We present new probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment, called Zλ- and lexλ-entailment, which are parameterized through a value λ ∈ [0,1] that describes the strength of the inheritance of purely probabilistic knowledge. In the special cases of λ = 0 and λ = 1, the notions of Zλ- and lexλ-entailment coincide with probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment that have been recently introduced by the author. We show (...) that the notions of Zλ- and lexλ-entailment have similar properties as their classical counterparts. In particular, they both satisfy the rationality postulates of System P and the property of Rational Monotonicity. Moreover, Zλ-entailment is weaker than lexλ-entailment, and both Zλ- and lexλ-entailment are proper generalizations of their classical counterparts. (shrink)

Rational Pavelka logic extends Lukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0, 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

Rational Pavelka logic extends Lukasiewicz infinitely valued logic $by adding truth constants \bar{r} for rationals in [0, 1].$ We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

In this paper, we consider multiplicative-additive fragments of affine propositional classical linear logic extended with n-contraction. To be specific, n-contraction (n 2) is a version of the contraction rule where (n+ 1) occurrences of a formula may be contracted to n occurrences. We show that expansions of the linear models for (n + 1)- valued ukasiewicz logic are models for the multiplicative-additive classical linear logic, its affine version and their extensions with n-contraction. We prove the finite axiomatizability (...) for the classes of finite models, as well as for the class of infinite linear models based on the set of rational numbers in the interval [0, 1]. The axiomatizations obtained in a Gentzen-style formulation are equivalent to finite and infinite-valued ukasiewicz logics. (shrink)

We consider first-order infinite-valued Łukasiewicz logic and its expansion, first-order rational Pavelka logic RPL∀. From the viewpoint of provability, we compare several Gentzen-type hypersequent calculi for these logics with each other and with Hájek’s Hilbert-type calculi for the same logics. To facilitate comparing previously known calculi for the logics, we define two new analytic calculi for RPL∀ and include them in our comparison. The key part of the comparison is a density elimination proof that introduces no (...) cuts for one of the hypersequent calculi considered. (shrink)

O logice Stoików pisali Prantl, Zeller, Brochard. Żaden z nich nie zrozumiał tej logiki, bo żaden z nich nie miał dostatecznego wykształcenia logicznego. Co prawda, nie można im nawet robić z tego zarzutu, bo skądże się mieli logiki nauczyć? Sądy ich o logice stoickiej nie mają żadnej wartości. Dopiero dziś wiemy, dzięki logice matematycznej, że logika Stoików jest systemem całkowicie różnym od syllogistyki Arystotelesa. Logika stoicka bowiem jest odpowiednikiem dzisiejszej "teorji dedu...

We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore how probabilistic reasoning under coherence is related to model- theoretic probabilistic reasoning and to default reasoning in System . In particular, we show that the notions of g-coherence and of g-coherent entailment can be expressed by combining notions in model-theoretic probabilistic logic with concepts from default reasoning. Moreover, we show that probabilistic reasoning under coherence is a generalization of default reasoning (...) in System . That is, we provide a new probabilistic semantics for System , which neither uses infinitesimal probabilities nor atomic bound (or big-stepped) probabilities. These results also provide new algorithms for probabilistic reasoning under coherence and for default reasoning in System , and they give new insight into default reasoning with conditional objects. (shrink)

The ability to conduct logical reasoning is a fundamental aspect of intelligent behavior, and thus an important problem along the way to human-level artificial intelligence. Traditionally, symbolic methods from the field of knowledge representation and reasoning have been used to equip agents with capabilities that resemble human reasoning qualities. More recently, however, there has been an increasing interest in applying alternative approaches based on machine learning rather than logic-based formalisms to tackle this kind of tasks. Here, we make use (...) of state-of-the-art methods for training deep neural networks to devise a novel model that is closely coupled to symbolic reasoning methods, and thus able to learn how to effectively perform basic ontology reasoning. This term describes an important and at the same time very natural kind of problem settings where the rules for conducting reasoning are specified alongside with the actual information. Many problems in practice may be viewed as such reasoning tasks, which is why the presented approach is applicable to a plethora of important real-world problems. To demonstrate the effectiveness of the suggested method, we present the outcomes of several experiments that have been conducted on both toy datasets as well as real-world data, which show that our model learned to perform precise reasoning on a number of diverse inference tasks that require comprehensive deductive proficiencies. Furthermore, it turned out that the suggested model suffers much less from different obstacles that prohibit symbolic reasoning. (shrink)

