Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors

Boundary algebra [BA] is a simpler notation for Spencer-Brown’s (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] consists of the atoms ‘()’ and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting the presence or absence of () into a PA formula yields a BA formula. The BA axioms are "()()=()" (A1), and "(()) [=?] may be written or erased at will” (A2). Repeated application of these axioms to a PA formula yields a member of B= {(),?} called its simplification. (a) has two intended interpretations: (a) ? a? (Boolean algebra 2), and (a) ? ~a (sentential logic). BA is self-dual: () ? 1 [dually 0] so that B is the carrier for 2, ab ? a?b [a?b], and (a)b [(a(b))] ? a=b, so that ?=() [()=?] follows trivially and B is a poset. The BA basis abc= bca (Dilworth 1938), a(ab)= a(b), and a()=() (Bricken 2002) facilitates clausal reasoning and proof by calculation. BA also simplifies normal forms and Quine’s (1982) truth value analysis. () ? true [false] yields boundary logic.G. Spencer Brown; boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; C.S. Peirce; existential graphs.

Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives

Boundary algebra [BA] is a algebra of type , and a simplified notation for Spencer-Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a lower bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs

Posession as an operational dimension of language

In this study I want to show, above all, that the linguistic expression of POSSESSION is not a given but represents a problem to be solved by the human mind. We must recognize from the outset that linguistic POSSESSION presupposes conceptual or notional POSSESSION, and I shall say more about the latter in Chapter 3. Certain varieties of linguistic structures in the particular languages are united by the fact that they serve the common purpose of expressing notional POS SESSION. But this cannot be their sole common denominator. How would we otherwise be able to recognize, to understand, to learn and to translate a particular linguistic structure as representing POSSESSION? There must be a properly linguistic common denominator, an invariant, that makes this possible. The invariant must be present both within a particular language and in cross-language comparison. What is the nature of such an invariant? As I intend to show, it consists in operational programs and functional principles corresponding to the purpose of expressing notional POSSESSION. The structures of possessivity which we find in the languages of the world represent the traces of these operations, and from the traces it becomes possible to reconstruct stepwise the operations and functions

Possessivity, subject and object

The basic question is whether POSSESSOR and POSSESSUM are on the same level as the roles of VALENCE, two additional roles as it were. My research on POSSESSION has shown (Seiler 1981:7 ff.) that this is not the case, that there is a difference in principle between POSSESSION and VALENCE. However, there are multiple interactions between the two domains, and these interactions shall constitute the object of the following inquiry. It is hoped that this will contribute to a better understanding both of POSSESSION and of VALENCE

Preliminares al estudio de la huella en lingüística

The present paper constitutes a brief advance of much longer and more detailed ongoing work on the concept of “trace” in contemporary linguistic theory, particularly in syntax. It is commonly believed that the idea was coined by Noam Chomsky. However, we already detect its use, with a very accurate value, in the early work of Zellig Harris on mathematical linguistics or, to be more precise, on mathematical structures of language. In its origins, rather than being an index responsible for marking the location occupied by a unit previous to its syntactic movement (which always takes the form of fronting ), the trace was the result of a matrix product between n-adic functions. Thus, in Harris the trace is primarily a concept anchored in matrix calculus, or, put it differently, an algebraic notion. Chomsky’s notion, on its turn, is closely related with the LISP programming language. This text seeks to provide a preliminary analysis of the conceptual complexity implied in the concept of trace, which linguists should become aware of, for otherwise they will be doomed to be entangled in misunderstandings unfruitful to our discipline for decades to come.El presente documento constituye un breve avance de una obra en curso mucho más larga y más detallada sobre el concepto de “huella” en la teoría lingüística contemporánea, particularmente en la sintaxis. Se cree, por lo común, que la idea fue acuñada por Noam Chomsky. Sin embargo, ya detectamos su uso, con un valor muy preciso, en los primeros trabajos de Zellig Harris sobre lingüística matemática o, para ser más exactos, sobre estructuras 2matemáticas del lenguaje. En sus orígenes, en lugar de ser un índice responsable de marcar la ubicación de una unidad antes de su movimiento sintáctico (que siempre toma la forma de fronting), la traza o huella era el resultado de un producto matricial entre funciones n-ádicas. Por lo tanto, en Harris la huella es principalmente un concepto anclado en el cálculo matricial o, dicho de otro modo, una noción algebraica. La noción de Chomsky, por su parte, está estrechamente relacionada con el lenguaje de programación LISP. EL presente texto busca proporcionar un análisis preliminar de la complejidad conceptual implícita en el concepto de huella, del cual los lingüistas deben tomar conciencia, porque de lo contrario estarán condenados a enredarse en malentendidos infructuosos para nuestra disciplina durante las próximas décadas

