中南大学哲学系
Central South University
Some Applications of Lawvere's Fixpoint Theorem
Frontiers of Philosophy in China, 2019, 14 (3): 490–510.
The famous diagonal argument plays a prominent role in set theory as well as in the proof of undecidability results in computability theory and incompleteness results in metamathematics. Lawvere (1969) brings to light the common schema among them through a pretty neat fixpoint theorem which generalizes the diagonal argument behind Cantor's theorem and and characterizes self-reference explicitly in category theory. Not until Yanofsky (2003) rephrases Lawvere's fixpoint theorem using sets and functions, Lawvere's work has been overlooked by logicians. This paper will continue Yanofsky's work, and show more applications of Lawvere's fixpoint theorem to demonstrate the ubiquity of the theorem. For example, this paper will use it to construct uncomputable real number, unnameable real number, partial recursive but not potentially recursive function, Berry paradox, and fast growing Busy Beaver function. Many interesting lambda fixpoint combinators can also be fitted into this schema. Both Curry's $Y$ combinator and Turing's $\Theta$ combinator follow from Lawvere's theorem, as well as their call-by-value versions. At last, it can be shown that the lambda calculus version of the fixpoint lemma also fits Lawvere's schema.
@article{lixi19,
author = {Xi Li},
doi = {10.3868/s030-008-019-0029-6},
eid = {490},
journal = {Frontiers of Philosophy in China},
keywords = {paradox;fixpoint;diagonalization;combinator},
number = {3},
numpages = {20},
pages = {490--510},
publisher = {Front. Philos. China},
title = {Some Applications of Lawvere's Fixpoint Theorem},
url = {http://journal.hep.com.cn/fpc/EN/10.3868/s030-008-019-0029-6},
volume = {14},
year = {2019},
bdsk-url-1 = {http://journal.hep.com.cn/fpc/EN/10.3868/s030-008-019-0029-6},
bdsk-url-2 = {https://doi.org/10.3868/s030-008-019-0029-6}
}
Why Inductive Logic Needs Solomonoff Prior?
Studies in Logic, 2014, 7 (4): 48–68.
In 1950s, Carnap develops inductive logic to express the degree of confirmation of some hypothesis relative to some evidence. In 1960s, Somlomonoff invents the universal induction method to make prediction. In this paper we integrate the two methods to extent universal induction's expressive power and to enhance inductive logic's predictive power. We introduce Solomonoff prior into inductive logic, and prove the monadic first order logic version of Solomonoff completeness theorem. Then we make some comparison of the two quite different induction methods in the framework of the modified "inductive logic". As is well known, Carnap's $\lambda$-continuum fails to confirm universal generalizations. However, in the modified "inductive logic", the proposition "all ravens are black" can be confirmed in any computable world as long as all ravens are really black in that world. If we want to prove the completeness theorem by the method of Solomonoff's universal induction, we have to keep the complete information of the past history in memory. In the modified "inductive logic", we can neglect all of the irrelevant information to concentrate only on some specific pattern, and prove similar convergence results. Even without complete record of relevant information of the specific pattern, we can still build our belief through random sampling.
@article{lixi2014en,
author = {Xi Li},
title = {Why Inductive Logic Needs Solomonoff Prior?},
journal = {Studies in Logic},
volume = {7},
number = {04},
pages = {48-68},
year = {2014},
issn = {1674-3202},
doi = {CNKI:SUN:LJXJ.0.2014-04-005}
}
自我升级智能体的逻辑与认知问题
中国社会科学, 2019, 12: 46–61.
自我升级智能体的建立使人们对自我意识的研究有了一个程序化的标准, 借助这种形式化的方法有可能弥合学界关于机器意识的分歧, 破解机器意识研究面临的困局. 但它也有逻辑上的局限. 生成主义为自我升级智能体的提出奠定了认知基础. 自我升级智能体的成功为生成主义提供了一个强有力的例证. 尽管自我升级智能体向机器真正具有自我意识前进了一大步, 但是人们只能说它具有了"功能意识". 造成机器意识困局的症结源自分析哲学传统与现象学传统的分歧和偏颇. 解决的出路在于: 从对立到相容, 从互斥到互补, 进而达到融通的新境界.
@article{lixi-renxiaoming2019,
author = {任晓明 and 李熙},
title = {自我升级智能体的逻辑与认知问题},
journal = {中国社会科学},
number = {12},
pages = {46-61},
year = {2019},
issn = {1002-4921},
doi = {CNKI:SUN:ZSHK.0.2019-12-003}
}
卡尔纳普式的归纳逻辑的局限与所罗门诺夫先验的优势
自然辩证法研究, 2018, 34 (12): 23–28.
