On the Attitude of Trust: A Formal Characterization of Trust, Distrust, and Associated Notions

Abstract

Using modal logics to represent an agent’s beliefs, knowledge and wants, an analysis is given of trust in terms of an agent’s certainty that a particular, desired state-of-affairs will be realized. Similarly, a corresponding analysis of distrust is given. Placing these formal representations of trust and distrust at each of the ends of a spectrum, four intermediary structures may be identified, representing hope, two species of anxiety, and fear. In this way the relationships between the attitudes of trust/distrust and some basic types of emotional state may be precisely articulated. Some suggestions are also made regarding the analysis of some more complex types, including regretting that one trusted, and being ashamed that one trusted.

The paper employs modalities of type KD and KT for, respectively, the logics of belief and knowledge. It is shown that use of stronger doxastic and epistemic logics – of the kind often favoured in Artificial Intelligence – containing the positive and negative introspection axioms, would make three of the spectrum’s four intermediary positions logically inconsistent. It is suggested that this result provides good reason for rejecting the stronger logics, and that their adoption in AI has often been motivated primarily by considerations of computational convenience, rather than by considerations of conceptual accuracy.

Dedicated to the memory of my former mentor and research colleague Ingmar Pörn, who died in February 2014.

1 Introduction

Consider the following examples in which one agent trusts another:

• x trusts y to fulfil a contractual obligation;

• x trusts y to fulfil properly a role;

• x trusts what y says.

In Jones (2002) it was argued that, in each of these examples, the content, or object, of x’s trusting attitude concerns trustee compliance: y’s conformity to some governing norms or conventions. For the first example, the case is obvious; the contractual obligation is specified by some norm or other, and what x trusts is that y will comply with that norm. In the second example, the case turns on the assumption that roles are characterized, in part, in terms of a set of norms that apply to the role-holder when he is acting in that role – cf. (Pörn 1977, pp. 61–63). Trusting one’s physician, for instance, amounts to trusting that he acts in ways that conform to the standards governing members of the medical profession. For the third example, the case turns on an assumption to the effect that indicative signalling, verbal or non-verbal, exploits conventions that correlate signalling act-types to types of states of affairs; when an instance of a given signalling act-type is performed, the conventionally correlated state of affairs ought then to hold. (For a detailed development of this approach to indicative signalling, see Jones and Kimbrough (2008); Jones and Parent (2007). The origin of the approach lies in Stenius (1967).)

In what follows, this ‘trustee-compliance’ view of the object of the trusting attitude will be presupposed.¹ The focus here will be not on the object of the trusting attitude, but rather on the trusting attitude itself. And in regard to the characterization of that attitude, Jones (2002) 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.

The second of these two points reflects the fact that, ordinarily, it matters to truster x that compliance is forthcoming; compliance is not an issue on which x is indifferent: compliance is something that he wants. The presence of the volitional component in the trusting attitude explains, at least in part, why trust is so often linked to the notion of risk.

In order to develop an improved account of the trusting attitude, capable of repairing the shortcomings of the earlier approach, the point of departure here will be Pörn’s modal-logical taxonomy of types of emotions (Pörn 1986), in which modal logics are used to represent the cognitive and volitional components alluded to above.² One consequence will be that it becomes possible to get a clearer picture of the relationship between trust and distrust, on the one hand, and the cognitive and volitional aspects of hope, anxiety and fear, on the other.

2 Cognitive and Volitional Positions

Pörn (1986) applied the combinatory method of maxi-conjunctions, developed by Kanger for classifying types of Hohfeldian rights-relations.³

For the logic of belief, a modality of type KD is used, with the operator relativized to individual agents. The system KD of modal logic is formed by adding to the smallest normal system K – as defined in Chellas (1980) – the schema D:

D:

B _x p → ¬B _x ¬p

which says that if an agent x believes that p, where p is any proposition, then he does not believe not-p. For the logic of knowledge, a modality of type KT is used, with the operator again relativized to individual agents. The system KT of modal logic is formed by adding to the smallest normal system K the schema T:

T:

K _x p → p

A central conjecture in Pörn (1986) is that an agent’s certainty that p may be represented as the agent’s believing that he knows that p. Accordingly, the following two certainty positions may be identified:

• B _x K _x p: x is certain that p

• B _x K _x ¬p: x is certain that ¬p

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 two certainty positions and other, weaker doxastic-epistemic 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

The class of doxastic-epistemic ‘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

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. 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 six positions are mutually exclusive, and their disjunction is a logical truth. So precisely one of (DE1)–(DE6) must hold for any given proposition p.

