Extensions of the significance test for one-parameter signal detection hypotheses

The basic models of signal detection theory involve the parametric measure,d, generally interpreted as a detectability index. Given two observers, one might wish to know whether their detectability indices are equal or unequal. Gourevitch and Galanter (1967) proposed a large sample statistical test that could be used to test the hypothesis of equald values. In this paper, their large two sample test is extended to aK-sample detection test. If the null hypothesisd ₁=d ₂=...=d _K is rejected, one can employ the post hoc confidence interval procedure described in this paper to locate possible statistically significant sources of variance and differences. In addition, it is shown how one can use the Gourevitch and Galanter statistics to testd=0 for a single individual.This paper was written while the author was associated with the Institute of Human Learning at the University of California at Berkeley.     

Maximization of sums of quotients of quadratic forms and some generalizations

Monotonically convergent algorithms are described for maximizing six (constrained) functions of vectors x, or matricesX with columns x₁, ..., x _r . These functions are h₁(x)= _k (xA _kx)(xC _kx)^–1, H₁(X)= _k tr (XA _k X)(XC _k X)^–1, h₁(X)= _{k l} (x _l A _kx _l ) (x _l C _kx _l )^–1 withX constrained to be columnwise orthonormal, h₂(x)= _k (xA _kx)²(xC _kx)^–1 subject to xx=1, H₂(X)= _k tr(XA _kX)(XA_kX)(XC_kX)^–1 subject toXX=I, and h₂(X)= _{k l} (x _l A _kx _l )² (x _l C _kX _l )^–1 subject toXX=I. In these functions the matricesC _k are assumed to be positive definite. The matricesA _k can be arbitrary square matrices. The general formulation of the functions and the algorithms allows for application of the algorithms in various problems that arise in multivariate analysis. Several applications of the general algorithms are given. Specifically, algorithms are given for reciprocal principal components analysis, binormamin rotation, generalized discriminant analysis, variants of generalized principal components analysis, simple structure rotation for one of the latter variants, and set component analysis. For most of these methods the algorithms appear to be new, for the others the existing algorithms turn out to be special cases of the newly derived general algorithms.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to Jos ten Berge for stimulating this research and for helpful comments on an earlier version of this paper.     

A consistent theory of attributes in a logic without contraction

This essay demonstrates proof-theoretically the consistency of a type-free theoryC with an unrestricted principle of comprehension and based on a predicate logic in which contraction (A (A B)) (A B), although it cannot holds in general, is provable for a wide range ofA's.C is presented as an axiomatic theoryCH (with a natural-deduction equivalentCS) as a finitary system, without formulas of infinite length. ThenCH is proved simply consistent by passing to a Gentzen-style natural-deduction systemCG that allows countably infinite conjunctions and in which all theorems ofCH are provable.CG is seen to be a consistent by a normalization argument. It also shown that in a senseC is highly non-extensional.     

A modification of kendall's tau for measuring association in contingency tables

A coefficient of association is described for a contingency table containing data classified into two sets of ordered categories. Within each of the two sets the number of categories or the number of cases in each category need not be the same.=+1 for perfect positive association and has an expectation of 0 for chance association. In many cases also has –1 as a lower limit. The limitations of Kendall's _a and _b and Stuart's _c are discussed, as is the identity of these coefficients to' under certain conditions. Computational procedure for is given.     

Perceptual scission of surface-lightness and illumination: An examination of the gelb effect

Summary An attempt was made to examine how the photometric equation: luminance (L)=albedo (A)×illuminance (I) could be solved perceptually when a test field (TF) was not seen as figure, but as ground. A gray disk with two black or white patches was used as the TF. Illuminance of the TF was changed over 2.3 log units and TF albedo was varied from 2.5 to 8.0 in Munsell value. Albedos of the black- and white-appearing patches were 1.5 and 9.5 in Munsell values, respectively. Two types of category judgments for apparent TF lightness (A) and apparent overall illumination (I) were made on the total of 40 TFs (5 illuminances×4 TF-albedos×2 patch-albedos). The results indicated that when the black patches were added to the TF, A was indistinguishable from I and when the white patches were placed on the TF, A and I could be distinguished from each other. The Gelb effect was interpreted as a manifestation of such A–I scission. It was concluded, therefore, that as far as the Gelb effect was observed, the perceptual system could solve the equation, L=A×I, in the sense that for a fixed L, the product of A and I would be constant.     

