Since zombies enjoy no conscious experiences and since empathy presupposes the having of conscious experiences; therefore, zombies cannot experience empathy (nor can CDs for blisteringly obvious reasons). You are proposing the existence of a mysterious entity, a zombie, which has not been shown to exist. No beings can empathize unless they exist, and zombies - by definition - cannot empathize, because it's bound-up in conscious experiences - which they utterly lack.
The appropriate conditional, symbolizing the proposition, that "consciousness is causally inert" is B → -C, i.e. "No behavior arises from conscious experience". This relation to logicians is what is known as a "universal negation". All one must do to defuse a universal negation is to haul-out one counter-example to it.
1 is what one must adopt to become a playa' in the non-interactionary dualism game.
The rubbish-notion that "consciousness is inert" is no doubt the root of the zombie disease. What does it mean to say that x is causally inert? If there is an existing x, how is it logically impossible for x to have no subsequent, causal efficacy on (or interaction with) the world. No one can just hack-off x off from the web of universal, causal interdependency - without offering compelling evidence for this piece of zombie-magic.
Perhaps, it was a mistake to seize-onto non-interactionary dualism, conjure-up imaginary zombie critters to populate a possible world, conceal the fact from yourself that you embraced this queer, zombie-involved dualism, then as the final insult, pretend that the ultimate nature of the human mind can be "revealed" by pondering possible zombies in a possible world scenario.
In this noxiously foul cult, zombies (and CDs) have been assigned the traits of an "intentional component" and "mime" behavior - despite that no branch of science has detected these numinous zombie entities. When a specific claim about something happening in the world is made, evidence for it must appear. Failing that, the claim is un-proven (or worse). Hence, no "intentional component" nor any "mime" act on the part of any zombie has been demonstrated to occur. Even if - per impossibile - such components and mime acts were observed, it does not follow that zombies could experience empathic states. To confound sentient creatures (humans) who have a conscious experience of identifying with someone else's pain and a zombie (or a CD) - which lacks conscious experience - is a category mistake - and/or the final dissolution of the intellect.
In a seedy bar on the upper West Side, two zombies are nursin' Harvey Wallbangers and chewin' on hot-wings. A human enters through a plate-glass door - slightly nonplussed.
bartender: What's yours?
human: A zombie please. Err...
zombie1: Hey dude, how's tricks?
human (sitting down on a bar-stool): Not so good. I've developed a serious condition. Whenever I consciously entertain a thought, a desire or a plan, I cannot execute it. It's as though I've lost control over my efferent nerves, muscles, internal organs, limbs, etc.
zombie1 and zombie2 (in unison): Oh my God! With a little imagination, it seems that you are turning into one of us! Have you consulted a doctor?
human: Yeah, but he said the problem was either psychological, or - as he muttered: "Your condition may have arisen - due to the wholesale depletion of your potent conscious experiences".
zombie1: These "conscious experiences" escape my understanding. But, what were you doing before you lapsed-into this interruption of volitional instrumentality?
human (shelling peanuts): Well, I was online, and some guy by the name of Khuno asserted that consciousness is causally inert. After that encounter, I could not pee - when consciously beholding the pee-act before my mind's eye, intending and trying to pee. The doctor prescribed massive doses of diuretics and diapers to counter-act the malady!
zombie1 and zombie2 (whispering in unison): Ah, this Khuno person was philosophizing!
zombie2: You have contracted what is known to us crude realists as a logic disorder. It comes from exposure to non-interactionary dualists. Stay away from these people. They are nothing but the slaughter of human comprehension. Give it a week or so, and you'll be up and peeing (when consciously intending and trying to do so) like a champ.
human (embarrassed): Thanks. It's as though a great reservoir has been released from my bladder...
