A 2-categorial Generalization of the Concept of Institution

After defining, for each many-sorted signature Σ = (S, Σ), the category Ter(Σ), of generalized terms for Σ (which is the dual of the Kleisli category for \mathbb T_S{\mathbb {T}_{\bf \Sigma}}, the monad in Set ^S determined by the adjunction T_S \dashv G_S{{\bf T}_{\bf \Sigma} \dashv {\rm G}_{\bf \Sigma}} from Set ^S to Alg(Σ), the category of Σ-algebras), we assign, to a signature morphism d from Σ to Λ, the functor d_à{{\bf d}_\diamond} from Ter(Σ) to Ter(Λ). Once defined the mappings that assign, respectively, to a many-sorted signature the corresponding category of generalized terms and to a signature morphism the functor between the associated categories of generalized terms, we state that both mappings are actually the components of a pseudo-functor Ter from Sig to the 2-category Cat. Next we prove that there is a functor Tr^Σ, of realization of generalized terms as term operations, from Alg(Σ) × Ter(Σ) to Set, that simultaneously formalizes the procedure of realization of generalized terms and its naturalness (by taking into account the variation of the algebras through the homomorphisms between them). We remark that from this fact we will get the invariance of the relation of satisfaction under signature change. Moreover, we prove that, for each signature morphism d from Σ to Λ, there exists a natural isomorphism θ ^d from the functor Tr^L °(Id ×d_à){{{\rm Tr}^{\bf {\bf \Lambda}} \circ ({\rm Id} \times {\bf d}_\diamond)}} to the functor Tr^S °(d^* ×Id){{\rm Tr}^{\bf \Sigma} \circ ({\bf d}^* \times {\rm Id})}, both from the category Alg(Λ) × Ter(Σ) to the category Set, where d* is the value at d of the arrow mapping of a contravariant functor Alg from Sig to Cat, that shows the invariant character of the procedure of realization of generalized terms under signature change. Finally, we construct the many-sorted term institution by combining adequately the above components (and, in a derived way, the many-sorted specification institution), but for a strict generalization of the standard notion of institution.     

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

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.     

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 ${\mathcal{B} \mathcal{A} \vee \mathcal{L} \mathcal{G}}$ .     

Priestley Duality for Paraconsistent Nelson’s Logic

The variety of N4^{^}{{\bf N4}^\perp}-lattices provides an algebraic semantics for the logic N4^{^}{{\bf N4}^\perp} , a version of Nelson’s logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of N4^{^}{{\bf N4}^\perp}-lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson algebras constructed by R. Cignoli and A. Sendlewski.     

On the dynamics of institutional agreements

In this paper we investigate a logic for modelling individual and collective acceptances that is called acceptance logic. The logic has formulae of the form A_G:x j{\rm A}_{G:x} \varphi reading ‘if the agents in the set of agents G identify themselves with institution x then they together accept that j{\varphi} ’. We extend acceptance logic by two kinds of dynamic modal operators. The first kind are public announcements of the form x!y{x!\psi}, meaning that the agents learn that y{\psi} is the case in context x. Formulae of the form [x!y]j{[x!\psi]\varphi} mean that j{\varphi} is the case after every possible occurrence of the event x!ψ. Semantically, public announcements diminish the space of possible worlds accepted by agents and sets of agents. The announcement of ψ in context x makes all \lnoty{\lnot\psi} -worlds inaccessible to the agents in such context. In this logic, if the set of accessible worlds of G in context x is empty, then the agents in G are not functioning as members of x, they do not identify themselves with x. In such a situation the agents in G may have the possibility to join x. To model this we introduce here a second kind of dynamic modal operator of acceptance shifting of the form G:x-y{G:x\uparrow\psi}. The latter means that the agents in G shift (change) their acceptances in order to accept ψ in context x. Semantically, they make ψ-worlds accessible to G in the context x, which means that, after such operation, G is functioning as member of x (unless there are no ψ-worlds). We show that the resulting logic has a complete axiomatization in terms of reduction axioms for both dynamic operators. In the paper we also show how the logic of acceptance and its dynamic extension can be used to model some interesting aspects of judgement aggregation. In particular, we apply our logic of acceptance to a classical scenario in judgment aggregation, the so-called ‘doctrinal paradox’ or ‘discursive dilemma’ (Pettit, Philosophical Issues 11:268–299, 2001; Kornhauser and Sager, Yale Law Journal 96:82–117, 1986).     

The Geometry of Enhancement in Multiple Regression

In linear multiple regression, “enhancement” is said to occur when R ²=b′r>r′r, where b is a p×1 vector of standardized regression coefficients and r is a p×1 vector of correlations between a criterion y and a set of standardized regressors, x. When p=1 then b≡r and enhancement cannot occur. When p=2, for all full-rank R _xx≠I, R _xx=E[xx′]=V Λ V′ (where V Λ V′ denotes the eigen decomposition of R _xx; λ ₁>λ ₂), the set B₁:={b_i:R²=b_i￠r_i=r_i￠r_i;0 < R² ￡ 1}\boldsymbol{B}_{1}:=\{\boldsymbol{b}_{i}:R^{2}=\boldsymbol{b}_{i}'\boldsymbol{r}_{i}=\boldsymbol{r}_{i}'\boldsymbol{r}_{i};0R² ￡ 1;R²l_p ￡ r_i￠r_i < R²}0p≥3 (and λ ₁>λ ₂>⋯>λ _p ), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B ₁ occurs at the intersection of two hyper-ellipsoids in ℝ ^p . Equations are provided for populating the sets B ₁ and B ₂ and for demonstrating that maximum enhancement occurs when b is collinear with the eigenvector that is associated with λ _p (the smallest eigenvalue of the predictor correlation matrix). These equations are used to illustrate the logic and the underlying geometry of enhancement in population, multiple-regression models. R code for simulating population regression models that exhibit enhancement of any degree and any number of predictors is included in Appendices A and B.     