Concerning (DE6), Pörn said (Pörn 1986), p. 208 that it “…is the epistemic null-position; it is doubtful whether it is relevant for the analysis of emotions since in this position the subject has no belief at all concerning p. (An epistemic null-position may of course be the object of an emotion, but that is another matter.)” And in the development of his analysis of atomic emotional types he chose to disregard (DE6). In what follows, however, the possibility will be left open that (DE6) might be of relevance, particularly in the context of comparing trust with the (doxastic-epistemic components of) emotions. So (DE6) will be retained.

Each of the six (DE) positions may be simplified by removing any conjuncts that are themselves logically implied by one or more other conjunct. The result of that simplification is as follows:

(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

As regards the logic of volition, let expressions of the form D _x p be read ‘x desires/wants that p’, where D _x is a (relativized) normal modality of type KD. It may then readily be shown that there are just three basic volitional positions for any given agent x and any proposition p. They are:

(V1):

D _x p

(V2):

D _x ¬p

(V3):

¬D _x p & ¬D _x ¬p

Following Pörn, (V3) may be said to be the position of ‘volitional indifference’. He was inclined to the view that it is irrelevant to the analysis of the emotions; given the present interest in the analysis of trust, and given what was said in the introductory remarks to the effect that it ordinarily matters to truster x that trustee y acts in a way that fulfils the trust bestowed upon him, the focus here will also be exclusively on volitional positions (V1) and (V2); indifference will be disregarded.

3 Cognitive and Volitional Positions Combined

The result of conjoining (V1) and (V2), respectively, to each of (SDE1)–(SDE6) is given in the following list of 12 doxastic-epistemic/volitional positions:

(DEV1):

B _x K _x p & D _x p

(DEV2):

B _x K _x p & D _x ¬p

(DEV3):

B _x K _x ¬p & D _x p

(DEV4):

B _x K _x ¬p & D _x ¬p

(DEV5):

B _x ¬K _x p & B _x ¬K _x ¬p & D _x p

(DEV6):

B _x ¬K _x p & B _x ¬K _x ¬p & D _x ¬p

(DEV7):

B _x ¬K _x p & ¬B _x K _x ¬p & ¬B _x ¬K _x ¬p & D _x p

(DEV8):

B _x ¬K _x p & ¬B _x K _x ¬p & ¬B _x ¬K _x ¬p & D _x ¬p

(DEV9):

B _x ¬K _x ¬p & ¬B _x K _x p & ¬B _x ¬K _x p & D _x p

(DEV10):

B _x ¬K _x ¬p & ¬B _x K _x p & ¬B _x ¬K _x p & D _x ¬p

(DEV11):

¬B _x ¬K _x p & ¬B _x ¬K _x ¬p & D _x p

(DEV12):

¬B _x ¬K _x p & ¬B _x ¬K _x ¬p & D _x ¬p

(DEV1)–(DEV10) are Pörn’s ten ‘atomic emotional types’. It is interesting to consider the labels he gave to them. (DEV1) and (DEV4) are both types of security, in the sense that, in each case, what the agent is certain of matches what he desires; by contrast, (DEV2) and (DEV3) represent despair (Pörn’s label), or hopelessness, since in each case what the agent is certain of is the opposite of that which he desires.⁴ (DEV5) and (DEV6) both represent a form of anxiety, in as much as the agent believes that he does not know whether p holds – and in the one case he wants p, whereas in the other he doesn’t. Consider next (DEV7): the agent desires p and (first conjunct) believes that he does not know that p; although (second conjunct) he does not believe that he knows not-p, his knowing not-p is compatible with all that he believes (third conjunct). (That description of the third conjunct follows (Hintikka 1962) by interpreting ‘¬B _x ¬’ as ‘compatibility with all that x believes’.) So (DEV7), and its counterpart (DEV10), represent fear. Parallel considerations lead to the conclusion that (DEV8) and (DEV9) represent hope.

