Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expressed as Kronecker polynomials. These polynomials allow us to show that many important tautologies of classical logic are generated from basic operators via the operations called Type I and Type (...) II products. Finally, it is described a matrix version of the Fredkin gate that extends its properties to the many-valued domain, and it is proved that the filtered Fredkin operators are second degree Kronecker polynomials that cannot be generated by Type I or Type II products. (shrink)

The new paradigm in the psychology of reasoning adopts a Bayesian, or prob- abilistic, model for studying human reasoning. Contrary to the traditional binary approach based on truth functional logic, with its binary values of truth and falsity, a third value that represents uncertainty can be introduced in the new paradigm. A variety of three-valued truth table systems are available in the formal literature, including one proposed by de Finetti. We examine the descriptive adequacy of these systems for natural language (...) indicative condi- tionals and bets on conditionals. Within our framework the so-called “defective” truth table, in which participants choose a third value when the antecedent of the indicative conditional is false, becomes a coherent response. We show that only de Finetti’s system has a good descriptive fit when uncer- tainty is the third value. (shrink)

Łukasiewicz three-valued logic Ł3 is often understood as the set of all 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic Ł3 is axiomatized as an extension of Routley (...) and Meyer’s basic positive logic following Brady’s strategy for axiomatizing many-valued logics by employing two-valued under-determined or over-determined interpretations. Finally, it is proved that “well determined” Łukasiewicz logics are paraconsistent. (shrink)

This essay describes a variety of contributions which relate to the connection of probability with logic. Some are grand attempts at providing a logical foundation for probability and inductive inference. Others are concerned with probabilistic inference or, more generally, with the transmittance of probability through the structure (logical syntax) of language. In this latter context probability is considered as a semantic notion playing the same role as does truth value in conventional logic. At the conclusion of the essay two fully (...) elaborated semantically based constructions of probability logic are presented. (shrink)

Epistemic modal operators give rise to something very like, but also very unlike, Moore's paradox. I set out the puzzling phenomena, explain why a standard relational semantics for these operators cannot handle them, and recommend an alternative semantics. A pragmatics appropriate to the semantics is developed and interactions between the semantics, the pragmatics, and the definition of consequence are investigated. The semantics is then extended to probability operators. Some problems and prospects for probabilistic representations of content and context are explored.

In the first part of this paper we indicate how Meredith's condensed detachment may be used to give a new proof of Belnap's theorem that if every axiom x of a calculus S has the two-property that every variable which occurs in x occurs exactly twice in x, then every theorem of S is a substitution instance of a theorem of S which has the two-property. In the remainder of the paper we discuss the use of mechanical theorem-provers, based either (...) on condensed detachment or on the resolution rule of J. A. Robinson, to investigate various calculi whose axioms all have the two-property. Particular attention is given to D-groupoids, i.e. sets of formulae which are closed under condensed detachment. (shrink)

This article explores the question of how the members of the Lvov-Warsaw School promoted values that can be regarded as components of so-called logical culture. The author argues that these values are strictly connected with science. With references to Łukasiewicz, Czeżowski, and Kotarbiński,the article explores how values shape the logical culture and determines society as directed towards values. The article connects the meta-philosophical perspective with the philosophical one.

In order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular ‘justifications’ are demonstrably without epistemic value. Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus non-classical logics. In this paper the question is answered positively, based on meaning-preserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic and quantum logics, (...) can be translated into classical logic by introducing additional intensional operators into the language. Based on this result it is argued that classical logic is representationally optimal. In sec. 6 it is investigated whether non-classical logics can be likewise representationally optimal. The answer is predominantly negative but partially positive. Nevertheless the situation is not symmetric, because classical logic has important ceteris paribus advantages as a unifying metalogic. (shrink)