In a ghetto behind a dilapidated warehouse - somewhere in the heartland of the USA, two zombies are gettin' up to no good.
cop: Halt! What are you zombies doing?
zombie1 and zombie2 (in unison): We are empathizing!
cop: No zombie can empathize - in this or in any possible world!
zombie1: But - if compact discs can, why can't we? This is an illegal act of invidious discrimination! We'll hire a lawyer, suing the powers that be!
cop (swinging his night stick): Lemme' tell you why. Imaginary beings, trivially, cannot empathize. Zombies - by definition - lack conscious minds, and empathizing presupposes a conscious state of mind. For the same reason, neither can compact discs empathize. No "intentional components" of empathy have been found inside of CDs - which "mime" empathic states. Only conscious beings enjoy such states, and - even if a behavioral aspect of empathy were detected, it would not equate to a state of conscious experience.
zombie1: But Khuno said -
cop (interrupting): Khuno is a fool. Our Western Civilization has undergone two Enlightenments. The ancient Greeks, in theirs, got rid of myth, trading up for rationality. The Europeans, in the 18th century, got rid of the priests, trading up for science. We once thought that the last Enlightenment would have put an end to Khuno's balderdash, rendering us forever immune. Sadly, we were mistaken.
zombie1: But, can a zombie try to empathize?
cop: No! A zombie cannot try to empathize, inevitably failing at the task. He cannot even try to try to empathize. Do you understand that you are hypothetical beings, and that - as far as anyone knows, you do not exist?
zombie1, zombie2 and the cop (singing Cole Porter):
The chimpanzees, in the zoos, do it,
Some courageous kangaroos do it,
Let's do it, let's em-path-ize.
I'm sure giraffes, on the sly, do it,
Heavy hippopotami do it,
Let's do it, let's em-path-ize.
zombie: This sudden explosion of interest in us zombies is positively disconcerting. Descartes - in your heyday - had you ever conceived of advancing a theory of the human mind, resting its final understanding on what imaginary beings like us zombies "do"?
Descartes (perplexed): Not really. I thought perhaps I could have contributed a lick or two to the understanding of mind. Exploring interactionary psycho-physical dualism - literally - tied me up in knots. With its outcome, I was never entirely satisfied. I studied anatomy of course - though not extremely proficiently. But, I never wanted to go off the "deep end". After all, I'm a product of the Age of Reason.
zombie (voice modulating to sarcasm): Quite correct. But, a philosophical associate of yours, one Mr. Leibniz, is "credited" with developing the possible world scenario. How do you reconcile its application to us zombies with your vaunted "Age of Reason"?
Descartes (annoyed): Bosh! Listen to me you god-damned zombie: Gotti and I were mathematicians! Perhaps, we pursued the "dark side" a bit, but we never confused a possible world with the freakin' world! Gotti's thesis was co-opted, re-purposed and sold "down the river" by modern irrationalists - these non-interactionary dualists!
zombie: You would think that - after all of the intellectual progress of your Age and subsequently, no one would self-consciously embrace non-interactionary dualism, graft it to a possible world scheme then proceed with it to confuse zombies with people and people with zombies without - in the process - becoming the laughing-stock to the entire world, would you?
Descartes (affecting a regal air): Plus ça change, plus c'est la meme chose.
zombie: We shall return after this message...
(commercial voice over): In our possible world - according to statistics, four out of five zombies suffer from real (but un-painful) hemorroids. Don't be embarrassed totin' 'round that 'roid ring anymore! Use Preparation Z, the anti-hemorroid ointment specially formulated for us god-damned zombies! The "Z" means zero swelling. Get yers' today! That's Preparation Z, available at fine, zombie-marts everywhere.
zombie: Locating a voice of reason in our possible world was a pointless adventure. So, we sent-out to the great beyond for Bertrand Russell. Tonight, I'll be grilling the Lord on the crucial matter of ontological refutation; that is, the view that something said to exist in the world need not exist in the world for the view to be held as true. Lord, could you expound on this? 
