Predictive probability and analogy by similarity in inductive logic

The-continuum of inductive methods was derived from an assumption, called-condition, which says that the probability of finding an individual having propertyx _j depends only on the number of observed individuals having propertyx _j and on the total number of observed individuals. So, according to that assumption, all individuals with properties which are different fromx _j have equal weight with respect to that probability and, in particular, it does not matter whether any individual was observed having some propertysimilar tox _j (the most complete proof of this result is presented in Carnap, 1980).The problem thus remained open to find some general condition, weaker than the-condition, which would allow for thederivation of probability functions which might be sensitive to similarity. Carnap himself suggested a weakening of the-condition which might allow for similarity sensitive probability functions (Carnap, 1980, p. 45) but he did not find the family of probability functions derivable from that principle. The aim of this paper is to present the family of probability functions derivable from Carnap's suggestion and to show how it is derived.In Section 1 the general problem of analogy by similarity in inductive logic is presented, Section 2 outlines the notation and the conceptual background involved in the proof, Section 3 gives the proof, Section 4 discusses Carnap's principle and the result, Section 5 is a brief review of the solutions which have previously been proposed.     

Special issue: Combining probability and logic

This editorial explains the scope of the special issue and provides a thematic introduction to the contributed papers.     

Using logic programming for theory representation and scientific inference

The aim of this paper is to show that logic programming is a powerful tool for representing scientific theories and for scientific inference. In a logic program it is possible to encode the qualitative and quantitative components of a theory in first order predicate logic, which is a highly expressive formal language. A theory program can then be handed to an algorithm that reasons about the theory. We discuss the advantages of logic programming with regard to building formal theories and present a novel software package for scientific inference: Theory Toolbox. Theory Toolbox can derive any conclusions that are entailed by a theory, explain why a certain conclusion follows from a theory, and evaluate a theory with regard to its internal coherence and generalizability. Because logic is, or should be, a cornerstone of scientific practice, we believe that our paper can make an important contribution to scientific psychology.     

When logic fails: Implicit transitive inference in humans

Transitive inference (TI) in animals (e.g., choosing A over C on the basis of knowing that A is better than B and B is better than C) has been interpreted by some as reflecting a declarative logical inference process. We invert this anthropomorphic interpretation by providing evidence that humans can exhibit TI-like behavior on the basis of simpler associative mechanisms that underlie many theories of animal learning. In this study, human participants were trained on a five-pair TI problem (A+B-, B+C-, C+D-, D+E-, E+F-) and, unlike in previous human TI studies, were prevented from becoming explicitly aware of the logical hierarchy, so they could not employ logical reasoning. They were then tested with three problems: B versus D, B versus E, and C versus E. Participants only reliably chose B over E, whereas the other test conditions yielded chance performance. This result is inconsistent with the use of logical reasoning and is instead consistent with an account developed to explain earlier TI studies with rats that found the same pattern of results. In this account, choice performance is based on differential associative strengths across the stimulus items that develop over training, despite equal overt reinforcement.     

SLAP: Specification logic of actions with probability

A logic for specifying probabilistic transition systems is presented. Our perspective is that of agents performing actions. A procedure for deciding whether sentences in this logic are valid is provided. One of the main contributions of the paper is the formulation of the decision procedure: a tableau system which appeals to solving systems of linear equations. The tableau rules eliminate propositional connectives, then, for all open branches of the tableau tree, systems of linear equations are generated and checked for feasibility. Proofs of soundness, completeness and termination of the decision procedure are provided.     

A quantum probability account of order effects in inference

Order of information plays a crucial role in the process of updating beliefs across time. In fact, the presence of order effects makes a classical or Bayesian approach to inference difficult. As a result, the existing models of inference, such as the belief-adjustment model, merely provide an ad hoc explanation for these effects. We postulate a quantum inference model for order effects based on the axiomatic principles of quantum probability theory. The quantum inference model explains order effects by transforming a state vector with different sequences of operators for different orderings of information. We demonstrate this process by fitting the quantum model to data collected in a medical diagnostic task and a jury decision-making task. To further test the quantum inference model, a new jury decision-making experiment is developed. Using the results of this experiment, we compare the quantum inference model with two versions of the belief-adjustment model, the adding model and the averaging model. We show that both the quantum model and the adding model provide good fits to the data. To distinguish the quantum model from the adding model, we develop a new experiment involving extreme evidence. The results from this new experiment suggest that the adding model faces limitations when accounting for tasks involving extreme evidence, whereas the quantum inference model does not. Ultimately, we argue that the quantum model provides a more coherent account for order effects that was not possible before.     

A frequentist interpretation of probability for model-based inductive inference

The main objective of the paper is to propose a frequentist interpretation of probability in the context of model-based induction, anchored on the Strong Law of Large Numbers (SLLN) and justifiable on empirical grounds. It is argued that the prevailing views in philosophy of science concerning induction and the frequentist interpretation of probability are unduly influenced by enumerative induction, and the von Mises rendering, both of which are at odds with frequentist model-based induction that dominates current practice. The differences between the two perspectives are brought out with a view to defend the model-based frequentist interpretation of probability against certain well-known charges, including [i] the circularity of its definition, [ii] its inability to assign ‘single event’ probabilities, and [iii] its reliance on ‘random samples’. It is argued that charges [i]–[ii] stem from misidentifying the frequentist ‘long-run’ with the von Mises collective. In contrast, the defining characteristic of the long-run metaphor associated with model-based induction is neither its temporal nor its physical dimension, but its repeatability (in principle); an attribute that renders it operational in practice. It is also argued that the notion of a statistical model can easily accommodate non-IID samples, rendering charge [iii] simply misinformed.     

Probability logic,logical probability,and inductive support

This paper seeks to defend the following conclusions: The program advanced by Carnap and other necessarians for probability logic has little to recommend it except for one important point. Credal probability judgments ought to be adapted to changes in evidence or states of full belief in a principled manner in conformity with the inquirer’s confirmational commitments—except when the inquirer has good reason to modify his or her confirmational commitment. Probability logic ought to spell out the constraints on rationally coherent confirmational commitments. In the case where credal judgments are numerically determinate confirmational commitments correspond to Carnap’s credibility functions mathematically represented by so—called confirmation functions. Serious investigation of the conditions under which confirmational commitments should be changed ought to be a prime target for critical reflection. The necessarians were mistaken in thinking that confirmational commitments are immune to legitimate modification altogether. But their personalist or subjectivist critics went too far in suggesting that we might dispense with confirmational commitments. There is room for serious reflection on conditions under which changes in confirmational commitments may be brought under critical control. Undertaking such reflection need not become embroiled in the anti inductivism that has characterized the work of Popper, Carnap and Jeffrey and narrowed the focus of students of logical and methodological issues pertaining to inquiry.     

Transitive inference of social dominance by human infants

It is surprising that there are inconsistent findings of transitive inference (TI) in young infants given that non‐linguistic species succeed on TI tests. To conclusively test for TI in infants, we developed a task within the social domain, with which infants are known to show sophistication. We familiarized 10‐ to 13‐month‐olds (M = 11.53 months) to a video of two dominance interactions between three puppets (bear > elephant; hippo > bear) consistent with a dominance hierarchy (hippo > bear > elephant; where ‘>’ denotes greater dominance). Infants then viewed interactions between the two puppets that had not interacted during familiarization. These interactions were either congruent (hippo > elephant) or incongruent (elephant > hippo) with the inferred hierarchy. Consistent with TI, infants looked longer to incongruent than congruent displays. Control conditions ruled out the possibility that infants’ expectations were based on stable behaviors specific to individual puppets rather than their inferred transitive dominance relations. We suggest that TI may be supported by phylogenetically ancient mechanisms of ordinal representation and visuospatial processing that come online early in human development.     

The influence of category coherence on inference about cross-classified entities

A critical function of categories is their use in property inference (Heit, 2000). However, one challenge to using categories in inference is that most entities in the world belong to multiple categories (e.g., Fido could be a dog, a pet, a mammal, or a security system). Building on Patalano, Chin-Parker, and Ross (2006), we tested the hypothesis that category coherence (the extent to which category features go together in light of prior knowledge) influences the selection of categories for use in property inference about cross-classified entities. In two experiments, we directly contrasted coherent and incoherent categories, both of which included cross-classified entities as members, and we found that the coherent categories were used more readily as the source of both property transfer and property extension. We conclude that category coherence, which has been found to be a potent influence on strength of inference for singly classified entities (Rehder & Hastie, 2004), is also central to category use in reasoning about novel cross-classified ones.