Here's a use of Leibniz's Law to argue against dualism.
This argument begins by observing that nature is continuous in various ways. Contrast the property of being bald to the property of having a soul:
On the other hand, it is not clear how having a soul could be a vague matter, or a matter of degree. A soul would have to be something such that either one's got it or one doesn't got it. Some souls may be better, or smarter, than others. But still, either you've got a soul or you don't.
According to this argument, though, the continuities in nature make it appear that having a mind is more like being bald, than it is an all-or-nothing affair. There does not seem to be any sharp point in the development of a human embryo where it first acquires the ability to have thoughts and feelings. That is, it is a vague matter when an embryo is first able to think. Similarly, there does not seem to be any sharp point in our evolutionary history where we first acquired the ability to have thoughts and feelings. It is a vague matter which animals are capable of thoughts and feelings. And this is also a matter of degree.
But if the extent to which one has a mind is vague and admits of degrees, whereas having a soul is all-or-nothing, then it's hard to see how having a mind could consist in having a soul.In other words, this argument says:
What do you think of this argument?
One possible response for the dualist would be to insist that it is a perfectly sharp matter when one first acquires a mind, and which animals have minds. But this is hard to believe.
It seems like our experiences and sensations are caused by events in the external world. It also seems like our thoughts and decisions in turn cause certain things to happen in the external world. For instance, they cause our bodies to move in certain ways. The dualists have some difficulty explaining these kinds of causal relations.
Here are three arguments pressing these difficulties.
This is the kind of situation we should expect if Alfred was something apart from his body. Why should hitting Alfred on the head cause him to become unconscious? Why shouldn't it instead produce the following effects: Alfred feels a blow to his head, notices his body falling to the ground and no longer doing what he tells it, then he's left floating in darkness but still conscious and able to muse about his condition.
The body and the mind don't interact as we'd expect them to interact if the mind was something apart from the body, "remotely controlling" the body, as it were. The connection between them seems to be more intimate than that. The dualists owe us an explanation of why that should be so, if the soul is the independent substance they claim it to be.
The dualists have some latitude about what kinds of causal connections they are going to allow between the soul and the body. Here are two possibilities:
The epiphenomenalist asks us to accept: the hurting quality of your pain does not cause you to yell. The itchiness of your back does not cause you to scratch. These claims are hard to believe.
One problem for interactionism comes from the fact that we have good empirical evidence that nothing happens in or to a person's body except what conforms to physical and chemical laws. The physical world seems to be causally closed: that is, for everything that happens in the physical world, there seems to be a wholly physical cause. There is no real scientific evidence that, in order to find a causal explanation of why your body moved in such-and-such ways, we have to appeal to anything other than physical causes.
Now, if they're going to defend the claim that the mental and the physical causally interact, the dualists are going to have to say that there are special non-physical facts about us that play a role in causing our bodies to move in the ways they do. Strictly speaking, this is compatible with the claim that our bodies' movements have a wholly physical cause. For it may be that our bodies' movements are causally overdetermined. That is, our bodies' movements may have two separate causes, each of which is sufficient to make those movements occur. This would be like a case where two sharpshooters simultaneously shot someone, and each bullet was sufficient to cause the victim's death. In such a case, his death would be causally overdetermined.
If interactionism is true, though, then every time the mental facts about you play a role in causing your body's movements, those movements would be causally overdetermined. This would be causal overdetermination on a systematic and wide-scale basis. The interactionist's opponents find it hard to believe that there is all this causal overdetermination going on.
What pre-ordained harmony keeps the mental causes and the physical causes in synch with each other?
What would happen if the mental causes and the physical causes got out of alignment? How would our bodies move then?
Many materialists believe this to be the most decisive objection to dualism.