Łukasiewicz logic and the foundations of measurement

The logic of inexactness, presented in this paper, is a version of the Łukasiewicz logic with predicates valued in [0, ∞). We axiomatize multi-valued models of equality and ordering in this logic guaranteeing their imbeddibility in the real line. Our axioms of equality and ordering, when interpreted as axioms of proximity and dominance, can be applied to the foundations of measurement (especially in the social sciences). In two-valued logic they provide theories of ratio scale measurement. In multivalued logic they enable us to treat formally errors arising in nominal and ordinal measurements.     

Interdependent preferences on finite sets

The degree of interdependence of a preference relation ? on a finite subset X of a product set X₁ × X₂ × … × X_n is defined in terms of the highest order of preference interaction among the X_i that must be taken into account in a real-valued, interdependent additive representation for ?. The degree is zero when indifference holds throughout X, and zero or one in the additive conjoint measurement case. A degree of n signifies complete preference interdependence among the X_i.     

Representing approximate ordering and equivalence relations

The central question considered is: given appropriate precisations of the ideas of an empirical system's approximately satisfying laws of measurement with error at most ? (for some ? ≥ 0), and of a real-valued function over its domain providing an approximate representation of its basic operations and relations with error at most δ, can it be shown that satisfaction of the laws with ‘sufficiently small’ error insures numerical representability with arbitrarily small error? Positive answers are given in the cases of ordinal and nominal measurement, together with some indications of the sizes of the errors involved. Problems of extending the theory to more complex types of measurement are discussed, some open problems and conjectures are formulated, and a relation between the ‘approximate representation’ and ‘stochastic choice model’ approaches to measurement with fallible data is established.     

Semiorders with separability properties

We analyze different separability conditions that characterize the numerical representability of semiorders through a real-valued function and a strictly positive threshold. Any necessary and sufficient condition for the numerical representability of an interval order by means of two real-valued functions is proved to also characterize the Scott–Suppes representability of semiorders provided that a key additional condition of regularity with respect to sequences holds.     

On the Epistemic Foundation for Iterated Weak Dominance: An Analysis in a Logic of Individual and Collective attitudes

This paper proposes a logical framework for representing static and dynamic properties of different kinds of individual and collective attitudes. A complete axiomatization as well as a decidability result for the logic are given. The logic is applied to game theory by providing a formal analysis of the epistemic conditions of iterated deletion of weakly dominated strategies (IDWDS), or iterated weak dominance for short. The main difference between the analysis of the epistemic conditions of iterated weak dominance given in this paper and other analysis is that we use a semi-qualitative approach to uncertainty based on the notion of plausibility first introduced by Spohn, whereas other analysis are based on a quantitative representation of uncertainty in terms of probabilities.     

Continuous representability of interval orders and biorders

We furnish a characterization of the representability of an interval order through a pair of continuous real-valued functions which in addition represent two total preorders associated to the given interval order. Our techniques lean on the key concept of a biorder. We introduce the concept of a natural topology for an interval order, and through such concept we extend the classical biorder approach to the continuous case.     

Stimulus-response with reciprocal kernels: The rise and fall of sensation

The paper presents a theory for constructing response scales based on the reciprocal property of paired comparisons of stimuli from the same sensory continuum. Reciprocal paired comparisons define the pair estimator function K(s, t), the kernel of a Fredholm integral equation of the second kind. The solution of this equation yields a response scale of the form s^αe^−βs, where s is the stimulus scale. The response scale provides a basis for a theory to describe how images and sensation patterns are formed from stimuli.     