The Poznań school methodology of idealization and concretization from the point of view of a revised structuralist theory conception

My thesis is that some methodological ideas of the Pozna school, i.e., the principles of idealization and concretization (factualization), and the correspondence principle can be represented rather successfully using the relations of theoretization and specialization of revised structuralism.Let <n(i), t(j)> (i=1,...m, j=1,...k) denote the conceptual apparatus of a theory T, and a class M={} (i=1,...m, j=1,...k) the models of T. The n-components refer to the values of dependent variables and t-components to the values of independent variables of the theory. The n- and t-components in turn represent appropriate concepts. Consider T ^* as a conceptual enrichment of T with concepts <n(i ^*), t(j ^*)> (i<i ^* or j<j ^*) and models M ^*={<D ^*, n(i ^*), t(j ^*)>}. If the classes M and M ^* are suitably related, then the situation illustrates both the case of the theoretization-relation of (revised) structuralism and of the factualization-principle of the Pozna school.Assume now that the concepts n(i), t(j) of T for some i, j are operationalized using some special assumptions generating appropriate empirical values n and t for these concepts. Let M denote the class {<D,...n,...t,...>} which is formed by substituting n and t for values of concepts n(i), t(j) in the elements of M. If the classes M and M are related in a suitable way then the situation is an example of both the specialization-relation of (revised) structuralism and the concretization-principle of the Pozna school. The correspondence principle in turn can be represented as a limiting case of the theoretization-relation of (revised) structuralism.Many thanks to my anonymous referees for critical and fruitful comments and special thanks to Dr. Carol Norris for correcting the language of this paper.     

Syntactical investigations intoBI logic andBB′I logic

In this note, we will study four implicational logicsB, BI, BB and BBI. In [5], Martin and Meyer proved that a formula is provable inBB if and only if is provable inBBI and is not of the form of » . Though it gave a positive solution to theP - W problem, their method was semantical and not easy to grasp. We shall give a syntactical proof of the syntactical relation betweenBB andBBI logics. It also includes a syntactical proof of Powers and Dwyer's theorem that is proved semantically in [5]. Moreover, we shall establish the same relation betweenB andBI logics asBB andBBI logics. This relation seems to say thatB logic is meaningful, and so we think thatB logic is the weakest among meaningful logics. Therefore, by Theorem 1.1, our Gentzentype system forBI logic may be regarded as the most basic among all meaningful logics. It should be mentioned here that the first syntactical proof ofP - W problem is given by Misao Nagayama [6].Presented byHiroakira Ono     

Axiomatizing logics closely related to varieties

Let V be a s.f.b. (strongly finitely based, see below) variety of algebras. The central result is Theorem 2 saying that the logic defined by all matrices (A, d) with d A V is finitely based iff the A V have 1^st order definable cosets for their congruences. Theorem 3 states a similar axiomatization criterion for the logic determined by all matrices (A, ^A), A V, a term which is constant in V. Applications are given in a series of examples.     

Undecidability and intuitionistic incompleteness

Let S be a deductive system such that S-derivability (s) is arithmetic and sound with respect to structures of class K. From simple conditions on K and s, it follows constructively that the K-completeness of s implies MP(S), a form of Markov's Principle. If s is undecidable then MP(S) is independent of first-order Heyting arithmetic. Also, if s is undecidable and the S proof relation is decidable, then MP(S) is independent of second-order Heyting arithmetic, HAS. Lastly, when s is many-one complete, MP(S) implies the usual Markov's Principle MP.An immediate corollary is that the Tarski, Beth and Kripke weak completeness theorems for the negative fragment of intuitionistic predicate logic are unobtainable in HAS. Second, each of these: weak completeness for classical predicate logic, weak completeness for the negative fragment of intuitionistic predicate logic and strong completeness for sentential logic implics MP. Beth and Kripke completeness for intuitionistic predicate or sentential logic also entail MP.These results give extensions of the theorem of Gödel and Kreisel (in [4]) that completeness for pure intuitionistic predicate logic requires MP. The assumptions of Gödel and Kreisel's original proof included the Axiom of Dependent Choice and Herbrand's Theorem, no use of which is explicit in the present article.     

