On the attitude of trust - a formal characterization of trust , distrust , and associated notions Andrew J I Jones, Dept of Informatics, King’s College London andrewji.jones@kcl.ac.uk 1
The object of the trusting attitude • Consider x trusts y to fulfil a contractual obligation x trusts y to fulfil properly a role x trusts what y says • In each case, the content , or object , of x ’s trusting attitude concerns trustee compliance: y ’s conformity to governing norms or conventions ( Decision Support Systems , 2002). 2
The trusting attitude itself • As regards the characterization of the trusting attitude itself, the 2002 paper fell short in at least two respects: it described the cognitive aspect of the truster’s attitude in terms of mere belief; but the fully trusting agent feels sure, certain, secure that trustee compliance will occur it overlooked the volitional component. 3
The trusting attitude • Ordinarily, it matters to truster x that compliance is forthcoming; compliance is not an issue on which x is indifferent. • The presence of the volitional component in the trusting attitude explains why trust is so often linked to the notion of risk . • Point of departure: Pörn’s formal-logical taxonomy of the emotions (1986). 4
Cognitive and volitional positions • Pörn applied the combinatory method of maxi-conjunctions, developed by Kanger for classifying types of rights-relations in the sense of Hohfeld. • This talk takes no stance on whether Pörn’s taxonomy of emotions is adequate, nor on the question whether trust is an emotion – although trust does seem to be related to, for instance, hope and fear . 5
Cognitive positions • Use (relativised) KD for the modal logic of belief, KT for the logic of knowledge (Chellas classification). • (T) K x p → p • (D) B x p → ¬ B x ¬p • Two certainty positions: • B x K x p : x is certain that p • B x K x ¬p : x is certain that ¬p 6
• In virtue of the logical properties of the two modalities B and K , as modalities of type KD and KT, respectively, the following relations of logical implication may be shown to hold between the certainty positions and other, weaker epistemic- doxastic positions: • B x K x p → ¬ B x ¬K x p → ¬ B x K x ¬p • B x K x p → B x ¬K x ¬p → ¬ B x K x ¬p • B x K x ¬p → ¬ B x ¬K x ¬p → ¬ B x K x p • B x K x ¬p → B x ¬K x p → ¬ B x K x p 7
• The class of doxastic-epistemic (DE) ‘positions’ may now be generated as follows: first take the four positive expressions B x K x p , B x K x ¬p , B x ¬K x p , B x ¬K x ¬p , and then the corresponding negative expressions ¬ B x K x p , ¬ B x K x ¬p , ¬ B x ¬K x p , ¬ B x ¬K x ¬p . These eight expressions can be arranged as four truth-functional tautologies: (1) B x K x p v ¬ B x K x p (2) B x K x ¬p v ¬ B x K x ¬p (3) B x ¬K x p v ¬ B x ¬K x p (4) B x ¬K x ¬p v ¬ B x ¬K x ¬p 8
• Obviously, for any given agent, and for any proposition p , precisely one of the disjuncts in each of (1) – (4) must hold. There are 16 ways of selecting precisely one disjunct from each of (1) – (4), to form 16 conjunctions of four conjuncts each. • Of these 16 conjunctions, just 6 are logically consistent, given the logical properties adopted for the two modal operators. 9
The 6 logically consistent conjunctions are: • (DE1) B x K x p & ¬B x ¬K x p & ¬B x K x ¬p & B x ¬K x ¬p • (DE2) ¬B x K x p & B x ¬K x p & B x K x ¬p & ¬B x ¬K x ¬p • (DE3) ¬B x K x p & B x ¬K x p & ¬B x K x ¬p & B x ¬K x ¬p • (DE4) ¬B x K x p & B x ¬K x p & ¬B x K x ¬p & ¬B x ¬K x ¬p • (DE5) ¬B x K x p & ¬B x ¬K x p & ¬B x K x ¬p & B x ¬K x ¬p • (DE6) ¬B x K x p & ¬B x ¬K x p & ¬B x K x ¬p & ¬B x ¬K x ¬p • It may be shown that these 6 positions are mutually exclusive, and their disjunction is a logical truth. So precisely one of (DE1) – (DE6) must hold for any given proposition p . 10
• Each of the 6 (DE) positions may be simplified by removing any conjuncts that are themselves logically implied by one or more other conjunct: • (SDE1) B x K x p • (SDE2) B x K x ¬p • (SDE3) B x ¬K x p & B x ¬K x ¬p • (SDE4) B x ¬K x p & ¬B x K x ¬p & ¬B x ¬K x ¬p • (SDE5) B x ¬K x ¬p & ¬B x K x p & ¬B x ¬K x p • (SDE6) ¬B x ¬K x p & ¬B x ¬K x ¬p 11