Russell (stroking his chin): The refutation smacks of a "counter-realism". From my understanding, zombies, CDs, rocks and thermostats are asserted to exhibit various "intentional components" and that these behavioral aspects "mime" the thoughts, beliefs and desires of sentient creatures, i.e. humans. Yet - baffling to me, neither an "intentional component" nor an instance of a "mime" act has been observed in the world - though their existences cannot be doubted; hence, the assertion is both counter-real and true. This notion makes Kurt Gödel crack a smile...
zombie: Astounding! You - famously - chronicled the history of Western Philosophy, diagnosing the major illness of modern philosophy as the "subjective madness" introduced by Descartes. Does the "counter-realism" tie-in - somehow?
Russell: No! Compared to counter-realism, even attempting to dogmatically doubt one's own existence is downright sane!
zombie: Should we not just allow geologists the final-say on the "question" as to whether rocks contain "intentional components" and "mime" ability in them and chemists on the "question" as to whether mercury-containing thermostats incorporate these properties in them?
Russell: That's where the screw-ball crosses the plate. Science - to the counter-realist - deals with what happens in the world; hence, it deals in falsity. According to counter-realism - even though I am dead and you are unreal, our existences are true - apodictically; whereas, the photons emitted by quasars and the gases and particles belching from super-novas are false because they are real.
zombie: For the first time, I begin to grasp why we zombies are objects of a warped fascination to certain people on a Myspace group. Because we are unreal! That's the nexus we have been struggling to locate up here in our possible world. Could you sum-up this counter-realism for the lay-zombie?
Russell (pensive): Well, should a person make a claim about something happening in the world, the claim is true, if only if, the something does not happen in the world. For example, the "intentional components" and "mime" behavior on the part of CDs are sworn to be true - despite their not existing. The traits of "intentional components" and "mime" concerning imaginary zombies are doubly true - because neither the traits nor their bearers exist.
zombie: Thanks for a slice of your eternal time Lord. Congrats on your Nobel!
Russell: Thanks. Great to be anywhere - while dead...
zombie (assuming the demeanor of a prosecutor): I accuse you, Plato. There are people back in the real world who actually give a shit about us zombies. They meditate on our primal perfection, hoping for a sign of truth to show them the truth about themselves. The word on the zombie-street is that you, Plato, are responsible for originating this lunacy.
Plato (defensive): Look. Listen. Lemme' explain. It happened during the Greek Enlightenment. I was broke an' suckin' up to royalty. That Pure Form crapola was just my shtick. I was a gutter-wino, aimlessly roaming the Athenian streets! Philosophy was my only ticket to score drachma for cheap, Mendean wine! Never had I surmised that I'd be taken seriously. For Jupiter's sake, my toga was about to be repossessed!
zombie: I think that Jupiter was a Roman god and the toga a Roman invention.
Plato (contemptuous): Whatever...
zombie: Do you realize that back in the real world the progress - which science has made over the last 400 years - has been staggering! It's as though philosophy has been at a full-halt, stand-still since you kicked-off. On a Myspace group, there remain certain dolts who think that by pondering our Platonic zombie essence, that it is the authority, regulating what can (and cannot) be said about how the human mind/brain functions! For too long, we have witnessed this madness. The madness must end!
Plato (reprising Brando's performance in "On the Waterfront"): Scary... I hang my head in shame, abhoring my involvement in the fiasco. Turning to philosophy was my fatal blunder. Socrates, he said he was my brother and mentor. Socrates, he shoulda taken care of me just a little bit. You don't understand! I coulda had class. I coulda been a wrestler. I coulda been somebody - instead of a Helot, which is what I am. Let's face it. It was Socrates' fault. May Jupiter help my soul transmigrate...
zombie: Khuno, don't you understand that we do not want you pondering our zombie essence? It's perverse! You fail to grasp the ontological distinction between your real self and our imaginary selves, confounding people with zombies and zombies with people!
zombie (praying): Hear me O, Lord! Lord, I petition thee with prayer.
God: WTF! You cannot petition me - as I've never created you. For Christ's sake, you are a zombie! You are imaginary! You do not exist!
zombie: But Lord, there are those in the real world who have been construing us as archetypes. They have been contemplating our zombie essentials! We are sore afraid! Save us O Lord!
