The logic of proofs was introduced by Artemov in order to analize the formalization of the concept of proof rather than the concept of provability. In this context, some operations on proofs play a very important role. In this paper, we investigate some very natural operations, paying attention not only to positive information, but also to negative information (i.e. information saying that something cannot be a proof). We give a formalization for a fragment of such a logic of (...) proofs, and we prove that our fragment is complete and decidable. (shrink)

The main result established in this paper is the following: If the base normed spaceV of completely additive weights is a norm-determining subspace of the space of finitely additive weights V acting on the order unit space spanning the operational logic, thenV has the ε-Jordan-Hahn property iff V has the approximate Jordan-Hahn property. Several examples illustrating the theory are given.

SummaryThe working logician begins with whatever operations are necessary to make computation possible. He does not inquire into the foundations which the carrying out of his operations assumes; no axioms, no assumptions, just the computations themselves. Yet in logic of all places the starting‐point should be defensible. After examining the logical assumptions, the constructions of proofs, individuals and classes, and the metaphysical assumptions, the conclusion is reached that the net effect of operational logic is to assimilate logic (...) to mathematics rather than to consider mathematics an extension of logic. The price to be paid for this preference is to leave them both unexplained. It would mean that the metaphysics of mathematics would have to proceed without logic. (shrink)

SummaryThe working logician begins with whatever operations are necessary to make computation possible. He does not inquire into the foundations which the carrying out of his operations assumes; no axioms, no assumptions, just the computations themselves. Yet in logic of all places the starting‐point should be defensible. After examining the logical assumptions, the constructions of proofs, individuals and classes, and the metaphysical assumptions, the conclusion is reached that the net effect of operational logic is to assimilate logic (...) to mathematics rather than to consider mathematics an extension of logic. The price to be paid for this preference is to leave them both unexplained. It would mean that the metaphysics of mathematics would have to proceed without logic. (shrink)

The concept of an operator is used in a variety of practical and theoretical areas. Operators, as both conceptual and physical entities, are found throughout the world as subsystems in nature, the human mind, and the manmade world. Operators, and what they operate, i.e., their substrates, targets, or operands, have a wide variety of forms, functions, and properties. Operators have explicit philosophical significance. On the one hand, they represent important ontological issues of reality. On the other hand, epistemological operators (...) form the basic mechanism of cognition. At the same time, there is no unified theory of the nature and functions of operators. In this work, we elaborate a detailed analysis of operators, which range from the most abstract formal structures and symbols in mathematics and logic to real entities, human and machine, and are responsible for effecting changes at both the individual and collective human levels. Our goal is to find what is common in physical objects called operators and abstract mathematical structures, with the name operator providing foundations for building a unified but flexible theory of operators. The paper concludes with some reflections on functionalism and other philosophical aspects of the ‘operation’ of operators. (shrink)

This new edition of work that has evolved over the past seven years completes the derivation of the form of The Standard Model from quantum theory and the extension of the Theory of Relativity to superluminal transformations. The much derided form of The Standard Model is established from a consideration of Lorentz and superluminal relativistic space-time transformations. So much so that other approaches to elementary particle theory pale in comparison. In previous work color SU(3) was derived from space-time considerations. This (...) book shows that the SU(2) U(1) Weak Interaction sector follows directly from the extension of Lorentz transformations to superluminal (faster-than-light) transformations. In fact, ElectroWeak symmetry is shown to have an SU(2) U(1) U(1) symmetry that naturally includes WIMPs (Weakly Interacting Massive Particles), a candidate for Dark Matter. Thus the form of the Standard Model is dictated by space-time considerations fulfilling a dream of Einstein. Using a new matrix formulation of Logic - Operator Logic - we show that Dirac-like equations can be constructed. This edition also derives broken SU(4) four fermion generations as well as a non-Higgsian (Dimensional Mass Generation) derivation of fermion and ElectroWeak gauge boson masses from GL(4). The program of the book is completed by reprinting the book Quantum Theory of the Third Kind, which describes Two-Tier quantum field theory. This theory yields finite results (No Feynman diagram calculations diverge!) in The Standard Model and Quantum Gravity calculations to all orders in perturbation theory. Lastly, the book also contains a description of Operator Logic, Probabilistic Operator Logic, Quantum Operator Logic (for q-number statements). A comprehensive derivation of the form of The Standard Model from geometric considerations. (shrink)

