeed von Wright’s paper is contemporary to Kripke’s seminal paper. Nevertheless, von Wright’s graphical use of state-description is prima facie not a tool to provide a semantics for his logic, and no completeness results are investigated. As we shall see below, the ceteris paribus requirement makes the semantics of von Wright’s logic hard to track. For the time being, we present the method and show how to use it to evaluate the consistency of formulas. Given an expression in normal form (i.e., obtained after the transformations postulates have been applied to it) one sees that the constituents to the right and the left of the P are state description. A state description is a conjunction of literals (i.e., propositional letters or their negation), all taken from the universe of discourse, and such that every member of the universe of discourse is used once. For example, if UD(ϕ) = {p, q, r}, then p ∧ ¬q ∧ r and 10

Figure 1: Graphical representation of ϕP ψ.

¬p ∧ ¬q ∧ r are both state descriptions. It is easy to see that the transformation postulates applied to expressions yield P -expressions whose constituents are state descriptions. In particular, the fifth postulate guarantees that every members of the universe of discourse occurs in the end formula. The procedure goes as follows (p.48): given a P -expressions in normal form, consider the unnegated P -expressions of the form ϕP ψ, take ϕ and ψ to be points in a graph, and draw an arrow from the ϕ-state to the ψ-state (cf. figure 5). Close the diagram under transitivity of the relation, i.e., if there is an arrow between a ϕ-state and a ψ-state, and if there is an arrow between the ψ-state and a γ-state, then draw an arrow between the ϕ-state and the γ-state. The resulting diagram displays a consistent interpretation of the original formula if the two following conditions are met: 1. There is no arrow going in reverse order between two states description, i.e., asymmetry of preference is respected. 2. If a constituent of the expression in normal form is a negated P expression of the form ¬(ϕP ψ), then no arrow in the completed diagram goes from a ϕ-state to a ψ-state. Example 3. Let γ ≡ pP ¬p ∧ ¬qP q ∧ ¬(pP q) (p.49). Here, γ is the negation of example 1 and expresses that “a bad state is preferred to a good state”. Applying the transformation postulates, one obtains: [(p ∧ q)P (¬p ∧ q)] ∧ [(p ∧ ¬q)P (¬p ∧ ¬q)] ∧[(p ∧ ¬q)P (p ∧ q)] ∧ [(¬p ∧ ¬q)P (¬p ∧ q)] ∧(¬[(p ∧ ¬q)P (¬P ∧ q)] Figure 2 displays the four unnegated constituents of the previous formula, with the doted line representing the closure under transitivity. But the dotted line between the p∧¬q-state and the ¬p∧q-state violates the second condition, since ¬[p∧¬qP ¬p∧q] is a negated constituent of the formula in normal form. Hence, γ is not consistent. Therefore, the formula (pP ¬p) ∧ (¬qP q) → pP q is a P -tautology. 11

p, q

p,¬q

¬p, q

¬p,¬q

Figure 2:

This completes our survey of von Wright’s preference logic, as presented in [26]. In the remaining sections of the book (sections 23-28) von Wright develops a logic of preference and indifference but we will not survey this extended logic, as it is a natural extension which does not import new special features in the logic. What we are now interested in is to see how the notion of ceteris paribus was received in the literature, and what preference logic became over the years. We shall see that the most recent form of preference logic, as studied in Amsterdam, are in essence faithful to von Wright’s preference logic, although it falls short of proving a notion of preferences ceteris paribus.

2

Ceteris Paribus

In this section, we focus our attention on the notion of ceteris paribus. We investigate how it has been received in the field, and we look at various attempts to provide a more standard semantics of the notion. As we have seen, von Wright’s approach is syntactical, and it is not obvious from the outset what kind of semantics should be given to his preference logic. It also seems 12

as though various incompatible semantics could satisfy von Wright’s postulates. Nevertheless, von Wright’s discussion implicitly suggest a semantics for a preference (modal) operator, and this was first investigated by Rescher in [19]. We will see that Rescher does shed some light on the notion of ceteris paribus, but that it still falls short of a complete description. We will then look at von Wright’s own attempt at elucidating the notion in [27]. Finally, we will look at Doyle and Wellman’s suggestion in [10].

