Agenda manipulation under doctrinal paradox

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of "revenge Liar" arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's "useful four-valued logic", one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a "bi-and-a-half" lattice which determines seven-valued logics different from Priest's Logic of Paradox.

In the present chapter, we discuss the possibility of generalizing the very notion of a truth value by constructing truth values as complex units which may possess a ramified inner structure. We consider some approaches to truth values as structured entities and summarize this point in the notion of a generalized truth value conceived as a subset of some basic set of initial truth values of a lower degree. We are essentially led by the idea which is at the heart of Belnap and Dunn’s useful four-valued logic, where the set \({{\mathbf{2}}} = \{T,F\}\) of classical truth values is generalized to the set \({{\mathbf{4}}} = {{\fancyscript{P}}}({{\mathbf{2}}}) = \{\varnothing,\{T\},\{F\},\{T,F\}\}.\) We argue in favor of extending this process to the set \({{\mathbf{16}}} = {{\fancyscript{P}}}({{\mathbf{4}}}).\) It turns out that this generalization is well-motivated and leads to a notion of a truth value multilattice. In particular, we proceed from the bilattice \(FOUR_{2}\) with both an information and truth-and-falsity ordering to another algebraic structure, namely the trilattice \(SIXTEEN_{3}\) with an information ordering together with a truth ordering \(and\) a (distinct) falsity ordering. We also consider another exemplification of essentially the same structure based on the set of truth values one can find in various constructive logics.

Normal forms for fuzzy relations and their contribution to universal approximation

We consider the problem of efficiently indexing Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF) Boolean expressions over a high-dimensional multi-valued attribute space. The goal is to rapidly find the set of Boolean expressions that evaluate to true for a given assignment of values to attributes. A solution to this problem has applications in online advertising (where a Boolean expression represents an advertiser's user targeting requirements, and an assignment of values to attributes represents the characteristics of a user visiting an online page) and in general any publish/subscribe system (where a Boolean expression represents a subscription, and an assignment of values to attributes represents an event). All existing solutions that we are aware of can only index a specialized sub-set of conjunctive and/or disjunctive expressions, and cannot efficiently handle general DNF and CNF expressions (including NOTs) over multi-valued attributes. In this paper, we present a novel solution based on the inverted list data structure that enables us to index arbitrarily complex DNF and CNF Boolean expressions over multi-valued attributes. An interesting aspect of our solution is that, by virtue of leveraging inverted lists traditionally used for ranked information retrieval, we can efficiently return the top-N matching Boolean expressions. This capability enables emerging applications such as ranked publish/subscribe systems [16], where only the top subscriptions that match an event are desired. For example, in online advertising there is a limit on the number of advertisements that can be shown on a given page and only the "best" advertisements can be displayed. We have evaluated our proposed technique based on data from an online advertising application, and the results show a dramatic performance improvement over prior techniques.

The problem is the transformation of a conjunctive normal form (CNF) into a minimized (for the inclusion operator) disjunctive normal form (DNF) and vice versa. This operation is called the unionist product. For a CNF (resp. DNF), one pass of the unionist product provides the prime implicants (resp. implicates); two passes provide the prime implicates (resp. implicants). An algorithm built upon the classical Davis and Putnam procedure is presented for calculating, without the explicit minimization for the inclusion, this unionist product.

