Dr.Dr. Niki Pfeifer
Progic 2013 Website
progic 2013: The Sixth Workshop on Combining Probability and Logic
"Combining probability and logic to solve philosophical problems"
Carl Friedrich von Siemens Stiftung (Munich)
hosted by the
Munich Center for Mathematical Philosophy
17-18 September 2013
Abstracts of invited speakers
Conditionals and inferential connections
Many have the intuition that for a conditional to be true, there must exist some kind of connection between its antecedent and its consequent. We investigated experimentally the prospects for spelling out the connection in terms of inference. Participants were shown or given by description a series of fourteen numbered color patches ranging from blue to green. They were then asked to evaluate conditionals pertaining to these patches which, in the context of the experiment, are naturally thought of as embodying inferential connections (such as 'If patch number 5 is blue, then so is patch number 4'). The data we obtained in this experiment are shown to be better explained by a semantics for conditionals that makes inferential connectedness a truth condition than by any of the currently more popular semantics for conditionals. (Joint work with Shira Elqayam, Janneke Huitink, and David Over.)
Probabilities of counterfactuals and counterfactual probabilities
Probabilities and counterfactuals interact in interesting and sometimes surprising ways. Edgington appeals to their interaction in arguing for her "no truth value" account of counterfactuals. While I will defend counterfactuals having truth conditions, I will appeal to their interaction for a nefarious purpose of my own: undermining the standard similarity accounts of those truth conditions (à la Lewis and Stalnaker).
Kevin T. Kelly and Hanti Lin
Qualitative reasoning that tracks Jeffrey conditioning
On our previous visit to the MCMP in 2012 , we showed that acceptance based on odds thresholds and belief revision based on partial orders has the property that acceptance followed by qualitative belief revision agrees perfectly with Bayesian conditioning followed by acceptance. The upshot is that qualitative beliefs can have a genuine cognitive role in rational reasoning. We also showed that getting qualitative reasoning to track conditioning in that sense imposes nontrivial constraints on both the acceptance rule and the belief revision rule. In particular, it is impossible to obtain our tracking result for AGM belief revision. In this talk, we extend our earlier tracking result to Jeffrey conditioning, using ideas from projective geometry to construct the qualitative probability order on possibilities. We illustrate how the tracking property imposes still more stringent geometrical constraints on the geometry of the acceptance rule. We also discuss a Jeffrey-conditioning version of the Ramsey test for indicative conditionals: A > B is accepted in Bayesian credal state p if and only if every Jeffrey conditioning of p on A that results in acceptance of A also results in acceptance of B. This research was supported by a generous research grant from the John Templeton Foundation.
References to our earlier work:
MCMP round table on acceptance: http://www.mcmp.philosophie.uni-muenchen.de/mcmp_events/add_act/r_t_accept/index.html
Tracking result: http://rd.springer.com/article/10.1007/s10992-012-9237-3
Ramsey test: http://www.springerlink.com/content/546tv6646318584w/
Belief and stable probability
We will defend a theory of belief that is based on the following thesis: It is rational to believe A if and only if the subjective probability of A if stably high. We will make this formally precise, we will give three different formal arguments for it, and we will present some history of it that goes back as far as Hume. The upshot will be: doxastic logic and belief revision theory can coexist in harmony with subjective probability theory.
Information, confirmation, and conditionals
Loosely speaking, a proposition is the more informative, given background b, the greater the proportion of possibilities left open by b that it rules out. Plausible qualitative constraints lead to the result that any measure of information added should be a rescaling of a conditional probability function, a strictly decreasing rescaling sending 1 to 0 (Milne, J. Logic, Language & Information, 2012). The two commonest rescalings are −log P and 1 − P.
In similar vein: e is favourable evidence for hypothesis h relative to background b if h rules out a smaller proportion of the possibilities left open by b and e jointly, than left open by b alone. In terms of the underlying probability measure, this secures the familiar: e confirms h iff P(h|eb) > P(h|b).
f is more favourable evidence for h than e iff h rules out a smaller proportion of the possibilities left open by b and f jointly than left open by b and e jointly. In these terms, a measure of confirmation should be a function of the information added by h to be and to b, decreasing with the first and increasing with the second. (Not all measures of confirmation proposed in the literature satisfy this condition.) When e = h, the possibilities that drop out as we narrow the focus with e are exactly the possibilities left open by b but excluded by h. Thus the extent to which h confirms h relative to b is a measure of the information h adds to b. (Some confirmation measures suggested in the literature yield constant measures of information-added.)
Given a measure inf of information added, we can think of inf(ac,b) − inf(a,b) as a measure of the "deductive gap", relative to b, between a and ac. When inf(a,b) = −log P(a|b), inf(ac,b) − inf(a,b) = −log P(c,ab), the amount of information the indicative conditional 'if a then c' adds to b on the Adams account of that conditional. When inf(a,b) = 1 − P(a|b), inf(ac,b) − inf(a,b) = P(a. ¬c|b) = inf(a⊃c,b) where a⊃c is a material conditional.
The aim is to investigate "information theoretic" conditionals obtained from other measures of information-added derived, in turn, from measures of confirmation that yield non-constant measures of information-added.