logic, mathematics, and philosophy

### Dogramaci: The Varieties of Validity Worth Wanting

COLUMBIA PHILOSOPHY
The Varieties of Validity Worth Wanting
Sinan Dogramaci  (Texas)
Thursday, April 11th, 4:10 PM
716 Philosophy Hall, Columbia University

Abstract. I ask whether the validity of a Modus Ponens inference is any part of the explanation of why the inference is, in any sense, a good or valuable inference. I argue for the following. On the orthodox understanding of validity, a semantic understanding, validity has no explanatory relevance to the reasoning’s value. On a metaphysical understanding of validity, one definable in terms of possible worlds, validity is relevant to a partial, but crucially incomplete, explanation of the reasoning’s value. The complete explanation of the reasoning’s value must also appeal to a substitutional understanding of validity, a notion once advocated by Quine.

### Hájek: Staying Regular?

COLUMBIA FORMAL PHILOSOPHY WORKSHOP
Staying Regular?
Alan Hájek (Australian National University)
Thursday, April 4, 2013, 4:00 – 5:30 PM
716 Philosophy Hall, Columbia University

Abstract.
‘Regularity’ conditions provide nice bridges between the various ‘box’/‘diamond’ modalities and various notions of probability. Schematically, they have the form:

If X is possible, then the probability of X is positive

(or equivalents). Of special interest are the conditions we get when ‘possible’ is understood doxastically (i.e. in terms of binary belief), and ‘probability’ is understood subjectively (i.e. in terms of degrees of belief). I characterize these senses of ‘regularity’—one for each agent—in terms of a certain internal harmony of the agent’s probability space  <Ω, F, P>. I distinguish three grades of probabilistic involvement. A set of possibilities may be recognized by such a probability space by being a subset of Ω; by being an element of F; and by receiving positive probability from P. These are non-decreasingly committal ways in which the agent may countenance a proposition. An agent’s space is regular if these three grades collapse into one.

I briefly review several of the main arguments for regularity as a rationality norm, due especially to Lewis and Skyrms. There are two ways an agent could violate this norm: by assigning probability zero to some doxastic possibility, and by not assigning probability at all to some doxastic possibility (a probability gap). Authors such as Williamson have argued for the rationality of the former kind of violation, and I give an argument of my own. So I think that the second and third grades of probabilistic involvement may come apart for a rational agent. I then argue for the latter kind of violation: the first and second grades may also come apart for such an agent.

Both kinds of violations of regularity have serious consequences for traditional Bayesian epistemology. I consider especially their ramifications for:

• conditional probability
• conditionalization
• probabilistic independence
• decision theory

### Butterfield: Morningside Story

COLUMBIA PHILOSOPHY NAGEL LECTURE
Morningside Story: Light, Matter and Ernest Nagel
Jeremy Butterfield (Cambridge)
Tuesday, April 2, 2013, at 4:10 PM
716 Philosophy Hall, Columbia University

Reception to follow

### Fujukawa: Intentionality and Plurality

NY PHILOSOPHICAL LOGIC GROUP
Intentionality and Plurality
Naoya Fujukawa (CUNY)
4pm, Monday, March 11,
2nd floor seminar room, NYU (5 Washington Place)

### Fine: Partial Content

NY PHILOSOPHICAL LOGIC GROUP
Partial Content
Kit Fine (NYU)
4pm, Monday, Feb 18,
2nd floor seminar room, NYU (5 Washington Place)

### Halpern: Constructive Decision Theory

CUNY SEMINAR IN LOGIC AND GAMES
Constructive decision theory: Decision theory with subjective states and outcomes
Joe Halpern (Cornell)
February 15, 2013, 4 PM,
CUNY Mathematics Lounge, 4th floor
Abstract. The standard approach in decision theory (going back to Savage) is to place a preference order on acts, where an act is a function from states to outcomes. If the preference order satisfies appropriate postulates, then the decision maker can be viewed as acting as if he has a probability on states and a utility function on outcomes, and is maximizing expected utility. This framework implicitly assumes that the decision maker knows what the states and outcomes are. That isn’t reasonable in a complex situation. For example, in trying to decide whether or not to attack Iraq, what are the states and what are the outcomes? We redo Savage viewing acts essentially as syntactic programs. We don’t need to assume either states or outcomes. However, among other things, we can get representation theorems in the spirit of Savage’s theorems; for Savage, the agent’s probability and utility are subjective; for us, in addition to the probability and utility being subjective, so is the state space and the outcome space. I discuss the benefits, both conceptual and pragmatic, of this approach. As I show, among other things, it provides an elegant solution to framing problems.

This is joint work with Larry Blume and David Easley. No prior knowledge of Savage’s work is assumed.