Aspects of analytic deduction

Let ? be the ordinary deduction relation of classical first-order logic. We provide an “analytic” subrelation ?³ of ? which for propositional logic is defined by the usual “containment” criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq Atom(\Gamma ),$$ whereas for predicate logic, ?^a is defined by the extended criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq ' Atom(\Gamma ),$$ where Atom(?) $ \subseteq '$ Atom(Γ) means that every atomic formula occurring in ? “essentially occurs” also in Γ. If Γ, ? are quantifier-free, then the notions “occurs” and “essentially occurs” for atoms between Γ and ? coincide. If ? is formalized by Gentzen's calculus of sequents, then we show that ?^a is axiomatizable by a proper fragment of analytic inference rules. This is mainly due to cut elimination. By “analytic inference rule” we understand here a rule r such that, if the sequent over the line is analytic, then so is the sequent under the line. We also discuss the notion of semantic relevance as contrasted to the previous syntactic one. We show that when introducing semantic sequents as axioms, i.e. when extending the pure logical axioms and rules by mathematical ones, the property of syntactic relevance is lost, since cut elimination no longer holds. We conclude that no purely syntactic notion of analytic deduction can ever replace successfully the complex semantico-syntactic deduction we already possess.     

Positive modal logic

We give a set of postulates for the minimal normal modal logicK ₊ without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are

Fregean logics with the multiterm deduction theorem and their algebraization

A deductive system $\mathcal{S}$ (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas $$\{ \left\langle {\alpha ,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \} ,$$ is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the deduction theorem of the classical and intuitionistic propositional calculi (IPC) in which a finite system of possibly compound formulas collectively plays the role of the implication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the first evidence of a significant difference between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective.     

A Routley-Meyer Type Semantics for Relevant Logics Including B<Subscript>r</Subscript> Plus the Disjunctive Syllogism

Routley-Meyer type ternary relational semantics are defined for relevant logics including Routley and Meyer’s basic logic B plus the reductio rule \( \vdash A\rightarrow \lnot A\Rightarrow \vdash \lnot A\) and the disjunctive syllogism. Standard relevant logics such as E and R (plus γ) and Ackermann’s logics of ‘strenge Implikation’ Π and Π^′ are among the logics considered.     

Theories with the Independence Property

A first-order theory T{{\mathcal T}} has the Independence Property provided T   \vdash (Q)(FT F₁ ú. . .úF_n){{{\mathcal T} \, \, \vdash (Q)(\Phi \Rightarrow {\Phi_1} \vee.\,.\,.\vee {\Phi_n})}} implies T   \vdash (Q)(FT F_i){{{\mathcal T} \, \, \vdash (Q)(\Phi \Rightarrow {\Phi_i})}} for some i whenever F,F₁, . . . ,F_n{{\Phi,\Phi_1,\,.\,.\,.\,,\Phi_n}} are formulae of a suitable type and (Q) is any quantifier sequence. Variants of this property have been noticed for some time in logic programming and in linear programming.     

Intermediate Logics Admitting a Structural Hypersequent Calculus

Studia Logica - We characterise the intermediate logics which admit a cut-free hypersequent calculus of the form $$\mathbf {HLJ} + \mathscr {R}$$ , where $$\mathbf {HLJ}$$ is the hypersequent...     

Countably Many Weakenings of Belnap–Dunn Logic

Studia Logica - Every Berman’s variety $$\mathbb {K}_p^q$$ which is the subvariety of Ockham algebras defined by the equation $${\sim ^{2p+q}}a = {\sim ^q}a$$ ($$p\ge 1$$ and $$q\ge 0$$)...     

A Deterministic Weakening of Belnap–Dunn Logic

Studia Logica - A deterministic weakening $$\mathsf {DW}$$ of the Belnap–Dunn four-valued logic $$\mathsf {BD}$$ is introduced to formalize the acceptance and rejection of a proposition at a...     

Simplified Kripke-Style Semantics for Some Normal Modal Logics

Studia Logica - Pietruszczak (Bull Sect Log 38(3/4):163–171, 2009. https://doi.org/10.12775/LLP.2009.013) proved that the normal logics $$\mathrm {K45}$$, $$\mathrm {KB4}$$ ($$=\mathrm...     

A Propositional Dynamic Logic for Instantial Neighborhood Semantics

Studia Logica - We propose a new perspective on logics of computation by combining instantial neighborhood logic $$\mathsf {INL}$$ with bisimulation safe operations adapted from $$\mathsf {PDL}$$ ....     

Axiomatizing collective judgment sets in a minimal logical language

We investigate under what conditions a given set of collective judgments can arise from a specific voting procedure. In order to answer this question, we introduce a language similar to modal logic for reasoning about judgment aggregation procedures. In this language, the formula expresses that is collectively accepted, or that is a group judgment based on voting. Different judgment aggregation procedures may be underlying the group decision making. Here we investigate majority voting, where holds if a majority of individuals accepts, consensus voting, where holds if all individuals accept, and dictatorship. We provide complete axiomatizations for judgment sets arising from all three aggregation procedures.     

