Truth-functional Research Articles

Theory of truth degrees of formulas in Łukasiewicz n-valued propositional logic and a limit theorem

The concept of truth degrees of formulas in Łukasiewicz n -valued propositional logic L n is proposed. A limit theorem is obtained, it says that the truth function τn induced by truth degrees converges to the integrated truth function τ when n converges to infinite, hence this limit theorem builds a bridge between discrete valued Lukasiewicz logic and continuous valued Lukasiewicz logic. Moreover, the results obtained in the present paper is a natural generalization of the corresponding results obtained in two-valued propositional logic.The concept of truth degrees of formulas in Łukasiewicz n -valued propositional logic Ln is proposed. A limit theorem is obtained, which says that the truth function τ n induced by truth degrees converges to the integrated truth function τ n converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Łukasiewicz logic and the continuous valued Łukasiewicz logic. Moreover, the results obtained in the present paper is a natural generalization of the corresponding results obtained in two-valued propositional logic.

The Expressive Unary Truth Functions of n-valued Logic

The expressive truth functions of two-valued logic have all been identified. This paper begins the task of identifying the expressive truth functions of n-valued logic by characterizing the unary ones. These functions have distinctive algebraic, semantic, and closure-theoretic properties.

Graded many-valued resolution with aggregation

The aim of this paper is to give a sound resolution deduction rule in many-valued logic with truth value range [0,1], and with arbitrary connectives and graded information. We introduce and study the resolution truth function which will evaluate the truth value of the result of resolution. Conditions under which this function yields an aggregation operator will be given.

A characterization of truth-functions in the nilpotent minimum logic

By introducing a new family of partitions into the n-cube [0,1] n , the problem of characterizing truth tables of formulas in the nilpotent minimum logic is solved and their normal forms are presented. So far, only this kind of fuzzy truth functions have normal forms among all fuzzy propositional calculi which are based on left-continuous but discontinuous t-norm.

Coordinating perspectives in context

Swindal seeks to incorporate temporality into the formal-pragmatic analysis of discourse by developing what he calls ‘event-determining’ reflection. After outlining his motivations for introducing this new form of reflection, I offer a critique, first, of his appeal to meta-discourse about when to engage in discourse and, second, of the function of truth in his account. Finally, I suggest that Swindal’s theory of reflective acceptability fruitfully complements Robert Brandom’s normative pragmatics.

The Expressive Truth Conditions of Two-Valued Logic

In a finitary closure space, irreducible sets behave like two-valued models, with membership playing the role of satisfaction. If f is a function on such a space and the membership of $fx_1 ,\ldots, x_n$ in an irreducible set is determined by the presence or absence of the inputs $x_1 ,\ldots, x_n$ in that set, then f is a kind of truth function. The existence of some of these truth functions is enough to guarantee that every irreducible set is maximally consistent. The closure space is then said to be expressive. This paper identifies the two-valued truth functional conditions that guarantee expressiveness.

Critical principles: on the negative side of rationality

A naturalized model of rationality is developed, with a focus on an important but largely neglected aspect: knowledge of error, or “negative” knowledge. The development of knowledge of what counts as error occurs via a kind of internal variation and selection, or quasi-evolutionary, process. Processes of reflection generate a hierarchy of principles of error, a hierarchy that frames and constrains positive rationality. The dynamics of rationality is an internalization of processes of reflected-upon variation and selection. The nature and origin of logic are addressed from within this framework, and the overall rationality model is applied to three central issues in the philosophy of science: the rational function of truth and realism in science, the nature of progress in science, and the rationality of certain induction-like considerations.

Decomposition of Multiuniverse Fuzzy Truth Functions

This paper focuses on the decomposition problem of fuzzy relations using the concepts of multiuniverse fuzzy propositional logic. Given two fuzzy propositions in different universes, it is always possible to construct a fuzzy relation in the common universe through a prescribed combination. However, the converse is not so obvious, if possible at all. In other words, given a fuzzy relation, how would we know if it really represents a certain relationship between some fuzzy propositions? It is important to recognize whether the given fuzzy relation is a meaningful representation of information according to certain criteria applicable to some fuzzy propositions that constitute the fuzzy relation itself. Two basic structures of decomposition are investigated. Necessary and sufficient conditions for decomposition of multiuniverse fuzzy truth functions in terms of one-universe truth functions are presented. An algorithm for decomposition is proposed.

DECOMPOSITION OF MULTIUNIVERSE FUZZY TRUTH FUNCTIONS

This paper focuses on the decomposition problem of fuzzy relations using the concepts of multiuniverse fuzzy propositional logic. Given two fuzzy propositions in different universes, it is always possible to construct a fuzzy relation in the common universe through a prescribed combination. However, the converse is not so obvious, if possible at all. In other words, given a fuzzy relation, how would we know if it really represents a certain relationship between some fuzzy propositions? It is important to recognize whether the given fuzzy relation is a meaningful representation of information according to certain criteria applicable to some fuzzy propositions that constitute the fuzzy relation itself. Two basic structures of decomposition are investigated. Necessary and sufficient conditions for decomposition of multiuniverse fuzzy truth functions in terms of one-universe truth functions are presented. An algorithm for decomposition is proposed.

Wittgenstein, Finitism, and the Foundations of Mathematics

1. Wittgenstein's Anti-Platonism 2. Logicism without Classes 3. Arbitrary Functions 4. Quantification and Finitism 5. From Truth-Functional Logic to a Logic of Equations 6. Philosophy and Logical Foundations 7. The Continuum 8. Strict Finitism Bibliography Index

Residuated fuzzy logics with an involutive negation

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive negation. However, for t-norms without non-trivial zero divisors, $\neg$ is Godel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation.

