Symmetry bias has been discussed often in the context of one on one relationship, that is the relationship between name and object. This paper presents other possibilities from the viewpoint of predicate logic, especially, in the usage of the Skolem function. In this paper, from the present viewpoint, a few types of new symmetry bias are provided. Section 2 introduces symmetry bias between subject and object, and symmetry bias of the Skolem function between antecedent and consequent in conditions.In section 3, a new type of symmetry bias between name and object is clarified from the standpoint of predicate logic and the cognitive mechanism. In section 5, first, an intentional predicate logic is defined, then possibilities of new symmetry bias are examined, depending on the intentional predicate logic.The final section discusses a particular predicate logic form which contains all the above mentioned cases of symmetry bias, based on a concrete story.