At present, the Chinese government is trying to resolve various social contradictions, such as people’s ever-growing need for a better life and unbalanced and inadequate development. To do so, urban governance practices including holistic governance, decentralized and interconnected governance, multiple participatory governance, and smart governance have been developed in China. Urban smart governance supported by mobile Internet, the Internet of Things, quantum computing, big data, artificial intelligence, and other information technologies has also entered the field of vision of academics and (...) administrators. However, the research and practice on the integration of organizational structure and smart governance technology for urban governance in China are still insufficient. Hence, this paper proposes the design of the organizational structure of an urban smart governance platform and presents the “1 + N” integrated smart information platform and the “power-sharing linkage, one-center and dual-track” platform construction command system. In this system, the functional module of the information platform for smart urban governance is elaborated. This paper contributes to promoting the modernization of China’s urban governance capacity, which can enhance social equity and social order, promote social democracy and rule of law, and enhance the efficiency of urban governance. (shrink)

We show that Leitgeb’s dependence operator of Leitgeb is a \-operator and that this is best possible.

This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest (...) that are covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory. This book is Open Access under a CC BY licence. (shrink)

I present a notion of invariance under arbitrary surjective mappings for operators on a relational finite type hierarchy generalizing the so-called Tarski-Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use it to characterize logicality.

Let K = (p, q...; &, ∨, ~) be a zeroth-order formal language with sentence variables p, q..., two place connectives & (and), ∨ (or) and negation sign ~, and let F be the formula algebra (set of well-formed formulas in K defined in the standard way by induction from the sentence variables). If v is an assignment of truth values 1(true), 0(f alse) to the sentence variables p, q..., then classical propositional logic is characterized by extending v by (...) induction from p, q... to the formula algebra F in such a manner that the extension (called interpretation and also denoted by v) be a “homomorphism” from F into the two-element Boolean algebra B 2 ≡ (1, 0, ⋂, ⋃, ⊥), where “homomorphism” means that the following hold υ(q&p)=υ(q)∩υ(p)υ(q∨p)=υ(q)∪υ(p)υ(∼q)=υ(q)⊥. (shrink)

Tarski and Mautner proposed to characterize the "logical" operations on a given domain as those invariant under arbitrary permutations. These operations are the ones that can be obtained as combinations of the operations on the following list: identity; substitution of variables; negation; finite or infinite disjunction; and existential quantification with respect to a finite or infinite block of variables. Inasmuch as every operation on this list is intuitively "logical", this lends support to the Tarski-Mautner proposal.

I consider how complex logical operations might self-assemble in a signalling-game context via composition of simpler underlying dispositions. On the one hand, agents may take advantage of pre-evolved dispositions; on the other hand, they may co-evolve dispositions as they simultaneously learn to combine them to display more complex behaviour. In either case, the evolution of complex logical operations can be more efficient than evolving such capacities from scratch. Showing how complex phenomena like these might evolve provides an additional path to (...) the possibility of evolving more or less rich notions of compositionality. This helps provide another facet of the evolutionary story of how sufficiently rich, human-level cognitive or linguistic capacities may arise from simpler precursors. 1Signalling and Self-assembly2Simple Unary Logic Games3Composing Unary Functions for Binary Inputs 3.1Utilizing pre-evolved dispositions3.2Co-evolving logical dispositions3.3Learning appropriate outputs3.4Taking account of the full state-space of unary games3.5Role-free composition4Discussion 4.1Efficacy and efficiency of learning complex dispositions4.2Other binary operations4.3To infinity and beyond5Conclusion. (shrink)