2.1

Rescher

In [19], Rescher considers various semantical approaches to preference logic. His semantics are value-based assignments on worlds from which preference relations that satisfy desirable properties such as Irreflexivity and transitivity are derived. His general approach does not directly give a semantics for preference, but instead some plausibility assignment on possible worlds from which notions of preference are derivable. For example (p.43-44), let W = {w1 , ..., wn } be a set of worlds, and let # : W −→ R be a function that assigns an index of merit to worlds. Then,to any formula is assigned the arithmetical mean of the values of the worlds at which it is true. One then obtains a semantics for the preference relation as: ϕP ψ iff #(ϕ) > #(ψ) This semantics relies on a plausibility assignment of possible worlds, and as Huang points out in [14], it is “based on the unrealistic assumption of numeric measures of goodness associated with possible worlds, and hence on a complete preference ordering.”(p.113) In any case, it is not clear how this semantics relates to more standard semantics in terms of possible worlds. For this reason, we will focus our attention on the semantics given by Rescher that do not depend on plausibility assignments to worlds. The semantics that he gives and in which we are interested in the perspective of that paper, is the semantics that conveys a notion of preferences ceteris paribus, `a la von Wright. Let there be a set of possible worlds {w1 , ..., wn } totally ordered by a preference order  allowing indifference between worlds. The semantics of αP w β is given by the following condition (p.48): for every γ (independent of α and β), and for any two worlds wi , wj , if α is true, β is false and γ is true at world wi , and if α is false, β is true and γ is true at world wj , then wi  wj . 13

Here, γ expresses the idea that the other aspects of the context of evaluation stay constant. It corresponds to the fifth preference postulates presented in section 1.3. The idea behind the independence of γ is that the context of evaluation of two statements must not influence one statement over the other. If one prefers P to Q, but the context is such that P cannot be obtained, or actually forces P to be false or undesirable, then the evaluation is not“all other things being equal” anymore. This seems like a natural constraint on the worlds of evaluation. We also see from the definition that α is preferred to β if the situations in which α is true and β is false are preferred to the situations in which α is false and β is true. This again corresponds to third postulate of section 1.3.

2.2

Von Wright, take 2

We now look at von Wright’s second attempt at defining the notion of preferences ceteris paribus in [27]. The paper is a simplification and clarification of what we presented in section 1. The newer version does not modify the logic drastically, although it presents a different set of preference principles. For example, instead of having transitivity as a primitive requirement, it is derived from the principle of (Value)-Comparability(p.151-152), which states that: (xP y) → [(xP z) ∨ (zP y)] i.e., that if x is preferred to y and z is an independent state, then either x is also preferred to z, or z is itself preferred to y. The main point of this postulate is that if x is preferred to y and the agent is indifferent between x and z, then we should expect that she also prefers z to y. Otherwise, how could she claim to be indifferent between x and z? But if x is preferred to both y and z, then one cannot infer from this consideration alone that either zP y or yP z. We now present von Wright’s updated definitions of the notion of preferences ceteris paribus. Let there be a finite set S = {p1 , ..., pn } of logically independent states of affairs. Let W = P(S) = {w1 , ..., w2n } be the set of possible total states of the world. The requirement that the states be independent is not essential for the definitions to follow. If there are some logical dependencies between states, then the number of total states will not be maximal (p.148). Consider two states s and t such that either 1) s, t ∈ S or 2) s, t are molecular compounds of a total of m elements in S. Then, there 14

