On the axiomatization of finiteK-frames

We find a short way to construct a formula which axiomatizes a given finite frame of the modal logicK, in the sense that for each finite frameA, we construct a formula ωA which holds in those and only those frames in which every formula true inA holds. To obtain this result we find, for each finite model \(\mathfrak{A}\) and each natural numbern, a formula ω \(\mathfrak{A}\) which holds in those and only those models in which every formula true in \(\mathfrak{A}\) , and involving the firstn propositional letters, holds.     

The representation of Takeuti's\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} -operator

Gaisi Takeuti has recently proposed a new operation on orthomodular latticesL, \(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \) :P(L)»L. The properties of \(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \) suggest that the value of \(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \) (A) (A) \( \subseteq \) L) corresponds to the degree in which the elements ofA behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular latticesL and the existence of two-valued homomorphisms onL.     

On Birkhoff’s Common Abstraction Problem

In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his readers to develop a ??common abstraction?? that includes Boolean algebras and latticeordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of ${\mathcal{B} \mathcal{A}}$ and ${\mathcal{L} \mathcal{G}}$ their join ${\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}$ in the lattice of subvarieties of ${\mathcal{F} \mathcal{L}}$ (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for ${\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}$ relative to ${\mathcal{F} \mathcal{L}}$ . Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing ${\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}$ .     

Relation Formulas for Protoalgebraic Equality Free Quasivarieties; Pałasińska’s Theorem Revisited

We provide a new proof of the following Pa?asińska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are ${\mathcal{Q}}$ Q -relation formulas for a protoalgebraic equality free quasivariety ${\mathcal{Q}}$ Q . They are the counterparts of the congruence formulas used for describing the generation of congruences in algebras. Having this tool in hand, we prove a finite axiomatization theorem for ${\mathcal{Q}}$ Q when it has definable principal ${\mathcal{Q}}$ Q -subrelations. This is a property obtained by carrying over the definability of principal subcongruences, invented by Baker and Wang for varieties, and which holds for finitely generated protoalgebraic relation distributive equality free quasivarieties.     

Disjunctions in closure spaces

The main result of this paper is the following theorem: a closure space X has an 〈α, δ, Q〉-regular base of the power \(\mathfrak{n}\) iff X is Q-embeddable in \(B_{\alpha ,\delta }^\mathfrak{n} \) It is a generalization of the following theorems:

1. Stone representation theorem for distributive lattices (α = 0, δ = ω, Q = ω),
2. universality of the Alexandroff's cube for T ₀-topological spaces (α = ω, δ = ∞, Q = 0),
3. universality of the closure space of filters in the lattice of all subsets for 〈α, δ〉-closure spaces (Q = 0).

By this theorem we obtain some characterizations of the closure space \(F_\mathfrak{m} \) given by the consequence operator for the classical propositional calculus over a formalized language of the zero order with the set of propositional variables of the power \(\mathfrak{m}\) . In particular we prove that a countable closure space X is embeddable with finite disjunctions preserved into F _ω iff X is a consistent closure space satisfying the compactness theorem and X contains a 〈0, ω〉-base consisting of ω-prime sets. This paper is a continuation of [7], [2] and [3].     

Semi-intuitionistic Logic

The purpose of this paper is to define a new logic ${\mathcal {SI}}$ called semi-intuitionistic logic such that the semi-Heyting algebras introduced in [4] by Sankappanavar are the semantics for ${\mathcal {SI}}$ . Besides, the intuitionistic logic will be an axiomatic extension of ${\mathcal {SI}}$ .     

What is Nominalistic Mereology?

Hybrid languages are introduced in order to evaluate the strength of “minimal” mereologies with relatively strong frame definability properties. Appealing to a robust form of nominalism, I claim that one investigated language $\mathcal {H}_{\textsf {m}}$ is maximally acceptable for nominalistic mereology. In an extension $\mathcal {H}_{\textsf {gem}}$ of $\mathcal {H}_{\textsf {m}}$ , a modal analog for the classical systems of Leonard and Goodman (J Symb Log 5:45–55, 1940) and Le?niewski (1916) is introduced and shown to be complete with respect to 0-deleted Boolean algebras. We characterize the formulas of first-order logic invariant for $\mathcal {H}_{\textsf {gem}}$ -bisimulations.     

The sampling distribution ofd′

The distribution of sample $\hat d's$ , although mathematically intractable, can be tabulated readily by computer. Such tabulations reveal a number of interesting properties of this distribution, including: (1) sample $\hat d's$ are biased, with an expected value that can be higher or lower than the true value, depending on the sample size, the true value itself, and the convention adopted for handling cases in which the sample $\hat d'$ is undefined; (2) the variance of $\hat d'$ also depends on the convention adopted for handling cases in which the sample $\hat d'$ is undefined and is in some cases poorly approximated by the standard approximation formula, (3) the standard formula for a confidence interval for $\hat d'$ is quite accurate with at least 50–100 trials per condition, but more accurate intervals can be obtained by direct computation with smaller samples.     