It is well known that the structure of honest elementary degrees is a lattice with rather strong density properties. Let $\mbox{\bf a} \cup \mbox{\bf b}$ and $\mbox{\bf a} \cap \mbox{\bf b}$ denote respectively the join and the meet of the degrees $\mbox{\bf a}$ and $\mbox{\bf b}$ . This paper introduces a jump operator ( $\cdot'$ ) on the honest elementary degrees and defines canonical degrees $\mbox{\bf 0},\mbox{\bf 0}', \mbox{\bf 0}^{\prime \prime },\ldots$ and low and high degrees analogous to the (...) corresponding concepts for the Turing degrees. Among others, the following results about the structure of the honest elementary degrees are shown: There exist low degrees, and there exist degrees which are neither low nor high. Every degree above $\mbox{\bf 0}'$ is the jump of some degree, moreover, for every degree $\mbox{\bf c}$ above $\mbox{\bf 0}'$ there exist degrees $\mbox{\bf a},\mbox{\bf b}$ such that $\mbox{\bf c}=\mbox{\bf a} \cup \mbox{\bf b} = \mbox{{\bf a}}'=\mbox{\bf b}'$ . We have $\mbox{\bf a}'\cup \mbox{\bf b}' \leq (\mbox{\bf a}\cup\mbox{\bf b})'$ and $\mbox{\bf a}'\cap \mbox{\bf b}' \geq (\mbox{\bf a}\cap \mbox{\bf b})'$ . The jump operator is of course monotonic, i.e. $\mbox{\bf a}\leq\mbox{{\bf b}}\Rightarrow \mbox{\bf a}'\leq \mbox{\bf b}'$ . We prove that every situation compatible with $\mbox{\bf a}\leq\mbox{\bf b}\Rightarrow \mbox{\bf a}'\leq \mbox{\bf b}'$ is realized in the structure, e.g. we have incomparable degrees $\mbox{\bf a},\mbox{\bf b}$ such that $\mbox{\bf a}'<\mbox{\bf b}'$ and incomparable degrees $\mbox{\bf a},\mbox{\bf b}$ such that $\mbox{\bf a}' = \mbox{\bf b}'$ etcetera. We are able to prove all these results without the traditional recursion theoretic constructions. Our proof method relies on the fact that the growth of the functions in a degree is bounded. This technique also yields a very simple proof of an old result, namely that the structure is a lattice. (shrink)

We show that logic has more to offer to ontologists than standard first order and modal operators. We first describe some operators of linear logic which we believe are particularly suitable for ontological modeling, and suggest how to interpret them within an ontological framework. After showing how they can coexist with those of classical logic, we analyze three notions of artifact from the literature to conclude that these linear operators allow for reducing the ontological commitment needed for (...) their formalization, and even simplify their logical formulation. (shrink)

In this paper, we approach the problem of classical recapture for LP and K3 by using normality operators. These generalize the consistency and determinedness operators from Logics of Formal Inconsistency and Underterminedness, by expressing, in any many-valued logic, that a given formula has a classical truth value (0 or 1). In particular, in the rst part of the paper we introduce the logics LPe and Ke3 , which extends LP and K3 with normality operators, and we establish a classical (...) recapture result based on the two logics. In the second part of the paper, we compare the approach in terms of normality operators with an established approach to classical recapture, namely minimal inconsistency. Finally, we discuss technical issues connecting LPe and Ke3 to the tradition of Logics of Formal Inconsistency and Underterminedness. (shrink)

The logic with independent truth and falsehood operators TFL is proposed. In TFL(→) standard truth-conditions for the implication are adopted. Nevertheless the laws of classical logic are not valid. In this language more then 107 different binary connectives can be defined. So this logic can be treated as universal logic relatively to the class of sentential logics.