Some counterexamples in multidimensional scaling

Given a set X with elements x, y,… which has a partial order < on the pairs of the Cartesian product X², one may seek a distance function ? on such pairs (x, y) which satisfies ?(x₁, y₁) < ?(x₂, y₂) precisely when (x₁, y₁) < (x₂, y₂), and even demand a metric space (X, ?) with some such compatible ? which has an isometric imbedding into a finite-dimensional Euclidean space or a separable Hilbert space. We exhibit here systems (X, <) which cannot meet the latter demand. The space of real m-tuples (ξ₁,…,ξ_m) with either the “city-block” norm Σ_i ∥ξ_i∥ or the “dominance” norm max_i, ∥ξ_i∥ cannot possibly become a subset of any finite-dimensional Euclidean space. The set of real sequences (ξ₁, ξ₂,…) with finitely many nonzero elements and the supremum norm sup_i, ∥ξ_i∥ cannot even become a subset of any separable Hilbert space.     

Analyzing proximity matrices: The assessment of internal variation in combinatorial structure

A data analysis strategy is discussed for evaluating the degree to which a subset D of a larger object set S satisfies a particular algebraic property. Based on a set measure f(D) and a proximity function on S × S, two separate evaluation tasks, referred to as confirmatory and exploratory, are considered. In a confirmatory task the subset D is identified a priori and f(D) is compared against the distribution of f(·) over all subsets containing the same number of objects. The exploratory task, on the other hand, treats f(·) as an objective function to be optimized over all subsets of a given size. Examples of these two notions include the assessment of symmetry, cluster compactness, and the extent to which D satisfies the error-free conditions for a hierarchical model or a unidimensional scale.     

Real Interval Representations

This paper presents several necessary and sufficient conditions for real interval representability of biorders, interval orders, and semiorders. Let A and X be nonempty sets. We consider two types of interval representations for P⊂A×X. The first concerns the existence of two mappings, F: A→J_↓ and F: X→J_↑, such that, for all (a, x)∈A×X, (a, x)∈P⇔F(a)∩F (x)= ∅, where J_↓ and J_↑ respectively denote the set of all real intervals that are unbounded below and the set of all real intervals that are unbounded above. The second yields two mappings, F: A→J_↓ and G: X→J_↓, such that, for all (a, x)∈A×X, (a, x)∈P⇔F(a)⊂G(x). Specializations of those representations include the cases of A=X for interval orders and semiorders.     

Ordered categories and additive conjoint measurement on connected sets

Suppose that a binary relation is given on a n-fold Cartesian product. The study of the conditions guaranteeing the existence of n value functions such that the binary relation can be additively represented is known as additive conjoint measurement. In this paper we analyze a related problem: given a partition of a Cartesian product into r ordered categories, what conditions do ensure the representability of the partition in an additive model?     

A concept approach to input/output logic

This paper provides a semantics for input/input output logic based on formal concept analysis. The central result shows that an input/output logic axiomatised by a relation R is the same as the logic induced by deriving pairs from the concept lattice generated by R using a ∧- and ∨-classical Scott consequence relation. This correspondence offers powerful analytical techniques for classifying, visualising and analysing input/output relations, revealing implicit hierarchical structure and/or natural clusterings and dependencies. The application of all formal developments are illustrated by a worked example towards the end.     

A Developmental Study of Judgments of Semantic Similarity

Inhelder and Piaget offer a single protocol as their only evidence that a fully developed formal operational thinker uses all 16 binary operations of truth-functional logic. The investigators attempted to replicate the Inhelder-Piaget results with a random sample of 57 subjects (18 9-year-olds, 19 12-year-olds, 20 16-year-olds). Not one of the subjects used more than five of the 16 operations, and there was no developmental trend with regard to the number of operations used (X² = .4917, p > .05). A trend was manifest, however, since the more developed reasoner used the same operations as the less developed reasoner, but in a more complex and sophisticated manner.     

Semiorders and thresholds of utility discrimination: Solving the Scott–Suppes representability problem

We find an internal characterization of the so-called Scott–Suppes representability of a semiorder, defined as the existence of a numerical representation for a semiorder through a real-valued utility function and a positive constant threshold of discrimination.     

