本文探讨了逻辑反事实在逻辑决策理论中的重要作用，核心在于理解当假设一个错误命题为真时，会对其他命题的真值产生怎样的影响。文章通过费马大定理的例子，探讨了在假设其为假的情况下，如2+2=4等基本数学真理是否依然成立，并进一步讨论了逻辑反事实如何处理“可能存在但未明确”的反例。此外，文章还提出了在囚徒困境中，逻辑决策理论代理如何进行反事实推理，以及如何处理与自身相似但不完全相同的代理人，引发了对逻辑反事实在复杂情境中应用的思考。

🤔 逻辑反事实的核心在于假设一个错误命题为真，并考察这对其他命题真值的影响，例如假设费马大定理为假，探讨其他数学真理是否依然成立。

🧐 在处理反例时，逻辑反事实可能不会给出一个明确的反例，而是将概率分布在多种可能性上，甚至可能涉及非标准数反例，这增加了计算的复杂性。

🤝 在囚徒困境中，逻辑反事实的构建面临挑战：是仅考虑自身背叛，还是考虑所有逻辑决策理论代理人的背叛，甚至是更广泛的历史情境，这直接影响了决策的制定。

Published on December 15, 2024 9:16 PM GMT

Logical counterfactuals are when you say something like "Suppose $\pi =3$ , what would that imply?"

They play an important role in logical decision theory.

Suppose you take a false proposition $P$ and then take a logical counterfactual in which $P$ is true. I am imagining this counterfactual as a function $f\left(\text{logical propositions}\right)\to [0,1]\subset R$ that sends counterfactually true statements to 1 and false statements to 0.

Suppose $P$ is " not Fermat's last theorem".  In the counterfactual where Fermat's last theorem is false, I would still expect 2+2=4. Perhaps not with measure 1, but close. So $f("2+2=4")>0.9$

On the other hand, I would expect trivial rephrasing of Fermat's last theorem to be false, or at least mostly false.  

But does this counterfactual produce a specific counter example? Does it think that ${14}^3+{26}^3={29}^3$? Or does it do something where the counterfactual insists a counter-example exists, but spreads probability over many possible counter-examples. Or does it act as if there is a non-standard number counterexample?

How would I compute the value of $f$ in general? 

 

Suppose you are a LDT agent trying to work out whether to cooperate or defect in a prisoners dilemma.

What does the defect counterfactual look like? Is it basically the same as reality except you in particular defect. (So exact clones of you defect, and any agent that knows your exact source-code and is running detailed simulations will defect.)

Or is it broader than that, is this a counterfactual world in which all LDT agents defect in prisoners dilemma situations in general. Is this a counterfactual world in which a bunch of homo-erectus defected on each other, and then all went extinct, leaving a world without humans? 

All of the thought about logical counterfactuals I have seen so far is on toy problems that divide the world into Exact-simulations-of-you and Totally-different-from-you. 

I can't see any clear idea about what to do with the vaguely similar but not identical to you agents. 

Discuss