Depth relevance of some paraconsistent logics

The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If _DR- AB thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the number of nested 's required to reach the occurrence ofB inA. The method of proof is to show that a model structureM consisting of {M ₀ , M₁, ..., M}, where theM _i s are all characterized by Meyer's 6-valued matrices (c. f, [2]), satisfies the depth relevance condition. Then, it is shown thatM is a model structure for the systemDR.     

Predicate provability logic with non-modalized quantifiers

Predicate modal formulas with non-modalized quantifiers (call them Q-formulas) are considered as schemata of arithmetical formulas, where is interpreted as the provability predicate of some fixed correct extension T of arithmetic. A method of constructing 1) non-provable in T and 2) false arithmetical examples for Q-formulas by Kripke-like countermodels of certain type is given. Assuming the means of T to be strong enough to solve the (undecidable) problem of derivability in QGL, the Q-fragment of the predicate version of the logic GL, we prove the recursive enumerability of the sets of Q-formulas all arithmetical examples of which are: 1) T-provable, 2) true. In. particular, the first one is shown to be exactly QGL and the second one to be exactly the Q-fragment of the predicate version of Solovay's logic S.     

Paraconsistent logic and model theory

The object of this paper is to show how one is able to construct a paraconsistent theory of models that reflects much of the classical one. In other words the aim is to demonstrate that there is a very smooth and natural transition from the model theory of classical logic to that of certain categories of paraconsistent logic. To this end we take an extension of da Costa'sC ₁ ⁼ (obtained by adding the axiom A A) and prove for it results which correspond to many major classical model theories, taken from Shoenfield [5]. In particular we prove counterparts of the theorems of o-Tarski and Chang-o-Suszko, Craig-Robinson and the Beth definability theorem.     

Axiomatization and completeness of uncountably valued approximation logic

A first order uncountably valued logicL _Q(0,1) for management of uncertainty is considered. It is obtained from approximation logicsL _T of any poset type (T, ) (see Rasiowa [17], [18], [19]) by assuming (T, )=(Q(0, 1), ) — whereQ(0, 1) is the set of all rational numbersq such that 0<q<1 and is the arithmetic ordering — by eliminating modal connectives and adopting a semantics based onLT-fuzzy sets (see Rasiowa and Cat Ho [20], [21]). LogicL _Q(0,1) can be treated as an important case ofLT-fuzzy logics (introduced in Rasiowa and Cat Ho [21]) for (T, )=(Q(0, 1), ), i.e. asLQ(0, 1)-fuzzy logic announced in [21] but first examined in this paper.L _Q(0,1) deals with vague concepts represented by predicate formulas and applies approximate truth-values being certain subsets ofQ(0, 1). The set of all approximate truth-values consists of the empty set ø and all non-empty subsetss ofQ(0, 1) such that ifqs andqq, thenqs. The setLQ(0, 1) of all approximate truth-values is uncountable and covers up to monomorphism the closed interval [0, 1] of the real line.LQ(0, 1) is a complete set lattice and therefore a pseudo-Boolean (Heyting) algebra. Equipped with some additional operations it is a basic plain semi-Post algebra of typeQ(0, 1) (see Rasiowa and Cat Ho [20]) and is taken as a truth-table forL _Q(0,1) logic.L _Q(0,1) can be considered as a modification of Zadeh's fuzzy logic (see Bellman and Zadeh [2] and Zadeh and Kacprzyk, eds. [29]). The aim of this paper is an axiomatization of logicL _Q(0,1) and proofs of the completeness theorem and of the theorem on the existence ofLQ(0, 1)-models (i.e. models under the semantics introduced) for consistent theories based on any denumerable set of specific axioms. Proofs apply the theory of plain semi-Post algebras investigated in Cat Ho and Rasiowa [4].Presented byCecylia Rauszer     

