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The probabilization of a logic system consists of enriching the language (the formulas) and the semantics (the models) with probabilistic features. Such an operation is said to be exogenous if the enrichment is done on top, without internal changes to the structure, and is called endogenous otherwise. These two different enrichments can be applied simultaneously to the language and semantics of a same logic. We address the problem of studying the transference of metaproperties, such as completeness and decidability, to the exogenous probabilization of an abstract logic system. First, we setup the necessary framework to handle the probabilization of a satisfaction system by proving transference results within a more general context. In this setup, we define a combination mechanism of logics through morphisms and prove sufficient condition to guarantee completeness and decidability. Then, we demonstrate that probabilization is a special case of this exogenous combination method, and that it fulfills the general conditions to obtain transference of completeness and decidability. Finally, we motivate the applicability of our technique by analyzing the probabilization of the linear temporal logic over Markov chains, which constitutes an endogenous probabilization. The results are obtained first by studying the exogenous semantics, and then by establishing an equivalence with the original probabilization given by Markov chains.

In Computer Science stepwise refinement of algebraic specifications is a well-known formal methodology for rigorous program development. This paper illustrates how techniques from Algebraic Logic, in particular that of interpretation, understood as a multifunction that preserves and reflects logical consequence, capture a number of relevant transformations in the context of software design, reuse, and adaptation, difficult to deal with in classical approaches. Examples include data encapsulation and the decomposition of operations into atomic transactions. But if interpretations open such a new research avenue in program refinement, (conceptual) tools are needed to reason about them. In this line, the paper’s main contribution is a study of the correspondence between logical interpretations and morphisms of a particular kind of coalgebras. This opens way to the use of coalgebraic constructions, such as simulation and bisimulation, in the study of interpretations between (abstract) logics., Fundação para a Ciência e a Tecnologia (FCT)

We present a new logic-based approach to the reasoning about knowledge which is independent of possible worlds semantics. $${\in_K}$$ (Epsilon-K) is a non-Fregean logic whose models consist of propositional universes with subsets for true, false and known propositions. Knowledge is, in general, not closed under rules of inference; the only valid epistemic principles are the knowledge axiom K i ? ? ? and some minimal conditions concerning common knowledge in a group. Knowledge is explicit and all forms of the logical omniscience problem are avoided. Various stronger epistemic properties such as positive and/or negative introspection, the K-axiom, closure under logical connectives, etc. can be restored by imposing additional semantic constraints. This yields corresponding sublogics for which we present sound and complete axiomatizations. As a useful tool for general model constructions we study abstract versions of some 3-valued logics in which we interpret truth as knowledge. We establish a connection between $${\in_K}$$ and the well-known syntactic approach to explicit knowledge proving a result concerning equi-expressiveness. Furthermore, we discuss some self-referential epistemic statements, such as the knower paradox, as relaxations of variants of the liar paradox and show how these epistemic "paradoxes" can be solved in $${\in_K}$$ . Every specific $${\in_K}$$ -logic is defined as a certain extension of some underlying classical abstract logic.

Psychology is not alone in its struggle with conceptualizing the dynamic relationship between space and individual or collective identity. This general epistemological issue haunts biology where it has a specific focus in evolutionary arguments. It arises because of the incompatibility between definitive logical systems of 'contradiction or unity', which can only apply to inert material systems, and natural evolutionary processes of cumulative energetic transformation. This incompatibility makes any attempt to apply definitive logic to evolutionary change unrealistic and paradoxical. It is important to recognise, because discrete perceptions of self and group, based on the supposition that any distinguishable identity can be completely cut free, as an 'independent singleness', from the space it inescapably includes and is included in, are a profound but unnecessary source of psychological, social and environmental conflict. These perceptions underlie Darwin's definition of 'natural selection' as 'the preservation of favoured races in the struggle for life'. They result in precedence being given to striving for homogeneous supremacy, through the competitive suppression of others, instead of seeking sustainable, co-creative evolutionary relationship in spatially and temporally heterogeneous communities. Here, I show how 'natural inclusion', a new, post-dialectic understanding of evolutionary process, becomes possible through recognising space as a limitless, indivisible, receptive (non-resistive) 'intangible presence' vital for movement and communication, not as empty distance between one tangible thing and another. The fluid boundary logic of natural inclusion as the co-creative, fluid dynamic transformation of all through all in receptive spatial context, allows all form to be understood as flow-form, distinctive but dynamically continuous, not singularly discrete. This simple move from regarding space and boundaries as sources of discontinuity and discrete definition to sources of continuity and dynamic distinction correspondingly enables self-identity to be understood as a dynamic inclusion of neighbourhood, through the inclusion of space throughout and beyond all natural figural forms as configurations of energy. Fully to appreciate and communicate the significance of this move, it is necessary to widen the linguistic, mathematical and imaginative remit of conventional scientific argument and explication so as to include more poetic, fluid and artistic forms of expression.