Adding volition • The object of the trusting attitude, p , is that the trustee will comply with the relevant rule; and trustee compliance is something that the truster desires. Read expressions of the form D x p as ‘ x desires that p ’ (desire modality is of type KD). Then the class of positions that needs to be considered is: (TR1) B x K x p & D x p [attitude of trust] (TR2) B x K x ¬p & D x p [attitude of distrust] (TR3) B x ¬K x p & B x ¬K x ¬p & D x p (TR4) B x ¬K x p & ¬B x K x ¬p & ¬B x ¬K x ¬p & D x p (TR5) B x ¬K x ¬p & ¬B x K x p & ¬B x ¬K x p & D x p (TR6) ¬B x ¬K x p & ¬B x ¬K x ¬p & D x p 12
Positions between trust ( B x K x p & D x p ) and distrust ( B x K x ¬p & D x p ) (TR5) B x ¬K x ¬p & ¬B x K x p & ¬B x ¬K x p & D x p 2 nd conj: x not certain y will comply; but (1 st conj) x believes it’s compatible with what he knows that y will comply and (3 rd conj) it’s compatible with what he believes that he knows y will comply. Attitude of hope . (Note that the 3rd conj implies ¬B x ¬p ) 13
• (TR4) B x ¬K x p & ¬B x K x ¬p & ¬B x ¬K x ¬p & D x p x is not certain that y won’t comply (2 nd conj); but (3 rd conj) it’s compatible with what x believes that he knows y won’t comply, and (1 st conj) x believes that he doesn’t know that y will comply. Attitude of fear . (Note that the 3rd conj implies ¬B x p ) 14
• The spectrum now looks like this: Trust – hope – [(TR3),(TR6)] – fear – distrust (TR1) – (TR5) – [(TR3),(TR6)] – (TR4) – (TR2) • How are the two middle positions to be understood ? 15
(TR3) and (TR6) (TR3) B x ¬K x p & B x ¬K x ¬p & D x p First two conjuncts say that x is of the opinion that he doesn’t know whether y will comply. (Given that B is a normal modality, the first two conjuncts of (TR3) may be equivalently expressed as B x ¬(K x p v K x ¬p) .) (TR6) ¬B x ¬K x p & ¬B x ¬K x ¬p & D x p First two conjuncts indicate that x hasn’t formed an opinion at all about what he knows about p : it is compatible with what he believes that he knows y will comply, and compatible with what he believes that he knows y won’t comply. 16
• (TR6) fits, it seems, the kind of situation that would arise in a ‘first - trade’ scenario if x totally lacks information about the trustworthiness of y , whereas (TR3) would fit the situation in which, on the basis of previous experience of y , x has come to the conclusion that he just doesn’t know whether or not y can be trusted. 17
• Another way of highlighting the difference between (TR3) and (TR6): in virtue of its first conjunct, (TR6) logically implies ¬B x ¬p , and in virtue of its second conjunct it logically implies ¬B x p . However, B x p may be consistently conjoined with (TR3), and B x ¬p may be consistently conjoined with (TR3) – but obviously not both, because of the D schema. • (TR6) is characterised by the agent’s lack of relevant information; only when that lack is remedied can he move to a position that is compatible either with the belief that p , or the belief that not p . • (TR3) and (TR6): two types of anxiety ? 18
Strengthening the logics of belief and knowledge • It is commonly accepted that knowledge implies belief. So now add the schema: (KB) K x p → B x p • It has been usual in AI to adopt KD45 for the logic of belief and KT5 for the logic of knowledge. Essentially, this amount to adding the so-called positive and negative introspection schemas: (B4) B x p → B x B x p (positive introspection) (B5) ¬B x p → B x ¬B x p (negative introspection) (K4) K x p → K x K x p (positive introspection) (K5) ¬K x p → K x ¬K x p (negative introspection) 19
• The most significant consequence of adopting these changes: (TR6) ceases to be a logically consistent position; so the ‘trust – distrust’ spectrum reduces to five positions. • The inconsistency of (TR6) turns essentially on the adoption of the schemas (T) and (B5), together with the adoption of (KB). In virtue of the latter, and the normality of the belief modality, the first conjunct of (TR6) implies ¬B x ¬B x p . But the second conjunct of (TR6) implies ¬B x p. Then, by the (B5) schema, it follows that B x ¬B x p , and a contradiction is derived. 20
• From the model-theoretic point of view, the inconsistency of the conjunction ¬B x ¬K x p & ¬B x ¬K x ¬p turns on the basic-truth conditions for B x and K x reflexivity of R K x that R B x is a sub-relation of R K x that R B x is euclidean [Note: comment on self-deception] 21
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