The properties of the ${\forall^{1}}$ quantifier defined by Kontinen and Väänänen in [13] are studied, and its definition is generalized to that of a family of quantifiers ${\forall^{n}}$ . Furthermore, some epistemic operators δ n for Dependence Logic are also introduced, and the relationship between these ${\forall^{n}}$ quantifiers and the δ n operators are investigated.The Game Theoretic Semantics for Dependence Logic and the corresponding Ehrenfeucht- Fraissé game are then adapted to these new connectives.Finally, it is proved that the (...) ${\forall^{1}}$ quantifier is not uniformly definable in Dependence Logic, thus answering a question posed by Kontinen and Väänänen in the above mentioned paper. (shrink)

We study a knowledge logic that assumes that to each set of agents, an indiscernibility relation is associated and the agents decide the membership of objects or states up to this indiscernibility relation. Its language contains a family of relative knowledge operators. We prove the decidability of the satisfiability problem, we show its EXPTIME-completeness and as a side-effect, we define a complete Hilbert-style axiomatization.

In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in [8] (D. Gregory, Completeness and decidability results for some propositional modal logics containing “actually” operators, Journal of Philosophical Logic 30(1): 57–78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in [8] are obtained. We then compare Gregory's modal language enriched with an “actually” operator with the work of Arthur Prior now (...) known under the name of hybrid logic. This analysis relates the “actually” axioms to standard hybrid axioms, yields the decidability results in [8], and provides a number of complexity results. Finally, we use a bisimulation argument to show that the hybrid language is strictly more expressive than Gregory's language. (shrink)

. In this paper, the significance of using general logic-systems and finite consequence operators defined on non-organized languages is discussed. Results are established that show how properties of finite consequence operators are independent from language organization and that, in some cases, they depend only upon one simple language characteristic. For example, it is shown that there are infinitely many finite consequence operators defined on any non-organized infinite language L that cannot be generated from any finite logic-system. On the (...) other hand, it is shown that for any nonempty language L, a set map is a finite consequence operator if and only if it is defined by a general logic-system. Simple logic-system examples that determine specific consequence operator properties are given. (shrink)

This paper deals with the proof theory of first-order applicative theories with non-constructive μ operator and a form of the bar rule, yielding systems of ordinal strength Γ0 and 20, respectively. Relevant use is made of fixed-point theories with ordinals plus bar rule.

We study operator equations within the Turing machine based framework for computability in analysis. Is there an algorithm that maps pairs to solutions of Tx = u ? Here we consider the case when T is a bounded linear mapping between Hilbert spaces. We are in particular interested in computing the generalized inverse T†u, which is the standard concept of solution in the theory of inverse problems. Typically, T† is discontinuous and hence no computable mapping. However, we will use (...) effective versions of theorems from the theory of regularization to show that the mapping ↦ T†u is computable. We then go on to study the computability of average-case solutions with respect to Gaussian measures which have been considered in information based complexity. Here, T† is considered as an element of an L2-space. We define suitable representations for such spaces and use the results from the first part of the paper to show that ↦ T† is computable. (shrink)

We show that the e-degree 0'e and the map u ↦ u' are definable in the upper semilattice of all e-degrees. The class of total e-degrees ≥0'e is also definable.

Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: BON (...) plus set induction is proof-theoretically equivalent to Peano arithmetic PA; BON plus formula induction is proof-theoretically equivalent to the system <0 of second-order arithmetic. (shrink)

Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of certain set-valued mappings between function spaces. This paper deals with the computational properties of these accretive and (generalized) monotone set-valued operators. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a program in mathematical logic that seeks to extract computational information from prima facie “non-computational” proofs from the mainstream literature. To this end, we establish (...) logical metatheorems that guarantee and quantify the computational content of theorems pertaining to accretive and (generalized) monotone set-valued operators. On the one hand, our results unify a number of recent case studies, while they also provide characterizations of central analytical notions in terms of proof theoretic ones on the other, which provides a crucial perspective on needed quantitative assumptions in future applications of proof mining to these branches. (shrink)