(DEV11) and (DEV12) of course do not figure among Pörn’s atomic types, since they are based on the epistemic null-position. But perhaps a case can be made for maintaining that they represent another type of anxiety. If it is compatible with all that an agent believes that he knows that p, but also compatible with all that he believes that he knows that not-p, then it would seem that he totally lacks any information that would enable him to decide the question of p’s truth/falsity. But then if he is not indifferent, either wanting p to be the case or wanting not-p to be the case, he has grounds for anxiety – albeit grounds of a cognitive type different from that expressed in (DEV5) and (DEV6). This point will be considered further below, in the discussion of trust and distrust.

It is important to note Pörn’s emphasis that – as he put it – the atomic types are ‘unrestricted’, in as much as their characterization is independent of any particular specification of the kind of state-of-affairs p describes. He then considers ((Pörn 1986), pp. 209–210), by way of contrast, some examples of emotions, such as anger, that are ‘restricted to objects of a certain kind’. It is at this point that the above account of the doxastic-epistemic/volitional positions can be linked to the introductory discussion of the object of the trusting attitude.

4 A Spectrum of Cases

Assume now that the proposition p in (DEV1)–(DEV12) is restricted to trustee compliance in the sense described in the first paragraph of this paper, and elaborated in (Jones 2002). It is appropriate then to confine attention to those six cases in which D _x p, rather than D _x ¬p, appears, since the assumption is that truster x desires compliance on the part of the trustee. The contracted list is this:

(DEV1):

B _x K _x p & D _x p

(DEV3):

B _x K _x ¬p & D _x p

(DEV5):

B _x ¬K _x p & B _x ¬K _x ¬p & D _x p

(DEV7):

B _x ¬K _x p & ¬B _x K _x ¬p & ¬B _x ¬K _x ¬p & D _x p

(DEV9):

B _x ¬K _x ¬p & ¬B _x K _x p & ¬B _x ¬K _x p & D _x p

(DEV11):

¬B _x ¬K _x p & ¬B _x ¬K _x ¬p & D _x p

A suitable label for (DEV1) is complete trust: x is certain that compliance, which he desires, will be forthcoming. Thus complete trust is a specific instance of what Pörn calls security. By contrast (DEV3) represents complete distrust: x is certain that compliance, which he desires, will not be forthcoming. Thus, plausibly enough, complete distrust is a specific instance of despair or hopelessness.

The six cases might in fact be considered to constitute a spectrum, with complete trust at the left-hand end and complete distrust at the right-hand end. Next to complete trust comes (DEV9), which represents hope-of-compliance, and immediately preceding complete distrust comes (DEV7), which represents fear-of-non-compliance. The middle of the spectrum is occupied by the positions (DEV5) and (DEV11), which represent two species of anxiety-about-whether-compliance-will-occur. So the spectrum looks like this:

• (DEV1) - (DEV9) - [(DEV5), (DEV11)] - (DEV7) - (DEV3)

One of the contexts in which matters of trust have lately been given a great deal of attention is the field of e-commerce. In that context, a distinction is often drawn between commercial interactions in which the traders have some previous experience of each other, and the so-called ‘first-trade scenario’, where the parties may be completely unfamiliar with one another. That distinction may perhaps be used to illustrate the differences between the two types of anxiety in the spectrum, (DEV5) and (DEV11). The latter fits, it seems, the kind of situation that would arise in a ‘first-trade scenario’ if the one party, x, totally lacks information relevant to assessing the trustworthiness of the other party, y, whereas (DEV5) would be a more appropriate description of 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.

Another way of highlighting the difference between (DEV5) and (DEV11) is as follows: in virtue of its first conjunct, (DEV11) 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 (DEV5), and B _x ¬p may be consistently conjoined with (DEV5) – but obviously not both, because of the D schema. (DEV11) is characterized by the agent’s lack of relevant information; only when that lack is remedied can he move to a position that would be compatible either with the belief that p, or with the belief that not-p.

Some may object to the description of the spectrum offered above, and indeed more generally to Pörn’s approach to the characterization of the emotions, on the grounds that there is more to an emotional state than the mere combination of epistemic-doxastic and volitional elements, making it inappropriate to use such terms as hope, fear, anxiety as labels. But nothing essential hinges on the use of those terms; the six positions, and their ordering on the spectrum, are clearly characterized by means of the component logics, and the entire account could thus be re-formulated without appeal to the emotion-terms. The key point is that a small set of modal building-blocks have been used to describe precisely and formally the attitudes of complete trust and complete distrust, and to exhibit their respective relationships to, and differences from, a set of intermediary attitudes.