The book provides a historical and systematic exposition of the semantic theory of truth formulated by Alfred Tarski in the 1930s. This theory became famous very soon and inspired logicians and philosophers. It has two different, but interconnected aspects: formal-logical and philosophical. The book deals with both, but it is intended mostly as a philosophical monograph. It explains Tarski’s motivation and presents discussions about his ideas as well as points out various applications of the semantic theory of truth to philosophical (...) problems. (shrink)

In this paper I formalise the falsificationist proposal omitting Popper’s requirement of consistency. This omission results in (i) trivial theories being falsifiable in an inappropriate sense of the term, but also in (ii) some inconsistent non-trivial theories being so in an appropriate one. This justifies a slight alteration of the definition of falsifiability that excludes (i) but allows (ii). Instead of requiring that a falsifiable theory be consistent, my proposal only requires that the intersection of its classes of potential corroborators (...) and falsifiers be empty. (shrink)

I shall elaborate more on the idea of anti-irrationalism proposed by the Polish analytic philosopher Kazimierz Ajdukiewicz, a prominent member of the Lvov-Warsaw School of philosophy and logic. In my reading, anti-irrationalism stands in opposition not only to overt irrationalism, which is made clear by the term itself, but also to all forms of rationalism that tip toward something like worship of reason. Having characterized anti-irrationalism as it originally appeared in Ajdukiewicz’s works, I shall propose a certain reformulation of it, (...) so as to capture a broader philosophical legacy in Poland, primarily, yet not exclusively the Lvov-Warsaw School. (shrink)

This volume examines the entire logical and philosophical production of Nikolai A. Vasil’ev, studying his life and activities as a historian and man of letters. Readers will gain a comprehensive understanding of this influential Russian logician, philosopher, psychologist, and poet. The author frames Vasil’ev’s work within its historical and cultural context. He takes into consideration both the situation of logic in Russia and the state of logic in Western Europe, from the end of the 19th century to the beginning of (...) the 20th. Following this, the book considers the attempts to develop non-Aristotelian logics or ideas that present affinities with imaginary logic. It then looks at the contribution of traditional logic in elaborating non-classical ideas. This logic allows the author to deal with incomplete objects just as imaginary logic does with contradictory ones. Both logics are objects of interesting analysis by modern researchers. (shrink)

En este artículo, analizo las principales respuestas que se han dado al argumento finitista de Etchemendy, y muestro que ninguna de ellas es exitosa. Primero, describo y critico las propuestas que intentan resolverlo apelando a consideraciones modales. Estas soluciones fallan porque presuponen un finitismo demasiado débil, donde se acepta la existencia de infinitos conjuntos o de mundos posibles con infinitos objetos. Pero hay versiones más fuertes del finitismo que reintroducen el problema. Luego considero las soluciones que apelan a categorías semánticas. (...) Una de ellas categoriza incorrectamente este problema como un desacuerdo sobre el significado de los cuantificadores. La otra solución falla porque, si fuera tomada en serio, tendría efectos muy nocivos para la lógica en general. Finalmente argumento que la mejor solución es morder la bala y aceptar que la lógica no debería ser fuertemente independiente de algunos asuntos que tradicionalmente fueron considerados como 'extralógicos'. In this paper, I analyze the main replies that have been given to Etchemendy's finitist argument, and I show that none of them is successful. First, I describe and criticize the proposals that try to solve the problem by appealing to modal considerations. These solutions fail because they presuppose a very weak finitism, where the existence of infinitely many sets, or possible words with infinitely many objects, is accepted. But there are stronger versions of finitism that reintroduce the problem. Then I consider the solutions which appeal to semantical categories. One of them incorrectly categorizes the problem as a disagreement in the meaning of the quantifiers. The other fails because, if taken seriously, it would have harmful effects on logic in general. Finally I argue that the best solution is to bite the bullet and accept that logic shouldn't be strongly independent from some issues which were traditionally considered as 'extra-logical'. (shrink)