are n − m remaining states in S that can be obtained in 2n−m combinations, denoted C1 , ..., C2n−m . Now, von Wright’s gives two definitions of ‘s is preferred to t under the circumstances Ci ’: Definition 2 (D1). s is preferred to t under the circumstances Ci , if and only if, every Ci -world which is also an s-world but not a t-world is preferred to every Ci -world which is also a t-world but not an s-world. Definition 3 (D2). s is preferred to t under the circumstances Ci , if and only if, some Ci -world which is also an s-world is preferred to some Ci -world which is also a t-world, and no Ci world which is a t-world is preferred to any Ci -world which is an s-world. Finally, if s is preferred to t under all circumstances Ci , according to either definition, then we say that s is preferred to t ceteris paribus. Definition D1 sticks to the conjunctive expansion principle (see 2a). The principle expresses that preferences be evaluated in worlds in which it is not the case that both states obtain, i.e., sP t ≡ s ∧ ¬tP ¬s ∧ t. But if there are no worlds such that both s and t obtain, then D1 reduces to D2. A demarcation between D1 and D2 is that the principle of contraposition (ϕP ψ → ¬ψP ¬ϕ) holds for the former, but not for the latter. An argument against the principle of contraposition widely quoted is found in [7]. For a discussion of the principle and for a separate counter-example, see Hansson [13], p. 349. According to von Wright though, the counterarguments are not sufficient to precludes the evaluation of preferences as in D1, and these considerations are outside the scope of logic. In other words, both definitions seems reasonable and worth investigating from a logical point of view, and the argument against conjunctive expansion and contraposition of preferences are external to the logic of preference. We now turn our attention to a more modern treatment of preferences ceteris paribus given by Doyle and Wellman [10].

2.3

Doyle and Wellman

In [10], Doyle and Wellman show how to lift preferences between individuals to preferences between classes of individuals, all other things being equal. The underlying relation is a total preorder (i.e., transitive, reflexive and connected), and propositions are taken to be sets of individual objects. There 15

is a close link to studies by Boutilier in [3], where Boutilier interprets ceteris paribus by “all other things being normal”, whereas this paper defines it as “all other things being equal”, by mean of contextual equivalence, i,e., equivalence relations among individuals that vary with the context of application. We will not pursue this comparison further. Our goal is to show how von Wright’s notion of preferences ceteris paribus may be understood. Doyle and Wellman’s formalism for preference is different from von Wright’s, although the essence displays strong similarities, and we will present it in detail. Let there be a given set of propositional letters prop. The (modal) language L built over this set of propositions is given by the following inductive rules: L := p | ¬ϕ | ϕ ∨ ψ | ϕP ψ. Notice that we take preference to be a binary operator between sentences, instead of von Wright’s primitive relation between them. Notice also that there are no restrictions on the inductive definitions of the formation of preference formulas, i.e., preferences are not restricted to first-order preferences. Thus, we relegate transitivity and asymmetry to the relation in the model. Furthermore, we shall identify propositions to sets of possible worlds, as is standardly done. We need the following preliminary definitions: Definition 4. Let Γ be a set, then ε(Γ) denotes the set of equivalence relations on Γ, i.e., the set of reflexive, transitive and symmetric relations on Γ. Definition 5 (Contextual equivalence). Let Γ be a set. A contextual equivalence function is a function η : P(P(Γ)) ≡ ε(Γ) that assigns equivalence relations to sets of propositions. The intuitive reading and the relevance of these definitions becomes clear in the definition of preference models: Definition 6. A preference model M =< W, ≺, V, η > is a quadruple in which W is a set of worlds, ≺ is a transitive and asymmetric (partial) order, V is a propositional valuation, and η is a contextual equivalence function on W. Here, the contextual equivalence function assigns equivalence relation to each set of propositions. The relevance of the equivalence function now becomes 16

