# Has there been any mathematical study of causality?

Causality seems to play an important role in physics. There also seems to be a close parallel between "$$P$$ causes $$Q$$" and "if $$P$$ then $$Q$$." Mathematical logic studies logical inference; has there been any formal mathematical study of causal inference?

• Someone more informed can say more, but I believe there have been attempts to axiomatize the rules of causality. Possibly this is considered philosophy, and not math. Apr 24, 2022 at 13:05
• The Stanford Encyclopedia of Philosophy article on causal models is a good place to start reading about this topic. By the way, note that causality in modern physics is a much more subtle topic than you might think at first. So the superficial similarity between logical inference and causality is not as straightforward as it might seem at first glance. Apr 24, 2022 at 21:52
• For a somewhat more philosophical discussion of the relationship between causality and if/then conditionals, see the article on counterfactual theories of causation. David Lewis tried to analyze causality in terms of logical conditionals, but his proposal was controversial, and in any case, he was using conditionals based on possible world semantics (modal logic) rather than the material conditional of classical mathematical logic. Apr 24, 2022 at 21:59
• I've tried to rewrite the question to be more suitable for MO. If it is reopened, I plan to expand my comments above into an answer. Apr 25, 2022 at 14:36
• Related: mathoverflow.net/q/28224/30186 Apr 25, 2022 at 15:26

Setting aside the alleged parallel between causation and inference for a moment, there has indeed been some mathematical investigation of cause and effect. The Stanford Encyclopedia of Philosophy article on causal models is a good starting point for reading about this topic. One of the theories mentioned there has been described in some detail in semi-popular terms in The Book of Why by Judea Pearl and Dana Mackenzie. This theory lies more in the domain of statistics than mathematics proper, and tries to address that perennial problem that statistical correlations do not in themselves prove causation. Pearl has also written a more technical monograph, Causality: Models, Reasoning, and Inference.

Returning to the word "inference," let us note there are several interpretations of what that word means. In mathematics, the flavors that most commonly arise are the material conditional $$P \Rightarrow Q$$, which is equivalent to $$(\neg P) \vee Q$$, and the provability relation $$T \vdash \phi$$. Though both of these relations might superficially resemble the relation "$$P$$ causes $$Q$$," most people who have thought about the analogy have concluded that causality more closely resembles a different kind of conditional statement, namely a counterfactual conditional. Roughly speaking, "$$P$$ caused $$Q$$" seems akin to the statement, "If $$P$$ had not occurred then $$Q$$ would not have occurred." A counterfactual conditional is a very different beast from the material conditional. In everyday speech, the material conditional rarely comes up, except when someone half-jokingly says something like, "If Chris is a good cook then I'm the king of England!" Conditionals in everyday speech are much more likely to be counterfactual, and nowadays, counterfactuals are usually analyzed using possible world semantics and modal logic. Again the Stanford Encyclopedia of Philosophy has a good article on David Lewis's attempt to analyze causation in terms of counterfactuals. Though the mathematics of modal logic is rigorous, the question of whether Lewis has successfully used it to analyze causality is a philosophical one, and is controversial.

Finally, although you might think that causality plays an essential role in modern physics, the truth is more subtle. The fundamental equations of physics make no explicit mention of causality, and are in fact time-reversible. Intuitively, causality involves an arrow of time, which is a notoriously difficult concept to explain in terms of modern physics. Causality does get mentioned sometimes in physics, e.g., in special relativity, but not in a way that suggests a direct connection with provability or the material conditional.

To summarize, while your intuition might suggest that mathematical logic should be intimately related to causality, closer inspection reveals that the relationship is not as tight as you might have expected.

• Thank you for your answer. Maybe, the study of time in physics is not clear, therefore the study of causality is so controversial. xiomatization of causality has to have axiomatization of time or entropy ahead. having axiomatized time or entropy ahead, I think that cause and effect is able to be expressed as inference(computability) or proposition($P\vee Q$ and$\neg P \vee \neg Q$). Apr 27, 2022 at 0:49
• @XL_At_Here_There I very much doubt that matters are so simple. For example, it is very common for $P\Rightarrow Q$ and $Q\Rightarrow R$ and $R\Rightarrow P$ to all hold simultaneously, but to say that $P$ causes $Q$ and $Q$ causes $R$ and $R$ causes $P$ violates our intuition about causality. Apr 27, 2022 at 1:05
• Axiomatization of time or entropy is not simple, and your example is not hard to explain, if we incorporate the time which is an order and linked to the material world. Apr 27, 2022 at 1:37
• sorry for mistakes, $P \vee Q$ and $\neg P \vee \neg Q$ have to be $P \rightarrow Q$ and $\neg P \rightarrow \neg Q$ Apr 27, 2022 at 13:56
• A definition for causality: causality is defined as, $P\rightarrow Q$ or $\neg P\rightarrow \neg Q$ where $P$ happens before $Q$ ($P \twoheadrightarrow Q$), and $P \twoheadrightarrow Q$ is invariant under every special relativistic transformation. Apr 28, 2022 at 3:52