Note on the Scope of Truth-functional Logic

A plausible and popular rule governing the scope of truth-functional logic is shown to be indequate. The argument appeals to the existence of truth-functional paraphrases which are logically independent of their natural language counterparts. A more adequate rule is proposed.

Knowledge Representation and Conversion for Hybrid Expert Systems

Many different knowledge representations, such as rules and frames, have been proposed for use with engineering expert systems. Every knowledge representation has certain inherent strengths and weaknesses. A knowledge engineer can exploit the advantages, and avoid the pitfalls, of different common knowledge representations if the knowledge can be mapped from one representation to another as needed. This paper derives the mappings between rules, logic diagrams, decision tables and decision trees using the calculus of truth-functional logic. The mappings for frames have also been derived by Chambers and Parkinson (1995). The logical mappings between these representations are illustrated through a simple example, the limitations of the technique are discussed, and the utility of the technique for the rapid-prototyping and validation of engineering expert systems is introduced. The technique is then applied to three engineering applications, showing great improvements in the resulting knowledge base.

The Juridical Matrix

... in opposition to the juridico-philosophical discourse that constructs itself around the problem of sovereignty and law, the discourse that interprets the permanent war in society is essentially politico-historical, an indeterminately critical and, at the same time, extremely mythical discourse in which truth func tions as a weapon to gain partisan victory. (Michel Foucault, 1989: 91)

Anscombe on the Tractatus

Setting the scene for the discussion she rehearses familiar doctrines from the Tractatus. A significant sentence is an arrangement of names of simple objects, or a truth function of such. This arrangement is called the structure of the sentence, structure being associated with actuality. The possibility of such a structure occurring is calledform. Thus the form of representation of a sentence is the possibility that the simple objects named by the names in the sentence, can have the same arrangement as those names. The arrangement doesn't actually have to be the same between names and objects. When they are the same there is a true sentence, when they are not there is a false, but nevertheless meaningful sentence. If the common form (possibility) were not shared between sentence and fact, then there would be no possibility of having a meaningful sentence (whether true or false). Thus the standard way to read the Tractatus is that propositions are logical pictures of facts. The elements of propositions stand for the elements of the fact pictured. The picturing occurs through isomorphism of the

Approximate solutions of fuzzy relational equations and a characterization of t-norms that define metrics for fuzzy sets

If one considers the membership degrees of fuzzy sets as truth values of a many-valued logic there is a canonical way to define a fuzzified equality for fuzzy sets. This generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation. Furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the “quality” to which it is an approximate solution. The background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets. Formally, the definition of this fuzzified equality contains as parameter some t-norm which acts as truth functions of a conjunction connective. For some such t-norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word. The paper gives a necessary and sufficient condition which characterizes all those t-norms which in that way yield a metric for fuzzy sets.

Wittgenstein, Truth-Functions, and Generality

Wittgenstein, Truth-Functions, and Generality

Incomplete Information and the Functional Data Model

This paper presents an experimental functional database language Fudal which is a further development of our group's work on persistent functional database languages. In this latest work we consider how unknown or partially known information can be treated in the functional context. The language we have implemented, Fudal, includes certainty and possibility operators. We outline the problems that are caused by the use of null values and truth functional logic in conventional database languages, and show how these problems can be overcome by defining the semantics of queries of a database containing partial information in terms of its ‘completions’. If D is a database containing partial information then a completion of D is a database which is consistent with D and contains no partial information. We demonstrate that, even when a database has a large number of completions, sensible queries can be constructed using certainty and possibility operators. Finally we show how these operators can be implemented and discuss the use of Fudal in practical contexts.

But and Negation

Many languages (e.g. Spanish, German, and Swedish) have more than one morpheme corresponding to the English “but”. It has been argued by Anscombre & Ducrot (1977) and by Horn (1989) that two of these morphemes covary with two different functions of natural language negation. Anscombre's and Ducrot's descriptions of these BUTs are examined, and it is shown that there is no such link. There is, however, an interesting connection between “but” and negation. One of the two morphemes imposes a restriction on the use of negation in the clause that follows it, which suggests that natural language negation does not correspond to a truth function.

From two- to four-valued logic

The purpose of this note is to show that a known and natural four-valued logic co-exists with classical two-valued logic in the familiar context of truth tables. The tool required is the power construction. The context of this paper is that of 2-valued truth tables in n propositional variables, say p1, p2, . . . , pn. The two classical truth values are 0 and 1. The truth value assignments (or valuations) are the 2 {0, 1}-vectors of length n. Propositional formulae are freely generated from the propositional variables by application of the usual classical connectives ∼ (not), & (and) and ∨ (or), each of which is defined by a given truth table. Accordingly, each propositional formula has a truth table. Formulae with the same truth table are logically equivalent, and we do not distinguish between them. The valuations form a Boolean algebra Val = (2,∧,∨, ′ ,0,1), where the Boolean operations ∧, ∨ and ′ are defined componentwise from their counterparts in the 2-element Boolean algebra 2 = {0, 1}, and 0 and 1 are respectively the vectors (0, 0, . . . , 0) and (1, 1, . . . , 1) of length n. The elements of 2 are denoted by x,y,z, . . . Up to equivalence, formulae may be regarded as truth functions from 2 to 2, and the connectives are characterised by operations in the image algebra 2. That is, if bool(A,−) : 2 → 2 is the truth function corresponding to a propositional formula A, then bool (∼A,x) = bool (A,x)′ , bool (A & B,x) = bool (A,x) ∧ bool (B,x) , (1) bool (A ∨ B,x) = bool (A,x) ∨ bool (B,x) . 1991 Mathematics Subject Classification: 03B46, 03B05. The paper is in final form and no version of it will be published elsewhere.