Rationalizability, originally proposed by Bernheim and Pearce, generalizes the notion of Nash equilibrium. Nash equilibrium requires common knowledge of strategies. Rationalizability only requires common knowledge of rationality. However, their original notion assumes that the payoffs are common knowledge. I.e. agents do know what world they are in, but may be ignorant of what other agents are playing.
We generalize the original notion of rationalizability to consider situations where agents do not know what world they are in, or where some know but others do not know. Agents who know something about the world can take advantage of their superior knowledge. It may also happen that both Ann and Bob know about the world but Ann does not know that Bob knows. How might they act?
We will show how a notion of rationalizability in the context of partial knowledge, represented by a Kripke structure, can be developed.
Gödel’s disjunction asserts that either “the mind cannot be mechanized” or “there are absolutely undecidable statements.” Arguments are examined for and against each disjunct in the context of precise frameworks governing the notions of absolute provability and truth. The focus is on Penrose’s new argument, which interestingly involves type-free truth. In order to reconstruct Penrose’s argument, a system, DKT, is devised for absolute provability and type-free truth. It turns out that in this setting there are actually two versions of the disjunction and its disjuncts. The first, fully general versions end up being (provably) indeterminate. The second, restricted versions end up being (provably) determinate, and so, in this case there is at least an initial prospect of success. However, in this case it will be seen that although the disjunction itself is provable, neither disjunct is provable nor refutable in the framework.
UNIVERSITY SEMINAR ON LOGIC, PROBABILITY, AND GAMES
Gödel’s Disjunction
Peter Koellner (Harvard University)
5:00 pm, Friday, April 7th, 2017
716 Philosophy Hall, Columbia University
In the first chapter of his Prasannapadā, Candrakīrti famously defended Buddhapālita against Bhāviveka’s criticism that he had failed to formulate Nāgārjuna’s critique of causality in terms of probative arguments, but rather left the arguments in the form of reductios. This debate is well known to be the starting point of the “Prāsaṅgika-Madhyamaka” interpretation in Tibet. Indeed, Tsongkhapa (1357-1419) has said that “an autonomous probative argument is not suitable to generate the view of thusness in an opponent” in the context of explaining Candrakīrti’s Prāsaṅgika (dgongs pa rab gsal, 226). In his critique of Tsongkhapa’s synthesis of Prāsaṅgika-Madhyamaka and pramāṇa, the fifteenth-century Sakya scholar, Daktsang (stag tsang), accused Tsongkhapa of “eighteen great contradictions,” including a contradiction that “the presence of inference contradicts the absence of probative arguments.” That is, Daktsang argued that a robust notion of inference – that is, inference qua pramāṇa – is antithetical to the logic behind Candrakīrti’s denial of probative arguments in this context. This paper discusses some of the issues driving this debate and shows how this debate sheds light on the place of epistemology in an anti-realist interpretation of Madhyamaka.
Please save the following dates for our upcoming talks:
Oct. 6: Jake Davis (New York University)
Nov. 3: Daniel Breyer (Illinois State University)
Dec. 8: Nico Silins (Cornell University) and Susanna Siegel (Harvard University)
I’ve argued that the Indian Buddhist tradition, broadly construed, has tended to endorse a unique view of freedom and responsibility, a view I’ve called Buddhist Perspectivalism. According to this view, we should always regard ourselves as genuinely free and responsible agents, because we have good reason to do so, while we should never regard others in this way, because we have equally good reason to see them as neither free nor responsible. In this talk, I clarify Buddhist Perspectivalism as a theory of moral responsibility and defend it against some concerns that scholars like Christopher Gowans and Charles Goodman have raised.
With a response from:
Rick Repetti (Kingsborough Community College, CUNY)
Columbia Society for Comparative Philosophy:
Oct. 6: Jake Davis (New York University)
Nov. 3: Daniel Breyer (Illinois State University)
Dec. 8: Nico Silins (Cornell University) and Susanna Siegel (Harvard University)
The umwelt is a notion invented by the Baltic-German biologist Jakob von Uexküll. It represents how a creature, an animal, a child or even an adult “sees” the world and is a precursor to the Wumpus world in contemporary AI literature. A fly is caught in a spider’s web because its vision is too coarse to see the fine threads of the web. Thus though the web is part of the world, it is not a part of the fly’s umwelt. Similarly a tick will suck not only on blood but also on any warm liquid covered by a membrane. In the tick’s umwelt, the blood and the warm liquid are “the same”.
We represent an umwelt as a homomorphic image of the real world in which the creature, whatever it might be, has some perceptions, some powers, and some preferences (utilities for convenience). Thus we can calculate the average utility of an umwelt and also the utilities of two creatures combining their umwelts into a symbiosis.
A creature may also have a “theory” which is a map from sets of atomic sentences to sets of atomic sentences. Atomic sentences which are observed may allow the creature to infer other atomic sentences not observed. This weak but useful notion of theory bypasses some of Davidson’s objections to animals having beliefs.