Disjunctive and conjunctive normal forms of pseudo-Boolean functions

Normal forms for fuzzy logic functions and their approximation ability

We describe the problem of mining set valued rules in large relational tables containing categorical attributes taking a finite number of values. Such rules allow for an interval of possible values to be selected for each attribute in condition instead of a single value for association rules, while conclusion contains a projection of the data restricted by the condition onto a target attribute. An example of such a rule might be “if HOUSEHOLDSIZE = {Two OR Tree} AND OCCUPATION={Professional OR Clerical} THEN PAYMENT_METHOD = {CashCheck (Max=249, Sum=4952) OR DebitCard (Max=175, Sum=3021)} WHERE Confidence=85%, Support=10%.}” We use an original conceptional and formal framework for representing multidimensional distribution induced from data by a number of so-called prime disjunctions upper bounding its surface. Each prime disjunction represents a wide multidimensional interval of impossible combinations of attribute values. This original formalism generalises the conventional boolean approach in two directions: (i) finite-valued attributes (instead of only 0 and 1), and (ii) continuous-valued semantics instead of (true and false). In addition, we describe an efficient algorithm, which carries out the generalised dual transformation from possibilistic disjunctive normal form (DNF) representing data into conjunctive normal form (CNF) representing knowledge and thus generates all the most interesting prime disjunctions. Once obtained they can be used to build different forms of rules or for other purposes (prediction, clustering etc.).

Fuzzy set theories allow us to represent our knowledge under various interpretations and axiomatic foundations from linguistic to computational representations. There are at least four levels of knowledge representation: (i) linguistic, (ii) meta-linguistic, (iii) propositional, and (iv) computational. There are three transformations, which depend on the particular interpretations put on a knowledge representation schema. There are various choices, corresponding to one's interpretation of: (a) type of set representation, (fuzzy or crisp), (b) type of propositional connectives and normal forms, and (c) type of computational connectives, i.e., weak or strong t-norms and co-norms. In the light of these selections, fuzzy disjunctive and conjunctive normal forms (FDNF, FCNF) are derived from fuzzy truth tables. It is shown that classical expressions such as excluded middle etc., when fuzzified, ought to be reinterpreted with a type II, second order, semantic uncertainty. The classical expressions should not be interpreted as to whether they are valid or not. One can only state that the well-known tautologies of classical logic are valid to many degrees specified by an interval defined by their FDNF and FCNF. FDNF and FCNF boundaries identify the nonspecificity measure associated with type II, second order, semantic uncertainty. Thus, those researchers who are not familiar or who are not concerned with type II semantic uncertainty work with a myopic understanding of fuzzy set and logic theories. >

In the article [4], a new constructive set theoretic principle called Refinement was introduced and analysed. While it seemed to be significantly weaker than its alternative, the more established axiom of Fullness (a constructive version of the Powerset axiom from classical set theory), it was shown to suffice to imply many of the mathematically important consequences. In this article, we will define for each set A a set of truth values which measures the complexity of the equality relation on A. Using these sets we will show that Refinement is actually equivalent to Fullness on the basis of the other axioms of constructive Zermelo‐Fraenkel set theory (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

It is shown that the infinite-valued first-order Godel logic G° based on the set of truth values {1/k: k e w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kroger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal logic of linear discrete time with gaps follows.

We explore different ways to generalize Hintikka’s classic game theoretic semantics to a many-valued setting, where the unit interval is taken as the set of truth values. In this manner a plethora of characterizations of Łukasiewicz logic arise. Among the described semantic games is Giles’s dialogue and betting game, presented in a manner that makes the relation to Hintikka’s game more transparent. Moreover, we explain a so-called explicit evaluation game and a ‘bargaining game’ variant of it. We also describe a recently introduced backtracking game as well as a game with random choices for Łukasiewicz logic.

We present some compositions of clones of Boolean functions that imply factorizations of Ω, the clone of all Boolean functions, into minimal clones. These can be interpreted as representation theorems, providing representations of Boolean functions analogous to the disjunctive normal form, the conjunctive normal form, and the Zhegalkin polynomial representations.

In this paper, we put forward a weak conjunctive normal form for G3, which is similar to the conjunctive normal form for classical propositional logic. We also give two algorithms to reduce a formula to weak conjunctive normal form, where one is based on charactering countermodels in model-based-formulas, and the other is based on rewriting formulas according to transformation rules. In the end, we show that there is no polynomial time algorithm to reduce an arbitrary formula to the weak conjunctive normal form.

Hedge algebras: An algebraic approach to structure of sets of linguistic truth values