Suszko's Problem: Mixed Consequence and Compositionality

Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko's problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that (intersective) mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various structural properties (reflexivity, transitivity, none, or both). We strengthen these results into exact rank results for non-permeable logics (roughly, those which distinguish the role of premises and conclusions). We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties.Comment: Keywords: Suszko's thesis; truth value; logical consequence; mixed consequence; compositionality; truth-functionality; many-valued logic; algebraic logic; substructural logics; regular connective

A chunking mechanism in a neural system for the parallel processing of a propositional production rules

The problem of extracting more compact rules from a rule-based knowledge base is approached by means of a chunking mechanism implemented via a neural system. Taking advantage of the parallel processing potentialities of neural systems, the computational problem normally arising when introducing chuncking processes is overcome. Also the memory saturation effect is coped with using some sort of "forgetting" mechanism which allows the system to eliminate previously stored, but less often used chunks. Even though some connection weights are changed in the process of storing or discarding chunks, we emphasize that this neural system cannot be regarded as a "connectionist" system, since a localist semantic interpretation is adopted and no classical learning algorithm is employed

A chunking mechanism in a neural system for the parallel processing of a propositional production rules

The problem of extracting more compact rules from a rule-based knowledge base is approached by means of a chunking mechanism implemented via a neural system. Taking advantage of the parallel processing potentialities of neural systems, the computational problem normally arising when introducing chuncking processes is overcome. Also the memory saturation effect is coped with using some sort of "forgetting" mechanism which allows the system to eliminate previously stored, but less often used chunks. Even though some connection weights are changed in the process of storing or discarding chunks, we emphasize that this neural system cannot be regarded as a "connectionist" system, since a localist semantic interpretation is adopted and no classical learning algorithm is employed

Classical logic, argument and dialectic

A well studied instantiation of Dung's abstract theory of argumentation yields argumentation-based characterisations of non-monotonic inference over possibly inconsistent sets of classical formulae. This provides for single-agent reasoning in terms of argument and counter-argument, and distributed non-monotonic reasoning in the form of dialogues between computational and/or human agents. However, features of existing formalisations of classical logic argumentation (Cl-Arg) that ensure satisfaction of rationality postulates, preclude applications of Cl-Arg that account for real-world dialectical uses of arguments by resource-bounded agents. This paper formalises dialectical classical logic argumentation that both satisfies these practical desiderata and is provably rational. In contrast to standard approaches to Cl-Arg we: 1) draw an epistemic distinction between an argument's premises accepted as true, and those assumed true for the sake of argument, so formalising the dialectical move whereby arguments\u2019 premises are shown to be inconsistent, and avoiding the foreign commitment problem that arises in dialogical applications; 2) provide an account of Cl-Arg suitable for real-world use by eschewing the need to check that an argument's premises are subset minimal and consistent, and identifying a minimal set of assumptions as to the arguments that must be constructed from a set of formulae in order to ensure that the outcome of evaluation is rational. We then illustrate our approach with a natural deduction proof theory for propositional classical logic that allows measurement of the \u2018depth\u2019 of an argument, such that the construction of depth-bounded arguments is a tractable problem, and each increase in depth naturally equates with an increase in the inferential capabilities of real-world agents. We also provide a resource-bounded argumentative characterisation of non-monotonic inference as defined by Brewka's Preferred Subtheories

Development And Validation Of The Extended Model Of Gifttedness And Talent (EMGT) Using Structural Equation Modeling

The concept of compounding of different abilities into a specific capability was demonstrated by Spearman (1905) and Mcgrew (1997) for the Cattell Horn Carroll Model using factor analysis but has remained unexplored since then and the effects of nurturing elements has also remained unresolved. This study investigated the Extended Model of Giftedness and Talent (EMGT) based on the compoundability hypothesis and the effects of nurturing elements such as environment and motivation on the development of giftedness and talent across the intellectual domain by developing new instruments and test items and employing statistical validation procedures using Structural Equation Modelling (SEM). The instrument consisted of culture-reduced test items that incorporated Cattell’s Culture-Fair Test (CCFT) items and tested for six standard subscales of intellectual aptitudes such as memory span, pattern recognition, sense of observation

Negotiating Socially Optimal Allocations of Resources with Argumentation

The resource allocation problem of multi-agent systems is the problem of deciding how to allocate resources, controlled by agents, to agents within a given system. Agents typically have preferences over alternative allocations of resources. These preferences may be derived from the agents’ goals, which can be fulfilled by different plans (sets of resources). The problem arises because agents may not be able to fulfil their goals without being re-allocated resources controlled by other agents and agents may have conflicting preferences over allocations. Examples of the resource allocation problem include electronic commerce (where resources are commodities equipped with prices), the grid (where resources are computational entities equipped with computational power), and scheduling and timetabling (where resources may be tasks with costs). The focus in this thesis is distributed decision-making amongst agents, whereby agents actively participate in computing re-allocations, starting from initial allocations which may or may not fulfil their goals. A re-allocation is arrived at by means of local negotiation steps wherein resources change hands between the agents involved in the negotiations. The negotiation method of choice in this thesis is argumentation-based negotiation supported by assumption-based argumentation. This method allows agents to work towards their goals despite incomplete information regarding the goals of and resources allocated to other agents, to share knowledge, thereby eliminating unknowns, and to resolve conflicts within themselves and between one another which may arise because of inconsistent information. Solutions generated by a resource allocation mechanism may be ranked according to how they affect the individual welfare of the agents as well as the overall social welfare of the agent society, according to different notions of social welfare borrowed from economics. The argumentation-based negotiation mechanism we propose guarantees, for the problem domain of interest in this thesis, that negotiations between agents always terminate converging to a solution. Moreover, the mechanism guarantees that solutions reached optimise the welfare of the individual agents as well as the agent society as a whole according to Pareto optimal and utilitarian notions of social welfare