We present a logic to model the behaviour of an agent trusting or not trusting messages sent by another agent. The logic formalizes trust as a consistency checking function with respect to currently available information. Negative trust is modelled in two forms: distrust as the rejection of incoming inconsistent information; mistrust, as revision of previously held information becoming undesirable in view of new incoming inconsistent information, which the agent wishes to accept. We provide a natural deduction calculus, a relational semantics and prove soundness and completeness results. We overview a number of applications which have been investigated for the proof-theoretical formulation of the logic. Consider the following modified example: Example 1.5 (Intentional Untrust Multiplication). Alice does not trust φ from Bob: she believes he sends her intentionally false information. Bob does not trust ¬φ from Carol: he believes she sends him intentionally false information. Assume Alice is aware of that, should she trust ¬φ from Carol? The question whether Alice is safe in assuming the original ¬φ to be valid is now specified more precisely, and the related epistemic action of distrust, reformulating the previous notion from Definition 1.2, has the following intuitive semantics: Definition 1.3 (Distrust). If Alice reads φ from Bob, and φ is inconsistent with Alice's profile, Alice distrusts φ and writes ¬φ. A distinct case for trust misplacement can be formulated as follows: Example 1.6 (Unintentional Untrust Multiplication). Alice reads φ from Bob, false in view of her current information: she believes she has unintentionally held false information ¬φ. Bob has received φ from Carol, who can confirm it to Alice. Should Alice trust φ from Carol? This operation has an intuitive semantic meaning which underpins an act of trust (of the new information) but leads to an act of negative trust (of the old information held): Definition 1.4 (Mistrust). If Alice reads φ from Bob, φ is inconsistent with Alice's profile and Alice wants to maintain consistency, then she mistrusts ¬φ. To accept or reject such contradicting information (i.e, to distrust or mistrust the received information) might depend on the number and role of other agents available for confirmation, or from selected parameters.