In the paper we argue that truth-relativism is potentially hostage to a problem of exhibiting witnesses of its own truth. The problem for the relativist stems from acceptance of a trumping principle according to which there is a dependency between ascriptions of truth of an utterance and ascriptions of truth to other ascriptions of truth of that utterance. We argue that such a dependency indeed holds in the case of future contingents and the case of epistemic modals and that, consequently, (...) the relativist about these domains cannot exhibit witnesses to his relativism. In the appendix we provide some results on the relation between trumping and multi-order relativism. (shrink)

This book is a survey of the most important developments in Austrian philosophy in its classical period from the 1870s to the Anschluss in 1938. Thus it is intended as a contribution to the history of philosophy. But I hope that it will be seen also as a contribution to philosophy in its own right as an attempt to philosophize in the spirit of those, above all Roderick Chisholm, Rudolf Haller, Kevin Mulligan and Peter Simons, who have done so much (...) to demonstrate the continued fertility of the ideas and methods of the Austrian philosophers in our own day. For some time now, historians of philosophy have been gradually coming to terms with the idea that post-Kantian philosophy in the German-speaking world ought properly to be divided into two distinct traditions which we might refer to as the German and Austrian traditions, respectively. The main line of the first consists in a list of personages beginning with Kant, Fichte, Hegel and Schelling and ending with Heidegger, Adorno and Bloch. The main line of the second may be picked out similarly by means of a list beginning with Bolzano, Mach and Meinong, and ending with Wittgenstein, Neurath and Popper. As should be clear, it is the Austrian tradition that has contributed most to the contemporary mainstream of philosophical thinking in the Anglo-Saxon world. For while there are of course German thinkers who have made crucial contributions to the development of exact or analytic philosophy, such thinkers were outsiders when seen from the perspective of native German philosophical culture, and in fact a number of them found their philosophical home precisely in Vienna. When, in contrast, we examine the influence of the Austrian line, we encounter a whole series of familiar and unfamiliar links to the characteristic concerns of more recent philosophy of the analytic sort. As Michael Dummett points out in his Origins of Analytic Philosophy, the newly fashionable habit of referring to analytic philosophy as "Anglo-American" is in this light a "grave historical distortion". If, he says, we take into account the historical context in which analytic philosophy developed, then such philosophy "could at least as well be called "Anglo-Austrian" (1988, p. 7). Much valuable scholarly work has been done on the thinking of Husserl and Wittgenstein, Mach and the Vienna Circle. The central axis of Austrian philosophy, however, which as I hope to show in what follows is constituted by the work of Brentano and his school, is still rather poorly understood. Work on Meinong or Twardowski by contemporary philosophers still standardly rests upon simplified and often confused renderings of a few favoured theses taken out of context. Little attention is paid to original sources, and little effort is devoted to establishing what the problems were by which the Austrian philosophers in general were exercised -- in spite of the fact that many of these same problems have once more become important as a result of the contemporary burgeoning of interest on the part of philosophers in problems in the field of cognitive science. (shrink)

Certain distributivity results for Lukasiewicz’s infinite-valued logic Lℵ0..

In the present paper, we study some properties of matrices for non-structural consequence operators. These matrices were introduced in a former work (see [3]). In sections 1. and 2., general definitions and theorems are recalled; in section 3. a correspondence is studied, among our matrices and Wójcicki's ones for structural operators. In section 4. a theorem is given about operators, induced by submatrices or epimorphic images, or quotient matrices of a given one.Such matrices are used to characterize lattices of non-structural (...) consequence operators, by constructing lattices, antiisomorphic to them (see section 5.). In the last section, a sufficient condition is given for a non-structural operator to be finite. (shrink)

In the paper, Jan Łukasiewicz’s program of the logicization of philosophy is presented and discussed. Łukasiewicz, known mostly for his invention of trivalent logic as well as his achievements in propositional calculus and metalogic, had always been concerned with the methodological condition of philosophy. He finally found “the measure of exactness” in mathematical logic. According to him, only the use of logical tools may provide philosophical investigations with an appropriate level of exactness. He expressed his views most firmly and directly (...) in the paper, “A call for the method of Philosophy”. Łukasiewicz proposed giving philosophical theories the form of axiomatic systems by indicating the primitive terms of their language, selecting suitable axioms, and explicitly determining the applied rules of inference. All the theses of these systems should be consequences of the accepted axioms and confronted with the data of experience and the results of science. Łukasiewicz’s program is presented together with its inspirations and prospects. (shrink)

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

This volume portrays the Polish or Lvov-Warsaw School, one of the most influential schools in analytic philosophy, which, as discussed in the thorough introduction, presented an alternative working picture of the unity of science.