We propose a combination of probabilistic reasoning from conditional constraints with approaches to default reasoning from conditional knowledge bases. In detail, we generalize the notions of Pearl's entailment in system Z, Lehmann's lexicographic entailment, and Geffner's conditional entailment to conditional constraints. We give some examples that show that the new notions of z-, lexicographic, and conditional entailment have similar properties like their classical counterparts. Moreover, we show that the new notions of z-, lexicographic, and conditional entailment are proper generalizations of (...) both their classical counterparts and the classical notion of logical entailment for conditional constraints. (shrink)

The ability to conduct logical reasoning is a fundamental aspect of intelligent behavior, and thus an important problem along the way to human-level artificial intelligence. Traditionally, symbolic methods from the field of knowledge representation and reasoning have been used to equip agents with capabilities that resemble human reasoning qualities. More recently, however, there has been an increasing interest in applying alternative approaches based on machine learning rather than logic-based formalisms to tackle this kind of tasks. Here, we make use (...) of state-of-the-art methods for training deep neural networks to devise a novel model that is closely coupled to symbolic reasoning methods, and thus able to learn how to effectively perform basic ontology reasoning. This term describes an important and at the same time very natural kind of problem settings where the rules for conducting reasoning are specified alongside with the actual information. Many problems in practice may be viewed as such reasoning tasks, which is why the presented approach is applicable to a plethora of important real-world problems. To demonstrate the effectiveness of the suggested method, we present the outcomes of several experiments that have been conducted on both toy datasets as well as real-world data, which show that our model learned to perform precise reasoning on a number of diverse inference tasks that require comprehensive deductive proficiencies. Furthermore, it turned out that the suggested model suffers much less from different obstacles that prohibit symbolic reasoning. (shrink)

In the discussion at hand I have attempted to pave the way for such a treatment of the principle of contradiction. In a number of respects it seems to me worthwhile to relate my critical exposition to Aristotle's train of thought. Indeed, every critique must be raised against something substantial, otherwise it generally becomes the critic's leisurely game with his own cerebral phantasies. Now Aristotle's intuitions regarding the principle of contradiction are, for the most part and clear down to the (...) present day, the usual and traditional ones; and arguments for and against the principle can be found together in the Stagirite in greater completeness than in any one modern textbook of logic. My investigation will proceed with the Aristotelian text at hand and with regard to the results of symbolic logic. The most important results shall be sketched very briefly in the following. (shrink)

It is a popular view thatpractical deliberation excludes foreknowledge of one's choice. Wolfgang Spohn and Isaac Levi have argued that not even a purely probabilistic self-predictionis available to thedeliberator, if one takes subjective probabilities to be conceptually linked to betting rates. It makes no sense to have a betting rate for an option, for one's willingness to bet on the option depends on the net gain from the bet, in combination with the option's antecedent utility, rather than on the offered (...) odds. And even apart from this consideration, assigning probabilities to the options among which one is choosing is futile since such probabilities could be of no possible use in choice. The paper subjects these arguments to critical examination and suggests that, appearances notwithstanding, practical deliberation need not crowd outself-prediction. (shrink)

Formulas of n variables of Łukasiewicz sentential calculus can be represented, via McNaughton’s theorem, by piecewise linear functions, with integer coefficients, from hypercube [0,1] n to [0,1], called McNaughton functions. As a consequence of the McNaughton representation of a formula it is obtained a canonical form of a formula. Indeed, up to logical equivalence, any formula can be written as an infimum of finite suprema of formulas associated to McNaughton functions which are truncated functions to $[0,1]$ of the restriction (...) to [0,1] n of single hyperplanes, for short, called simple McNaughton functions. In the present paper we will concern with the problem of presenting formulas of Lukasiewicz sentential calculus in normal form. Here we list the main results we obtained: a) we give an axiomatic description of some classes of formulas having the property to be canonically mapped one-to-one onto the class of simple Mc Naughton functions; b) we provide normal forms for Lukasiewicz sentential calculus, making use of formulas defined in a); c) we prove the polynomial complexity of formulas, in normal form, coming from a certain class described as in a); d) we extend the results described in a), b) and c) to Rational Lukasiewicz logic. (shrink)

In this paper we provide a finite axiomatization (using two finitary rules only) for the propositional logic (called $L\Pi$ ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from $L \Pi$ by the adding of a constant symbol and of a defining axiom for $\frac{1}{2}$ , called $L \Pi\frac{1}{2}$ . We show that $L \Pi \frac{1}{2}$ contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Gödel's Fuzzy (...) Logic, Takeuti and Titani's Propositional Logic, Pavelka's Rational Logic, Pavelka's Rational Product Logic, the Lukasiewicz Logic with $\Delta$ , and the Product and Gödel's Logics with $\Delta$ and involution. Standard completeness results are proved by means of investigating the algebras corresponding to $L \Pi$ and $L \Pi \frac{1}{2}$ . For these algebras, we prove a theorem of subdirect representation and we show that linearly ordered algebras can be represented as algebras on the unit interval of either a linearly ordered field, or of the ordered ring of integers, Z. (shrink)

