Ukasiewicz Logic Research Articles

Repetition-free and infinitary analytic calculi for first-order rational Pavelka logic

We present a hypersequent calculus $\text{G}^3\text{\L}\forall$ for first-order infinite-valued {\L}ukasiewicz logic and for an extension of it, first-order rational Pavelka logic; the calculus is intended for bottom-up proof search. In $\text{G}^3\text{\L}\forall$, there are no structural rules, all the rules are invertible, and designations of multisets of formulas are not repeated in any premise of the rules. The calculus $\text{G}^3\text{\L}\forall$ proves any sentence that is provable in at least one of the previously known hypersequent calculi for the given logics. We study proof-theoretic properties of $\text{G}^3\text{\L}\forall$ and thereby provide foundations for proof search algorithms.

On a Convex Logic Fragment for Learning and Reasoning

In this paper we introduce the convex fragment of {\L}ukasiewicz Logic and discuss its possible applications in different learning schemes. Indeed, the provided theoretical results are highly general, because they can be exploited in any learning framework involving logical constraints. The method is of particular interest since the fragment guarantees to deal with convex constraints, which are shown to be equivalent to a set of linear constraints. Within this framework, we are able to formulate learning with kernel machines as well as collective classification as a quadratic programming problem.

Infinitary logic and basically disconnected compact Hausdorff spaces

We extend \L ukasiewicz logic obtaining the infinitary logic $\mathcal{IR}\L$ whose models are algebras $C(X,[0,1])$, where $X$ is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in $\sigma$-complete Riesz spaces with strong unit. The Lindenbaum-Tarski algebra of $\mathcal{IR}\L$ is, up to isomorphism, an algebra of $[0,1]$-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval $[0,1]$.

SOFT TOPOLOGY AND SOFT PROXIMITY AS FUZZY PREDICATES BY FORMULAE OF LUKASIEWICZ LOGIC

In this paper, based in the L ukasiewicz logic, the definition offuzzifying soft neighborhood structure and fuzzifying soft continuity areintroduced. Also, the fuzzifying soft proximity spaces which are ageneralizations of the classical soft proximity spaces are given. Severaltheorems on classical soft proximities are special cases of the theorems weprove in this paper.

Multimodal epistemic Łukasiewicz logics with application in immune system

We offer a new logic, a multimodal epistemic ?ukasiewicz logic, which is an extension of the infinitely valued ?ukasiewicz logic, the language of the logic is an extended by unary connectives that are interpreted as modal operators (knowledge operators). We propose the use such a logic in studying immune system. A relational system is developed as a semantic of this logic. The relational systems represent the immune system which in its turn is a part of relational biology.

Can Many-Valued Logic Help to Comprehend Quantum Phenomena?

Following {\L}ukasiewicz, we argue that future non-certain events should be described with the use of many-valued, not 2-valued logic. The Greenberger-Horne-Zeilinger `paradox' is shown to be an artifact caused by unjustified use of 2-valued logic while considering results of future non-certain events. Description of properties of quantum objects before they are measured should be performed with the use of propositional functions that form a particular model of infinitely-valued {\L}ukasiewicz logic. This model is distinguished by specific operations of negation, conjunction, and disjunction that are used in it.

Faulty sets of Boolean formulas and ukasiewicz logic

Faulty sets of Boolean formulas and ukasiewicz logic

Non-deterministic Conditionals and Transparent Truth

Theories where truth is a naive concept fall under the following dilemma: either the theory is subject to Curry's Paradox, which engenders triviality, or the theory is not trivial but the resulting conditional is too weak. In this paper we explore a number of theories which arguably do not fall under this dilemma. In these theories the conditional is characterized in terms of (infinitely-valued) non-deterministic matrices. These non-deterministic theories are similar to infinitely-valued ?ukasiewicz logic in that they are consistent and their conditionals are quite strong. The difference is the following: while ?ukasiewicz logic is $${\omega}$$?-inconsistent, the non-deterministic theories might turn out to be $${\omega}$$?-consistent.