Russell, Stuart J., and Peter Norvig. “Artificial intelligence: a modern approach (International Edition).” (2002).
Von Uexküll, J., von Uexküll, M., & O’Neil, J. D. (2010). A foray into the worlds of animals and humans: With a theory of meaning. U of Minnesota Press.
The Pratyabhijñā Śaiva tradition, as first systematized by Utpaladeva (10th century) and elaborated by Abhinavagupta (10th-11th century), follows a number of other classical Indian philosophical traditions in 1) fully acknowledging that Buddhist Vijñānavādins claim to account for the diversity manifest in the conventional world through an appeal to beginningless karmic imprints; and 2) utterly rejecting that this solution avoids circularity. To sum up the Śaiva critique: a Vijñānavādin cannot avoid the question of what causes the diversity of experiences in the conventional world by appealing to beginningless causal processes because these processes themselves require the existence of some kind of real stuff that has the capacity to manifest in diverse forms. These Śaivas hone their argument in relation to a Dharmakīrtian view of ultimate consciousness as utterly beyond causal relations—a view that Dharmakīrti (7th century) uses to brush aside any questions about the real relationship between conventional and ultimate reality as incoherent. These Śaivas argue that the question of how the variegation of a specific moment of awareness arises if no part of this variegation—including the variegation of the causes that produce it—is inherent to what is ultimately real is philosophically salient. Moreover, this question cannot be addressed simply by an appeal to beginningless ignorance. While an appeal to beginningless karmic imprints is perfectly sufficient to account for the differences between various karmic streams within the conventional world, it is not sufficient to account for the mere fact that there is differentiated stuff capable of entering into causal relations. The Pratyabhijñā Śaivas offer a complex and distinctive solution to this problem: while they affirm that ultimate reality is beginningless in the sense that it is beyond time, they also claim that time itself has a “beginning” in the expression of the nondual differentiation inherent to the ultimate itself. They further link the expression of time with the creation of the subject/object pairs that define conventional worlds—and use Dharmakīrti’s own apoha (exclusion) theory of concept formation to explain how this happens.
THE COLUMBIA SOCIETY FOR COMPARATIVE PHILOSOPHY
Welcomes:
Catherine Prueitt (George Mason University)
With a response from:
Andrew Nicholson (Stony Brook University, SUNY)
Please save the following dates for our upcoming talks:
March 30: Kin Cheung (Moravian College)
April 13: Lara Braitstein (McGill University)
May 11: David Cummiskey (Bates College)
We examine general decision problems with loss functions that are bounded below. We allow the loss function to assume the value ∞. No other assumptions are made about the action space, the types of data available, the types of non-randomized decision rules allowed, or the parameter space. By allowing prior distributions and the randomizations in randomized rules to be finitely-additive, we find very general complete class and minimax theorems. Specifically, under the sole assumption that the loss function is bounded below, every decision problem has a minimal complete class and all admissible rules are Bayes rules. Also, every decision problem has a minimax rule and a least-favorable distribution and every minimax rule is Bayes with respect to the least-favorable distribution. Some special care is required to deal properly with infinite-valued risk functions and integrals taking infinite values. This talk will focus on some examples and the major differences between finitely-additive and countably-additive decision theory. This is joint work with Teddy Seidenfeld, Jay Kadane, and Rafael Stern.
UNIVERSITY SEMINAR ON LOGIC, PROBABILITY, AND GAMES
Finitely-Additive Decision Theory
Mark Schervish (Carnegie Mellon)
4:10 pm, Friday, February 16th, 2018
Faculty House, Columbia University
The circa 9th century CE poet-saint Saraha enjoys a close association with spontaneity in both his reported actions and recorded works. This association leads him to be popularly read and remembered as a freewheeling antinomian sacred figure opposed to institutions, rituals, and even social norms. His appears to be a call to joyful chaos. But many of his verses invite readers towards a different kind of chaos, towards mental/perceptual chaos as the path towards correct conventional perception.
What does it mean to correctly perceive an object? Many Buddhist sources describe how perception functions, and theorize the differences between correct and incorrect perception. A related important distinction is made between conventional and ultimate truth in the discussion of the reality of phenomena even when correctly perceived. But this arguably epistemic distinction may also be understood as the difference between an ordinary person’s correct perception and a Buddha’s perception. I am not here exploring ultimate truth. I am interested in conventional truth, in what makes it true. Broadly, correct conventional perception is associated with the product of a rational mind processing sense perceptions fed to it by functioning sense organs, and conventional truth then is the experience of that reality. Correct conventional perception can be contrasted with incorrect conventional perception, which would be perception based on an irrational or deluded mind, or an experience based on damaged or non-functioning sense organs.
