On Endomorphisms of Ockham Algebras with Pseudocomplementation

A pO-algebra ${(L; f, \, ^{\star})}$ is an algebra in which (L; f) is an Ockham algebra, ${(L; \, ^{\star})}$ is a p-algebra, and the unary operations f and ${^{\star}}$ commute. Here we consider the endomorphism monoid of such an algebra. If ${(L; f, \, ^{\star})}$ is a subdirectly irreducible pK _1,1- algebra then every endomorphism ${\vartheta}$ is a monomorphism or ${\vartheta^3 = \vartheta}$ . When L is finite the endomorphism monoid of L is regular, and we determine precisely when it is a Clifford monoid.     

Algebraic Functions

Let A be an algebra. We say that the functions f ₁, . . . , f _m : A ⁿ ?? A are algebraic on A provided there is a finite system of term-equalities ${{\bigwedge t_{k}(\overline{x}, \overline{z}) = s_{k}(\overline{x}, \overline{z})}}$ satisfying that for each ${{\overline{a} \in A^{n}}}$ , the m-tuple ${{(f_{1}(\overline{a}), \ldots , f_{m}(\overline{a}))}}$ is the unique solution in A ^m to the system ${{\bigwedge t_{k}(\overline{a}, \overline{z}) = s_{k}(\overline{a}, \overline{z})}}$ . In this work we present a collection of general tools for the study of algebraic functions, and apply them to obtain characterizations for algebraic functions on distributive lattices, Stone algebras, finite abelian groups and vector spaces, among other well known algebraic structures.     

Completeness in Hybrid Type Theory

We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret \(@_i\) in propositional and first-order hybrid logic. This means: interpret \(@_i\alpha _a\) , where \(\alpha _a\) is an expression of any type \(a\) , as an expression of type \(a\) that rigidly returns the value that \(\alpha_a\) receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and (as is usual inhybrid logic) we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logicover Henkin’s logic.     

Continuous Orthogonal Complement Functions and Distribution-Free Goodness of Fit Tests in Moment Structure Analysis

It is shown that for any full column rank matrix X ₀ with more rows than columns there is a neighborhood $\mathcal{N}$ of X ₀ and a continuous function f on $\mathcal{N}$ such that f(X) is an orthogonal complement of X for all X in $\mathcal{N}$ . This is used to derive a distribution free goodness of fit test for covariance structure analysis. This test was proposed some time ago and is extensively used. Unfortunately, there is an error in the proof that the proposed test statistic has an asymptotic χ ² distribution. This is a potentially serious problem, without a proof the test statistic may not, in fact, be asymptoticly χ ². The proof, however, is easily fixed using a continuous orthogonal complement function. Similar problems arise in other applications where orthogonal complements are used. These can also be resolved by using continuous orthogonal complement functions.     

Inverse Images of Box Formulas in Modal Logic

We investigate, for several modal logics but concentrating on KT, KD45, S4 and S5, the set of formulas B for which ${\square B}$ is provably equivalent to ${\square A}$ for a selected formula A (such as p, a sentence letter). In the exceptional case in which a modal logic is closed under the (‘cancellation’) rule taking us from ${\square C \leftrightarrow \square D}$ to ${C \leftrightarrow D}$ , there is only one formula B, to within equivalence, in this inverse image, as we shall call it, of ${\square A}$ (relative to the logic concerned); for logics for which the intended reading of “ ${\square}$ ” is epistemic or doxastic, failure to be closed under this rule indicates that from the proposition expressed by a knowledge- or belief-attribution, the propositional object of the attitude in question cannot be recovered: arguably, a somewhat disconcerting situation. More generally, the inverse image of ${\square A}$ may comprise a range of non-equivalent formulas, all those provably implied by one fixed formula and provably implying another—though we shall see that for several choices of logic and of the formula A, there is not even such an ‘interval characterization’ of the inverse image (of ${\square A}$ ) to be found.     

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.     

