Monotonicity and Necessary Connections

Monotonicity is a logical feature of propositions, and thus of facts (which are just true propositions). Monotonicity says :

Let’s say we have the  proposition If P, then Q. If this is so, then given P, Q must be true (this move is often called modus ponens). Next, given any conjunction that has P as a conjunct, Q must be true. This is because, given P and some other fact, R let’s say, well P is still true. This feature of P to entail Q even when conjoined with any other proposition, given if P, then Q is called monotonicity. The entailment of Q from P in this case is said to be monotonic.

Now, monotonicity is a feature of any proposition in the form of the material conditional (e.g. If P, then Q). In terms of necessity more broadly speaking, another way to put monotonicity would be to say that if Q is necessary from P (where its necessity from P from a material conditional is a special case), then Q will be the case given P, no matter what else is so (e.g. R).

A black hole whose event horizon is one light-year in diameter will necessarily pull toward its center anything that crosses it. If this is so, then this will be true even given any other fact, even the fact that there is a stronger black hole whose event horizon is 10 light-years in diameter that just crossed the first black hole and an object that just crossed it. This illustration may cast into doubt the original claim that the first black hole necessarily pulls anything toward its center anything that crosses it, since a more powerful black hole just pulled it in toward itself. With this illustration, a pressing question is: given the violation of monotonicity here, are necessary connections in causal relations to be doubted, given that anything stronger could always jump into a causal relation and prevent the expected effect?

There’s a quick way I will answer in the negative and then a slightly longer way. The quick No involves noting how the illustration I provided above works. That illustration works, I would press, because the bigger black hole that swoops onto the scene is readily seen to necessarily lead to the defeat the expected effect. Such an interpretation shows that the original proposition that expressed necessity was incorrect because monotonicity, and with it necessity, is readily violated. The longer No involves being more specific about the original proposition.

The proposition, again, was: A black hole whose event horizon is one light-year in diameter will necessarily pull anything toward its center anything that crosses it.

Such a black hole may necessarily do such a thing, within a certain context. In my previous post, I said that such a context can be fixed in such a way that introducing some other object to the causal context that would defeat the expected outcome would contradict the context, and thus be barred from introduction to it. Thus we may modify the original proposition: a black hole whose event horizon is one light-year in diameter, in the context of being within the vacuum of space and objects of the size of Earth or smaller and of the density of Earth or less for 100 light-years or greater would necessarily pull anything toward its center that crosses it.

Such a statement is, I think, true. Is it monotonic? Sure: let’s add any number of facts for things greater than 100 light-years away from such a black hole. The result that anything that crosses it is pulled toward its center would seem to be preserved.

Can the result be defeated for the monotonic addition of a fact? I think not. Adding a larger black hole to the scene would contradict the context I specified. However, we may of course be very theoretical with our facts. We may posit that there could (in a loose, logical sense) be some black hole crusher that affects black holes instantaneously at a distance well outside the 100 light-year bubble I’ve placed the black hole in. However, I can counter by simply expanding the bubble of the black hole to encompass the entire universe. Now there is no hope for the appearance of a fact that both upholds the context and defeats the expected effect.

Furthermore, back to the short answer, such a black hole crusher intuitively works well to defeat the expected effect precisely because it is thought to necessitate the defeat of that effect. If we didn’t think so, then we might question whether or  not it did defeat the expected effect this time around whenever a causal defeater is brought up. We don’t tend to ever doubt the power of an introduced causal defeater to bring about its swift determinations. That’s their purpose. This highlights once more that causal thinking exhibits necessary connections. More than this, if I’ve been successful in my last two posts, causal relations themselves also exhibit necessary connections. At least, there’s no reason to think they should be denied for the possibility of causal defeaters, or for the fact that monotonicity must be preserved if necessity is preserved. Necessary connections in causal relations is still viable given these considerations.

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