Leibniz's law states that if two things are identical, than everything about them must be identical. If identity theory is to remain consistent with Leibniz's law, then mental processes must be spatial because brain processes are spatial. Brain scans consistently show specific activity in the brain when different mental activities are performed. However, it is possible that the observed neurological activity is a response to the mental activity, not the activity itself. Consider for example that a person's perception of a tree is not the same as the tree itself. If this is the case, then mental activity is not spatial, and therefore it cannot be synonymous with the brain.
The second argument states that first-person accounts demonstrate the difference between mental states and the brain. For example, it is possible to experience pain that a neuroscientist cannot see on a brain scan. Assuming the technology exists to always see pain on a scan, either the patient is lying or the mental state of pain is not the same as the brain's perception of pain. According to Leibniz's law, one instance of different properties is sufficient to reject that two things are the same.
Multiple realizability arguments use biological studies to argue against identity theory. Identity theory states there is a unique chemical state in the brain for every mental state. However, multiple realizability theory observes that many species, with many different brain chemistries, experience the same mental states, such as fear and pain. As a result, the experience of fear and pain cannot be confined to a single neuro-chemical structure and therefore brain and mind cannot be the same.
In his 1980 book, Naming and Necessity, Saul Kripke proposes that the concept of rigid designators makes comparisons more precise. Variables are rigid designators if they are the same in all possible contexts. Kripke further proposes that two rigid designators are the same if they cannot exist independently of each other. He then demonstrates that identity theory must be false because brain states and mental states are both rigid designators but they can exist independently of each other. For example, C-fiber stimulation (brain) does not always elicit pain (mental state) in the same patient.