clear. An equivalence class for a set of proposition is the set of worlds which we will use to capture the ceteris paribus clause in the semantic definition of the preference operator. We introduce one more piece of notation: Definition 7. If worlds w and w ′ are equivalent modulo a set of propositions Γ = {p, q, ...}, i.e. if wη(Γ)w ′, we write w ≡ w ′ modη (Γ). Definition 8 (Ceteris paribus preferences). The semantic definition for the propositions and the Boolean operators is standard. We focus on the semantics of the preference modality. We say that ϕP ψ is satisfied in the model M at world w, written M, w  ϕP ψ iff ∀w1 , w2 ∈ W, 1. M, w1  ϕ ∧ ¬ψ, 2. M, w2  ¬ϕ ∧ ψ, and 3. w1 ≡ w2 mod(ϕ ∩ ψ, ϕ ∩ ψ) ⇒ w1 ≺ w2 . The notation ϕ stands for the set-theoretical complement of ϕ with respect to W . Clauses 1 and 2 together correspond to von Wright’s conjunctive expansion. The third clause is the important one, and expresses that the comparison is made between ϕ ∩ ψ-worlds and ϕ ∩ ψ-worlds and equivalent in every other respect. For example, the equivalence relation may be defined in a model with respect to the propositional valuation such that two worlds are equivalent if the propositional valuation to each world is the same except for the assignment of propositions of the universe of discourse of ϕP ψ. In other words, if pP q and the propositional valuation for two worlds w and w ′ is the same for every propositional letter except that it assigns p to w and q to w ′ , we say that w ≡ w ′ mod(p ∩ q, p ∩ q). It is not too difficult to see that definition 8 expresses that ϕ is preferred to ψ ceteris paribus, if ϕ ∧ ¬ψ is preferred to ¬ϕ ∧ ψ in situations that are equivalent in every other respects, as indicated by the equivalence relation. This seems very close to the intended semantics behind von Wright’s principles. At the very least, a mathematical representation of “all other things being equal” in terms of equivalence classes seems the most natural. Unfortunately, Doyle and Wellman do not provide explicit ways of building the equivalence relation in a way that is intuitive and which captures von Wright’s intentions. This question is further pursued in [11].

17

3

Semantics for preference logic in a standard modal language

For the remainder of this paper, we will consider contemporary work in preference logic. Most of this work came out of computer science considerations. For example, in [8], Cui and Swift introduce a fixed-point semantics embedded in the so-called well-founded semantics. The system is applied to the problem of data standardization, the extraction of useful information from a poorly structured textual data. Having preferences underlying extractions simplifies the task of the computer. Doyle and Wellman’s [9] displays a kinship between preferences and goal, and show how this may be applied in artificial intelligence when goals are assigned to robots; the robots will fulfil the goals in the most favorable way, as guided by its preferences.

3.1

Boutilier

In [5], Boutilier shows how decision making under uncertainty may be studied qualitatively with preferences. The resulting logical calculus allows to reason about decision making systems that use quantitative notions such as probabilities, and extracts qualitative information to derive goals. His language is a basic modal language augmented with a backward looking modality along the accessibility relation. This allows him to express a conditional notion of preference: the most preferred ϕ-worlds are also ψ-worlds. Boutilier’s models have an underlying total preorder, and the unary modality’s semantics is given by: M w α iff for each v such that v ≤ w, M v α. In this language may be defined a global modality, saying that ϕ is true everywhere. Adding a global modality to the basic modal language is the starting point of the recent work of van Benthem, Otterloo and Roy (the Dutch squad below) in [23]. We shall focus our attention on this latter work, but we stress that Boutilier is the pioneer of this new way of looking at preference logic.

3.2

Preferences with a global modality

Von Wright’s semantic definition has a universal flavor, in the sense that the truth of ϕP ψ depends on what occurs at every world of the model, accessible 18

or not from w. This is actually the crucial point of departure in the Dutch approach. As we will see, they reduce preference logic to a basic (multi)modal language augmented with the so-called existential modality Eϕ, which is true at a world if ϕ is true somewhere in the model, independently of the accessibility relation. The Full language is given by: L = p | ¬ϕ | ϕ ∨ ψ | ♦i ϕ | Eϕ and models are of the following form: M = hW, N, {i }i∈N , V i where W is a set of worlds, N is a set of agents, i∈N is a family of reflexive and transitive relations for every agent, and V is a propositional valuation. The preorder assigned to every agent is interpreted as the preferential hierarchy of worlds in a model. Notice that the relation is not strict and allows for indifference between worlds, if both w  w ′ and w ′  w. The hierarchy is introduced in the model instead of the object language to relieve the latter from the burden of expressing all there is to preference. By making the accessibility relation transitive and symmetric, we capture the essential features of a preferential hierarchy, as they were introduced by von Wright and widely used by his followers. This approach gives a lot of freedom to express conditional preferences in a unary modal language, thus situating the logic of preference in a well-known and developed landscape of modal logics. Furthermore, as was shown by Boutilier, the approach also allows to relate the logic of preference to a big sample of nonmonotonic logics and also belief revision logics (cf. [2, 4]). The semantics for the Boolean connectives and for the family of diamond operators is standard. The semantics of the modality ‘E’, know as the existential modality, is as follows: M, w |= Eϕ iff ∃w ′ s.t. M, w ′ |= ϕ Notice that the existential modality comes with its dual Uϕ = ¬E¬ϕ, called the universal modality. As was already realized by Boutilier in [5], a conditional ϕP ψ expressing that “the preferred ϕ worlds are at least as good as the preferred ψ-worlds”, can be expressed in this language by the following: ϕP ψ := U(ϕ → ♦i ψ) 19