20世纪50年代, 卡尔纳普发展了归纳逻辑来表示证据相对于假设的"确证度". 随后,利斯塔、古德-图灵等人提出了各种平滑方法, 这些平滑方法可以看作广义的卡尔纳普式的归纳逻辑. 这些方法虽然都可以从某个层次的"无差别原则"导出, 但这并不能构成其理论基础. "无差别原则"无论作用在这里的哪一层都不合适, 根据机器学习领域的无免费午餐定理, 都不具有通用性, 只有作用在可能世界的产生方式这一层次上导出的所罗门诺夫先验才具有通用性,能够逼近任何可计算的模式. 而且, 不仅如此, 在同时满足奥卡姆剃刀原则和最大熵原则的意义上, 所罗门诺夫先验具有最优性.
@article{lixi2018,
author = {李熙},
title = {卡尔纳普式的归纳逻辑的局限与所罗门诺夫先验的优势},
journal = {自然辩证法研究},
volume = {34},
number = {12},
pages = {23-28},
year = {2018},
issn = {1000-8934},
doi = {10.19484/j.cnki.1000-8934.2018.12.005}
}
通用智能框架下的纽康姆难题
逻辑学研究, 2019, 12 (04): 52–63.
面对纽康姆难题, 主流的决策理论出现了分歧, 采用期望效用最大化原则和占优原则分别会导向两种不同的选择, 这对决策理论构成了一个挑战. 通过合适的概率计算, 占优原则可以看作一种极端的期望效用最大化原则, 所以决策理论的基础——追求效用最大化并不与占优原则冲突, 问题是——如何计算期望效用. 纽康姆难题背后的关键也是如何通过概率把握因果, 其核心是归纳预测问题. 通用人工智能领域的通用归纳、通用智能模型刻画的恰恰是因果预测问题, 本文认为, 为了解决纽康姆难题发展各种新奇的决策理论是不必要的, 而借助通用智能模型AIXI探讨纽康姆难题中涉及到的因果性问题.
@article{lixi-newcomb2019,
author = {李熙},
title = {通用智能框架下的纽康姆难题},
journal = {逻辑学研究},
volume = {12},
number = {04},
pages = {52-63},
year = {2019},
issn = {1674-3202},
doi = {CNKI:SUN:LJXJ.0.2019-04-004}
}
莱布尼茨、计算主义与两个哲学难题
科学技术哲学研究, 2019, 36 (06): 26–31.
莱布尼茨的单子是无形的自动机, 如果把它看作抽象的图灵机, 莱布尼茨单子论式的可能世界就是当下计算主义关心的数字宇宙. 在这种抽象的"单子"宇宙中, 数学的可应用性"谜题"以"不足道"的方式解决, 但另一个哲学难题——世界的可知性"谜题"却凸显出来. 借助算法信息论的话语, 可以精确界定几类"可知的"可能世界, 继而可以严格的探讨它们之间的相互关系. 至于具体如何认识的问题, 所罗门诺夫的通用归纳模型直接体现了方法论自然主义"假设——验证"的试错过程和诸如"简单性"的先验原则, 抛开它不可计算的缺陷不谈, 借助它, 任何可计算的可能世界都是可以被近似认识的.
@article{lixi-leibniz2019,
author = {李熙},
title = {莱布尼茨、计算主义与两个哲学难题},
journal = {科学技术哲学研究},
volume = {36},
number = {06},
pages = {26-31},
year = {2019},
issn = {1674-7062},
doi = {CNKI:SUN:KXBZ.0.2019-06-005}
}
莱布尼茨哲学的一种现代阐释及其对通用人工智能的启示
科学技术哲学研究, 2020, 37 (04): 27–32.
莱布尼茨的形而上学的核心是单子论,单子是无形的自动机. 如果把单子解释成抽象的图灵机,甚至可以形式化地研究它们的决策行为. 根据莱布尼茨的通用文字、理性演算, 可以构造一种能够进行信念更新的概率归纳逻辑, 其中的关键是如何给出合理的"先验信念", 而"先验信念"的赋予方式可借助莱布尼茨的形而上学, 所罗门诺夫的通用归纳模型可以看作这种思想的一种体现. 但这是一种"单子"把自身孤立出世界之外的超然的"第三人称"视角, 如果把"单子"置身于"可共存的单子集团"中, 那么可以对单子论作出一种博弈论的解释, 这与莱布尼茨关于游戏的哲学一致, 胡特尔的通用智能模型AIXI可看作这种思想的反映. 单子所角逐的效用不是别的, 恰是莱布尼茨的"完满性". 在这种阐释下, 对莱布尼茨"前定和谐"的合理刻画可以为通用智能主体进行理性决策的"先验信念"的赋予方式提供有益的启示.
@article{lixi-leibniz2020,
author = {李熙},
title = {莱布尼茨哲学的一种现代阐释及其对通用人工智能的启示},
journal = {科学技术哲学研究},
volume = {37},
number = {04},
pages = {27-32},
year = {2020},
issn = {1674-7062},
doi = {CNKI:SUN:KXBZ.0.2020-04-005}
}
主持的国家社科基金项目: