A Set Theory with Support for Partial Functions

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.     

Deflecting Arrow by Aggregating Rankings of Multiple Correlated Criteria According to Borda

This paper has two objectives. The first is to review and address concerns raised by Hazelrigg that Arrow's impossibility theorem prevents the selection of rational aggregation methods for use in engineering trade studies. In addressing these concerns, the work of Saari is cited to establish the fact that the Borda count is the only ‘non‐dictatorial’ positional voting method that satisfies the criteria for a rational decision procedure while using complete information. Hence, the resulting rank ordering of the alternatives is the most reliable outcome. Several previous studies that use other aggregation methods are critiqued, and Borda is applied to examples to illustrate the differences in the outcomes. The second objective is to extend the applicability of Borda to include attributes such as cost, schedule duration, and certain technical and performance measures that are generally more reasonably described as correlated random variables. Exact Probabilities by Simulation with Borda, a method introduced by Hulkower that improves a technique by Book for determining which candidate in a trade study is the probable lowest‐cost alternative, is generalized to include multiple correlated criteria, each of which is expressed as a random variable and thus incorporates probabilistic uncertainty. Copyright © 2016 John Wiley & Sons, Ltd.     

Seeing is believing: Priors,trust, and base rate neglect

Kahneman and Tversky (1973) described an effect they called ‘insensitivity to prior probability of outcomes’, later dubbed base rate neglect, which describes people’s tendency to underweight prior information in favor of new data. As probability theory requires that prior probabilities be taken into account, via Bayes’ theorem, the fact that most people fail to do so has been taken as evidence of human irrationality and, by others, of a mismatch between our cognitive processes and the questions being asked (Cosmides & Tooby, 1996). In contrast to both views, we suggest that simplistic Bayesian updating using base rates is not necessarily rational. To that end, we present experiments in which base rate neglect is often the right strategy, and show that people’s base rate usage varies systematically as a function of the extent to which the data that make up a base rate are perceived as trustworthy.     

Prover9's Simplification Explained Away

This note discusses P. Oppenheimer and E. Zalta's ‘A Computationally-Discovered Simplification of the Ontological Argument’ [this journal, 2011]. I try to explain why the simplification presented there was successful and comment on the technical aspects of the method they applied.     

Adaptation to Reversal of Retinal Feedback of Eyemovements

Both optical and hybrid-computer methods were devised to reverse the visual feedback which the eye gets of its own movements. Optical reversal by a contact lens device caused persisting, unstable, skittered, blurred vision, to which the subject made no effective adaptation. Using a hybrid analog-digital-analog computer, eyemovement-retinal feedback was reversed by yoking oscilloscope targets to the transduced electrical signals of ocular movements and reversing the sign of input-output control of the computer system. Vision with such computer-reversed feedback from ocular movements duplicated the experiences with the contact-lens method. Practice over five sessions of 50 trials indicated that three Ss showed no significant improvement in eye-tracking accuracy with the reversed vision. Perception of direction in vision and the basic signal processes of vision were thus found to be dependent on built-in direction specificity of eyemovement-retinal feedback interactions which also govern dynamic specialization and control of pursuit, saccadic, and fixation movements.     

Identical Twins,Deduction Theorems,and Pattern Functions: Exploring the Implicative BCSK Fragment of S5

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we show that for the pure logic of one of these implicational connectives two – in general distinct – consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the ‘propositional operations’ associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives.     

Sequent Calculi for Some Strict Implication Logics

We introduce various sequent systems for propositional logicshaving strict implication, and prove the completeness theoremsand the finite model properties of these systems.The cut-eliminationtheorems or the (modified) subformula properties are provedsemantically.     

Connection Tableau Calculi with Disjunctive Constraints

Automated theorem proving amounts to solving search problems in usually tremendous search spaces. A lot of research therefore focuses on search space reductions. Our approach reduces the search space which arises when using so-called connection tableau calculi for first-order automated theorem proving. It uses disjunctive constraints over first-order equations to compress certain parts of this search space. We present the basics of our constrained-connection-tableau calculi, a constraint extension of connection tableau calculi, and deal with the efficient handling of constraints during the search process. The new techniques are integrated into the automated connection tableau prover Setheo.     

A Note on Goodman's Theorem

Goodman's theorem states that intuitionistic arithmetic in all finite types plus full choice, HA + AC, is conservative over first-order intuitionistic arithmetic HA. We show that this result does not extend to various subsystems of HA, HA with restricted induction.     

Causality assessment in epidemiology

Epidemiology relies upon a broad interpretation of determinism. This paper discusses analogies with the evolution of the concept of cause in physics, and analyzes the classical nine criteria proposed by Sir Austin Bradford Hill for causal assessment. Such criteria fall into the categories of enumerative induction, eliminative induction, deduction and analogy. All of these four categories are necessary for causal assessment and there is no natural hierarchy among them, although a deductive analysis of the study design is preliminary to any assessment.