Dynamics of lying

We propose a dynamic logic of lying, wherein a ‘lie that $\varphi $ ’ (where $\varphi $ is a formula in the logic) is an action in the sense of dynamic modal logic, that is interpreted as a state transformer relative to the formula $\varphi $ . The states that are being transformed are pointed Kripke models encoding the uncertainty of agents about their beliefs. Lies can be about factual propositions but also about modal formulas, such as the beliefs of other agents or the belief consequences of the lies of other agents. We distinguish two speaker perspectives: (Obs) an outside observer who is lying to an agent that is modelled in the system, and (Ag) an agent who is lying to another agent, and where both are modelled in the system. We distinguish three addressee perspectives: (Cred) the credulous agent who believes everything that it is told (even at the price of inconsistency), (Skep) the skeptical agent who only believes what it is told if that is consistent with its current beliefs, and (Rev) the belief revising agent who believes everything that it is told by consistently revising its current, possibly conflicting, beliefs. The logics have complete axiomatizations, which can most elegantly be shown by way of their embedding in what is known as action model logic or in the extension of that logic to belief revision.     

IF Modal Logic and Classical Negation

The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as ‘Simple IF modal logic’ (IFML _s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML _s is not equivalent to any formula of basic modal logic (ML). We generalize the notion of bisimulation familiar from ML; the resulting asymmetric simulation concept is used to prove that IFML _s is not closed under complementation. In fact we obtain a much stronger result: the only IFML _s formulas admitting their classical negation to be expressed in IFML _s itself are those whose truth-condition is in fact expressible in ML.     

Finite axiomatization for some intermediate logics

LetN. be the set of all natural numbers (except zero), and letD _n ^* = {k ∈N ∶k|n} ∪ {0} wherek¦n if and only ifn=k.x f or somex∈N. Then, an ordered setD _n ^* = 〈D _n ^* , ? _n , wherex? _ny iffx¦y for anyx, y∈D _n ^* , can easily be seen to be a pseudo-boolean algebra. In [5], V.A. Jankov has proved that the class of algebras {D _n ^* ∶n∈B}, whereB =,{k ∈N∶ ? \(\mathop \exists \limits_{n \in N} \) (n > 1 ≧n ² k)is finitely axiomatizable. The present paper aims at showing that the class of all algebras {D _n ^* ∶n∈B} is also finitely axiomatizable. First, we prove that an intermediate logic defined as follows: $$LD = Cn(INT \cup \{ p_3 \vee [p_3 \to (p_1 \to p_2 ) \vee (p_2 \to p_1 )]\} )$$ finitely approximatizable. Then, defining, after Kripke, a model as a non-empty ordered setH = 〈K, ?〉, and making use of the set of formulas true in this model, we show that any finite strongly compact pseudo-boolean algebra ? is identical with. the set of formulas true in the Kripke modelH _B = 〈P(?), ?〉 (whereP(?) stands for the family of all prime filters in the algebra ?). Furthermore, the concept of a structure of divisors is defined, and the structure is shown to beH D _n ^* = 〈P (D _n ^* ), ?〉for anyn∈N. Finally, it is proved that for any strongly compact pseudo-boolean algebraU satisfying the axiomp ₃∨ [p ₃→(p₁→p₂)∨(p₂→p₁)] there is a structure of divisorsD _* ⁿ such that it is possible to define a strong homomorphism froomiH D _n ^* ontoH D _U . Exploiting, among others, this property, it turns out to be relatively easy to show that \(LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )\) .     

The variance of d′ estimates obtained in yes—no and two-interval forced choice procedures

In studies of detection and discrimination, data are often obtained in the form of a 2 × 2 matrix and then converted to an estimate of d′, based on the assumptions that the underlying decision distributions are Gaussian and equal in variance. The statistical properties of the estimate of d′, $\hat d'$ , are well understood for data obtained using the yes—no procedure, but less effort has been devoted to the more commonly used two-interval forced choice (2IFC) procedure. The variance associated with $\hat d'$ is a function of trued′ in both procedures, but for small values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be less than the variance of $\hat d'$ obtained using yes—no; for large values of trued′, the variance of $\hat d'$ obtained using the 2IFC procedure is predicted to be greater than the variance of $\hat d'$ from yes—no. These results follow from standard assumptions about the relationship between the two procedures. The present paper reviews the statistical properties of $\hat d'$ obtained using the two standard procedures and compares estimates of the variance of $\hat d'$ as a function of trued′ with the variance observed in values of $\hat d'$ obtained with a 2IFC procedure.     

