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The Strength of Desires: A Logical Approach

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Abstract

The aim of this paper is to propose a formal approach to reasoning about desires, understood as logical propositions which we would be pleased to make true, also acknowledging the fact that desire is a matter of degree. It is first shown that, at the static level, desires should satisfy certain principles that differ from those to which beliefs obey. In this sense, from a static perspective, the logic of desires is different from the logic of beliefs. While the accumulation of beliefs tend to reduce the remaining possible worlds they point at, the accumulation of desires tends to increase the set of states of affairs tentatively considered as satisfactory. Indeed beliefs are expected to be closed under conjunctions, while, in the positive view of desires developed here, one can argue that endorsing \(\varphi \vee \psi\) as a desire means to desire \(\varphi\) and to desire \(\psi\). However, desiring \(\varphi\) and \(\lnot \varphi\) at the same time is not usually regarded as rational, since it does not make much sense to desire one thing and its contrary at the same time. Thus when a new desire is added to the set of desires of an agent, a revision process may be necessary. Just as belief revision relies on an epistemic entrenchment relation, desire revision is based on a hedonic entrenchment relation satisfying other properties, due to the different natures of belief and desire. While epistemic entrenchment relations are known to be qualitative necessity relations (in the sense of possibility theory), hedonic relations obeying a set of reasonable postulates correspond to another set-function in possibility theory, called guaranteed possibility, that drive well-behaved desire revision operations. Then the general framework of possibilistic logic provides a syntactic setting for encoding desire change. The paper also insists that desires should be carefully distinguished from goals.

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Notes

  1. According to Dretske (1988), desire is also a necessary condition for reward. In particular, desire determines what counts as a reward for an agent. For example, a person can be rewarded with water only if she is thirsty and she desires to drink.

  2. This is an example where the logical disjunction symbol \(\vee\) should be read “and”, in natural language, as it translates into a set union, because the interpretations of \(\varphi\) understood as a desire are not mutually exclusive, contrary to when \(\varphi\) represents a belief. In the case where \(\varphi\) is a belief, the models of \(\varphi\) are mutually exclusive as candidates to be the real state of the world. So this set is a disjunction of situations, while if \(\varphi\) is a desire, any model of \(\varphi\) is a desirable situation, and the set of models of \(\varphi\) now collects all desirable situations, thus making the set of models a conjunction of situations. See Dubois and Prade (2012) and Zadeh (1978) for the importance of the distinction between the conjunctive and disjunctive interpretations of sets in another area.

  3. Remember that the set of models of \(\bot\) is \(\emptyset\).

  4. Called anhedonia as pointed out by a referee.

  5. As noticed by a referee.

  6. For simplicity, we assume that W does not contain unbearable interpretations.

  7. The term guaranteed comes from the fact that in possibility theory, \(\Delta (A) = 1\) means that all instances of A can actually be observed. It contrasts with potential possibility, which takes value 1 as soon as one instance of A is observed.

  8. This equality is the same as the one that Spohn ranking functions \(\kappa\) (Spohn 2012) satisfy, but identifying \(\Delta\) and \(\kappa\) would be misleading: \(\Delta\) and \(\kappa\) are antimonotonic with inclusion, but the ranges of \(\Delta\) and \(\kappa\), respectively [0, 1] and \({\mathbb {N}}\) (non-negative integers) are directed in opposite ways: \(\kappa ( A) = 0\) expresses full possibility, and the higher \(\kappa ( A)\) the more A is impossible, while \(\Delta (A) = 1\) expresses full possibility. When mapping \({\mathbb {N}}\) to [0, 1] via an order-reversing function (e.g., \(f(A) = 2^{-\kappa (A)})\), we obtain an increasing maxitive function: \(f(A\cup B) = \max (f(A),\, f(B))\), which departs from the characteristic property of \(\Delta\).

  9. Note that wine and cheese are complementary, while beer and wine can be viewed as redundant, and thus less jointly desirable.

  10. This distinction leads to the identification of two different kinds of moral dilemmas. The first kind of moral dilemma is the one which is determined by the logical conflict between two moral values. The paradigmatic example is the situation of a soldier during a war. As a member of the army, the soldier feels obliged to kills his enemies, if this is the only way to defend his country. But, as a catholic, he thinks that human life should be respected. Therefore, he feels morally obliged not to kill other people. The other kind of moral dilemma is the one which is determined by the logical conflict between desires and moral values. The paradigmatic example is that of Adam and Eve in the garden of Eden. They are tempted by the desire to eat the forbidden fruit and, at the same time, they have a moral obligation not to do it.

  11. At least this is so in the book (Gärdenfors 1988) as the epistemic entrenchment is a consequence of the axioms.

  12. In the paper, we use a very restrictive definition of epistemic entrenchment, which is the one in Gärdenfors (1988), that is, a total preorder relation on the language that obeys specific properties, which can only be justified if we take the two last AGM axioms for granted. Clearly one could envisage a less restrictive framework for desire revision, similar to the ones studied by Rott (2001) without (D*7) and (D*8).

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Acknowledgements

The authors are grateful to the referees for a careful reading and insightful remarks, that led us to clarify a number of issues in the paper, in particular, the shaping of the proof of the completeness Theorem 1.

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  1. IRIT-CNRS, Université Paul Sabatier, 31062, Toulouse Cedex 09, France

    Didier Dubois, Emiliano Lorini & Henri Prade

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  1. Didier Dubois
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  2. Emiliano Lorini
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Correspondence to Henri Prade.

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Dubois, D., Lorini, E. & Prade, H. The Strength of Desires: A Logical Approach. Minds & Machines 27, 199–231 (2017). https://doi.org/10.1007/s11023-017-9426-5

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