Journal of Applied Logic最新文献

A translation technique is presented which transforms a class of First Order Logic formulas, called Restricted formulas, into ground formulas. For the formulas in this class the range of quantified variables is restricted by Domain formulas.

If we have a complete knowledge of the predicates involved in the Domain formulas their extensions can be evaluated with the Relational Algebra and these extensions are used to transform universal (respectively existential) quantifiers into finite conjunctions (respectively disjunctions).

It is assumed that the complete knowledge is represented by Completion Axioms and Unique Name Axioms à la Reiter. These axioms involve the equality predicate. However, the translation allows to remove the equality in the ground formulas and for a large class of formulas their consequences are the same as the initial First Order formulas. This result open the door for the design of efficient deduction techniques.

We investigate the notion of a signature in Polyadic Inductive Logic and study the probability functions satisfying the Principle of Signature Exchangeability. We prove a representation theorem for such functions on binary languages and show that they satisfy a binary version of the Principle of Instantial Relevance. We discuss polyadic versions of the Principle of Instantial Relevance and Johnson's Sufficientness Postulate.

We present a novel approach, which is based on multiple-valued logic (MVL), to the verification and analysis of digital hardware designs, which extends the common ternary or quaternary approaches for simulations. The simulations which are performed in the more informative MVL setting reveal details which are either invisible or harder to detect through binary or ternary simulations. In equivalence verification, detecting different behavior under MVL simulations may lead to the discovery of a genuine binary non-equivalence or to a qualitative gap between two designs. The value of a variable in a simulation may hold information about its degree of truth and its “place of birth” and “date of birth”. Applications include equivalence verification, initialization, assertions generation and verification, partial control on the flow of data by prioritizing and block-oriented simulations. Much of the paper is devoted to theoretical aspects behind the MVL approach, including the reason for choosing a specific algebra for computations and the introduction of the notions of De Morgan Canonical Form and of verification complexity of Boolean expressions. Two basic simulation-based algorithms are presented, one for satisfying and verifying combinational designs and the other for equivalence verification of sequential designs.

Logical theories have been developed which have allowed temporal reasoning about eventualities (à la Galton) such as states, processes, actions, events and complex eventualities such as sequences and recurrences of other eventualities. This paper presents the problem of coincidence within the framework of a first order logical theory formalizing temporal multiple recurrence of two sequences of fixed duration eventualities and presents a solution to it.

The coincidence problem is described as: if two complex eventualities (or eventuality sequences) consisting respectively of component eventualities $x_0,x_1,\dots ,x_r$ and $y_0,y_1,\dots ,y_s$ both recur over an interval k and all eventualities are of fixed durations, is there a subinterval of k over which the incidence $x_t$ and $y_u$ for $0\le t\le r$ and $0\le u\le s$ coincide? The solution presented here formalizes the intuition that a solution can be found by temporal projection over a cycle of the multiple recurrence of both sequences.

It is known that the logic BI of bunched implications is a logic of resources. Many studies have reported on the applications of BI to computer science. In this paper, an extension BIS of BI by adding a sequence modal operator is introduced and studied in order to formalize more fine-grained resource-sensitive reasoning. By the sequence modal operator of BIS, we can appropriately express “sequential information” in resource-sensitive reasoning. A Gentzen-type sequent calculus SBIS for BIS is introduced, and the cut-elimination and decidability theorems for SBIS are proved. An extension of the Grothendieck topological semantics for BI is introduced for BIS, and the completeness theorem with respect to this semantics is proved. The cut-elimination, decidability and completeness theorems for SBIS and BIS are proved using some theorems for embedding BIS into BI.

In the companion article “The eco-cognitive model of abduction” [66] I illustrated the main features of my eco-cognitive model of abduction (EC-Model). With the aim of delineating further aspects of that “naturalization of logic” recently urged by John Woods [94] I will now set out to further analyze some properties of abduction that are essential from a logical standpoint. When dealing with the so-called “inferential problem”, I will opt for the more general concepts of input and output instead of those of premisses and conclusions, and show that in this framework two consequences can be derived that help clarify basic logical aspects of abductive reasoning: 1) it is more natural to accept the “multimodal” and “context-dependent” character of the inferences involved, 2) inferences are not merely conceived of in the terms of the process leading to the “generation of an output” or to the proof of it, as in the traditional and standard view of deductive proofs, but rather, from this perspective abductive inferences can be seen as related to logical processes in which input and output fail to hold each other in an expected relation, with the solution involving the modification of inputs, not that of outputs. The chance of finding an abductive solution still appears to depend on the Aristotelian concept of “leading away” (ἀπαγωγή) I dealt with in the companion article, that is, on the starting of the application of a supplementary logic implementing an appropriate formal inference engine. An important result I will emphasize is that irrelevance and implausibility are not always offensive to reason. In addition, we cannot be sure, more broadly, that our guessed hypotheses are plausible (even if we know that looking – in advance – for plausibility is a human good and wise heuristic), indeed an implausible hypothesis can later on result plausible. In the last part of the article I will describe that if we wish to naturalize the logic of the abductive processes and its special consequence relation, we should refer to the following main aspects: “optimization of situatedness”, “maximization of changeability” of both input and output, and high “information-sensitiveness”. Furthermore, I will point out that a logic of abduction must acknowledge the importance of keeping record of the “past life” of abductive inferential praxes, contrarily to the fact that traditional demonstrative ideal systems are prototypically characterized by what I call “maximization of memorylessness”.

Let us write $\ell G_u^f$ for the category whose objects are lattice-ordered abelian groups (l-groups for short) with a strong unit and finite prime spectrum endowed with a collection of Archimedean elements, one for each prime l-ideal, which satisfy certain properties, and whose arrows are l-homomorphisms with additional structure. In this paper we show that a functor which assigns to each object $(A,\hat{u})\in \ell G_u^f$ the prime spectrum of A, and to each arrow $f:(A,\hat{u})\to (B,\hat{v})\in \ell G_u^f$ the naturally induced p-morphism, has a left adjoint.

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Moreover, we prove the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases. Finally, we apply our results to study selected probabilistic versions of classical categorical syllogisms and construct a new version of the square of opposition in terms of defaults and negated defaults.

This paper presents a progic, or probabilistic logic, in the sense of Haenni et al. [8]. The progic presented here is based on Bayesianism, as the progic discussed by Williamson [15]. However, the underlying generalised Bayesianism differs from the objective Bayesianism used by Williamson, in the calibration norm, and the liberalisation and interpretation of the reference probability in the norm of equivocation. As a consequence, the updating dynamics of both Bayesianisms differ essentially. Whereas objective Bayesianism is based on a probabilistic re-evaluation, orthodox Bayesianism is based on a probabilistic revision. I formulate a generalised and iterable orthodox Bayesian revision dynamics. This allows to define an updating procedure for the generalised Bayesian progic. The paper compares the generalised Bayesian progic and Williamson's objective Bayesian progic in strength, update dynamics and with respect to language (in)sensitivity.

I will propose an alternative philosophical approach to the representation of uncertain doxastic states. I will argue that the current account of measuring inaccuracy of uncertain doxastic states is inadequate for Belnap's four-valued logic. Specifically, a situation can be found in which either an inaccuracy measure returns a completely wrong result or an agent's inaccuracy score is inadequate relative to the mistake in her doxastic attitude. This will motivate an alternative representation of uncertain doxastic states based on ordered pairs. I will describe a possible inaccuracy measure that is suitable for ordered pairs, and I will show that it has all the qualities that are required for an inaccuracy measure to be legitimate. Finally, I will introduce conditions of rationality for uncertain doxastic states represented by ordered pairs.