What Are Degrees of Belief?

Probabilism is committed to two theses:

Improving Accuracy on Bayesian Inference Problems Using a Brief Tutorial

Research suggests that most people struggle when asked to interpret the outcomes of diagnostic tests such as those presented as Bayesian inference problems. To help people interpret these difficult problems, we created a brief tutorial, requiring less than 10 minutes, that guided participants through the creation of an aid (either graph or table) based on an example inference problem and then showed the correct way to calculate the positive predictive value of the problem (i.e., likelihood that positive tests correctly indicate presence of condition). Approximately 70% of those in each training condition found the correct response on at least one problem in the format for which they were trained. Just under 55% of those in the control condition (i.e., no training) were able to find the correct response on at least one table or graph problem. We demonstrated a relationship between numeracy and performance on both problem formats, although we did not find evidence for a relationship between graph literacy and performance for either problem format. Potential improvements to and applications of the tutorial are discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

Löb's Theorem in a Set Theoretical Setting

We present a semantic proof of Löb's theorem for theories T containing ZF. Without using the diagonalization lemma, we construct a sentence AUT _T, which says intuitively that the predicate autological with respect to T (i.e. applying to itself in every model of T) is itself autological with respect to T. In effect, the sentence AUT _T states I follow semantically from T. Then we show that this sentence indeed follows from T and therefore is true.     

The Role of Semantic Clustering in Optimal Memory Foraging

Recent studies of semantic memory have investigated two theories of optimal search adopted from the animal foraging literature: Lévy flights and marginal value theorem. Each theory makes different simplifying assumptions and addresses different findings in search behaviors. In this study, an experiment is conducted to test whether clustering in semantic memory may play a role in evidence for both theories. Labeled magnets and a whiteboard were used to elicit spatial representations of semantic knowledge about animals. Category recall sequences from a separate experiment were used to trace search paths over the spatial representations of animal knowledge. Results showed that spatial distances between animal names arranged on the whiteboard were correlated with inter‐response intervals (IRIs) during category recall, and distributions of both dependent measures approximated inverse power laws associated with Lévy flights. In addition, IRIs were relatively shorter when paths first entered animal clusters, and longer when they exited clusters, which is consistent with marginal value theorem. In conclusion, area‐restricted searches over clustered semantic spaces may account for two different patterns of results interpreted as supporting two different theories of optimal memory foraging.     

Bunched sequential information

It is known that the logic BI of bunched implications is a logic of resources. Many studies have reported on the applications of BI to computer science. In this paper, an extension BIS of BI by adding a sequence modal operator is introduced and studied in order to formalize more fine-grained resource-sensitive reasoning. By the sequence modal operator of BIS, we can appropriately express “sequential information” in resource-sensitive reasoning. A Gentzen-type sequent calculus SBIS for BIS is introduced, and the cut-elimination and decidability theorems for SBIS are proved. An extension of the Grothendieck topological semantics for BI is introduced for BIS, and the completeness theorem with respect to this semantics is proved. The cut-elimination, decidability and completeness theorems for SBIS and BIS are proved using some theorems for embedding BIS into BI.     

Some Remarks on Dualization in Vector Optimization

In recent years there have been several reports on duality in vector optimization. However, there seems to be no unified approach to dualization. In a previous paper by the author a geometric consideration was given to duality in non-linear vector optimization. In this paper a relationship between duality, stability (normality) and condition of the alternative will be reported on the basis of some geometric consideration. In addition, Iserman's duality in linear cases will be derived from the stated geometric approach.