The logic of how-questions

Philosophers and scientists are concerned with the why and the how of things. Questions like the following are so much grist for the philosopher’s and scientist’s mill: How can we be free and yet live in a deterministic universe?, How do neural processes give rise to conscious experience?, Why does conscious experience accompany certain physiological events at all?, How is a three-dimensional perception of depth generated by a pair of two-dimensional retinal images?. Since Belnap and Steel’s pioneering work on the logic of questions, Van Fraassen has managed to apply their approach in constructing an account of the logic of why-questions. Comparatively little, by contrast, has been written on the logic of how-questions despite the apparent centrality of questions such as How is it possible for us to be both free and determined? to philosophical enterprise.¹ In what follows I develop a logic for how-questions of various sorts including how-questions of cognitive resolution, how-questions of manner, how-questions of method, of means, and of mechanism.     

Elementary categorial logic,predicates of variable degree,and theory of quantity

Developing some suggestions of Ramsey (1925), elementary logic is formulated with respect to an arbitrary categorial system rather than the categorial system of Logical Atomism which is retained in standard elementary logic. Among the many types of non-standard categorial systems allowed by this formalism, it is argued that elementary logic with predicates of variable degree occupies a distinguished position, both for formal reasons and because of its potential value for application of formal logic to natural language and natural science. This is illustrated by use of such a logic to construct a theory of quantity which is argued to be scientifically superior to existing theories of quantity based on standard categorial systems, since it yields realvalued scales without the need for unrealistic existence assumptions. This provides empirical evidence for the hypothesis that the categorial structure of the physical world itself is non-standard in this sense.I would like to thank my collegue Mark Brown and an anonymous referee for helpful comments on an earlier draft of this paper.     

Weight for Stephen Finlay

According to Stephen Finlay, ‘A ought to X’ means that X-ing is more conducive to contextually salient ends than relevant alternatives. This in turn is analysed in terms of probability. I show why this theory of ‘ought’ is hard to square with a theory of a reason’s weight which could explain why ‘A ought to X’ logically entails that the balance of reasons favours that A X-es. I develop two theories of weight to illustrate my point. I first look at the prospects of a theory of weight based on expected utility theory. I then suggest a simpler theory. Although neither allows that ‘A ought to X’ logically entails that the balance of reasons favours that A X-es, this price may be accepted. For there remains a strong pragmatic relation between these claims.     

A conjunction in closure spaces

This paper is closely related to investigations of abstract properties of basic logical notions expressible in terms of closure spaces as they were begun by A. Tarski (see [6]). We shall prove many properties of ω-conjunctive closure spaces (X is ω-conjunctive provided that for every two elements of X their conjunction in X exists). For example we prove the following theorems:

1. For every closed and proper subset of an ω-conjunctive closure space its interior is empty (i.e. it is a boundary set).
2. If X is an ω-conjunctive closure space which satisfies the ω-compactness theorem and \(\hat P\) [X] is a meet-distributive semilattice (see [3]), then the lattice of all closed subsets in X is a Heyting lattice.
3. A closure space is linear iff it is an ω-conjunctive and topological space.
4. Every continuous function preserves all conjunctions.

     

The Structure Group of a Generalized Orthomodular Lattice

Orthomodular lattices with a two-valued Jauch–Piron state split into a generalized orthomodular lattice (GOML) and its dual. GOMLs are characterized as a class of L-algebras, a quantum structure which arises in the theory of Garside groups, algebraic logic, and in connections with solutions of the quantum Yang–Baxter equation. It is proved that every GOML X embeds into a group G(X) with a lattice structure such that the right multiplications in G(X) are lattice automorphisms. Up to isomorphism, X is uniquely determined by G(X), and the embedding \(X\hookrightarrow G(X)\) is a universal group-valued measure on X.     

Trichotomous preferences for gambles

Let G_X be the set of all gambles on a set X of decision outcomes. Trichotomous preferences divide G_X into three disjoint subsets, viz., the set P⁺ of preferable gambles, the set P⁰ of indifferent gambles, and the set P^- of unpreferable gambles. Those preference comparisons are made against the status quo, i.e., the do-nothing alternative. This paper presents and discusses dyadic representations of such trichotomous preferences and identifies necessary and sufficient axioms for the existence of those representations.