Commensurability, incommensurability, and cumulativity in scientific knowledge

Until the middle of the present century it was a commonly accepted opinion that theory change in science was the expression of cumulative progress consisting in the acquisition of new truths and the elimination of old errors. Logical empiricists developed this idea through a deductive model, saying that a theory T superseding a theory T must be able logically to explain whatever T explained and something more as well. Popper too shared this model, but stressed that T explains the old known facts in its own new way. The further pursual of this line quickly led to the thesis of the non-comparability or incommensurability of theories: if T and T are different, then the very concepts which have the same denomination in both actually have different meanings; in such a way any sentence whatever has different meanings in T and in T and cannot serve to compare them. owing to this, the deductive model was abandoned as a tool for understanding theory change and scientific progress, and other models were proposed by people such as Lakatos, Kuhn, Feyerabend, Sneed and Stegmüller. The common feature of all these new positions may be seen in the claim that no possibility exists of interpreting theory change in terms of the cumulative acquisition of truth. It seems to us that the older and the newer positions are one-sided, and, in order to eliminate their respective shortcomings, we propose to interpret theory change in a new way.The starting point consists in recognizing that every scientific discipline singles out its specific domain of objects by selecting a few specific predicates for its discourse. Some of these predicates must be operational (that is, directly bound to testing operations) and they determine the objects of the theory concerned. In the case of a transition from T to T, we must consider whether or not the operational predicates remain unchanged, in the sense of being still related to the same operations. If they do not change in their relation to operations, then T and T are comparable (and may sometimes appear as compatible, sometimes as incompatible). If the operational predicates are not all identical in T and T, the two theories show a rather high degree of incommensurability, and this happens because they do not refer to the same objects. Theory change means in this case change of objects. But now we can see that even incommensurability is compatible with progress conceived as the accumulation of truth. Indeed, T and T remain true about their respective objects (T does not disprove T), and the global amount of truth acquired is increased.In other words, scientific progress does not consist in a purely logical relationship between theories, and moreover it is not linear. Yet it exists and may even be interpreted as an accumulation of truth, provided we do not forget that every scientific theory is true only about its own specific objects.It may be pointed out that the solution advocated here relies upon a limitation of the theory-ladeness of scientific concepts, which involves a reconsideration of their semantic status and a new approach to the question of theoretical concepts. First of all, the feature of being theoretical is attributed to a concept not absolutely, but relatively, yet in a sense different from Sneeds's: indeed every theory is basically characterized by its operational concepts, and the non-operational are said to be theoretical, this distinction clearly depending on every particular theory. For the operational concepts it happens that their mean-     

On a certain class ofm-address machines

Conclusion It follows from the proved theorems that ifM =Q, (whereQ={0,q ₁,q ₂,...,q }) is a machine of the classM _F then there exist machinesM _i such thatM _i(1,c)=M (q _i,c) andQ _i={0, 1, 2, ..., +1} (i=1, 2, ..., ).And thus, if the way in which to an initial function of content of memorycC a machine assigns a final onecC is regarded as the only essential property of the machine then we can deal with the machines of the formM ={0, 1, 2, ..., }, and processes (t) (wheret=1,c,cC) only.Such approach can simplify the problem of defining particular machines of the classM _F , composing and simplifying them.Allatum est die 19 Januarii 1970     

On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

On defining necessity in terms of entailment

In their book Entailment, Anderson and Belnap investigate the consequences of defining Lp (it is necessary that p) in system E as (pp)p. Since not all theorems are equivalent in E, this raises the question of whether there are reasonable alternative definitions of necessity in E. In this paper, it is shown that a definition of necessity in E satisfies the conditions { _E Lpp, _EL(pq)(LpLq), _E pLp} if and only if its has the form C ₁.C₂ .... C_np, where each C _iis equivalent in E to either pp or ((pp)p)p.     

Least-squares theory based on general distributional assumptions with an application to the incomplete observations problem

The linear regression modely=x+ is reanalyzed. Taking the modest position that x is an approximation of the best predictor ofy we derive the asymptotic distribution ofb andR ², under mild assumptions.The method of derivation yields an easy answer to the estimation of from a data set which contains incomplete observations, where the incompleteness is random.     

A new formulation of discussive logic

S. Jakowski introduced the discussive prepositional calculus D ₂as a basis for a logic which could be used as underlying logic of inconsistent but nontrivial theories (see, for example, N. C. A. da Costa and L. Dubikajtis, On Jakowski's discussive logic, in Non-Classical Logic, Model Theory and Computability, A. I. Arruda, N. C. A da Costa and R. Chuaqui edts., North-Holland, Amsterdam, 1977, 37–56). D ₂has afterwards been extended to a first-order predicate calculus and to a higher-order logic (cf. the quoted paper). In this paper we present a natural version of D ₂, in the sense of Jakowski and Gentzen; as a consequence, we suggest a new formulation of the discussive predicate calculus (with equality). A semantics for the new calculus is also presented.