5 Strengthening the Logics of Belief and Knowledge

It is commonly accepted that knowledge implies belief. So now add to the logics described above the schema:

KB:

K _x p → B _x p

Furthermore, it has been usual in Artificial Intelligence to adopt KD45 for the logic of belief and KT5 for the logic of knowledge. Essentially, this amount to adding to the logic KD (for belief), and the logic KT (for knowledge), 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)

From the semantical point of view, this strengthened logic of knowledge and belief can be characterized by means of a standard model (in the sense of Chellas (1980)) in which there are two binary accessibility relations R^K _x and R^B _x satisfying the following properties:

• R^K _x is both reflexive and Euclidean

• R^B _x is serial, transitive and Euclidean

• R^B _x is a sub-relation of R^K _x.

The basic truth condition for sentences of the form K _x p is given as follows:

(TCK):

At any world w in any standard model M, K _x p is true at w iff p itself is true at every world w ₁ such that <w, w ₁ > є R^K _x

Similarly, the basic truth condition for sentences of the form B _x p is given by:

(TCB):

At any world w in any standard model M, B _x p is true at w iff p itself is true at every world w ₁ such that <w, w ₁ > є R^B _x.

Adoption of this strengthened logic of knowledge and belief would have significant consequences for the ‘trust-distrust’ spectrum described above, since three of the six component positions – (DEV7), (DEV9) and (DEV11) – would become inconsistent. In terms of the semantics, the inconsistencies may be demonstrated in the following way, starting from (DEV11).

Suppose that each of the first two conjuncts of (DEV11), ¬B _x ¬K _x p and ¬B _x ¬K _x ¬p, is true at some world w in a model M of the kind just outlined above. Since ¬B _x ¬K _x p holds at w, it follows by (TCB) that there must be some world w ₁ such that <w, w ₁ > є R^B _x and such that K _x p holds at w ₁ . Similarly, since ¬B _x ¬K _x ¬p holds at w, it follows by (TCB) that there must be some world w ₂ such that <w, w ₂ > є R^B _x and such that K _x ¬p holds at w ₂ . Since R^B _x is a sub-relation of R^K _x, it now follows that <w, w ₁ > є R^K _x and that <w, w ₂ > є R^K _x. But the relation R^K _x is Euclidean, so it now follows that <w ₁ , w ₂ > є R^K _x. But then, since K _x p holds at w ₁ , it follows by (TCK) that p itself must hold at w ₂ . However, since R^K _x is also reflexive, it follows that <w ₂ , w ₂ > є R^K _x and thus, since K _x ¬p holds at w ₂ , it follows by (TCK) that ¬p also holds at w ₂ . This reduces to absurdity the initial assumption that each of ¬B _x ¬K _x p and ¬B _x ¬K _x ¬p holds at w.

Consider next (DEV9), and suppose that its second and third conjuncts, ¬B _x K _x p and ¬B _x ¬K _x p, are both true at some world w in a model M of the kind under consideration. Since ¬B _x ¬K _x p holds at w, it follows by (TCB) that there must be some world w ₁ such that <w, w ₁ > є R^B _x and such that K _x p holds at w ₁ . Similarly, since ¬B _x K _x p holds at w, it follows by (TCB) that there must be some world w ₂ such that <w, w ₂ > є R^B _x and such that ¬K _x p holds at w ₂ . Since R^B _x is a sub-relation of R^K _x, it now follows that <w, w ₁ > є R^K _x and that <w, w ₂ > є R^K _x. But the relation R^K _x is Euclidean, so it now follows that <w ₂ , w ₁ > є R^K _x. Now, since ¬K _x p holds at w ₂ , it follows by (TCK) that there must be some world w ₃ such that <w ₂ , w ₃ > є R^K _x and such that ¬p holds at w ₃ . Since it has been established that <w ₂ , w ₁ > є R^K _x and that <w ₂ , w ₃ > є R^K _x, it now follows from a further application of the Euclidean property of R^K _x that <w ₁ , w ₃ > є R^K _x. But K _x p holds at w ₁ ; so it follows by (TCK) that p itself must hold at w ₃ . This reduces to absurdity the initial assumption that each of ¬B _x K _x p and ¬B _x ¬K _x p holds at w. (By means of the same pattern of argument it may also be demonstrated that (DEV7) is inconsistent.)