(9)

Unwinding the truth definition of this latter sentence, we get: M, w |= U(ϕ → ♦i ψ) iff ∀w ′ , M, w ′ |= ϕ → ♦i ψ iff ∀w ′ , M, w ′ |= ϕ ⇒ ∃w ′′ s.t. w ′ i w ′′ and M, w ′′ |= ψ and this says that for every ϕ-world, there is a ψ-world which is preferred to it. A complete axiomatization for this logic can be found in [24]. The Dutch squad starts with a (natural) semantics for preference logic, in a nowadays standard modal tradition, and then proceed to give a completeness proof. It is interesting to see how much of von Wright’s preference logic is kept in this new version. Firstly, conjunctive expansion is not included in the semantics, but this can quite easily be adapted. One would simply define a conditional preference in the following way: ϕP ψ := U(ϕ ∧ ¬ψ → ♦i (ψ ∧ ¬ϕ)). For the sake of generality, we neglect the conjunctive expansion principle. Secondly, and more importantly, it is clear that preferences here are not ceteris paribus. Indeed, it might be the case that for some p independent from ϕ and from ψ, p is false in the closest ϕ-worlds, but true in the closest ψ-world. This is harder to accommodate, and requires an extension of the language with a ceteris paribus diamond. This is further investigated in the follow-up paper [11]. Nevertheless, some comparison is still possible, and we shall do so by revisiting the examples presented in section 1.3.1. Example 4. Transitivity of preferences is preserved under translation (9). The translation yields: U(p → ♦q) → [U(q → ♦r) → U(p → ♦r)] Assume that U(p → ♦q) and U(q → ♦r), and let w be arbitrary such that M, w |= p. Since U(p → ♦q), M, w |= p → ♦q, hence there is a world w ′ with w  w ′ such that M, w ′ |= q. Since U(q → ♦r), M, w ′ |= q → ♦r, thus there is a world w ′′ with w ′  w ′′ such that M, w ′′ |= r. By transitivity of , we have w  w ′′ . Hence, M, w |= p → ♦r. Finally, since w was chosen arbitrarily, we see that U(p → ♦r) is valid. 20

Figure 3: Countermodel to U(p → ♦(q ∨ r) → U(p → ♦q) ∧ U(p → ♦r).

Example 5. pP q → ¬(qP p) is only preserved under translation (9) for the strict part of , i.e. for the pairs of worlds w, w ′ such that w  w ′ but w ′ 6 w. Example 6. (1) p ∨ qP r ≡ pP r ∧ qP r and (2) U(p → ♦q) ∧ U(p → ♦r) → U(p → ♦(q ∨ r)) are valid, but (3) U(p → ♦(q ∨ r)) → U(p → ♦q) ∧ (p → ♦r) is not. A countermodel to (3) is given in picture 6. We show that (2) is valid and leave (1) to the reader. To show that (2) is valid, assume that (a) M, w |= U(p → ♦q) and (b) M, w |= U(p → ♦r). Let w ′ be arbitrary such that M, w ′ |= p. By assumption (b), M, w ′ |= ♦q, thus there exists a w ′′ with w ′  w ′′ such that M, w ′′ |= q. But then M, w ′′ |= p ∨ q. Hence, M, w ′ |= ♦(q ∨ r). But w’ was chosen arbitrarily. Therefore, M, w |= U(p → ♦(q ∨ r)). Example 7. (pP ¬p) ∧ (¬qP q) → (pP q) is valid in preference logic. The translation of the formula in the global modality language is given by: U(p → ♦¬p) ∧ U(¬q → ♦q) → U(p → ♦q). Assume that M, w |= U(p → ♦¬p) ∧ U(¬q → ♦q). Consider an arbitrary world w ′ such that M, w ′ |= p (if there are none, the claim is vacuously true). By the truth-definition, there is a world w ′′ with w ′  w ′′ such that M, w ′′ |= ¬p. If M, w ′′ |= q, then we are done. If M, w ′′ 6|= q, then there is a world w ′′′ with w ′′  w ′′′ such that M, w ′′′ |= q. By transitivity of , we have that w ′  w ′′′ , and thus M, w ′ |= ♦q. But w ′ was chosen arbitrarily, hence for all world u, M, u |= p → ♦q. Therefore, M, w |= U(p → ♦q). Example 8. The formula (p ∨ ¬p)P q is not valid in preference logic, since our preference relation  is assumed to be reflexive.