God: Go on, my son.
zombie: Realworld people on a Myspace group have been ruminating on us as disembodied Forms - as patterns of themselves. Make it stop! Make it stop! Make it stop! O Lord, we beseech thee.
God: So, ya wan'me ta send a couple of plagues down? Heh. I got a tyhoon or two I ain't doin' nothin' with.
zombie: Lord, thou speaketh crudely as one from South Jersey speaketh!
God: Woodbury, precisely. Heh. I'm holdin' onto a toliet chain. If I pull this baby, the realwolders and the pathetic civilizations they've created will be flushed-down the out-house crapper.
zombie: Oh, would'st thou O Lord? Though the complete and utter annihilation of all realworlders seems a bit disproportionate to the offense. Could'st thou extinguish only those realworders on the Myspace group who have been ogling us zombies with impure thoughts?
zombie Sinatra (singing "My Way"): I traveled each and ev'ry highway. And more, much more than this, I did it my waaay!
zombie Wittgenstein (interposing): NO! You did nothing. You can do nothing! You and I are a god-damned zombies! Events happen to us. In the world, we can make nothing happen. From my knowledge of logic and physics, even things happening to us cannot happen to us - as we are imaginary! We are not real!
zombie Sinatra (offended): Yeah, but in 1948, I won a supporting Oscar for my stellar performance in the pic' "From here to eternity"!
zombie Wittgenstein: NO! You performed nothing. Throughout the filming, something mysterious programmed your gestures, locomotion, facial expressions, verbal patterns, etc. We have no volitional efficacy that could causally link our thoughts to our actions - as we have no thoughts!
zombie Sinatra (furious): But, but - what about that doll Ava Gardner, "the most perfect face in cinematic history"? I never consciously courted, seduced and married her?
zombie Wittgenstein: NO! "Arkle is not a horse", and you never entertained a conscious desire for Ava! You (mechanistically) went-through the motions - of dating, of seducing and of marrying her. But, you never consciously beheld the object of your desire before your mind's eye, then acted upon it. We zombies are ontological losers. We're the sludge of all possible worlds!
zombie Sinatra (singing "New York, New York"): Start spreadin' the news. I'm leavin' today. I want to be a part of it - New York, New York. If I can make it here, I'll make it anywhere. Its up to you - New York, New York!
(Cult-leader Khuno needs to reduce us all to zombie-like creatures - in order to leverage his non-interactionary, pan-psychic dualism. Humans, do not let this happen to you! The Chairman of Board would never have permitted it! Start spreadin' the news...)
human (alone on stage): According to non-interactionary dualism - since my body cannot mediate my consciousness, my conscious thoughts cannot reach out, causally modifying any event in the world. When I have a conscious desire to masturbate and since my conscious states cannot inform the enactment of any subsequent masturbating-behavior, how is it that I masturbate? Masturbation seems the stuff of the most esoteric magicks!
zombie (appearing out of nowhere): Yes, magic it is - numinous and occult... Hey brother, you seem to masturbate just like me! And - according to any coherent interpretation of the laws of physics - unlike you, I do not exist!
human: But, how can non-interactionary dualism not appear self-evidently self-refuting to all thinking beings - if no magicks are involved?
zombie: You remember your Wittgenstein, don't you?
"Philosophical problems arise when language goes on holiday."
You have consciousness. We have none. The words "intention" and "consciousness" have been elastically redefined by Khuno to make it appear as though we zombies possess human-levels of intentionality and consciousness. Two words have been illicitly redefined into the world - not a new discovery made in it.
human: Aha... So true my brother...
zombie: Wittgenstein also remarked:
"Willing is not a thing which happens to me; it is something I do."
"The word willing is used with connection to phenomena bound up in our bodies."
So, we zombies, should we exist, have things happen to us. You humans make things happen. Your consciousness is causally potent - not inert. You humans masturbate - as a result of your conscious desire to masturbate.
human: Aha... So true my brother... So true...
