A Gentzen Calculus for Nothing but the Truth

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic (ETL), an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof.     

A New–old Characterisation of Logical Knowledge

We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ‘sortal terms’, two theories that will feature prominently. Second, we propose that logic comprises four ‘momental sectors’: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (in)valid deduction, inference or substitution. Third, we elaborate on two neglected features of logic: the various modes of negating some part(s) of a proposition R, not only its ‘external’ negation not-R; and the assertion of R in the pair of propositions ‘it is (un)true that R’ belonging to the neglected logic of asserted propositions, which is usually left unstated. We also address the overlooked task of testing the asserted truth-value of R. Fourth, we locate logic among other foundational studies: set theory and other theories of collections, metamathematics, axiomatisation, definitions, model theory, and abstract and operator algebras. Fifth, we test this characterisation in two important contexts: the formulation of some logical paradoxes, especially the propositional ones; and indirect proof-methods, especially that by contradiction. The outcomes differ for asserted propositions from those for unasserted ones. Finally, we reflect upon self-referring self-reference, and on the relationships between logical and mathematical knowledge. A subject index is appended.     

Arrovian Aggregation of Generalised Expected-Utility Preferences: (Im)possibility Results by Means of Model Theory

Cerreia-Vioglio et al. (Econ Theory 48(2–3):341–375, 2011) have proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences—and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory (in the sense of mathematical logic), is employed.     

Partial Reinforcement in a Bar-Pressing Situation with Preschool Children

Forty fifth- and sixth-grade children were given a list of names which they had prerated for Reinforcement Value in a free recall format. The prediction was made that high self-esteem children will learn their liked names more rapidly than their disliked names, but that low self-esteem children would show the reverse tendency. Although an RV-positive effect is noted in the sample, the expected reversal did not occur. A confounding factor proved to be intelligence, since it was impossible to divorce self-esteem from IQ in the sample investigated. The presentation closes with a discussion of the “logical learning theory” on which RV study is based, and the implications that this theory has for the role of self-induced behavior.     

Reasoning About Collectively Accepted Group Beliefs

A proof-theoretical treatment of collectively accepted group beliefs is presented through a multi-agent sequent system for an axiomatization of the logic of acceptance. The system is based on a labelled sequent calculus for propositional multi-agent epistemic logic with labels that correspond to possible worlds and a notation for internalized accessibility relations between worlds. The system is contraction- and cut-free. Extensions of the basic system are considered, in particular with rules that allow the possibility of operative members or legislators. Completeness with respect to the underlying Kripke semantics follows from a general direct and uniform argument for labelled sequent calculi extended with mathematical rules for frame properties. As an example of the use of the calculus we present an analysis of the discursive dilemma.     

A generalized family of coefficients of relational agreement for numerical scales

A family of coefficients of relational agreement for numerical scales is proposed. The theory is a generalization to multiple judges of the Zegers and ten Berge theory of association coefficients for two variables and is based on the premise that the choice of a coefficient depends on the scale type of the variables, defined by the class of admissible transformations. Coefficients of relational agreement that denote agreement with respect to empirically meaningful relationships are derived for absolute, ratio, interval, and additive scales. The proposed theory is compared to intraclass correlation, and it is shown that the coefficient of additivity is identical to one measure of intraclass correlation.The author thanks the Editor and anonymous reviewers for helpful suggestions.     

Cues to referent gender in randomly constructed names.

50 volunteers' assignments of gender to 32 randomly constructed names confirmed previous findings with respect to the structure of actual names (men's names are shorter, women's names tend to end in vowels, especially the schwa vowel) and with respect to the emotionality of sounds in actual names (men's names sound more Active, Nasty, and Unpleasant; women's names sound softer). Assignment of sex to random names was predictable (R>.8) on the basis of combinations of these variables.     

Absolute Probability Functions for Intuitionistic Propositional Logic

Provided here is a characterisation of absolute probability functions for intuitionistic (propositional) logic L, i.e. a set of constraints on the unary functions P from the statements of L to the reals, which insures that (i) if a statement A of L is provable in L, then P(A) = 1 for every P, L's axiomatisation being thus sound in the probabilistic sense, and (ii) if P(A) = 1 for every P, then A is provable in L, L's axiomatisation being thus complete in the probabilistic sense. As there are theorems of classical (propositional) logic that are not intuitionistic ones, there are unary probability functions for intuitionistic logic that are not classical ones. Provided here because of this is a means of singling out the classical probability functions from among the intuitionistic ones.     

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer??s calculus of ??conditional density operators??. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.     

一个多值逻辑的一阶谓词系统

鞠实儿曾提出一个开放类三值命题逻辑系统,这一逻辑也可以推广到任意m值逻辑情形,成为一个联结词函数完全的逻辑。本文将对推广的命题逻辑系统L^＊建立一种一阶谓词系统,并证明其可靠性、完全性。     

Encoding of colors in short-term memory

A theory is proposed to explain results from prior experiments on release from proactive inhibition and the recall of colors or color names in short-term memory. It is assumed that colors are encoded in two ways, verbally and perceptually, while color names are encoded only verbally. Assuming that the release occurs when a new and different encoding is performed on changed material, it follows that release from proactive inhibition should occur with shifts from color names to colors but not with shifts in the opposite direction. These results were obtained in prior experiments. In the present experiment ambiguous colors were used to minimize the verbal encoding. As predicted, release from proactive inhibition was found with shifts from ambiguous colors to names as well as with shifts from names to the ambiguous colors.     

