Hyper-Contradictions, Generalized Truth Values and Logics of Truth and Falsehood

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.

In this chapter, we reconsider the notion of an \(n\)-valued propositional logic. In many-valued logic, sometimes a distinction is made not only between designated and undesignated (not designated) truth values, but also between designated and antidesignated truth values. Even if the set of truth values is, in fact, tripartitioned, usually only a single semantic consequence relation is defined that preserves the possession of a designated value from the premises to the conclusions of an inference. We argue that if in the set of semantical values the sets of designated and antidesignated truth values are not complements of each other, it is natural to define at least \(two\) entailment relations, a “positive” one that preserves the possession of a designated value from the premises to the conclusions of an inference, and a “negative” one that preserves the possession of an antidesignated value from the conclusions to the premises. Once this distinction has been drawn, it is quite natural to reflect it in the logical object language and to contemplate many-valued logics \(\Uplambda\) whose language is split into a positive and a matching negative logical vocabulary. If the positive and the negative entailment relations do not coincide, if the interpretations of matching pairs of connectives are distinct, and if the positive entailment relation restricted to the positive vocabulary is nevertheless isomorphic to the negative entailment relation restricted to the negative vocabulary, then we say that \(\Uplambda\) is a \(harmonious\) many-valued logic. We reconstruct some of the logical systems considered in this book as harmonious, finitely-valued logics. At the end of the chapter, we outline some possible ways of generalizing the notion of a harmonious \(n\)-valued propositional logic.

Статья посвящена замкнутым классам функций четырехзначной логики. Мы представляем следующие результаты:
 (1) Базовые операции логики, полученной расширением четырехзначной алгебры Де Моргана оператором конфляции, порождают замкнутый класс всех функций, которые одновременно сохраняют классические истинностные значения и самодвойственны относительно конфляции. Этот класс предполон в классе всех функций, сохраняющих классические истинностные значения.
 (2) Между замкнутым классом, порожденным базовыми операциями логики истины фон Вригта и классом всех функций, сохраняющих классические истинностные значения, лежит в точности два замкнутых класса. Каждый из них представляет собой класс всех функций, одновременно сохраняющих классические истинностные значения и одно из трехэлементных надмножеств множества классических истинностных значений.
 (3) Базовые операции тетравалентной модальной логики, полученной расширением четырехзначной алгебры Де Моргана оператором необходимости, порождают замкнутый класс всех функций, которые одновременно сохраняют классические истинностные значения, самодвойственны относительно конфляции, а также сохраняют оба трехэлементных надмножества множества классических истинностных значений. Мы показываем, что данный класс предполон в классе всех функций, которые одновременно сохраняют классические истинностные значения и самодвойственны относительно конфляции. Кроме того, мы демонстрируем, что между этим классом и замкнутым классом, порожденным операциями логики истины фон Вригта, находится в точности один замкнутый класс.
 Таким образом мы получаем семиэлементную решетку, состоящую из всех возможных четырехзначных расширений тетравалентной модальной логики, которые сохраняют классические истинностные значения.

Many types of fuzzy truth values have been proposed, such as numerical truth values, interval truth values, triangular truth values and trapezoid truth values and so on. Recently, different type of fuzzy truth values (which we call multi-interval truth values) have been proposed and discussed. A characteristic feature of multi-interval truth values is that some of the truth values are not convex. This fact makes us difficult to understand algebraic properties on multi-interval truth values. This paper first shows that a set of multi-interval truth values is de Morgan bi-lattices when the conventional logical operations are introduced. Next, this paper focuses on functions over the set of the simplest multi-interval truth values, i.e., we define a multi-interval truth value as a non-empty subset of {0, 1, 2}. Then, this paper discusses mathematical properties of functions over multi-interval truth values.

In Chapter 4, many—valued logics were discussed. In that case, sets of truth values were finite and in Chapter 12 we have demonstrated that rough sets may be interpreted as Łukasiewicz algebras of 3—valued Łukasiewicz calculi. However, the Łukasiewicz semantics, introduced in [Łukasiewiczl8] and discussed also in [Łukasiewicz20, 30], and presented in Chapter 4, does allow for infinite sets of truth values. In this case, one may admit as a set of truth values any infinite set T ⊑ [0, 1] which is closed under functions EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864! ]></EquationSource><EquationSource Format="TEX"><![CDATA[$$c:\min \left( 1,1-x+y \right),n:1-x$$ in the sense that c(x, y), n(x) ∈ T whenever x, y ∈ T; we recall that the function c is the meaning of implication C and the function n is the meaning of negation N in the Łukasiewicz semantics. One shows (cf. [McNaughton5l] quoted in [Rose—Rosser58]) that any such T needs to be dense in [0, 1]; it is manifest that 0,1 ∈ T for each admissible T. Thus, sets Q 0 = Q ∩ [0, 1] of rational numbers in [0,1] as well as the whole unit interval [0, 1] may be taken as sets of truth values leading to respective infinite valued logical calculi. The additional merit of these two sets of truth values is that both Q, R l are from algebraic point of view, algebraic fields so they may be regarded as vector spaces over themselves, and we will make use of this fact later on.KeywordsFuzzy LogicInference RuleInductive StepResiduated LatticeInductive AssumptionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Resumen

