The concept of dual attitudes is the reality that a person can have more than one set of beliefs about about the same object or subject. These two points of view are known as implicit (non-conscious and uncontrolled) and explicit (conscious and controlled) attitudes. Both can coexist within a subject's mind and the subject may not even be fully aware that this duality of attitudes exists.
This might be easiest explained by an examination of racial prejudice. Depending on a subject's upbringing (regional, ethnicity, exposure to the attitudes of others, etc.) they may have been exposed to prejudiced attitudes to the point of having deeply prejudiced implicit attitudes towards certain other ethnic groups. However, with age, education, exposure, and shifts in social attitudes the subject may choose to embrace an explicit attitude of non-prejudice in order to not feel social judgments or ramifications for their prejudiced beliefs. This subject can be described as having dual attitudes. In this type of case, the subject would need to engage in serious self-examination to acknowledge this duality.