In this paper we deepen Mundici's analysis on reducibility of the decision problem from infinite-valued ukasiewicz logic to a suitable m-valued ukasiewicz logic m , where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in if and only if it is also valid in m. We also reduce the notion of logical consequence in to (...) the same notion in a suitable finite set of finite-valued ukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued ukasiewicz logic. (shrink)

In Rationality and Logic, Robert Hanna argues that logic is intrinsically psychological and that human psychology is intrinsically logical. He claims that logic is cognitively constructed by rational animals and that rational animals are essentially logical animals. In order to do so, he defends the broadly Kantian thesis that all rational animals possess an innate cognitive "logic faculty." Hanna 's claims challenge the conventional philosophical wisdom that sees logic as a fully formal (...) or "topic-neutral" science irreconcilably separate from the species- or individual-specific focus of empirical psychology.Logic and psychology went their separate ways after attacks by Frege and Husserl on logical psychologism--the explanatory reduction of logic to empirical psychology. Hanna argues, however, that--despite the fact that logical psychologism is false--there is an essential link between logic and psychology. Rational human animals constitute the basic class of cognizers or thinkers studied by cognitive psychology; given the connection between rationality and logic that Hanna claims, it follows that the nature of logic is significantly revealed to us by cognitive psychology. Hanna 's proposed "logical cognitivism" has two important consequences: the recognition by logically oriented philosophers that psychologists are their colleagues in the metadiscipline of cognitive science; and radical changes in cognitive science itself. Cognitive science, Hanna argues, is not at bottom a natural science; it is both an objective or truth-oriented science and a normative human science, as is logic itself. (shrink)

ABSTRACT Reiter's default logic is one of the most prominent and well-studied approaches to nonmonotonic reasoning. Its evolution has resulted in diverse variants enjoying many interesting properties. This process however seems to be diverging because it has led to default logics that are difficult to compare due to different formal characterizations—sometimes even dealing with different objects of discourse. This problem is addressed in this paper in two ways. One the one hand, we elaborate on the relationships between different types (...) of default logics and discuss their differences wrt some basic properties. In particular, we show how two recently proposed variants, namely rational and CA-default logic, are related to each other and existing default logics. As an interesting byproduct, we obtain a cumulative counterpart to Lukasiewicz' variant of default logic. The resulting system is insofar surprising since it necessitates to refine the canonical way of assuring cumulativity in default logics. On the other hand, we pursue an integrating approach. We give a translation embedding rational and CA-default logic into a recently developed framework for default logics. This allows us not only to classify these variants, but it moreover provides us with semantical underpinnings that have been lacking so far. (shrink)

Does rationality require logical omniscience? Our best formal theories of rationality imply that it does, but our ordinary evaluations of rationality seem to suggest otherwise. This paper aims to resolve the tension by arguing that our ordinary evaluations of rationality are not only consistent with the thesis that rationality requires logical omniscience, but also provide a compelling rationale for accepting this thesis in the first place. This paper also defends an account of apriori justification for logical beliefs that is designed (...) to explain the rational requirement of logical omniscience. On this account, apriori justification for beliefs about logic has its source in logical facts, rather than psychological facts about experience, reasoning, or understanding. This account has important consequences for the epistemic role of experience in the logical domain. In a slogan, the epistemic role of experience in the apriori domain is not a justifying role, but rather an enabling and disabling role. (shrink)

This text, serving as an introduction to Aristotelian, symbolic, inductive, and practical logic, presents the techniques of these approaches to logic, some of the philosophical problems of logic, and some of the attempts to solve philosophical problems by means of various logical techniques. It discusses the problem of universals and null classes; briefly introduces the theory of types; and presents Lukasiewicz's formalization of Aristotelian syllogistic logic as an example of a formal system. Classical logic and (...) propositional logic are carefully and fully presented. The text, however, is weak in its presentation of quantificational logic. After a correct, but brief and very inadequate introduction to quantificational logic, the readers are referred to Quine, Methods of Logic, or to Copi, Symbolic Logic, for a more adequate development. The chapter on inductive logic discusses the uses and dangers of generalizations and introduces the probability calculus; the chapter on informal logic discusses criteria of rationality and presents the usual informal fallacies. There are problems and exercises after many of the sections, and answers in the back of the book.—R. L. S. (shrink)