One of the major problems of intrusion detection concerns the large amount of alerts that intrusion detection systems (IDS) produce. Security operator who analyzes alerts and takes decisions, is often submerged by the high number of alerts to analyze. In this paper, we present a new alert correlation approach based on knowledge and preferences of security operators. This approach, which is complementary to existing ones, allows to rank-order produced alerts on the basis of a security operator knowledge about (...) the system, used IDS and his preferences about alerts that he wants to analyze or to ignore. Our approach is based on the development of a new non-classical logic for representing preferences, called FO-MQCL (First Order - Minimal Qualitative Choice Logic). Our logic extends a fragment of the first order logic by adding a new logical connective. The general idea is to present only alerts that fully fit security operator's preferences and knowledge. And if needed, less preferred alerts can also be presented. (shrink)

One of the major problems of intrusion detection concerns the large amount of alerts that intrusion detection systems (IDS) produce. Security operator who analyzes alerts and takes decisions, is often submerged by the high number of alerts to analyze. In this paper, we present a new alert correlation approach based on knowledge and preferences of security operators. This approach, which is complementary to existing ones, allows to rank-order produced alerts on the basis of a security operator knowledge about (...) the system, used IDS and his preferences about alerts that he wants to analyze or to ignore. Our approach is based on the development of a new non-classical logic for representing preferences, called FO-MQCL (First Order - Minimal Qualitative Choice Logic). Our logic extends a fragment of the first order logic by adding a new logical connective. The general idea is to present only alerts that fully fit security operator's preferences and knowledge. And if needed, less preferred alerts can also be presented. (shrink)

I present a notion of invariance under arbitrary surjective mappings for operators on a relational finite type hierarchy generalizing the so-called Tarski-Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use it to characterize logicality.

We shall describe the set of strongly meet irreducible logics in the lattice ϵLin.t of normal tense logics of weak orderings. Based on this description it is shown that all logics in ϵLin.t are independently axiomatizable. Then the description is used in order to investigate tense logics with respect to decidability, finite axiomatizability, axiomatization problems and completeness with respect to Kripke semantics. The main tool for the investigation is a translation of bimodal formulas into a language talking about partitions of (...) general frames into intervals so that relative to both Kripke frames and descriptive frames the expressive power of both languages coincides. (shrink)

Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expressed as Kronecker polynomials. These polynomials allow us to show that many important tautologies of classical logic are generated from basic operators via the operations called Type (...) I and Type II products. Finally, it is described a matrix version of the Fredkin gate that extends its properties to the many-valued domain, and it is proved that the filtered Fredkin operators are second degree Kronecker polynomials that cannot be generated by Type I or Type II products. (shrink)

Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. This accounting is made possible by removing the ‘structural’ rules of contraction and weakening, adding a modal operator and adding finer versions of the propositional connectives. Linear logic has fundamental logical interest and applications to computer science, particularly to Petri nets, concurrency, storage allocation, garbage collection and the control structure of logic programs. In addition, there is a (...) direct correspondence between polynomial-time computation and proof normalization in a bounded form of linear logic. In this paper we show that unlike most other propositional logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes PSPACE-complete. We also establish membership in NP for the multiplicative fragment, NP-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for fragments of noncommutative propositional linear logic. (shrink)

A characterisation of a type of weak-operator continuous linear functional on certain linear subsets of B, where H is a Hilbert space, is derived within Bishop-style constructive mathematics.

The logic of quantum mechanical propositions—called quantum logic—is constructed on the basis of the operational foundation of logic. Some obvious modifications of the operational method, which come from the incommensurability of the quantum mechanical propositions, lead to the effective quantum logic. It is shown in this paper that in the framework of a calculization of this effective quantum logic the negation of a proposition is uniquely defined (Theorem I), and that a weak form of the (...) quasimodular law can be derived (Theorem II). Taking account of the definiteness of truth values for quantum mechanical propositions, the calculus of full quantum logic can be derived (Theorem III). This calculus represents an orthocomplemented quasimodular lattice which has as a model the lattice of subspaces of Hilbert space. (shrink)

In this note we introduce and study algebras of type such that is a bounded distributive lattice and ⌝ is an operator that satisfies the condition ⌝ = a ⌝ b and ⌝ 0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi-Stone algebras.