21

4

Conclusion

The Dutch way of approaching preference logic is not dissonant with most of von Wright’s preference logic. The ceteris paribus clause is lost, but this seems to be amendable. The clear advance of the Dutch approach is in providing a (standard) semantics for preference logic, and this helps us to situate it in the landscape of modal logics. But more to the point, preference logic in this setting turns out to be very helpful in applications, as we noted above in the work of Boutilier, but also in the extension of the language offered by the Dutch squad to cope with game theoretic concepts such as the Nash equilibrium and backward induction. The addition of the existential modality to the basic modal language allows to situate applications of modal logics, for example in belief revision or in preference logic, within a hierarchy of decidable, yet expressive logics. The Dutch approach also allows the dynamification of preference logic (cf. [25]). This is briefly outlined in the appendix, along with alternative perspectives on preferences. We have surveyed in detailed the origin of preference logic in the work of von Wright, and we have experienced that the difficulty in understanding this logic lies in the syntactical approach, which leaves the underlying semantics for the ceteris paribus clause opaque.

References [1] A. Baltag, L. Moss, and S. Solecki. The logic of public announcements, common knowledge and private suspicions. Preceedings TARK 1998, pages 43–56, 1998. [2] C. Boutilier. Conditional logics of normality: a modal approach. Artificial Intelligence, 68:87–154, 1993. [3] C. Boutilier. A modal characterization of defeasible deontic conditionals and conditional goals. AAAI Spring Symposium on Reasoning about Mental States, pages 30–39, 1993. [4] C. Boutilier. Unifying default reasoning and belief revision in a modal framework. Artificial Intelligence, 68:33–85, 1993.

22

[5] C. Boutilier. Toward a logic for qualitative decision theory. In Proceedings of the Fourth International Conference on Principles of Knowledge Representation and Reasoning (KR-94), 1994. [6] H. N. Castaneda. On the logic of ‘better.’ review. Philosophy and Phenomenological Research, 19(2):266, December 1958. [7] R.M. Chisholm and E. Sosa. On the logic of intrinsically better. American Philosophical Quarterly, 3:244–249, 1966. [8] B. Cui and T. Swift. Preference logic grammars: fixed point semantics and application to data standardization. Artificial Intelligence, 138:117– 147, May 2002. [9] J. Doyle and M. P. Wellman. Preferential semantics for goals. In Proceedings of the Ninth National Conference on Artificial Intelligence (AAAI91), July 1991. [10] J. Doyle and M. P. Wellman. Representing preferences as Ceteris Paribus comparatives. Working Notes of the AAAI Symposium on Decision-Theoretic Planning, March 1994. [11] Patrick Girard. A ceteris paribus diamond. Work in progress, 2006. [12] S. Halld´en. On the logic of ‘better’. Lund, 1957. [13] S. O. Hansson. Preference logic. In D. Gabbay and F. Guenthner, editors, Handbook of philosophical logic, volume 4, chapter 4, pages 319– 393. Kluwer, 2 edition, 2001. [14] Z. Huang. Logics for agents with bounded rationality. PhD thesis, University of Amsterdam, 1994. [15] R. E. Jennings. Preference and choice as logical correlates. Mind, 76(304):556–567, October 1967. [16] D. Lewis. Counterfactuals. Harvard University Press, 1973. [17] D. Makinson. Bridges from Classical to Nonmonotonic Logic, volume 5 of Texts in Computing. King’s College Publications, London, 2005.