This text, serving as an introduction to Aristotelian, symbolic, inductive, and practical logic, presents the techniques of these approaches to logic, some of the philosophical problems of logic, and some of the attempts to solve philosophical problems by means of various logical techniques. It discusses the problem of universals and null classes; briefly introduces the theory of types; and presents Lukasiewicz's formalization of Aristotelian syllogistic logic as an example of a formal system. Classical logic and (...) propositional logic are carefully and fully presented. The text, however, is weak in its presentation of quantificational logic. After a correct, but brief and very inadequate introduction to quantificational logic, the readers are referred to Quine, Methods of Logic, or to Copi, Symbolic Logic, for a more adequate development. The chapter on inductive logic discusses the uses and dangers of generalizations and introduces the probability calculus; the chapter on informal logic discusses criteria of rationality and presents the usual informal fallacies. There are problems and exercises after many of the sections, and answers in the back of the book.—R. L. S. (shrink)

This paper is concerned with decision proceedures for the 0-valued ukasiewicz logics,. It is shown how linear algebra can be used to construct an automated theorem checker. Two decision proceedures are described which depend on a linear programming package. An algorithm is given for the verification of consequence relations in, and a connection is made between theorem checking in two-valued logic and theorem checking in which implies that determing of a -free formula whether it takes the value one is (...) NP-complete problem. (shrink)

We present an abstract model of rationality that focuses on structural properties of attitudes. Rationality requires coherence between your attitudes, such as your beliefs, values, and intentions. We define three 'logical' conditions on attitudes: consistency, completeness, and closedness. They parallel the familiar logical conditions on beliefs, but contrast with standard rationality conditions like preference transitivity. We establish a formal correspondence between our logical conditions and standard rationality conditions. Addressing John Broome's programme 'rationality through reasoning', we formally characterize how you can (...) (not) become more logical by reasoning. Our analysis connects rationality with logic, and enables logical talk about multi-attitude psychology. (shrink)

cognitive psychology; given the connection between rationality and logic that Hanna claims, it follows that the nature of logic is significantly revealed to us by cognitive psychology. Hanna's proposed "logical cognitivism" has two important consequences: the recognition by logically oriented philosophers that psychologists are their colleagues in the metadiscipline of cognitive science; and radical changes in cognitive science itself. Cognitive science, Hanna argues, is not at bottom a natural science; it is both an objective or truth-oriented science and (...) a normative human science, as is logic itself. (shrink)

Preface for instructors -- Preface for students (you really should read it) -- What is logic about? -- Sentential logic basics -- Sentential logic proofs -- More sentential logic : contradictions, tautologies and assumptions -- Predicate logic basics -- Proofs in predicate logic -- Probability : the basic rules of life -- The theorem of Dr. Bayes -- Probability illusions : why we are so bad at inductive reasoning -- Studies have shown ... or (...) have they? -- Inference to the best explanation -- Rhetoric : the art of persuasion -- Won't get fooled again (fallacies and other foibles) -- Spin, spin, spin (or how not to be a sucker) -- Everything you know is wrong! : pseudoscience and bogus scholarship. (shrink)

Chang's MV algebras are the algebras of the infinite-valued sentential calculus of ukasiewicz. 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.

In this brief article, I review the main argument's of Robert Hanna's <em>Rationality and Logic</em>.

The equivalence connective in ukasiewicz logic has its algebraic counterpart which is the distance function d(x,y) =|x–y| of a positive cone of a commutative -group. We make some observations on logically motivated algebraic structures involving the distance function.

In [9] it is proved the categorical isomorphism of two varieties: bounded commutative BCK-algebras and MV -algebras. The class of MV -algebras is the algebraic counterpart of the infinite valued propositional calculus L of Lukasiewicz . The main objective of the present paper is to study that isomorphism from the perspective of logic. The B-C-K logic is algebraizable and the quasivariety of BCKalgebras is the equivalent algebraic semantics for that logic . We call commutative B-C-K logic, (...) briefly cBCK, to the extension of B-C-K logic associated to the variety of commutative BCK–algebras. Moreover, we present the extension Boc of cBCK obtained by adding the axiom of “boundness”. We prove that the deductive system Boc is equivalent to L. We observe that cBCK admits two interesting extensions: the logic Boc, treated in this paper, which is equivalent to the system L of Lukasiewicz, and the logic Co that is naturally associated to the system Balo of `-groups . This constructions establish a link between L and Balo , that would be a logical approach to the categorical relationship between MV–algebras and `-groups. (shrink)

The infinite-valued logic of ukasiewicz was originally defined by means of an infinite-valued matrix. ukasiewicz took special forms of negation and implication as basic connectives and proposed an axiom system that he conjectured would be sufficient to derive the valid formulas of the logic; this was eventually verified by M. Wajsberg. The algebraic counterparts of this logic have become know as Wajsberg algebras. In this paper we show that a Wajsberg algebra is complete and atomic (as a (...) lattice) if and only if it is a direct product of finite Wajsberg chains. The classical characterization of complete and atomic Boolean algebras as fields of sets is a particular case of this result. (shrink)