Cold Empathy is the ability to read others emotional states, but then use that to harm them.
Dr. Tara J. Palmatier
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Cold Empathy is the ability to read others emotional states, but then use that to harm them.
Dr. Tara J. Palmatier

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Sam Vaknin, arguably the world's foremost expert on NPD:
“Trump lacks empathy and clearly enjoys embarrassing and hurting other people gratuitously. Such antisocial bullying misconduct makes him feel (and, in his mind, actually renders him) all-powerful and God-like ("omnipotent"). He seems to regard himself as an intellectual and entrepreneurial Gulliver, trapped among contemptuous, pusillanimous, and retarded Lilliputians who he despises;
Trump has an inordinately developed "cold empathy": the kind of an "x-ray vision" that allows him to immediately spot the vulnerabilities, weaknesses, and chinks in the armor of his interlocutors, adversaries, partners, and enemies and to leverage this knowledge to his benefit by penetrating their defenses, taunting, and tormenting them, often with aggressive and cruel “humor”. This capacity for instant penetration allows him to resonate powerfully and intimately with the hidden hopes, dreams, fantasies, delusions, and negative emotions (rage, hatred, fear) of his "constituencies". He is a consummate predator.
To my mind, Trump is the most perfect example I have ever come across of a malignant and, probably, psychopathic narcissist.”
Scientists Find That Feeling Cold is Contagious
Peter Hale, Uncanny Valley, and Cold Empathy
Uncanny Valley is a hypothesis in the field of human aesthetics which holds that when human features look and move almost, but not exactly, like natural human beings, it causes a negative response (fear, repulsion, etc.) among human observers. Some examples of this include robotics and reconstructive surgery techniques.
The same reactive trajectory applies to narcissists and psychopaths: they are near-perfect imitations of humans, but lacking empathy and emotions. They are so close to being real, but they are just this shade of wrong.
Peter’s version of empathy, "cold empathy", has the cognitive element of empathy, but not its emotional correlate. Consequently, it’s a detached and cerebral kind of intrusive gaze, devoid of compassion and a feeling of affinity with one's fellow humans. When new characters are introduced to Peter the Uncanny Valley reaction kicks in: people get spooked. They feel revolted, scared, and repelled. As we have seen with pretty much everyone who has dealt with Peter aside from Derek, they can't put their finger on what it is that provokes these negative reactions, but, after a few initial encounters, they tend to keep their distance.
Peter may appear to be "empathizing" with his extended social group, but it’s not real. You can tell, because this kind of displayed "empathy" usually vanishes the minute its subject ceases to play a role in the Peter's psychodynamic processes.