This paper examines the two properties of the discourse marker but: non-truth-conditionality and coherence. Non-truth-conditionality is the feature that has brought but into the center of the research on Relevance Theory, but coherence has been paid much less attention. This is due to the cognitive nature of the underpinnings of Relevance Theory. Relevance Theory claims that utterance interpretation involves decoding of linguistic stimuli into conceptual representations and inferential manipulation of conceptual representations, so that linguistic meaning comes into two varieties: conceptual and procedural meaning. This paper analyzes data from 31 British adults` narrative discourse and demonstrates that but is a discourse marker that encodes a procedural meaning which leads to contradict an assumption inferred from the previous context. This paper also shows how a variety of interpretations of but, such as the denial of expectation and contrast can be derived pragmatically from its encoded meaning, and how but is interpreted as performing a ``boundary marking`` function on the ``global level`` in Relevance-theoretic terms.