Intermediate Logics Research Articles

Prior’s OIC nonconservativity example revisited

In his 1964 note, ‘Two Additions to Positive Implication’, A. N. Prior showed that standard axioms governing conjunction yield a nonconservative extension of the pure implicational intermediate logic OIC (order implicational calculus) of R. A. Bull. Here, after reviewing the situation with the aid of an adapted form of the Kripke semantics for intuitionistic and intermediate logics, we proceed to illuminate this example by transposing it to the setting of modal logic, and then relate it to the propositional logic of what have been called ‘Hilbert algebras with infimum’.

Topologies for intermediate logics

We investigate the problem of characterizing the classes of Grothendieck toposes whose internal logic satisfies a given assertion in the theory of Heyting algebras, and introduce natural analogues of the double negation and De Morgan topologies on an elementary topos for a wide class of intermediate logics.

Irreflexive modality as a new logical connective in the Dummett logic

We give an example of a countable family of extensions of Dummett’s superintuitionistic logic LC in a language with one extra unary logical connective each of them defining a new logical connective in LC in the sense of Novikov, and every two logics in this family are incompatible over LC.

The Lyndon property and uniform interpolation over the Grzegorczyk logic

We consider versions of the interpolation property stronger than the Craig interpolation property and prove the Lyndon interpolation property for the Grzegorczyk logic and some of its extensions. We also establish the Lyndon interpolation property for most extensions of the intuitionistic logic with Craig interpolation property. For all modal logics over the Grzegorczyk logic as well as for all superintuitionistic logics, the uniform interpolation property is equivalent to Craig’s property.

Completeness of intermediate logics with doubly negated axioms

Let denote a first‐order logic in a language that contains infinitely many constant symbols and also containing intuitionistic logic . By , we mean the associated logic axiomatized by the double negation of the universal closure of the axioms of plus . We shall show that if is strongly complete for a class of Kripke models , then is strongly complete for the class of Kripke models that are ultimately in .

New constants in the superintuitionistic logic L2

A classification of all Novikov complete extensions of the superintuitionistic logic L2 in a language with several extra constants is given. The decidability of the (algorithmic) conservativeness problem for extensions of the form \(L2 + A\left( {\bar \phi } \right)\) over L2 is established.

On unification and admissible rules in Gabbay–de Jongh logics

In this paper we study the admissible rules of intermediate logics. We establish some general results on extensions of models and sets of formulas. These general results are then employed to provide a basis for the admissible rules of the Gabbay–de Jongh logics and to show that these logics have finitary unification type.

AGGREGATION AND IDEMPOTENCE

AbstractA 1-ary sentential context is aggregative (according to a consequence relation) if the result of putting the conjunction of two formulas into the context is a consequence (by that relation) of the results of putting first the one formula and then the other into that context. All 1-ary contexts are aggregative according to the consequence relation of classical propositional logic (though not, for example, according to the consequence relation of intuitionistic propositional logic), and here we explore the extent of this phenomenon, generalized to having arbitrary connectives playing the role of conjunction; among intermediate logics, LC, shows itself to occupy a crucial position in this regard, and to suggest a characterization, applicable to a broader range of consequence relations, in terms of a variant of the notion of idempotence we shall call componentiality. This is an analogue, for the consequence relations of propositional logic, of the notion of a conservative operation in universal algebra.

A short proof of Glivenko theorems for intermediate predicate logics

We give a simple proof-theoretic argument showing that Glivenko's theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko's theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic.

Algebraic semantics for modal and superintuitionistic non-monotonic logics

The paper provides a preliminary study of algebraic semantics for modal and superintuitionistic non-monotonic logics. The main question answered is: how can non-monotonic inference be understood algebraically?

Should Anti-Realists be Anti-Realists About Anti-Realism?

On the Dummettian understanding, anti-realism regarding a particular discourse amounts to (or at the very least, involves) a refusal to accept the determinacy of the subject matter of that discourse and a corresponding refusal to assert at least some instances of excluded middle (which can be understood as expressing this determinacy of subject matter). In short: one is an anti-realist about a discourse if and only if one accepts intuitionistic logic as correct for that discourse. On careful examination, the strongest Dummettian arguments for anti-realism of this sort fail to secure intuitionistic logic as the single, correct logic for anti-realist discourses. Instead, antirealists are placed in a situation where they fail to be justified in asserting monism (that intuitionistic logic is the unique correct logic). Thus, antirealists seem forced either to accept pluralism (i.e. one or more intermediate logic is at least as `correct’ as intuitionistic logic–an option I take to be unattractive from the anti-realist perspective), or they must be anti-realists about the realism/anti-realism debate (and, in particular, must refuse to assert the instance of excluded middle equivalent to logical monism or logical pluralism).