Is ZF a hack?: Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics

This paper presents Automath encodings (which are also valid in LF/λP) of various kinds of foundations of mathematics. Then it compares these encodings according to their size, to find out which foundation is the simplest.

The systems analyzed in this way are two kinds of set theory (ZFC and NF), two systems based on Church's higher order logic (Isabelle/Pure and HOL), three kinds of type theory (the calculus of constructions, Luo's extended calculus of constructions, and Martin-Löf's predicative type theory) and one foundation based on category theory.

The conclusions of this paper are that the simplest system is type theory (the calculus of constructions), but that type theories that know about serious mathematics are not simple at all. In that case the set theories are the simplest. If one looks at the number of concepts needed to explain such a system, then higher order logic is the simplest, with twenty-five concepts. On the other side of the scale, category theory is relatively complex, as is Martin-Löf's type theory.

(The full Automath sources of the contexts described in this paper are one the web at http://www.cs.ru.nl/~freek/zfc-etc/.)     

Axiomatising first-order temporal logic: Until and since over linear time

We present an axiomatisation for the first-order temporal logic with connectives Until and Since over the class of all linear flows of time. Completeness of the axiom system is proved.We also add a few axioms to find a sound and complete axiomatisation for the first order temporal logic of Until and Since over rational numbers time.The author would like to thank Dov Gabbay and Ian Hodkinson for helpful discussions on this material. The work was supported by the U.K. Science and Engineering Research Council under the Metatem project (GR/F/28526).Presented by Dov Gabbay     

Group Cancellation and Resolution

We establish a connection between the geometric methods developed in the combinatorial theory of small cancellation and the propositional resolution calculus. We define a precise correspondence between resolution proofs in logic and diagrams in small cancellation theory, and as a consequence, we derive that a resolution proof is a 2-dimensional process. The isoperimetric function defined on diagrams corresponds to the length of resolution proofs.     

On the bounded rationality of gender stereotyping in fame judgments

The false‐fame effect is the phenomenon that familiar names are falsely judged famous more often than unfamiliar names. M.R. Banaji and A.G. Greenwald ( 1995 ) demonstrated a gender bias in the false‐fame effect: In line with existing gender stereotypes, the false‐fame effect was larger for male than for female names. A more general explanation for gender biasing in fame judgments is based on cognitive availability. Name gender could be used as an ecologically valid cue when making fame judgments. If the relevant universe of famous names contained more male than female names, a gender bias in fame judgments should be observed, if it contained more female names, the gender bias should be reversed. Indeed, this pattern could be demonstrated experimentally, and we argue that it is not compatible with an account that draws on gender stereotyping but with one based on cognitive availability. Copyright © 2004 John Wiley & Sons, Ltd.     

Notes on Logics of Metric Spaces

In [14], we studied the computational behaviour of various first-order and modal languages interpreted in metric or weaker distance spaces. [13] gave an axiomatisation of an expressive and decidable metric logic. The main result of this paper is in showing that the technique of representing metric spaces by means of Kripke frames can be extended to cover the modal (hybrid) language that is expressively complete over metric spaces for the (undecidable) two-variable fragment of first-order logic with binary pred-icates interpreting the metric. The frame conditions needed correspond rather directly with a Boolean modal logic that is, again, of the same expressivity as the two-variable fragment. We use this representation to derive an axiomatisation of the modal hybrid variant of the two-variable fragment, discuss the compactness property in distance logics, and derive some results on (the failure of) interpolation in distance logics of various expressive power. Presented by Melvin Fitting     

Handling Inconsistencies in the Early Calculus

The early calculus is a popular example of an inconsistent but fruitful scientific theory. This paper is concerned with the formalisation of reasoning processes based on this inconsistent theory. First it is shown how a formal reconstruction in terms of a sub-classical negation leads to triviality. This is followed by the evaluation of the chunk and permeate mechanism (C&P) proposed by Brown and Priest in (Journal of Philosophical Logic, 33(4), 379–388, 2004) to obtain a non-trivial formalisation of the early infinitesimal calculus. Different shortcomings of this application of C&P as an explication of inconsistency tolerant reasoning are pointed out, both conceptual and technical. To remedy these shortcomings, an adaptive logic is proposed that allows for conditional permeations of formulas under the assumption of consistency preservation. First the adaptive logic is defined and explained and thereafter it is demonstrated how this adaptive logic remedies the defects C&P suffered from.     

Proper names and how they are learned

Abstract

Proper names function in our conceptual lives as means for denoting individuals in kinds. Kinds are denoted by common names, more precisely count nouns, and so there are important interrelations between proper names and common nouns. All of this shows up in the way we interpret proper names and employ them in everyday inferences. For example, an airline may count three passengers in relation to a single person Jane, if Jane takes three trips with the airline. Each of the three passengers is Jane, but there is only one Jane. To handle such operations we propose a theory of proper names as part of the theory of kinds. This enables us to specify certain resources (some of them unlearned) that are necessary for the learning of proper names and also a theory of how they are learned. We review the experimental literature on the learning of proper names from the standpoint of the theory. We do not extend the theory to cover recognition or recall.