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.     

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.     

On Provability Logics with Linearly Ordered Modalities

We introduce the logics GLP _Λ, a generalization of Japaridze’s polymodal provability logic GLP _ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP _ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP _Λ and the decidability of GLP _Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment of GLP _Λ.     

Reasoning About Games

A mixture of propositional dynamic logic and epistemic logic that we call PDL + E is used to give a formalization of Artemov??s knowledge based reasoning approach to game theory, (KBR), [4, 5]. Epistemic states of players are represented explicitly and reasoned about formally. We give a detailed analysis of the Centipede game using both proof theoretic and semantic machinery. This helps make the case that PDL + E can be a useful basis for the logical investigation of game theory.     

Extensions of Priest-da Costa Logic

In this paper, we look at applying the techniques from analyzing superintuitionistic logics to extensions of the cointuitionistic Priest-da Costa logic daC (introduced by Graham Priest as “da Costa logic”). The relationship between the superintuitionistic axioms- definable in daC- and extensions of Priest-da Costa logic (sdc-logics) is analyzed and applied to exploring the gap between the maximal si-logic SmL and classical logic in the class of sdc-logics. A sequence of strengthenings of Priest-da Costa logic is examined and employed to pinpoint the maximal non-classical extension of both daC and Heyting-Brouwer logic HB . Finally, the relationship between daC and Logics of Formal Inconsistency is examined.     

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.     

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.     

Shen Dao��s Own Voice in the Shenzi Fragments

Shen Dao ???? is known as one of the Legalists who influenced Han Feizi ?n???? in terms of the concept of shi ?? (circumstantial advantage, power, or authority). This argument is based on the ??A Critique of Circumstantial Advantage (Nanshi ?y??)?? chapter of the Hanfeizi, where Han Feizi advances his own idea of shi after criticizing both Shen Dao and an anonymous Confucian. However, there are other primary sources to contain Shen Dao??s thought, namely, seven incomplete Shenzi ???? chapters of the Essentials on Government from the Assemblage of Books (Qunshu zhi yao ??????) and other fragments preserved in other Chinese texts. This article examines the Shenzi fragments in order to ask whether Shen Dao stresses the concept of shi.     

Fatal Heyting Algebras and Forcing Persistent Sentences

Hamkins and L?we proved that the modal logic of forcing is S4.2. In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H _ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.     

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

Causal essentialism and mereological monism

Several philosophers have recently defended Causal Essentialism—the view that every property confers causal powers, and whatever powers it confers, it confers essentially. I argue that on the face of it, Causal Essentialism implies a form of Monism, and in particular, the thesis I call ‘Mereological Monism’: that there is some concretum that is a part of every concretum. However, there are three escape routes, three views which are such that if one of them is true, Causal Essentialism does not imply any form of Monism at all. I survey the costs associated with taking these escape routes along with the costs associated with accepting Mereological Monism.     

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 \(LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )\) .     

Propositional discourse logic

A novel normal form for propositional theories underlies the logic pdl, which captures some essential features of natural discourse, independent from any particular subject matter and related only to its referential structure. In particular, pdlallows to distinguish vicious circularity from the innocent one, and to reason in the presence of inconsistency using a minimal number of extraneous assumptions, beyond the classical ones. Several, formally equivalent decision problems are identified as potential applications: non-paradoxical character of discourses, admissibility of arguments in argumentation networks, propositional satisfiability, and the existence of kernels of directed graphs. Directed graphs provide the basis for the semantics of pdl and the paper concludes by an overview of relevant graph-theoretical results and their applications in diagnosing paradoxical character of natural discourses.     

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.     

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 .     

Bohrification of operator algebras and quantum logic

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set ${\mathcal{C}(A)}$ of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that ${\mathcal{C}(A)}$ consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski?CMulvey) of the ??Bohrification?? ${\underline A}$ of A, which is a commutative Rickart C*-algebra in the topos of functors from ${\mathcal{C}A}$ to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns?CLakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of ${\underline A}$ for A?=?B(H).     

Further evidence for phonological constraints on visual lexical access: TOWED primes FROG

If the phonological codes of visually presented words are assembled-rapidly and automatically for use in lexical access, then words that sound alike should induce similar activity within the internal lexicon.Towed is homophonous with TOAD, which is semantically related tofrog, andbeach is homophonous withbeech, which is semantically related totree. Stimuli such as these were used in a priming-of-namingtask, in which words homophonous with associates of the target words preceded the targets at an onset asynchrony of 100 msec. Relative to spelling controls (trod, bench), the low-frequencytowed and the high-frequencybeach speeded up the naming offrog andtree, respectively, to the same degree. This result was discussed in relation to the accumulating evidence for the primacy of phonological constraints in visual lexical access. nt]mis|This research was supported in part by National Institute of Child Health and Human Development Grants HD-08945 and HD-0 1994 to the first author and Haskins Laboratories, respectively.     

Expansions of Semi-Heyting Algebras I: Discriminator Varieties

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [48] and [50] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH. We also use it to prove that the variety DQSSH of dually quasi-Stone semi- Heyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH. Finally, we apply these results to give bases for ??small?? subvarieties of BDQDSH, DQSSH, and DblSH.     

Axiomatic extensions of the constructive logic with strong negation and the disjunction property

We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B εV, thenA×B is a homomorphic image of some well-connected algebra ofV. We prove:

• each varietyV of Nelson algebras with PQWC lies in the fibre σ^?1(W) for some varietyW of Heyting algebras having PQWC,
• for any varietyW of Heyting algebras with PQWC the least and the greatest varieties in σ^?1(W) have PQWC,
• there exist varietiesW of Heyting algebras having PQWC such that σ^?1(W) contains infinitely many varieties (of Nelson algebras) with PQWC.