To sentential language we add an operator C to be read as ‘it changes that…’ and present an axiomatic system in the frame of classical logic to catch some meaning of the term ‘change’. A typical axiom is e.g.: CA implies, a basic rule is: from A it may be inferred (theorems do not change). So this system is not regular. On the semantic level we introduce stages (of the development of some world, of some agents’ convictions or (...) of some argumentation) at which a sentence may be true or false. It turns out that with the help of C, an operator N can be defined whose intuitive meaning is ‘on the next occasion…’ and which behaves like A.N. Prior’s F (and also like the T operator of G.H. von Wright’s read ‘… and next…’). In a book by Świetorzecka (2008) cited below, the philosophical background is described which is the Aristotelian theory of substantial change. The author of this book shows also some metalogical properties of this logic. The aim of this text now is to present a formal extraction of Świetorzecka (2008) with a shortened axiomatisation and to describe some metalogical results. (shrink)

We describe here a simple method in order to obtain programs from proofs in second-order classical logic. Then we extend to classical logic the results about storage operators proved by Krivine for intuitionistic logic. This work generalizes previous results of Parigot.

In this paper the language of first-order modal logic is enriched with an operator @ ('actually') such that, in any model, the evaluation of a formula @A at a possible world depends on the evaluation of A at the actual world. The models have world-variable domains. All the logics that are discussed extend the classical predicate calculus, with or without identity, and conform to the philosophical principle known as serious actualism. The basic logic relies on the system (...) K, whereas others correspond to various properties that the actual world may have. All the logics are axiomatized. (shrink)

In the paper, there is presented the theory of logical consequence operators indexed with taboo functions. It describes the mechanisms of logical inference in the environment of forbidden sentences. This kind of processes take place in ideological discourses within which their participants create various narrative worlds (mental worlds). A peculiar feature of ideological discourses is their association with taboo structures of deduction which penalize speech acts. The development of discourse involves, among others, transforming its deduction structure towards the proliferation of (...) consequence operators and modifying penalty functions. The presented theory enables to define various processes of these transformations in the precise way. It may be used in analyses of conflicts between competing elm experts acting within a discourse. (shrink)

This paper describes Peirce's systems of logic diagrams, focusing on the so-called ''existential'' graphs, which are equivalent to the first-order predicate calculus. It analyses their implications for the nature of mental representations, particularly mental models with which they have many characteristics in common. The graphs are intended to be iconic, i.e., to have a structure analogous to the structure of what they represent. They have emergent logical consequences and a single graph can capture all the different ways in which (...) a possibility can occur. Mental models share these properties. But, as the graphs show, certain aspects of propositions cannot be represented in an iconic or visualisable way. They include negation, and the representation of possibilities qua possibilities, which both require representations that do not depend on a perceptual modality. Peirce took his graphs to reveal the fundamental operations of reasoning, and the paper concludes with an analysis of different hypotheses about these operations. (shrink)

We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of n conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan’s Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; in particular (...) we examine the Fréchet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction Cn+1 of n + 1 conditional events to the family {Cn,En+1|Hn+1}. We consider the relation with the notion of quasi-conjunction and we examine in detail the coherence of the prevision assessments related with the conjunction of three conditional events. Based on conjunction, we also give a characterization of p-consistency and of p-entailment, with applications to several inference rules in probabilistic nonmonotonic reasoning. Finally, we examine some non p-valid inference rules; then, we illustrate by an example two methods which allow to suitably modify non p-valid inference rules in order to get inferences which are p-valid. (shrink)

The general aim of this book is to provide an elementary exposition of some basic concepts in terms of which both classical and non-dassicallogirs may be studied and appraised. Although quantificational logic is dealt with briefly in the last chapter, the discussion is chiefly concemed with propo gjtional cakuli. Still, the subject, as it stands today, cannot br covered in one book of reasonable length. Rather than to try to include in the volume as much as possible, I have (...) put emphasis on some selected topics. Even these could not be roverrd completely, but for each topic I have attempted to present a detailed and precise t’Xposition of several basic results including some which are non-trivial. The roots of some of the central ideas in the volume go back to J. Luka siewicz’s seminar on mathematicallogi. (shrink)