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Randomized acts play a marginal role in traditional Bayesian decision theory, essentially only one of tie-breaking. Meanwhile, rationales for randomized decisions have been offered in a number of areas, including game theory, experimental design, and machine learning. A common and plausible way of accommodating some (but not all) of these ideas from a Bayesian perspective is by appeal to a decision maker’s bounded computational resources. Making this suggestion both precise and compelling is surprisingly difficult. We propose a distinction between interesting and uninteresting cases where randomization can help a decision maker, with the eventual aim of achieving a unified story about the rational role of randomization. The interesting cases, we claim, all arise from constraints on memory.

UNIVERSITY SEMINAR ON LOGIC, PROBABILITY, AND GAMES
On the Rational Role of Randomization
Thomas Icard (Stanford)

A standard way to challenge convergence-based accounts of inductive success is to claim that they are too weak to constrain inductive inferences in the short run. We respond to such a challenge by answering some questions raised by Juhl (1994). When it comes to predicting limiting relative frequencies in the framework of Reichenbach, we show that speed-optimal convergence—a long-run success condition—induces dynamic coherence in the short run. This is joint work with Eric Wofsey.

Michael Nielsen (Columbia University).
4:10 pm, Friday, November 16th, 2018
Faculty House, Columbia University

What does it mean that an event C “actually caused” event E? The problem of defining actual causation goes beyond mere philosophical speculation.  For example, in many legal arguments, it is precisely what needs to be established in order to determine responsibility.   (What exactly was the actual cause of the car accident or the medical problem?) The philosophy literature has been struggling with the problem of defining causality since the days of Hume, in the 1700s. Many of the definitions have been couched in terms of counterfactuals. (C is a cause of E if, had C not happened, then E would not have happened.) In 2001, Judea Pearl and I introduced a new definition of actual cause, using Pearl’s notion of structural equations to model counterfactuals.  The definition has been revised twice since then, extended to deal with notions like “responsibility” and “blame”, and applied in databases and program verification.  I survey the last 15 years of work here, including joint work with Judea Pearl, Hana Chockler, and Chris Hitchcock. The talk will be completely self-contained.

The seminar is concerned with applying formal methods to fundamental issues, with an emphasis on probabilistic reasoning, decision theory and games. In this context “logic” is broadly interpreted as covering applications that involve formal representations. The topics of interest have been researched within a very broad spectrum of different disciplines, including philosophy (logic and epistemology), statistics, economics, and computer science. The seminar is intended to bring together scholars from different fields of research so as to illuminate problems of common interest from different perspectives. Throughout each academic year, meetings are regularly presented by the members of the seminar and distinguished guest speakers.

details tba

02/08/2019 Faculty House, Columbia University
4:00 PM

03/22/2019 Faculty House, Columbia University
4:00 PM

04/19/2018 Faculty House, Columbia University
4:00 PM

There are two approaches to life. The first one, which we are identifying with Sir Karl Popper, is to think before we act and to let our hypotheses die in our stead when the overall outcome is likely to be negative. We act now for a better future, and we think now which action will bring the best future. Both decision theory and backward induction are technical versions of this train of thought.  The second approach, which we will identify with the Buddha, is to live in the present and not allow the future to pull us away from living in the ever present  Now. The Buddha’s approach is echoed in many others who came after him, Jelaluddin Rumi, Kahlil Gibran, and even perhaps Jesus.  It occurs in many contemporary teachers like Eckhart Tolle and Thich Nhat Hanh.  We may call Popper’s approach “futurism” and the Buddha’s approach “presentism.”

In this talk, we will discuss various aspects of the discourse on presentism and futurism. The purpose is to contrast one with the other. We will not attempt to side with one against the other, and instead leave it as a future project to find a prescriptive action-guiding choice between the two. We merely conjecture that a better optimal choice between these two positions may be somewhere in between. (This is joint work with Jongjin Kim.)

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.

Propositional dynamic logic (PDL) is a framework for reasoning about nondeterministic program executions (or, more generally, nondeterministic actions). In this setting, nondeterminism is taken as a primitive: a program is nondeterministic iff it has multiple possible outcomes. But what is the sense of “possibility” at play here? This talk explores an epistemic interpretation: working in an enriched logical setting, we represent nondeterminism as a relationship between a program and an agent deriving from the agent’s (in)ability to adequately measure the dynamics of the program execution. More precisely, using topology to capture the observational powers of an agent, we define the nondeterministic outcomes of a given program execution to be those outcomes that the agent is unable to rule out in advance. In this framework, determinism coincides exactly with continuity: that is, determinism is continuity in the observation topology. This allows us to embed PDL into (dynamic) topological (subset space) logic, laying the groundwork for a deeper investigation into the epistemology (and topology) of nondeterminism.

The seminar is concerned with applying formal methods to fundamental issues, with an emphasis on probabilistic reasoning, decision theory and games. In this context “logic” is broadly interpreted as covering applications that involve formal representations. The topics of interest have been researched within a very broad spectrum of different disciplines, including philosophy (logic and epistemology), statistics, economics, and computer science. The seminar is intended to bring together scholars from different fields of research so as to illuminate problems of common interest from different perspectives. Throughout each academic year, meetings are regularly presented by the members of the seminar and distinguished guest speakers.

details tba

02/08/2019 Faculty House, Columbia University
4:00 PM

03/29/2019 Faculty House, Columbia University
4:00 PM

04/19/2018 Faculty House, Columbia University
4:00 PM

Convergence to the truth is viewed with some ambivalence in philosophy of science. On the one hand, methods of inquiry that lead to the truth in the limit are prized as marks of scientific rationality. But an agent who, by using some method, expects to always converge to the truth seems to fail a minimum standard of epistemic modesty. This point was recently brought home by Gordon Belot in his critique of Bayesian epistemology. In this paper I will study convergence to the truth theorems within the framework of Edward Nelson’s radically elementary probability theory. This theory provides an enriched conceptual framework for investigating convergence and gives rise to an appropriately modest from of Bayesianism.

The seminar is concerned with applying formal methods to fundamental issues, with an emphasis on probabilistic reasoning, decision theory and games. In this context “logic” is broadly interpreted as covering applications that involve formal representations. The topics of interest have been researched within a very broad spectrum of different disciplines, including philosophy (logic and epistemology), statistics, economics, and computer science. The seminar is intended to bring together scholars from different fields of research so as to illuminate problems of common interest from different perspectives. Throughout each academic year, meetings are regularly presented by the members of the seminar and distinguished guest speakers.

details tba

02/08/2019 Faculty House, Columbia University
4:00 PM

03/22/2019 Faculty House, Columbia University
4:00 PM

04/19/2018 Faculty House, Columbia University
4:00 PM