"Supposing that by an act of stipulative imagination you can remove consciousness while leaving all cognitive systems intact - a quite standard but entirely bogus feat of imagination - is like supposing that by an act of stipulative imagination, you can remove health while leaving all bodily functions and powers intact. Health isn't that sort of thing, and neither is consciousness." -- Dan Dennett
zombie (alone on a pitch-dark stage with a single baby-spot illuminating his visage): Suppose that a model of a nuclear power-plant in a possible world - minus one gizmo - is devised by the "Theorists of Fission". No physicist would pay even scant attention to any hypothesis, derived from a model that a possible reactor plausibly explains how the real ones work. Nuclear scientists know exactly how the real ones work, or reactors would either not split atoms in controlled chain-reactions or they would melt-down. Insofar as the human mind, the model of a zombie in a possible world seems compelling (or at least seductive) - because the mind is terra incognito, still conceived of as a mysterious entity. But - what if the mind is an abstraction - not a complicated neuro-physical contraption?
If the mind turns-out an abstraction, we spin the possible wheels of our possible car - in vain, accelerating aimlessly through possible landscape after possible landscape, anticipating that the destination is around the next bend - though such a final destination remains forever elusive.
However, the mind appears to be a part of furniture of the world; thus, the house, itself, cannot be ignored, bull-dozed or relocated to a possible block in a possible world. The mind needs to fit into the decor of its own house - not into a possible house in a possible world, inhabited by possible home owners, i.e. zombies.
The fundamental error committed by most of these "philosophers of mind" is that they violate a sound injunction, laid-down by Wittgenstein:
"Philosophers should not play at being scientists."
Philosophers can no better know what the mind is (nor how it functions) by thinking about it - than could geolgists know that the earth contained a mantle by thinking about a possible earth in a possible world. The world matters more than what humans conjecture about what subsists in a possible world. If the human mind is a concrete entity, then it should be approached and treated with the proper respect that it deserves.
Let's dispell this conceptual-fog with a more appropriate analogy - than the latter. From the 13th century - until recently, conscious experiences included thoughts, beliefs, desires, wishes, etc. Of late, the phrase "conscious experiences" has been narrowly redefined to exclude the foregoing "attitudes", only applying to "what it is like to experience x", i.e. qualia. One might construct an analogy with the phrase "conscious experiences", contrasting the old and new phrase-meanings with that of the intensity of illumination, radiated by differing light sources.
Suppose that there's a football stadium, ringed by high intensity discharge-lamps, and that the light, emitted from the lamps, floods the surrounding countryside for miles. The lamps and their illumination under the old phrase-meaning refer to conscious experiences. Under the new definition - however - where the intentional attitudes have been stripped-away from the phrase-meaning and confined to "what it is like to experience x", the light source of qualia is analogous to one person on the 50 yard line, holding flickering candle. A pack of teenage vandals, (i.e. philosophers of the mind) had long since shot-out the high intensity discharge-lamps of the previously understood conscious experiences.
To conclude: A zombie in a possible world is a dream. Inferences, drawn from pondering austere, zombie perfection to the ultimate nature of the human mind, are the contents of the dream. If I dream that I date a super-model, it's not the case that I have dated a supermodel - because the truth-conditions under which the super-model and I "dated" were only mind-dependent illusions, dreamed up by me. What are taken as "True" and "False" in a dream (or in a possible world scenario) should not be confused with mapping reality. Hence, we zombies just ain't real.
E(x) E(y) [ Px ^ -My ^ Axy ] / E(y)My
There is at least one x and there is at least one y; such that, x is a person and y is not a zombie mime act and x asserts y; therefore, there is at least one zombie mime act. N.B. E(y)-My in the premise and E(y)My in the conclusion.
1. E(x) E(y) [ Px ^ -My ^ Axy ] / E(y)My 2. E(y) [ Pa ^ -My ^ Aay ] EI, 1 3. Pa ^ -Mb ^ Aab EI, 2 4. -Mb ^ Pa ^ Aab Com, 3 5. -Mb Simp, 4 6. E(y)-Mb EG, 5