A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ≥ 3. The purpose of this paper is to improve the form of Miyama's version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzen's logical calculus LK. Let T = {1,…, M} be the set of truth values. An M-tuple (Γ1,…, ΓM) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value μ Є T such that the set Γμ contains a formula of the value μ with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyama's result is as follows (in representative form):(I) If a sequent ({A}, ∅,…, ∅, {B}) is valid, then there is a formula D such that(i) every predicate or propositional variable occurring in D occurs in A and B, and(ii) the sequents {{A}, ∅,…, ∅, {D}) and (D}, ∅,…, ∅, {B}) are both valid.What shall be proved in this paper is the following (in representative form):(II) If a sequent ({A1}, {A2}, …, {AM}) is valid, then there is a formula D such that(i) every predicate or propositional variable occurring in D occurs in at least two of the formulas A1,…, AM, and(ii) the following M sequents are valid:({A1},{D},…,{D}),({D},{A2},…,{D}),…,({D},{D},…,{AM}).Clearly the former can be obtained as a corollary of the latter.

A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ≥ 3. The purpose of this paper is to improve the form of Miyama's version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzen's logical calculus LK. Let T = {1,…, M} be the set of truth values. An M-tuple (Γ1,…, ΓM) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value μ Є T such that the set Γμ contains a formula of the value μ with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyama's result is as follows (in representative form):(I) If a sequent ({A}, ∅,…, ∅, {B}) is valid, then there is a formula D such that(i) every predicate or propositional variable occurring in D occurs in A and B, and(ii) the sequents {{A}, ∅,…, ∅, {D}) and (D}, ∅,…, ∅, {B}) are both valid.What shall be proved in this paper is the following (in representative form):(II) If a sequent ({A1}, {A2}, …, {AM}) is valid, then there is a formula D such that(i) every predicate or propositional variable occurring in D occurs in at least two of the formulas A1,…, AM, and(ii) the following M sequents are valid:({A1},{D},…,{D}),({D},{A2},…,{D}),…,({D},{D},…,{AM}).Clearly the former can be obtained as a corollary of the latter.

Now we are going to start our work with many-valued logic. We have already made several choices. First, as indicated in the title of the present chapter, we shall deal with propositional logic. Second, we take the real unit interval [0, 1] for our set of truth values, 1 being absolute truth, 0 absolute falsity. The natural ordering ≤ of reals will play a very important role; thus our truth values are linearly ordered, the ordering is dense and complete (each set of truth values has its supremum and infimum). In some basic considerations we shall deal with lattice valued logics too.

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work.

An approach to the treatment of inference in the presence of uncertain truth values is described, based on representing uncertainties by sets of ordinary (certain) truth values. Both the algebraic and the logical aspects are studied for a variety of lattices used as truth value spaces in the domain of many-valued logic.

The study of canonically complete attribute-based access control (ABAC) languages is relatively new. A canonically complete language is useful as it is functionally complete and provides a "normal form" for policies. However, previous work on canonically complete ABAC languages requires that the set of authorization decisions is totally ordered, which does not accurately reflect the intuition behind the use of the allow, deny and not-applicable decisions in access control. A number of recent ABAC languages use a fourth value and the set of authorization decisions is partially ordered. In this paper, we show how canonical completeness in multi-valued logics can be extended to the case where the set of truth values forms a lattice. This enables us to investigate the canonical completeness of logics having a partially ordered set of truth values, such as Belnap logic, and show that ABAC languages based on Belnap logic, such as PBel, are not canonically complete. We then construct a canonically complete four-valued logic using connections between the generators of the symmetric group (defined over the set of decisions) and unary operators in a canonically suitable logic. Finally, we propose a new authorization language $\text{PTaCL}_{\sf 4}^{\leqslant}$, an extension of PTaCL, which incorporates a lattice-ordered decision set and is canonically complete. We then discuss how the advantages of $\text{PTaCL}_{\sf 4}^{\leqslant}$ can be leveraged within the framework of XACML.

Bilattices for deductions in multi-valued logic

This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including Łukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson's "constructible falsity".

In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [7] distinguish between an ontological and an epistemic interpretation of classical truth values. By taking the Cartesian product of the two disjoint sets of values thus obtained, they arrive at four generalized truth values and consider two "semi-classical negations" on them. The resulting semantics is used to define three novel logics which are closely related to Belnap's well-known four valued logic. A syntactic characterization of these logics is left for further work. In this paper, based on our previous work on a functionally complete extension of Belnap's logic, we present a sound and complete tableau calculus for these logics. It crucially exploits the Cartesian nature of the four values, which is reflected in the fact that each proof consists of two tableaux. The bi-facial notion of truth of Z&S is thus augmented with a bi-facial notion of proof. We also provide translations between the logics for semi-classical negation and classical logic and show that an argument is valid in a logic for semi-classical negation just in case its translation is valid in classical logic.