What invalidates baptism?

Notes on Thornwell’s The Invalidity of Roman Catholic Baptism:

The form of baptism is that by which the sacrament is distinguished from other instances of ablution with water. Only by adhering to the restrictions Christ placed upon the sacrament of baptism can we expect that the promises or grace attached to it will indeed take effect. That said:

The relations which an ordinance’s material elements sustain to the covenant of grace is essential to the ordinance itself, because the ordinance was instituted by Christ’s authority: if an ordinance is administered according to conditions other than those instituted by Christ, the administration is not sacramental by definition.

The correctness of the perception of the covenant of grace is also a necessary component to a valid baptism, as, for instance, one who invokes the names of the Trinity without believing in them – that is, in their co-equal authoritativeness et. al. – profane the sacrament. The administration of the sacrament is, in this instance, only analogous to Christian baptism, for there can be no Christian baptism wherein the essence of Christianity is not preserved. 

Finally, the intention of the sacrament is integral to its essence. The relation of the covenant of grace to the material elements is the very purpose of the sacrament, the very means by which grace is bestowed. A church which misunderstands the purpose of baptism voids the ordinance itself.