Confluence Proofs of Lambda-Mu-Calculi by Z Theorem

Studia Logica - This paper applies Dehornoy et al.’s Z theorem and its variant, called the compositional Z theorem, to prove confluence of Parigot’s $$\lambda \mu $$ -calculi extended...     

Wanting what’s not best

Philosophical Studies - In this paper, we propose a novel account of desire reports, i.e. sentences of the form $$\ulcorner$$ S wants p $$\urcorner$$ . Our theory is partly motivated by...     

Completeness theorems for some intermediate predicate calculi

We give completeness results — with respect to Kripke's semantic — for the negation-free intermediate predicate calculi: (1) $$\begin{gathered} BD = positive predicate calculus PQ + B:(\alpha \to \beta )v(\beta \to \alpha ) \hfill \\ + D:\forall x\left( {a\left( x \right)v\beta } \right) \to \forall xav\beta \hfill \\ \end{gathered}$$ (2) $$T_n D = PQ + T_n :\left( {a_0 \to a_1 } \right)v \ldots v\left( {a_n \to a_{n + 1} } \right) + D\left( {n \geqslant 0} \right)$$ and the superintuitionistic predicate calculus: (3) $$B^1 DH_2^ \urcorner = BD + intuitionistic negation + H_2^ \urcorner : \urcorner \forall xa \to \exists x \urcorner a.$$ The central point is the completeness proof for (1), which is obtained modifying Klemke's construction [3]. For a general account on negation-free intermediate predicate calculi — see Casari-Minari [1]; for an algebraic treatment of some superintuitionistic predicate calculi involving schemasB andD — see Horn [4] and Görnemann [2].     

Expanding Quasi-MV Algebras by a Quantum Operator

We investigate an expansion of quasi-MV algebras ([10]) by a genuine quantum unary operator. The variety of such quasi-MV algebras has a subquasivariety whose members—called cartesian—can be obtained in an appropriate way out of MV algebras. After showing that cartesian . quasi-MV algebras generate ,we prove a standard completeness theorem for w.r.t. an algebra over the complex numbers. Presented by Heinrich Wansing     

Empirical bayes estimates of domain scores under binomial and hypergeometric distributions for test scores

We introduce two simple empirical approximate Bayes estimators (EABEs)— and —for estimating domain scores under binomial and hypergeometric distributions, respectively. Both EABEs (derived from corresponding marginal distributions of observed test scorex without relying on knowledge of prior domain score distributions) have been proven to hold -asymptotic optimality in Robbins' sense of convergence in mean. We found that, where and are the monotonized versions of and under Van Houwelingen's monotonization method, respectively, the convergence rate of the overall expected loss of Bayes risk in either or depends on test length, sample size, and ratio of test length to size of domain items. In terms of conditional Bayes risk, and outperform their maximum likelihood counterparts over the middle range of domain scales. In terms of mean-squared error, we also found that: (a) given a unimodal prior distribution of domain scores, performs better than both and a linear EBE of the beta-binomial model when domain item size is small or when test items reflect a high degree of heterogeneity; (b) performs as well as when prior distribution is bimodal and test items are homogeneous; and (c) the linear EBE is extremely robust when a large pool of homogeneous items plus a unimodal prior distribution exists.The authors are indebted to both anonymous reviewers, especially Reviewer 2, and the Editor for their invaluable comments and suggestions. Thanks are also due to Yuan-Chin Chang and Chin-Fu Hsiao for their help with our simulation and programming work.     

Correction to: The Hahn Embedding Theorem for a Class of Residuated Semigroups

Studia Logica - Let $${\mathcal {O}}$$ be the class of odd involutive $$\hbox {FL}_e$$ -chains which admit finitely many idempotent elements. The price for describing the structure of such a rich...     

An optimal property of least squares weights in prediction models

In predicting scores fromp > 1 observed scores in a sample of sizeñ, the optimal strategy (minimum expected loss), under certain assumptions, is shown to be based upon the least squares regression weights computed from a previous sample. Letting represent the correlation between and the predicted values , and letting represent the correlation between and a different set of predicted values , where w is any weighting system which is not a function of , it is shown that the probability of being less than cannot exceed .50. The relationship of this result to previous research and practical implications are discussed.     

Equivalents for a Quasivariety to be Generated by a Single Structure

We present some equivalent conditions for a quasivariety of structures to be generated by a single structure. The first such condition, called the embedding property was found by A.I. Mal′tsev in [6]. It says that if are nontrivial, then there exists such that A and B are embeddable into C. One of our equivalent conditions states that the set of quasi-identities valid in is closed under a certain Gentzen type rule which is due to J. Łoś and R. Suszko [5]. Presented by Jacek Malinowski