This dialogue between Achilles and the Tortoise – in the spirit of those of Carroll and Hofstadter – argues against the idea, identified with the “traditional” interpretation of Aristotle’s “sea battle argument”, that future contingents are an exception to the Principle of Bivalence. It presents examples of correct everyday predictions, without which one would not be able to decide and to act; however, doing this is incompatible with the belief that the content of these predictions lacks a truth-value. The cost (...) of using a non-classical logic to cope with that may be too high for Stagirite’s defenders, and they would still need to explain why our ordinary predictions seem to have a binary truth-value. In the end, the paper suggests that the problem of future contingents – and of free will – is not a logical problem at all, but rather a limit on what an agent can believe before taking a decision. (shrink)

Of the many problems which evolutionary theodicy tries to address, the ones of animal suffering and extinction seem especially intractable. In this essay, I show how C. D. Broad's growing block conception of time does much to ameliorate the problems. Additionally, I suggest it leads to another way of understanding the soul. Instead of it being understood as a substance, it is seen as a history—a history which is resurrected in the end times. Correspondingly, redemption, I argue, should not be (...) seen as an event which redeems some future portion of time. God's triumph is over all of history, not just some future temporal portion. (shrink)

The aim of this work is to develop a declarative semantics for N-Prolog with negation as failure. N-Prolog is an extension of Prolog proposed by Gabbay and Reyle, which allows for occurrences of nested implications in both goals and clauses. Our starting point is an operational semantics of the language defined by means of top-down derivation trees. Negation as finite failure can be naturally introduced in this context. A goal-G may be inferred from a database if every top-down derivation of (...) G from the database finitely fails, i.e., contains a failure node at finite height.Our purpose is to give a logical interpretation of the underlying operational semantics. In the present work we take into consideration only the basic problems of determining such an interpretation, so that our analysis will concentrate on the propositional case. Nevertheless we give an intuitive account of how to extend our results to a first order language. A full treatment of N-Prolog with quantifiers will be deferred to the second part of this work. (shrink)

Condensed detachment is usually regarded as a notation, and defined by example. In this paper it is regarded as a rule of inference, and rigorously defined with the help of the Unification Theorem of J. A. Robinson. Historically, however, the invention of condensed detachment by C. A. Meredith preceded Robinson's studies of unification. It is argued that Meredith's ideas deserve recognition in the history of unification, and the possibility that Meredith was influenced, through ukasiewicz, by ideas of Tarski going back (...) at least to 1939, and possibly to 1930 or earlier, is discussed. It is proved that a term is derivable by substitution and ordinary detachment from given axioms if and only if it is a substitution instance of a term which is derivable from these axioms by condensed detachment, and it is shown how this theorem enables the ideas of ukasiewicz and Tarski mentioned above to be formalized and extended. Finally, it is shown how condensed detachment may be subsumed within the resolution principle of J. A. Robinson, and several computer studies of particular Hilbert-type propositional calculi using programs based on condensed detachment or on resolution are briefly discussed. (shrink)