An AC 1-complete model checking problem for intuitionistic logic

We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform AC 1. For superintuitionistic logics with one variable, we obtain NC 1-completeness for the model checking problem.

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.

On some Classes of Heyting Algebras with Successor that have the Amalgamation Property

In this paper we shall prove that certain subvarieties of the variety of Salgebras (Heyting algebras with successor) has amalgamation. This result together with an appropriate version of Theorem 1 of [L. L. Maksimova, Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras, Algebra i Logika, 16(6):643-681, 1977] allows us to show interpolation in the calculus IPC S (n), associated with these varieties. We use that every algebra in any of the varieties of S-algebras studied in this work has a canonical extension, to show completeness of the calculus IPC S (n) with respect to appropriate Kripke models.

Buffer-on-board memory systems

The design and implementation of the commodity memory architecture has resulted in significant performance and capacity limitations. To circumvent these limitations, designers and vendors have begun to place intermediate logic between the CPU and DRAM. This additional logic has two functions: to control the DRAM and to communicate with the CPU over a fast and narrow bus. The benefit provided by this logic is a reduction in pin-out to the memory system and increased signal integrity to the DRAM, allowing faster clock rates while maintaining capacity. While the few vendors utilizing this design have used the same general approach, their implementations vary greatly in their nontrivial details. A hardware-verified simulation suite is developed to accurately model and evaluate the behavior of this buffer-onboard memory system. A study of this design space is used to determine optimal use of the resources involved. This includes DRAM and bus organization, queue storage, and mapping schemes. Various constraints based on implementation costs are placed on simulated configurations to confirm that these optimizations apply to viable systems. Finally, full system simulations are performed to better understand how this memory system interacts with an operating system executing an application with the goal of uncovering behaviors not present in simple limit case simulations. When applying insights gleaned from these simulations, optimal performance can be achieved while still considering outside constraints (i.e., pin-out, power, and fabrication costs).

Characteristic Formulas of Partial Heyting Algebras

The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with every well-connected Heyting algebra we associate a set of characteristic formulas that correspond to each finite relative subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular characteristic formulas. In the last section we outline an approach how to generalize these obtained results to the broad classes of algebras.

Theorem proving for prenex G\"odel logic with Delta: checking validity and unsatisfiability

G\"odel logic with the projection operator Delta (G_Delta) is an important many-valued as well as intermediate logic. In contrast to classical logic, the validity and the satisfiability problems of G_Delta are not directly dual to each other. We nevertheless provide a uniform, computational treatment of both problems for prenex formulas by describing appropriate translations into sets of order clauses that can be subjected to chaining resolution. For validity a version of Herbrand's Theorem allows us to show the soundness of standard Skolemization. For satisfiability the translation involves a novel, extended Skolemization method.

Fatal Heyting Algebras and Forcing Persistent Sentences

Abstract Hamkins and Löwe 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 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.

Possibilistic intermediate logic

We define what we call 'possibilistic intermediate logic (PIL)'; we present results analogous to those of the well-known intermediate logic, such as a deduction theorem, a generalised version of the deduction theorem, a cut rule, a weak version of a refutation theorem, a substitution theorem and Glivenko's theorem. Also, we present a definition for 'possibilistic safe beliefs'. This definition allows us to establish a relation between safe beliefs, as presented on 'applications of intuitionistic logic in answer set programming' by Osorio et al., and our version for possibilistic intermediate logic (PILX). We also present a characterisation of possibilistic safe beliefs for possibilistic normal logic programs.

Energy-Efficient Pipeline Templates for High-Performance Asynchronous Circuits

We present two novel energy-efficient pipeline templates for high throughput asynchronous circuits. The proposed templates, called N-P and N-Inverter pipelines, use a single-track handshake protocol. There are multiple stages of logic within each pipeline. The proposed techniques minimize handshake overheads associated with input tokens and intermediate logic nodes within a pipeline template. Each template can pack a significant amount of logic in a single stage, while still maintaining a fast cycle time of only 18 transitions. Noise and timing robustness constraints of our pipelined circuits are quantified across all process corners. We present completion detection scheme based on wide NOR gates, which results in significant latency and energy savings especially as the number of outputs increase. To fully quantify all design trade-offs, three separate pipeline implementations of an 8x8-bit Booth-encoded array multiplier are presented. Compared to a standard QDI pipeline implementation, the N-Inverter and N-P pipeline implementations reduced the energy-delay product by 38.5% and 44% respectively. The overall multiplier latency was reduced by 20.2% and 18.7%, while the total transistor width was reduced by 35.6% and 46% with N-Inverter and N-P pipeline templates respectively.