Apart from the basic truth conditions for the knowledge and belief modalities, these proofs of inconsistency turn on three properties: that R^K _x is Euclidean; that R^K _x is reflexive; and that R^B _x is a sub-relation of R^K _x. The second and third properties are unproblematic: reflexivity is the key to guaranteeing validity of the T schema, and thus that knowledge implies truth, and the sub-relation property guarantees the KB schema, and thus that knowledge implies belief. So the problem lies in the adoption of the Euclidean property for the epistemic accessibility relation – the very property that plays the key role in validating the positive and negative introspection schemas for the knowledge modality. Given the intuitive plausibility of (DEV7), (DEV9) and (DEV11) as representations of, respectively, the fear-of-non-compliance position, the hope-of-compliance position, and one of the anxiety positions, the conclusion to be drawn is that the Euclidean property, and the corresponding introspection schemas, should be rejected.⁵

Why was KT5 often the epistemic logic of choice in AI ? Part of the answer to that question, perhaps, lies in the fact that KT5 has properties that are attractive from the point of view of computational tractability. The practice of allowing computational considerations to play a significant role in determining choice of conceptual model has been quite widespread in AI and in the closely allied research field of Agents and Multi-agent Systems. The problematic nature of that practice is discussed in some detail in (Jones et al. 2013), which outlines an approach to the design of intelligent socio-technical systems in which conceptual and computational models are properly integrated.

6 Non-atomic Types

Pörn (1986, pp. 210–213), discusses ways in which atomic types of emotions can be combined to form complex types. Pörn considers, for instance, envy; suppose

• (i) x is envious of y because y got the job.

Here, he suggests, we have a situation in which two instances of despair are combined, in that

• (ii) x is certain that y got the job, but wishes that he (y) had not; and

• (iii) x is certain that he himself did not get the job, but wishes that he had.

So, where p = ‘y got the job’ and q = ‘x got the job’, the logical form of (i) becomes:

• (iv) B _x K _x p & D _x ¬p & B _x K _x ¬q & D _x q

Another way in which complex emotional types can be formed, Pörn suggests, is when the object of an emotion is itself also an emotion. Consider this in relation to the case of complete trust, as analysed above⁶:

• (v) x is certain that p (i.e., that y will comply), and x desires that p.

And now consider how to interpret

• (vi) x regrets putting complete trust in y.

What, according to (vi), is x’s attitude (doxastic-epistemic and volitional) towards the object of his regret, as expressed by (v) ? A natural answer is that x is certain that (v) and desires that it had not been the case that (v). Expressed formally:

• (vii) B _x K _x (B _x K _x p & D _x p) & D _x ¬(B _x K _x p & D _x p)

Consider a specific example: x trusted Nick Clegg and the Liberal Democrats at the 2010 UK General Election; x was certain (believed that he knew) that they would deliver on their (manifold) promises, and desired that they should do so. x now regrets trusting: he is quite sure that he had that trust, and he wishes that he hadn’t !

What then would be the difference between that situation and one in which x is ashamed of having trusted Clegg and the Liberal Democrats ? One suggestion would be that x’s shame combines his regret with a conviction that he ought not to have trusted in the first place: he should have known better, should have been able to see through the pretence……If a suggestion of that sort is accepted, then the modal-logical language needs to be supplemented with an appropriate normative modality to represent ‘ought’. When that component is supplied, the sentence

• (viii) x is ashamed of putting complete trust in y

may be rendered as

• (ix) B _x K _x (B _x K _x p & D _x p) & D _x ¬(B _x K _x p & D _x p) & B _x K _x Ought¬(B _x K _x p & D _x p)

7 Concluding Remark

The paper has indicated a way of placing the concepts of trust and distrust in relation to a broader class of attitudes, in which doxastic-epistemic and volitional components are combined. It is evident that much work remains to be done on exploring the cognitive, volitional and perhaps normative aspects of the structure of complex types of emotions. Hopefully, however, the discussion presented here provides grounds for thinking that, in the spirit of Pörn’s work, these phenomena are amenable to rigorous and systematic analysis by means of application of the tools of modal logic.