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.     

Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

A scaled difference test statistic [(T)\tilde]_d\tilde{T}{}_{d} that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic [(T)\tilde]_d\tilde{T}_{d} is asymptotically equivalent to the scaled difference test statistic [`(T)]<sub>d</sub>\bar{T}_{d} introduced in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecker, pp. 233–247, 2000), which requires more involved computations beyond standard output of SEM software. The test statistic [(T)\tilde]<sub>d</sub>\tilde{T}_{d} has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic [`(T)]_d\bar{T}_{d} that avoids negative chi-square values.     

Finite basis problems and results for quasivarieties

Let be a finite collection of finite algebras of finite signature such that SP( ) has meet semi-distributive congruence lattices. We prove that there exists a finite collection ₁ of finite algebras of the same signature, , such that SP( ₁) is finitely axiomatizable.We show also that if , then SP( ₁) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates.While working on this paper, the first author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant no. T37877 and the second author was supported by the US National Science Foundation grant no. DMS-0245622.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

A note on the ω-incompleteness formalization

The paper studies two formal schemes related to -completeness.LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where (x) is a formula with the only free variable x): (a) (for anyn) ( _T (n)), (b) _{T x} Pr _T (^–(x)^–) and (c) _T x(x) (the notational conventions are those of Smoryski [3]). The aim of this paper is to examine the meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) (c) and (a) (b), where ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) (c) is equivalent to ¬Cons _T and 2. the schema corresponding to (a) (b) is not consistent with 1-CON _T. The former result follows from a simple adaptation of the -incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma.Presented byMelvin Fitting     

Actual truth,possible knowledge

The well-known argument of Frederick Fitch, purporting to show that verificationism (= Truth implies knowability) entails the absurd conclusion that all the truths are known, has been disarmed by Dorothy Edgington's suggestion that the proper formulation of verificationism presupposes that we make use of anactuality operator along with the standardly invoked epistemic and modal operators. According to her interpretation of verificationism, the actual truth of a proposition implies that it could be known in some possible situation that the proposition holds in theactual situation. Thus, suppose that our object language contains the operatorA — it is actually the case that ... — with the following truth condition: _vA iff _w0, wherew ₀ stands for the designated world of the model — the actual world. Then we can formalize the verificationist claim as follows:

Specified Meet Contraction

Specified meet contraction is the operation ?\div defined by the identity K ?p = K   ~  f(p)K \div p = K \,{\sim}\, f(p) where ∼ is full meet contraction and f is a sentential selector, a function from sentences to sentences. With suitable conditions on the sentential selector, specified meet contraction coincides with the partial meet contractions that yield a finite-based contraction outcome if the original belief set is finite-based. In terms of cognitive realism, specified meet contraction has an advantage over partial meet contraction in that the selection mechanism operates on sentences rather than on temporary infinite structures (remainders) that are cognitively inaccessible. Specified meet contraction provides a versatile framework in which other types of contraction, such as severe withdrawal and base-generated contraction, can be expressed with suitably chosen properties of the sentential selector.     

On fragments of Medvedev's logic

Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectivesΦ such that \(\{ \to , \vee , \urcorner \} \not \subseteq \Phi \subseteq \{ \to , \wedge , \urcorner \} \) theΦ-fragment ofMV equals theΦ fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, namely: is an intermediate logic based on the axiom (?a→b∨c) →(?a→b)∨(?a → c) separable?     

Valuation Semantics for First-Order Logics of Evidence and Truth

This paper introduces the logic QLET_F, a quantified extension of the logic of evidence and truth LET_F, together with a corresponding sound and complete first-order non-deterministic valuation semantics. LET_F is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment (FDE) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘A entails that A behaves classically, ∙A follows from A’s violating some classically valid inferences. The semantics of QLET_F combines structures that interpret negated predicates in terms of anti-extensions with first-order non-deterministic valuations, and completeness is obtained through a generalization of Henkin’s method. By providing sound and complete semantics for first-order extensions of FDE, K3, and LP, we show how these tools, which we call here the method of anti-extensions + valuations, can be naturally applied to a number of non-classical logics.

     

Dominions in quasivarieties of universal algebras

The dominion of a subalgebra H in an universal algebra A (in a class ) is the set of all elements such that for all homomorphisms if f, g coincide on H, then a^f = a^g. We investigate the connection between dominions and quasivarieties. We show that if a class is closed under ultraproducts, then the dominion in is equal to the dominion in a quasivariety generated by . Also we find conditions when dominions in a universal algebra form a lattice and study this lattice.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

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.     

Decomposability of the Finitely Generated Free Hoop Residuation Algebra

In this paper we prove that, for n > 1, the n-generated free algebra in any locally finite subvariety of HoRA can be written in a unique nontrivial way as Ł₂ × A′, where A′ is a directly indecomposable algebra in . More precisely, we prove that the unique nontrivial pair of factor congruences of is given by the filters and , where the element is recursively defined from the term introduced by W. H. Cornish. As an additional result we obtain a characterization of minimal irreducible filters of in terms of its coatoms. Presented by Daniele Mundici