Dependence and Independence

We introduce an atomic formula ${\vec{y} \bot_{\vec{x}}\vec{z}}$ intuitively saying that the variables ${\vec{y}}$ are independent from the variables ${\vec{z}}$ if the variables ${\vec{x}}$ are kept constant. We contrast this with dependence logic ${\mathcal{D}}$ based on the atomic formula = ${(\vec{x}, \vec{y})}$ , actually equivalent to ${\vec{y} \bot_{\vec{x}}\vec{y}}$ , saying that the variables ${\vec{y}}$ are totally determined by the variables ${\vec{x}}$ . We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ can be used to give partially ordered quantifiers and IF-logic an alternative interpretation without some of the shortcomings related to so called signaling that interpretations using = ${(\vec{x}, \vec{y})}$ have.     

The degrees of maximality of the intuitionistic propositional logic and of some of its fragments

Professor Ryszard Wójcicki once asked whether the degree of maximality of the consequence operationC determined by the theorems of the intuitionistic propositional logic and the detachment rule for the implication connective is equal to \(2^{2^\aleph 0} \) ? The aim of the present paper is to give the affirmative answer to the question. More exactly, it is proved here that the degree of maximality ofC _Ψ — theΨ — fragment ofC, is equal to \(2^{2^\aleph 0} \) , for every such that → εΨ.     

Canonical Extensions and Discrete Dualities for Finitely Generated Varieties of Lattice-based Algebras

The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety ${\mathcal {A}}$ is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory to show that topological and discrete dualities for ${\mathcal {A}}$ exist in partnership.     

The Church–Fitch knowability paradox in the light of structural proof theory

Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$ , by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$ (KP). The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$ (OP). A Gentzen-style reconstruction of the Church–Fitch paradox is presented following a labelled approach to sequent calculi. First, a cut-free system for classical (resp. intuitionistic) bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism.     

Indistinguishability, Choices, and Logics of Agency

This paper deals with structures ${\langle{\bf T}, I\rangle}$ in which T is a tree and I is a function assigning each moment a partition of the set of histories passing through it. The function I is called indistinguishability and generalizes the notion of undividedness. Belnap’s choices are particular indistinguishability functions. Structures ${\langle{\bf T}, I\rangle}$ provide a semantics for a language ${\mathcal{L}}$ with tense and modal operators. The first part of the paper investigates the set-theoretical properties of the set of indistinguishability classes, which has a tree structure. The significant relations between this tree and T are established within a general theory of trees. The aim of second part is testing the expressive power of the language ${\mathcal{L}}$ . The natural environment for this kind of investigations is Belnap’s seeing to it that (stit). It will be proved that the hybrid extension of ${\mathcal{L}}$ (with a simultaneity operator) is suitable for expressing stit concepts in a purely temporal language.     

Resolution of Algebraic Systems of Equations in the Variety of Cyclic Post Algebras

There is a constructive method to define a structure of simple k-cyclic Post algebra of order p, L _p,k , on a given finite field F(p ^k ), and conversely. There exists an interpretation ??₁ of the variety ${\mathcal{V}(L_{p,k})}$ generated by L _p,k into the variety ${\mathcal{V}(F(p^k))}$ generated by F(p ^k ) and an interpretation ??₂ of ${\mathcal{V}(F(p^k))}$ into ${\mathcal{V}(L_{p,k})}$ such that ??₂??₁(B) =  B for every ${B \in \mathcal{V}(L_{p,k})}$ and ??₁??₂(R) =  R for every ${R \in \mathcal{V}(F(p^k))}$ . In this paper we show how we can solve an algebraic system of equations over an arbitrary cyclic Post algebra of order p, p prime, using the above interpretation, Gröbner bases and algorithms programmed in Maple.     

Probability Propagation in Generalized Inference Forms

Probabilistic inference forms lead from point probabilities of the premises to interval probabilities of the conclusion. The probabilistic version of Modus Ponens, for example, licenses the inference from \({P(A) = \alpha}\) and \({P(B|A) = \beta}\) to \({P(B)\in [\alpha\beta, \alpha\beta + 1 - \alpha]}\) . We study generalized inference forms with three or more premises. The generalized Modus Ponens, for example, leads from \({P(A_{1}) = \alpha_{1}, \ldots, P(A_{n})= \alpha_{n}}\) and \({P(B|A_{1} \wedge \cdots \wedge A_{n}) = \beta}\) to an according interval for P(B). We present the probability intervals for the conclusions of the generalized versions of Cut, Cautious Monotonicity, Modus Tollens, Bayes’ Theorem, and some SYSTEM O rules. Recently, Gilio has shown that generalized inference forms “degrade”—more premises lead to less precise conclusions, i.e., to wider probability intervals of the conclusion. We also study Adam’s probability preservation properties in generalized inference forms. Special attention is devoted to zero probabilities of the conditioning events. These zero probabilities often lead to different intervals in the coherence and the Kolmogorov approach.     

Acquisition and decision in visual same-different search of letter displays

From 2 to 23 capital As, Bs, and Cs were positioned randomly over visual displays varying in sizefrom5 to 10deg square and in luminance from7 to 250cd/m², The task was to decide whether all letters were the same or one was different from the rest. Instructions stressed accuracy, and responses were 97% correct. Three experiments with 50 observers varied amount of practice, number of letters (N), and size and luminance of the display. All experiments produced a linear invariance between mean “same” \((\bar S)\) and mean “different” \((\bar D)\) response times in seconds with N as the parameter: \(\bar D\) ? \(\bar S\) /2+4. The data are consistent with Krueger’s same-different decision theory, and with the separation of acquisition from decision processes.