23

[18] J. M. McCarthy. Circumscription - a form of non monotonic reasoning. Artificial Intelligence, 13:27–39, 1980. [19] editor N. Resher. The Logic of Decision and Action. University of Pitsburgh Press, 1967. [20] R. Reiter. A logic for default reasoning. Artificial Intelligence, 13:81– 132, 1980. [21] Y. Shoham. Nonmonotonic logics: meaning and utility. In Proceedings of the 10th IJCAL, 388-393, Milan, 1987. [22] Y. Shoham. Reasoning about change: time and causation from the standpoint of artificial intelligence. MIT Press, Cambridge, MA, USA, 1988. [23] J. v. Benthem, S. v. Otterloo, and O. Roy. Preference logic, conditionals and solution concepts in games. ILLC Prepublication Series PP-200528, 2005. [24] S. v. Otterloo and O. Roy. Preference logic and applications. September 2005. [25] J. van Benthem and F. Liu. Dynamic logic of preference upgrade. [26] G. H. von Wright. The logic of preference. Edinburgh, 1963. [27] G. H. von Wright. The logic of preference reconsidered. Theory and Decision, 3:140–169, 1972. [28] Y.Murakami. Logic and social choice. Monographs in modern logic. Dover publications, 1968.

A

Appendix: Some other perspectives on preferences

In this appendix, we present alternative approaches and applications of preferences to the study of logics.

24

Figure 4: Initial model.

Figure 5: Upgraded model after the suggestion to take a trip.

A.1

The logic of preference dynamified

In [25], Liu and van Benthem show how preferences may be dynamified. In the spirit of the work by Baltag, Moss and Solecki ([1]), they use a update mechanism which transforms the models. The actions studied by Baltag and al. are public announcements, which remove worlds from the model. For dynamic preferences, the actions become suggestion, of the form ‘Let’s take a trip!’, and these actions remove links in the models. This is best exemplified with the following example:. Example 9. Suppose both s i t and t i s, i.e., agent i is indifferent between worlds s and t. Suppose that the proposition p := Let′ stakeatrip! is in s, but that s is not in t. The initial model is given in figure 9, and the upgraded model after the suggestion to take a trip has been performed is given in figure 9. The language for dynamic preference logic is given by the following: ϕ := ⊥ | ¬ϕ | ϕ ∧ ψ | [pref ]iϕ | Uϕ | [π] π := #ϕ. Here the modality [pref ]i is the preference modality, defined over the relation 25

i , and the action # is a suggestion action, whose action on models is to remove links. Formally, their semantics is given by the following: M, s |= [prefi ]ϕ iff ∀t : s i t ⇒ M, t |= ϕ M, s |= [#ϕ]ψ iff M#ϕ , s |= ϕ where 1. (M#ϕ , s) has the same domain, valuation, epistemic relations, and actual world as (M, s), but 2. the new preference relations are now ∗i =i −{(s, t) : M, s |= ϕ, M, t |= ¬ϕ}

A.2

Preference logic and nonmonotonic logic, Yoav Shoham

Let M be the set of models that satisfy a given set of formula Γ. Classically, one says that Γ entails a sentence γ, written Γ |= γ, if every member of M satisfies γ, i.e., if every model of Γ is also a model of γ. Nonmonotonic logics are logics that evaluate the entailment relation for a subset M ′ ⊆ M of models chosen from M. Another variant of nonmonotonic reasoning, which turns out to be discussed quite a lot in preference logic, is the logic of default reasoning, first introduced by Reiter in [20]. For a good survey of nonmonotonic logic, see Makinson [17]. It is natural to see the set M ′ as picking up the preferred models from M. This is what Shoham does in [21]. Using a notion of preference between models, he gives a unifying framework to think about nonmonotonic logics.1 We will outline Shoham’s framework and then see how it may be applied to a case example, McCarthy’s logic of circumscription as first presented in [18]. Let L be either the propositional or the first-order calculus and let ⊏ be a strict partial order on the interpretations of L. L and ⊏ together define a new preference logic, denoted L⊏. We read M1 ⊏ M2 as “the interpretation M2 is preferred over the interpretation M1 ”. Satisfiability and entailment are defined in the following way: 1

We will follow [21] closely here, and refer the reader to [22] for a more detailed treatment.