Classical Modal De Morgan Algebras

In this note we introduce the variety ${{\mathcal C}{\mathcal D}{\mathcal M}_\square}$ of classical modal De Morgan algebras as a generalization of the variety ${{{\mathcal T}{\mathcal M}{\mathcal A}}}$ of Tetravalent Modal algebras studied in [11]. We show that the variety ${{\mathcal V}_0}$ defined by H. P. Sankappanavar in [13], and the variety S of Involutive Stone algebras introduced by R. Cignoli and M. S de Gallego in [5], are examples of classical modal De Morgan algebras. We give a representation theory, and we study the regular filters, i.e., lattice filters closed under an implication operation. Finally we prove that the variety ${{{\mathcal T}{\mathcal M}{\mathcal A}}}$ has the Amalgamation Property and the Superamalgamation Property.     

On the Algebraizability of the Implicational Fragment of Abelian Logic

In this paper we consider the implicational fragment of Abelian logic \({{{\sf A}_{\rightarrow}}}\) . We show that although the Abelian groups provide an semantics for the set of theorems of \({{{\sf A}_{\rightarrow}}}\) they do not for the associated consequence relation. We then show that the consequence relation is not algebraizable in the sense of Blok and Pigozzi (Mem Am Math Soc 77, 1989). In the second part of the paper, we investigate an extension of \({{{\sf A}_{\rightarrow}}}\) in the same language and having the same set of theorems and show that this new consequence relation is algebraizable with the Abelian groups as its equivalent algebraic semantics. Finally, we show that although \({{{\sf A}_{\rightarrow}}}\) is not algebraizable, it is order-algebraizable in the sense of Raftery (Ann Pure Appl Log 164:251–283, 2013).     

Actuality in Propositional Modal Logic

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5.     

From constants to consequence, and back

Bolzano??s definition of consequence in effect associates with each set X of symbols (in a given interpreted language) a consequence relation ${\Rightarrow_X}$ . We present this in a precise and abstract form, in particular studying minimal sets of symbols generating ${\Rightarrow_X}$ . Then we present a method for going in the other direction: extracting from an arbitrary consequence relation ${\Rightarrow}$ its associated set ${C_\Rightarrow}$ of constants. We show that this returns the expected logical constants from familiar consequence relations, and that, restricting attention to sets of symbols satisfying a strong minimality condition, there is an isomorphism between the set of strongly minimal sets of symbols and the set of corresponding consequence relations (both ordered under inclusion).     

Compatible Operations on Residuated Lattices

This work extend to residuated lattices the results of [7]. It also provides a possible generalization to this context of frontal operators in the sense of [9]. Let L be a residuated lattice, and f : L ^k ?? L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L. We use this characterization of compatible functions in order to prove that the variety of residuated lattices is locally affine complete. We study some compatible functions on residuated lattices which are a generalization of frontal operators. We also give conditions for two operations P(x, y) and Q(x, y) on a residuated lattice L which imply that the function ${x \mapsto min\{y \in L : P(x, y) \leq Q(x, y)\}}$ when defined, is equational and compatible. Finally we discuss the affine completeness of residuated lattices equipped with some additional operators.     

