Analytical Decomposition of Trust in Terms of Mental and Social Attitudes

Abstract

Trust is defined as a truster’s belief in some properties. At the beginning they are to reach a goal and then they are refined in trust in some trustee’s property from which the truster can infer that his goal will be reached. This property may be the trustee’s ability to bring it about that the goal is reached which can itself be derived from the trustee’s intention to reach this goal. Then, we show that this intention may be adopted by the trustee depending on three kinds of social relationships: compliance of norms, mutual commitment with another agent or willingness to act without any compensation.

This analytical decomposition is formalized in a modal logic with a conditional connective. However, the technical details that could prevent an intuitive reading are omitted.

Annex

The axiomatics, in addition to the axiomatics of classical propositional calculus, is defined as follows.

The modal operators Bel _i and □ obey the axiomatics of a normal modal logic of system K.

For the conditional operator we have the following axiom schemas and inference rules:

(EQUIV):

If $\vdash \phi \leftrightarrow \phi '$ and $\vdash \psi \leftrightarrow \psi '$, then $\vdash (\phi \Rightarrow \psi ) \rightarrow (\phi ' \Rightarrow \psi ')$

(TRANS):

$(\phi _{1} \Rightarrow \phi _{2}) \wedge (\phi _{2} \Rightarrow \phi _{3}) \rightarrow (\phi _{1} \Rightarrow \phi _{3})$

(DIST):

$(\phi _{1} \Rightarrow \phi _{2}) \rightarrow (\phi _{1} \rightarrow \phi _{2})$

Property RS2.

We have: (R2) $Bel_{i}(\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow Attempt_{j}\phi )$ and (S2) $Bel_{i}(Attempt_{j}\phi \Rightarrow \lozenge \phi )$ entail (R1) $Bel_{i}(\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow \lozenge \phi )$.

Proof.

From the properties of a system K, from (R2) and (S2) we have:

• (1) $Bel_{i}((\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow Attempt_{j}\phi ) \wedge (Attempt_{j}\phi \Rightarrow \lozenge \phi ))$

  From (TRANS), we have:

• (2) $(\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow Attempt_{j}\phi ) \wedge (Attempt_{j}\phi \Rightarrow \lozenge \phi ) \rightarrow (\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow \lozenge \phi )$

  From Necessitation applied to Bel _i and (2) we have:

• (3) $Bel_{i}((\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow Attempt_{j}\phi ) \wedge (Attempt_{j}\phi \Rightarrow \lozenge \phi ) \rightarrow (\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow \lozenge \phi ))$

  From K and (1) and (3) we have:

  (R1):

  $Bel_{i}(\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow \lozenge \phi )$

Property RS23.

We have: (R3) $Bel_{i}(\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow Int_{j}\phi )$,

(S3) $Bel_{i}(Int_{j}\phi \Rightarrow Attempt_{j}\phi )$ and (S2) $Bel_{i}(Attempt_{j}\phi \Rightarrow \lozenge \phi )$ entail (R1) $Bel_{i}(\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow \lozenge \phi )$.

Proof.

With the same kind of proof as for Property RS2, from (R3) and (S3) we have:

• (1) $Bel_{i}(\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow Attempt_{j}\phi )$

  With the same kind of proof, from (1) and (S2) we have:

  (R1):

  $Bel_{i}(\neg \phi \wedge Goal_{i}\lozenge \phi \Rightarrow \lozenge \phi )$.

Property MN2.

We have the logical theorem: $Bel_{i}((Able_{x}\neg \phi ) \wedge (Attempt_{x}\neg \phi ) \rightarrow \lozenge \neg \phi )$.

Proof.

From (DIST) and Able definition we have:

• (1) $(Able_{x}\neg \phi ) \rightarrow (Attempt_{x}\neg \phi ) \rightarrow \lozenge \neg \phi )$

  Therefore, we have:

• (2) $(Able_{x}\neg \phi ) \wedge (Attempt_{x}\neg \phi ) \rightarrow \lozenge \neg \phi )$

Since Bel _i obeys a system K from (2) we have:

$$\displaystyle{Bel_{i}((Able_{x}\neg \phi ) \wedge (Attempt_{x}\neg \phi ) \rightarrow \lozenge \neg \phi ).}$$

Property MN3.

We have the logical theorem: $Bel_{i}((Determined_{x}\neg \phi ) \wedge (Able_{x}\neg \phi ) \wedge (Int_{x}\neg \phi ) \rightarrow \lozenge \neg \phi )$.

Proof.

From Determined and Able definitions, $(Determined_{x}\neg \phi ) \wedge (Able_{x}\neg \phi )$ is an abbreviation for:

• (1) $(Int_{x}\neg \phi \Rightarrow Attempt_{x}\neg \phi ) \wedge (Attempt_{x}\neg \phi \Rightarrow \lozenge \neg \phi )$

  From (TRANS), (1) entails:

• (2) $(Determined_{x}\neg \phi ) \wedge (Able_{x}\neg \phi ) \rightarrow (Int_{x}\neg \phi \Rightarrow \lozenge \neg \phi )$

  From (2) and (DIST) we have:

• (3) $(Determined_{x}\neg \phi ) \wedge (Able_{x}\neg \phi ) \rightarrow (Int_{x}\neg \phi \rightarrow \lozenge \neg \phi )$

  And from classical logic (3) entails:

• (4) $(Determined_{x}\neg \phi ) \wedge (Able_{x}\neg \phi ) \wedge (Int_{x}\neg \phi ) \rightarrow \lozenge \neg \phi$

  Since Bel _i obeys a system K, from (4) we have:

• (5) $Bel_{i}((Determined_{x}\neg \phi ) \wedge (Able_{x}\neg \phi ) \wedge (Int_{x}\neg \phi ) \rightarrow \lozenge \neg \phi )$