Justification of theorems plays a vital role in any rational human activity. It is indispensable in science. The deductive method of justifying theorems is used in all sciences and it is the only method of justifying theorems in deductive disciplines. It is based on the notion of proof, thus it is a method of proving theorems. In the Warsaw School of Logic (WSL) – the famous branch of the Lvov-Warsaw School (LWS) – two types of the method: axiomatic deduction method (...) and natural deduction method were developed and practiced. In this paper, both of these methods are briefly discussed with an emphasis on their historical, groundbreaking significance for logic. The axiomatic method by means of rejection (proposed by Jan Łukasiewicz – a co-creator of the WSL), which is the method of the so-called rejection proof (rejection/refutation method) in logical systems and the proving method of generalized natural deduction, which is a hybrid deduction–refutation method of proving theorems, are also outlined in the paper. The author discusses their historical significance. This paper also contains a brief mention of the most significant results which the application of the discussed methods introduced into contemporary scientific research, not only logical one. (shrink)

As the final component of a chain of reasoning intended to take us all the way to logical nihilism, Russell (2018) presents the atomic sentence ‘prem’ which is supposed to be true when featuring as premise in an argument and false when featuring as conclusion in an argument. Such a sentence requires a non-reflexive logic and an endnote by Russell (2018) could easily leave the reader with the impression that going non-reflexive suffices for logical nihilism. This paper shows how one (...) can obtain non-reflexive logics in which ‘prem’ behaves as stipulated by Russell (2018) but which nonetheless has valid inferences supporting uniform substitution of any formula for propositional variables such as modus tollens and modus ponens. (shrink)

Fundamental properties of N-valued logics are compared and eleven theorems are presented for their Logic Algebras, including Łukasiewicz–Moisil Logic Algebras represented in terms of categories and functors. For example, the Fundamental Logic Adjunction Theorem allows one to transfer certain universal, or global, properties of the Category of Boolean Algebras,, (which are well-understood) to the more general category \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal L}$$\end{document}Mn of Łukasiewicz–Moisil Algebras. Furthermore, the relationships of LMn-algebras to other many-valued logical structures, (...) such as the n-valued Post, MV and Heyting logic algebras, are investigated and several pertinent theorems are derived. Applications of Łukasiewicz–Moisil Algebras to biological problems, such as nonlinear dynamics of genetic networks – that were previously reported – are also briefly noted here, and finally, probabilities are precisely defined over LMn-algebras with an eye to immediate, possible applications in biostatistics. (shrink)

En la historia de la sociedad moderna –basada en la discriminación económica y en ideas competitivas de poder– la teoría de la evolución darwiniana, por medio de la selección natural de las especies, llegó como una justificación de aptitudes biológicas para su estructura socioeconómica. Pareció que el papel del individuo era contribuir a perpetuar la especie; todo lo que se tenía que hacer era dejar que los fenómenos naturales siguieran su curso. La ciencia, en especial la biología, aparecía justificando la (...) premisa “cualquier cosa por el bien de la humanidad”. (shrink)

A simple complete axiomatic system is presented for the many-valued propositional logic based on the conjunction interpreted as product, the coresponding implication (Goguen's implication) and the corresponding negation (Gödel's negation). Algebraic proof methods are used. The meaning for fuzzy logic (in the narrow sense) is shortly discussed.

We compare Jan ?ukasiewicz's and Karl Popper's views on induction. The English translation of the two ?ukasiewicz's papers is included in the Appendix.

The Warsaw School of Logic (WSL) was the famous branch of the Lviv-Warsaw School (LWS) – the most important movement in the history of Polish philosophy. Logic made the most important field in the activities of the WSL. The aim of this work is to highlight the role and significance of the WSL in the history of logic in the 20th century.

Charles Peirce incorporates modality into his Existential Graphs by introducing the broken cut for possible falsity. Although it can be adapted to various modern modal logics, Zeman demonstrates that making no other changes results in a version that he calls Gamma-MR, an implementation of Jan Łukasiewicz's four-valued Ł-modal system. It disallows the assertion of necessity, reflecting a denial of determinism, and has theorems involving possibility that seem counterintuitive at first glance. However, the latter is a misconception that arises from overlooking (...) the distinction between the intermediate truth values that are assigned to possibly true propositions as either X-contingent or Y-contingent. Any two propositions having the same ITV are possible together, while any two propositions having different ITVs, including those that are each other's negation, are possible individually yet not possible together. Porte shows that Ł-modal can be translated into classical logic by defining a constant for each ITV such that its implication of another proposition asserts the latter's possibility, while its conjunction with another proposition asserts the latter's necessity. These are expressed in the Alpha part of EG without broken cuts, simplifying derivations and shedding further light on Łukasiewicz's system, as long as graphs including either of the constants are properly interpreted. Ł-modal and Gamma-MR thus capture the two-sided nature of possibility as the limit between truth and falsity in Peirce's triadic conception. (shrink)