Łukasiewicz logic and Riesz spaces

We initiate a deep study of Riesz MV-algebras which are MV-algebras endowed with a scalar multiplication with scalars from \([0,1]\). Extending Mundici’s equivalence between MV-algebras and \(\ell \)-groups, we prove that Riesz MV-algebras are categorically equivalent to unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent to the class of commutative unital C\(^*\)-algebras. The propositional calculus \({\mathbb R}{\mathcal L}\) that has Riesz MV-algebras as models is a conservative extension of Łukasiewicz \(\infty \)-valued propositional calculus and is complete with respect to evaluations in the standard model \([0,1]\). We prove a normal form theorem for this logic, extending McNaughton theorem for Ł ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in \({\mathbb R}{\mathcal L},\) and relate them with the analogue of de Finetti’s coherence criterion for \({\mathbb R}{\mathcal L}\).

Łukasiewicz mu-Calculus

The paper explores properties of {\L}ukasiewicz mu-calculus, a version of the quantitative/probabilistic modal mu-calculus containing both weak and strong conjunctions and disjunctions from {\L}ukasiewicz (fuzzy) logic. We show that this logic encodes the well-known probabilistic temporal logic PCTL. And we give a model-checking algorithm for computing the rational denotational value of a formula at any state in a finite rational probabilistic nondeterministic transition system.

Dialogue games for Dishkant’s quantum modal logic

Recently some elaborations were made concerning the game theoretic semantic of $\L_{\aleph_0}$ and its extension. In the paper this kind of semantics is developed for Dishkant’s quantum modal logic $\L$Q which is also, in fact, the specific extension of $\L_{\aleph_0}$. As a starting point some game theoretic interpretation for the S$\L$ system (extending both $\L$ukasiewicz logic $\L_{\aleph_0}$ and modal logic S5) was exploited which has been proposed in 2006 by C. Fermuller and R. Kosik. They, in turn, based on ideas already introduced by Robin Giles in the 1970th to obtain a characterization of $\L_{\aleph_0}$ in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion.

Curry’s Paradox and ω -Inconsistency

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes1. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the ω-inconsistency in Łukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naive truth theory. On this basis I identify two natural subsystems of ukasiewicz logic which individually, but not jointly, lack the problematic feature.

N-Fold implicative basic logic is Gödel logic

We prove that Haveshki’s and Eslami’s n-fold implicative basic logic is Godel logic and n-fold positive implicative basic logic is a fragment of ukasiewicz logic.

Unification of Two Approaches to Quantum Logic: Every Birkhoff – von Neumann Quantum Logic is a Partial Infinite-Valued Łukasiewicz Logic

In the paper it is shown that every physically sound Birkhoff --- von Neumann quantum logic, i.e., an orthomodular partially ordered set with an ordering set of probability measures can be treated as partial infinite-valued ?ukasiewicz logic, which unifies two competing approaches: the many-valued, and the two-valued but non-distributive, which have co-existed in the quantum logic theory since its very beginning.

On copulas, quasicopulas and fuzzy logic

We investigate (quasi)copulas as possible truth functions of fuzzy conjunction which is not necessarily associative and present some axiom systems for such fuzzy logics. In particular, we study an expansion of ?ukasiewicz (infinite valued propositional) logic by a new connective interpreted as an arbitrary quasicopula (and also by a new connective interpreted as the residuum of the copula). Main results concern standard completeness.

Rule based fuzzy classification using squashing functions

In this paper we are dealing with the construction of a fuzzy rule based classifier. A three-step method is proposed based on ukasiewicz logic for the description of the rules and the fuzzy members...

Quantum logic as partial infinite-valued ?ukasiewicz logic

It is shown that an old and neglected Łukasiewicz (1913) paper contains construction of a many-valued logic which can be in an almost straightforward way used to describe physical experiments. The logic is infinite-valued and its truth values are interpreted as probabilities of experimental confirmation of propositions about results of future experiments. In the case of experiments on quantum objects the logic is partial, i.e. the existence of conjunctions and disjunctions cannot be guaranteed for all pairs of propositions. An outline of previous attempts of using many-valued logics in the description of quantum phenomena is given.

Averaging the truth-value in ?ukasiewicz logic

Chang's MV algebras are the algebras of the infinite-valued sentential calculus of Łukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of ‘average degree of truth’ of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.