"This sentence is false."
To assert that the sentence above is true is to also assert that it is false at the same time. Similarly, to assert that it is false is to also assert that it is true at the same time.
|The statement in this box is false.|
Other motivations include the paradoxes of Zeno in which both reason and experience seem true (The arrow seems to be moving, empirically speaking; but, rationally, an arrow that occupies any set of points in space-time isn't moving at all).
This refers to semantic issues in dialetheism. In the two cases cited above, of course, the Liar's paradox is about the meaningfulness of the statement; we ask the question: Does it make any sense? Does it communicate anything? If it doesn't convey any sense of meaning, then it is nonsensical. Falsehood can only be predicated of some assertion being made, but in the statement "This sentence is false", the sentence is empty of the assertion that needs to be falsified. It is a vacuous statement, devoid of meaning.
Of course, dialetheism is not the use of ambiguous statements (statements that carry two meanings because of the many meanings of the terms used). For example, in the statement, "I like tablets", one can ask for clarification regarding what "tablet" means (pill, gadget, or tile).
Phenomenal versus Essential Statements
To say that the sun rises in the east would be both true and false at the same time. It is true epistemically, phenomenally. But, does it really rise in the east? Phenomena must not be confused with essential form. To call a stone red is a claim regarding phenomenal truth (it appears red). But, whether the stone really possesses any colors is an essentialist (ontological) issue. While statements must be interpreted in the context of their literary genre (poetic statements are to be interpreted as poetical, for instance), the various contexts of the use of words also cannot be ignored.
The phenomenal versus essential problem is an epistemological issue. This doesn't mean that reality is ambiguous and open to any interpretation. However, paradoxes don't prove that the Law of Non-Contradiction doesn't apply in some situations. They only prove that the paradox arises because the Law is evidently not-contradicted.