In this paper we study an alternative approach to the concept of abstract logic and to connectives in abstract logics. The notion of abstract logic was introduced by Brown and Suszko (Diss Math 102:9–42, 1973)—nevertheless, similar concepts have been investigated by various authors. Considering abstract logics as intersection structures we extend several notions to their κ-versions (κ ≥ ω), introduce a hierarchy of κ-prime theories, which is important for our treatment of infinite connectives, and study different concepts of κ-compactness. We are particularly interested in non-topped intersection structures viewed as semi-lattices with a minimal meet-dense subset, i.e., with a minimal generator set. We study a chain condition which is sufficient for a minimal generator set, implies compactness of the logic, and in regular logics is equivalent to (κ-) compactness of the consequence relation together with the existence of a (κ-)inconsistent set, where κ is the cofinality of the cardinality of the logic. Some of these results are known in a similar form in the context of closure spaces, we give extensions to (non-topped) intersection structures and to big cardinals presenting new proofs based on set-theoretical tools. The existence of a minimal generator set is crucial for our way to define connectives. Although our method can be extended to further non-classical connectives we concentrate here on intuitionistic and infinite ones. Our approach leads us to the concept of the set of complete theories which is stable under all considered connectives and gives rise to the definition of the topological space of the logic. Topological representations of (non-classical) abstract logics by means of this space remain to be further investigated.

This paper examines the contribution of stimulus processing to animal logics. In the classic functionalist S-O-R view of learning (and cognition), stimuli provide the raw material to which the organism applies its cognitive processes-its logic, which may be taxon-specific. Stimuli may contribute to the logic of the organism's response, and may do so in taxon-specific ways. Firstly, any non-trivial stimulus has an internal organization that may constrain or bias the way that the organism addresses it; since stimuli can only be defined relative to the organism's perceptual apparatus, and this apparatus is taxon-specific, such constraints or biases will often be taxon-specific. Secondly, the representation of a stimulus that the perceptual system builds, and the analysis it makes of this representation, may provide a model for the synthesis and analysis done at a more cognitive level. Such a model is plausible for evolutionary reasons: perceptual analysis was probably perfected before cognitive analysis in the evolutionary history of the vertebrates. Like stimulus-driven analysis, such perceptually modelled cognition may be taxon-specific because of the taxon-specificity of the perceptual apparatus. However, it may also be the case that different taxa are able to free themselves from the stimulus logic, and therefore apply a more abstract logic, to different extents. This thesis is defended with reference to two examples of cases where animals' cognitive logic seems to be isomorphic with perceptual logic, specifically in the case of pigeons' attention to global and local information in visual stimuli, and dogs' failure to comprehend means-end relationships in string-pulling tasks.

Suszko's Thesis maintains that many-valued logics do not exist at all. In order to support it, R. Suszko offered a method for providing any structural abstract logic with a complete set of bivaluations. G. Malinowski challenged Suszko's Thesis by constructing a new class of logics (called q-logics by him) for which Suszko's method fails. He argued that the key for logical two-valuedness was the "bivalent" partition of the Lindenbaum bundle associated with all structural abstract logics, while his q-logics were generated by "trivalent" matrices. This paper will show that contrary to these intuitions, logical two-valuedness has more to do with the geometrical properties of the deduction relation of a logical structure than with the algebraic properties embedded on it.

A notion of factorizability for vector-valued measures on a quantum logic L enables us to pass from abstract logics to Hilbert space logics and thereby to construct tensor products. A claim by Kruszynski that, in effect, every orthogonally scattered measure is factorizable is shown to be false. Some criteria for factorizability are found.

An abstract logic 〈A, C〉 consists of a finitary algebraA and a closure systemC onA. C induces two other closure systems onA, CP andCI, by projective and inductive generation respectively. The various relations amongC, CP andCI are determined. The special case thatC is the standard equational closure system on monadic terms is studied in detail. The behavior of Boolean logics with respect to projective and inductive generation is determined.

The objective of the present paper is to introduce a new approach to the study of the logical structure of scientific theories. We shall consider, in broad outline, a basic framework for handling theories and for analyzing a number of key metatheoretical concepts. The framework itself may be formally captured within a suitable system of set theory. However, its major innovative and heuristic thrust derives from some notions em ployed in modern abstract logic; in particular, it encourages the application of model-theoretic concepts and results to questions con cerning the structure and dynamics of empirical theories. The viewpoint adopted here shares some features in common with previous attempts at framing metascientific concepts, especially with the 'logistic' or model-theoretic, and with the so-called structuralist ap proaches. Like the former, it stresses the use of formal semantics in metascience; with the latter, it also appeals to some of the more abstract and structural characteristics of theories. However, it differs from these and other traditional approaches, notably in the role in which logic is portrayed. The nature and function of logic in scientific theorizing is here construed in an unusually liberal fashion: it is neither seen as a constraint on the syntactical form of theories, nor viewed as imposing limitations on the kinds of inferences permitted in science. This flexible and versatile conception of logic will be seen to amount to a somewhat radical departure from traditional usage. It also constitutes what the authors believe to be the clearest grounds for upholding the present framework. It should be mentioned at the outset that no new formal results are brought in this paper;1 nor do we pretend to offer a complete and polished semantical machinery whose fine details are fully developed at each point. Instead, we shall attempt to present something of the flavour of this framework with a minimum of technical fuss.2 We shall also try to provide some arguments in its support, and to indicate how it may lead to some fresh insights on a range of familiar, and sometimes controversial