The Law of Causality is a corollary of the Law of Identity.
Causality is a direct corollary of the three inarguable axioms of Objectivism; there is simply no grounds to argue against it, as any alternative to causality is fundamentally impossible.
All things that exist have a single, finite identity.
That is, all things that exist have a finite set of characteristics that describe them. This is not to say that humans will ever know this exhaustive list, but it is a fundamental necessity -- an inarguable fact -- that every single existent possesses a single, finite identity, composed of everything that it is, and lacking everything that it is not. If an existent had no identity, then it wouldn't exist -- there wouldn't be a single thing that it is. If an existent had an infinite identity, then it would possess the characteristic 'visibly impaled through my right hand at this very moment,' which nothing possesses. Multiple identities ascribed to an existent would either combine to form one identity, or would contradict each other. A contradiction is impossible, because all contradictions can be reduced to claiming that something both exists and does not exist at the same time, which is likewise impossible. To see how this reduction works, take the example "this ball is all red and all green." Implicit to the concept 'red' is 'not green,' and so this example can be rephrased as "this ball is all not green and all green;" the characteristic 'green' is both existing and not existing. Assuming you can get your opponent to agree that something cannot both exist and not exist at the same time, this proves that contradictions are impossible (and that multiple identities are either impossible, or become one identity). So, an entity cannot possess no identity, infinite identity or multiple identities: All entities possess a single, finite identity.
Actions are actions of entities.
Actions don't exist separate from the entities that act. There is no 'walking' separate from the entity that walks. All actions are performed by entities.
Entities must act in accordance with their identities.
This follows from the above point. If actions are actions of entities, then the action must adhere to the single, finite identity of the entity acting. The only alternative is for an entity to act in contradiction to its identity, which is impossible -- contradictions cannot exist. The only possibility is that entities act in accordance with their identity.
This leads to causality. Given any set of circumstances, there is only one course of action available to an existent (well, aside from the choices made by a volitional consciousness, but that's a unique instance of causality and should be tackled separately): The course of action that is in accordance with its identity. If a billiard ball is smacked by another billiard ball, the identities of all of the circumstances involved (forces, motion, the balls themselves) permit only a single course of action for each existent -- the action that is in accordance with its single, finite identity.
At the heart of cause-and-effect is the assertion that any given cause can only have a single given effect -- and that's exactly what is being said above. Multiple possible effects would absolutely necessitate multiple identities, or contradicting identities, which simply cannot be. The opposite, that effects cannot happen without causes, is implicit as well -- a billiard ball with no forces acting on it is has only one action available to it, as with any other cause: Its only possible action is to remain still, which is in accordance with its identity and the identities of everything around it.
Scientific versus Philosophic Proofs
Now, the problem you're experiencing is that these people are linking our scientific understanding of physical causality (mechanical cause-and-effect) to the fundamental absolute of philosophical causality (action applied to identity). The former is based on induction and is a matter of science (and a matter of the philosophical defense of induction); causality is absolutely necessitated, but our understanding of the nature of this causality is limited by the context of our knowledge. The latter, philosophical causality, is rooted in fundamental axioms and is utterly inarguable. Your opponents are not supporting an argument against causality (no one can). They are claiming to do so, while actually formulating an argument against the validity of scientific induction -- and you should treat their argument in that way.
|Axioms: Existence, Consciousness, Identity • Corollary: Causality, Primacy of Existence|
|Reality: Primacy of Existence, Existents, the Metaphysically Given versus the Man Made|