Łukasiewicz presented two different analyses of modal notions by means of many-valued logics: the linearly ordered systems Ł3,..., Open image in new window,..., \; the 4-valued logic Ł he defined in the last years of his career. Unfortunately, all these systems contain “Łukasiewicz type paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following Łukasiewicz’s strategy for defining truth-functional (...) modal logics. The systems we define lack “Łukasiewicz type paradoxes”. Following Brady, we endow them with Belnap–Dunn type bivalent semantics. (shrink)

Classical logic rests on the assumption that there are two mutually exclusive and jointly exhaustive truth values. This assumption has always been surrounded by philosophical controversy. Doubts have been raised about its legitimacy, and hence about the legitimacy of classical logic. Usually, the assumption is stated in the form of a general principle, namely the principle that every proposition is either true or false. Then, the philosophical controversy is often framed in terms of the question whether every proposition is either (...) true or false. The main purpose of the paper is to show that there is something wrong in this way of putting things. The point is that the common way of understanding the controversial assumption is misconceived, as it rests on a wrong picture of propositions. In the first part of the paper I outline this picture and I argue against it. In the second part I sketch a different picture of propositions and I suggest how this leads to conceive the issue of classical logic in different terms. (shrink)

The aim of this paper is to show that it is the explicativecharacter of Tarski's semantic definition of truth given in his study of 1933 that allows forconsideration of a philosophical background of this definition in the proper sense. Given the explicativecharacter of this definition it is argued that the philosophical tradition that should be taken intoaccount with regard to this philosophical background is the tradition of the Lvov-Warsaw Schoolin its connections with the School of Brentano. As an example of (...) the explanatory power ofconsidering this tradition as far as Tarski's philosophical choices are concerned I use here thenotion of sentence-inscription, i.e., the notion of that entity of which truth is predicated inthe definition in question. One of the consequences of these statements is that philosophicaldiscussions concerning the semantic definition of truth can be regarded from two points ofview. On the one hand, they may take the perspective of its explicational function, i.e., theperspective of its philosophical background. On the other hand, they might consider the philosophicalconsequences of the definition with respect to the goal of the explication, i.e., they may considerits philosophical content independently of its historical background. (shrink)

The questions od determinism, causality, and freedom have been the main philosophical problems debated since the beginning of temporal logic. The issue of the logical value of sentences about the future was stated by Aristotle in the famous tomorrow sea-battle passage. The question has inspired Łukasiewicz’s idea of many-valued logics and was a motive of A. N. Prior’s considerations about the logic of tenses. In the scheme of temporal logic there are different solutions to the problem. In the paper we (...) consider indeterministic temporal logic based on the idea of temporal worlds and the relation of accessibility between them. (shrink)

In contemporary science uncertainty is often represented as an intrinsic feature of natural and of human phenomena. As an example we need only think of two important conceptual revolutions that occurred in physics and logic during the first half of the twentieth century: (1) the discovery of Heisenberg’s uncertainty principle in quantum mechanics; (2) the emergence of many-valued logical reasoning, which gave rise to so-called ‘fuzzy thinking’. I discuss the possibility of applying the notions of uncertainty, developed in the framework (...) of quantum mechanics, quantum information and fuzzy logics, to some problems of political and social sciences. (shrink)

Summary of the talk given to the 22nd Conference on the History of Logic, Cracow (Poland), July 5–9, 1976.