It's often said that causal connections aren't logically necessary – not even necessary causal connections. This is the central gist of Hume’s position and from which he derived so many of his arguments about causation as a whole. Popper, on the other, did think that causal connections are logically necessary... but not so quick! They're only logically necessary
‘in the sense that they follow deductively once we assert the appropriate natural law’. (11)
If we don't assert the appropriate natural law, or any natural law, then we can't say that they are logically necessary. This is a classic case of the principle that no matter if a logical axiom or premise is true (that is, even if it is false), what follows from it will still follow deductively and validly from it (providing one’s inferences are valid).
Did Popper only mean ‘logically necessary’ in this amended sense? Could he have used the words ‘logically necessary’ in any other way? That is, any logical necessity that causal connections have are simply inherited from the natural laws from which they are derived or deductively inferred. We can't expect anything more about causal logical necessity than this.
Now we can express an example of a scientific law that is expressed in causal terms:
An event of type C has occurred. But whenever an event of type C occurs, an event of type E later occurs.
Again, this will only happen of necessity if we also assert the appropriate natural law relating cause C to effect E. Without that natural law (or its assertion/assumption), an event of type C could be followed by an event of type F, or by anything for that matter!
However, the Humean can still express his problems with Popper’s position. Dale Jacquette writes:
"A defender of Hume on the contingency of causal connections might nevertheless object that although the inference is deductively valid, and to that extent carries necessity from assumptions to conclusion, the conclusion itself is not necessary unless the assumptions are also logically necessary, and that no scientific laws correlating causes to effects are logically necessary." (11)
This means that it doesn't matter if the move from the assumptions to the conclusion is deductively valid if the assumptions themselves aren't logically necessary (they may not even be true to bring about deductively validity). After all, causal matters are about the world. They have nothing to do with deductively validity or deductive inference.
A Humean would argue that although the inference from the assumptions to the conclusion is logically necessary (in that if the assumptions are taken to be true this truth is passed on to the conclusion), the assumptions themselves aren't logically necessary.
Does this mean that the passage from event type C to event type E isn't logically necessary? Or that event type C, taken by itself, isn't logically necessary?
If event type C isn't logically necessary, then how can it pass on logical necessity to event type E?
So now we can say that not even the move from event type C to event type E is logically necessary if event type C isn't itself logically necessary.
Again, we aren't talking about formal or subject-less deductively validity or inference here, but an aspect of the world – causation!
To put the conclusion at its most basic. Hume argued that no scientific laws correlating causes to effects are logically necessary. That is the gist of Hume’s argument. Not that there is no such thing as causation or causal connection. Not even that there are no such things as causal regularities. Of course there are! No. His point is simple. There are no scientific laws correlating causes to effects. A scientific law is required to be both universal and exceptionless. On Hume’s empiricist grounds, we have no way of observing the universal or truly knowing that something is indeed exceptionless. It follows that no causal relation - say between event type C and event type E - can be deemed to be universal and exceptionless. Thus it can't instantiate or fall under a genuine natural scientific law. That is Hume’s point against scientific and rationalist views of necessary causal relations.
Jacquette puts much of the above in the following way:
"Again, it might be questioned whether the logical necessity obtaining between the assumptions and conclusion of a valid inference about real world events necessarily qualifies or attaches to the events themselves… " (11)
What we have here are two things:
i) The logical necessity obtaining between the assumptions and conclusion of a valid inference about real world events.
ii) The logical necessity obtaining between events in the real world or between event type C and event type E.
Clearly they aren't the same thing.
We can even say that i) is analogous to a de dicto modality, whereas ii) is analogous to a de re modality. One kind of necessity is about the form of an inference from assumptions to a conclusion. The other kind of necessity is said to obtain between events in the actual world. There is a world of difference between the two. The latter would be metaphysical or ontological necessity, whereas the former would be strict logical necessity. It appears, then, that Popper attempted to fuse logical necessity with metaphysical necessity.