26

Definition 9 (Preferential satisfiability). Let A be any sentence in L⊏, and let M be an interpretation in L. We say that M preferentially satisfy A, M written |=M A in the classical sense, and if there is no other inter⊏ A if |= ′ ′ pretation M such that M ⊏ M ′ and |=M A. Definition 10 (Preferential entailment). Let A and B be two sentences of L⊏, we say that A preferentially entails B, written A |=⊏ B, if for any M interpretation M ∈ L, if |=M B. ⊏ A, then |= In this setting, one may precisely define monotonicity: Definition 11 (Monotonicity). A logic L⊏ is monotonic if for all A, B, C ∈ L⊏, if A ⊢⊏ C, then also A ∧ B |=⊏ C. As a case example, we present how Shoham subsumes McCarthy’s logic of circumscription under his framework. Circumscription is the addition of a second order axiom to a theory. A circumscription axiom is of the form: A(p) ∧ ∀p¬(A(p) ∧ p < P ) where p is any predicate with x as a free variable, and p < P stand for: ∀x(p(x) → P (x)) ∧ ¬∀x(P (x) → p(x) As Shoham notes, the circumscription operated by the axioms is tantamount to the following preference relation M1 ⊑ M2 over interpretations of the theory: 1. For all x, M1 and M2 agree on the interpretation of function symbols and all relation symbols other than P , 2. for all x, if M2 |= P (x), then M1 |= P (x), and 3. there exists a y such that M1 |= P (y), but M2 6|= P (y).

A.3

Preference and conditional logics

In [16] (p.98-99), Lewis considers a preference logic from sphere system point of view. The notion of preference he uses is stated as “world j is better than world k if and only if some sphere contains j but not k”. The underlying system of sphere is nested, closed under unions and closed under nonempty intersections. For a preference logic, no limit assumptions should be made, 27

since it might be the case that the goodness of an alternative ϕ is unbounded, or there may be worlds that are better than any of the ϕ-worlds. Some typical notions of sphere systems, which are not assumed from the outset, but which might be desirable, depending on the notion of preference S S at hand, are the notions of 1) centering ( $ = ∅), 2) universality ( $i = universe) and i S 3) absoluteness ( $i = O, for some given unique sphere O). Centering might not be desirable if one wants to include abnormal worlds, in which everything is permissible. Universality (which implies centering) accounts for worlds that are not evaluable from a given world i. Finally, absoluteness makes sense if, for example, the preference is a pleasure measure on some utility function, in which case some state of affair would obtain absolute preference over any other state of affairs. In [14], Huang suggests a semantics for ceteris paribus preferences in terms of a closest world function. Drawing on von Wright’s idea that a preference ceteris paribus is obtained when an agent favors a change of the world to a ϕ ∧ ¬ψ-world over a change to a ¬ϕ ∧ ψ-world, so long as the world does not change in any other respect. Her proposal truth-definition is the following: M, w |= ϕPcp ψ iff (∀w ′ ∈ W )cw(w ′, |ϕ ∧ ¬ψ|) ≺ cw(w ′, |ψ ∧ ¬ϕ|) Prima facie, Huang’s proposal does not capture a notion of preferences ceteris paribus as von Wright intended it, since it might be that the closest ϕ ∧ ¬ψworlds have p also true, but that the closest ¬ϕ ∧ ψ-worlds have p false, for some independent p. If p stands for “I have my boots on”, then this falls directly in von Wright’s counter-example, that I prefer my raincoat to my umbrella, in the sense that I prefer loosing my umbrella to my raincoat, unless I loose my boots along with the lost of my raincoat. Nevertheless, Huang’s proposed axiomatization is very similar to von Wright’s principles, and this might suggest a stronger link. However, as she does not provide any completeness proof, we leave the question open here for further investigations.

28