An Inconsistency-Adaptive Deontic Logic for Normative Conflicts

We present the inconsistency-adaptive deontic logic DP ^r , a nonmonotonic logic for dealing with conflicts between normative statements. On the one hand, this logic does not lead to explosion in view of normative conflicts such as O A?∧?O ～A, O A?∧?P ～A or even O A?∧?～O A. On the other hand, DP ^r still verifies all intuitively reliable inferences valid in Standard Deontic Logic (SDL). DP ^r interprets a given premise set ‘as normally as possible’ with respect to SDL. Whereas some SDL-rules are verified unconditionally by DP ^r , others are verified conditionally. The latter are applicable unless they rely on formulas that turn out to behave inconsistently in view of the premises. This dynamic process is mirrored by the proof theory of DP ^r .     

Velocity perception for sounds moving in frequency space

In three experiments, we considered the relative contribution of frequency change (Δf) and time change (Δt) to perceived velocity (Δf/Δt) for sounds that moved either continuously in frequency space (Experiment 1) or in discrete steps (Experiments 2 and 3). In all the experiments, participants estimated “how quickly stimuli changed in pitch” on a scale ranging from 0 (not changing at all) to 100 (changing very quickly). Objective frequency velocity was specified in terms of semitones per second (ST/s), with ascending and descending stimuli presented on each trial at one of seven velocities (2, 4, 6, 8, 10, 12, and 14 ST/s). Separate contributions of frequency change (Δf) and time change (Δt) to perceived velocity were assessed by holding total Δt constant and varying Δf or vice versa. For tone glides that moved continuously in frequency space, both Δf and Δt cues contributed approximately equally to perceived velocity. For tone sequences, in contrast, perceived velocity was based almost entirely on Δt, with surprisingly little contribution from Δf. Experiment 3 considered separate judgments about Δf and Δt in order to rule out the possibility that the results of Experiment 2 were due to the inability to judge frequency change in tone sequences.     

Frontal Operators in Weak Heyting Algebras

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation ${\tau(a) \leq b \vee (b \rightarrow a)}$ , for all ${a, b \in A}$ . These operators were studied from an algebraic, logical and topological point of view by Leo Esakia in [10]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces ${\langle X, \leq, T, R \rangle}$ where ${\langle X, \leq, T \rangle}$ is a WH-space [6], and R is an additional binary relation used to interpret the modal operator. We will also study the WH-algebras with successor and the WH-algebras with gamma. For these varieties we will give two topological dualities. The first one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH-spaces.     

On the notions of indiscernibility and indeterminacy in the light of the Galois–Grothendieck theory

We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to \(K\) -algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois–Grothendieck duality between \(G\) -spaces and the minimal observable algebras that discern (or separate) their points. According to the natural epistemic interpretation of the original Galois theory, the possible \(K\) -indiscernibilities between the roots of a polynomial \(p(x)\in K[x]\) result from the limitations of the field \(K\) . We discuss the relation between this epistemic interpretation of the Galois–Grothendieck duality and Leibniz’s principle of the identity of indiscernibles. We then use the conceptual framework provided by Klein’s Erlangen program to propose an alternative ontologic interpretation of this duality. The Galoisian symmetries are now interpreted in terms of the automorphisms of the symmetric geometric figures that can be placed in a background Klein geometry. According to this interpretation, the Galois–Grothendieck duality encodes the compatibility condition between geometric figures endowed with groups of automorphisms and the ‘observables’ that can be consistently evaluated at such figures. In this conceptual framework, the Galoisian symmetries do not encode the epistemic indiscernibility between individuals, but rather the intrinsic indeterminacy in the pointwise localization of the figures with respect to the background Klein geometry.     

Margaret R. Miles’ Augustine and the Fundamentalist’s Daughter: An Eriksonian Perspective

My perspective on Margaret R. Miles’s Augustine and the Fundamentalist’s Daughter is informed by Erik H. Erikson’s life cycle model (Erikson 1950, 1959, 1963, 1964, 1968a, b, 1982; Erikson and Erikson 1997) and, more specifically, by my relocation of his life stages and their accompanying human strengths (Erikson 1964) according to decades (Capps 2008). I interpret Miles’s account of her life from birth to age forty as revealing the selves that comprise the composite Self (Erikson 1968a) that come into their own during the first four decades of the life cycle, i.e., the hopeful, willing, purposeful, and competent selves