Saraha, however, sings a different tune. His work tells us that the very process of identifying and recognizing objects – what most of us would associate with the basic skills necessary to get around in the world – is itself deluded, its objects adventitious. He calls this kind of perception drenpa (dran pa; usually translated as mindfulness, memory, or recollection). In other words, he claims that our very perception of objects – no matter how carefully or clearly experienced – is evidence of our being deluded. In contrast, correct conventional perception is the undoing of that object-making, what he refers to as drenmé (dran med). Drenmé is an uncommon term the contexts of the two truths and meditation. Usually denoting a swoon or a coma, here it refers to a reversal or undoing of drenpa. What does it mean – if it means anything – to describe our perception of objects itself as evidence of our delusion? Can perception as object-making ever produce truth?
The Columbia Society for Comparative Philosophy (CSCP) is a University Seminar dedicated to the advancement of projects that draw on both western and non-western philosophy. The CSCP meets monthly on the campus of Columbia University and occasionally hosts conferences.
Please save the following dates for our upcoming talks:
March 30: Kin Cheung (Moravian College)
April 13: Lara Braitstein (McGill University)
May 11: David Cummiskey (Bates College)
At the core of Kantian liberalism is a conception of the independent autonomous subject. On the other hand, the most central and distinguishing feature of Buddhist philosophy is the doctrine of no-self. It thus seems that Buddhists should reject Kantian liberalism. My larger project develops the connections between Buddhist perfectionism, liberalism, and principles of justice. In this paper, I focus on Buddhist and Kantian conceptions of self-constitution, but my ultimate concern is the significance of the doctrine of no-self to theories of justice.
Buddhists need some conception of a minimal self to account for the karmic-continuity of persons and also to provide an adequate account of the subjectivity of experience. I argue that we should reject the (Abhidharma) reductionist view of the self as a mere fiction that is reducible to its simpler and more basic parts. As is often noted, the Buddhist reductionist approach is similar to Derek Parfit’s view. Parfit also argues that there is no deep metaphysical self and that relations of personal identity are reducible to relations of psychological connectedness and causal continuity in a series of experiences. Christine Korsgaard has responded to Parfit’s reductionist view by developing a non-metaphysical account of Kantian agency and self-constitution. I argue that the Buddhist doctrine of no-self is consistent with a more minimal, non-substantial, emergent, view of the self. This approach, which is more fully developed by Evan Thomson, Matthew MacKenzie, Georges Dreyfus, and others, is surprisingly similar to Korsgaard’s practical conception of the self. As a result, the non-reductionist Buddhist approach is also not vulnerable to Korsgaard’s objection to reductionist views. In addition, I argue that the process of self-constitution is embedded in a recursive nexus of dependent origination, and reject Korsgaard’s conception of the independent autonomous subject, which she refers to as “over and above” its ends. In short, a Buddhist can accept Korsgaard’s basic account of self-constitution but nonetheless reject the Kantian idea of the independent autonomous subject. For Buddhists, the Kantian autonomous subject is instead part of the “primal confusion” that projects a reified subject-other division on experience. This confusion is the source of existential suffering, anxiety and stress, which characterizes too much of the human condition. The goal is to transcend the Kantian subject and internalize the pervasive interdependence of persons. Instead of the autonomous self, Buddhism embraces a perfectionist ideal, of a non-egocentric reorientation and re-constitution of the self.
Buddhists thus have reason to reject Kantian liberalism, if it is based on the autonomy and independence of persons. In his shift to Political Liberalism, John Rawls recasts the conception of the person, as “a self-originating source of valid claims,” and emphasizes that this conception is restricted to the political domain. It is part of a narrow conception of the “moral powers” of a free and equal citizen; it is not a metaphysical conception or comprehensive ideal. I conclude by exploring the contrast between Buddhist Perfectionism and Political Liberalism.
With a Response From:
Carol Rovane (Columbia University)
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Also, please visit our website:
http://www.cbs.columbia.edu/cscp/
Co-Chairs
Professor Jonathan Gold
Associate Professor, Princeton University, Department of Religion
Professor Hagop Sarkissian
Associate Professor, The City University of New York, Baruch College | Graduate Center, Department of Philosophy
hagop.sarkissian@baruch.cuny.edu
Rapporteur
Jay Ramesh
Two proof-systems P and P* are said to be complementary when one proves exactly the non-theorems of the other. Complementary systems come as a particular kind of refutation calculi whose patterns of inference always work by inferring unprovable conclusions form unprovable premises. In the first part of my talk, I will focus on LK*, the sequent system complementing Gentzen’s system LK for classical logic. I will show, then, how to enrich LK* with two admissible (unary) cut rules, which allow for a simple and efficient cut-elimination algorithm. In particular, two facts will be highlighted: 1) for any given provable sequent, complementary cut-elimination always returns one of its simplest proofs, and 2) provable LK* sequents turn out to be “deductively polarized” by the empty sequent. In the second part, I will observe how an alternative complementary sequent system can be obtained by slightly modifying the Gentzen-Schütte system G3. I will finally show how this move could pave the way for a novel approach to multi-valuedness and proof-theoretic semantics for classical logic.