Information structure and the referential status of linguistic expression : workshop as part of the 23th annual meetings of the Deutsche Gesellschaft für Sprachwissenschaft in Leipzig, Leipzig, February 28 - March 2, 2001

This volume comprises papers that were given at the workshop Information Structure and the Referential Status of Linguistic Expressions, which we organized during the Deutsche Gesellschaft für Sprachwissenschaft (DGfS) Conference in Leipzig in February 2001. At this workshop we discussed the connection between information structure and the referential interpretation of linguistic expressions, a topic mostly neglected in current linguistics research. One common aim of the papers is to find out to what extent the focus-background as well as the topic-comment structuring determine the referential interpretation of simple arguments like definite and indefinite NPs on the one hand and sentences on the other

Detotaliseerimine ja tagasiulatuv jõud: musta püramiidi semiootika

Väitekirja üldiseks probleemiks on semiootika integreeritavus. Detotalisatsioon kirjeldab semiootikatraditsiooni, mille kohaselt suletud terviklikkus pole võimalik, ning mis oma põhiliste teoreetiliste koordinaatidena näeb psühhoanalüüsi, ideoloogia kriitikat ja strukturaalset semioloogiat. Oluliseks analüüsivahendiks on autori poolt välja töötatud nn “musta püramiidi” skeem-mudel, mille abil otsitakse vastust küsimusele: kuidas saab puhtdiferentsiaalne, erinevustel põhinev (internaalne) süsteem suhestuda välisega (eksternaalsega)? Järgnevalt jõutakse semiootikas esineva subjektiivse relativismi kriitikani ja võetakse kasutusele retroaktiivsuse mõiste, mille kaudu kirjeldatakse väliseid mõjusid. Semiootika osavaldu vaadeldakse retroaktiivsuse toimimise aspektist. “Musta püramiidi” skeem-mudel ühendab hübriidselt Peirce’i ja Hjelmslev’ semiootikat, integreerides Peirce’i detotalisatsiooniga. Skeem eristab märgifunktsiooni ja märgiproduktsiooni ala ning selle jaotuse kaudu sulandab Peirce’i trihhotoomia kokku Saussure’i dihhotoomiaga. Taolisel sünteesil on kaks eelist. Esmalt on detotalisatsiooni subjektivistlik relativism ankurdatud kognitiivsemiootika ja biosemiootika empiiriliste ja loogiliste rakenduste poolt. Teisalt on kognitiivsemiootika ja biosemiootika rikastatud retroaktiivsuse tekstiliste protseduuridega, mis võimaldab ligipääsu välisele ilma märgi määratlust kahjustamata. Seeläbi on olemas artikulatoorse alusmaatriksi teaduslik seletus, kuid samuti vajadus teaduslikus semiootikas detotalisatsioonile iseloomuliku tekstuaalse eksperimenteerimise järele. Just retroaktiivsus on see ühendav mõiste, mis seob kaks semiootika lahusolevat valda. Integreerides ka kognitiivsemiootika ja biosemiootika detotaliseeritud semiootika pildile, pakub väitekiri kokkuvõttes mittereduktiivse ja empiirilise vastuse relativismi probleemile semiootikas, säilitades seejuures semiootika teoreetilise terviklikkuse ja pakkudes välja ühtse metakeele killustatud sotsiaalteaduste tarbeks.  Detotalization describes the tradition of semiotics which takes psychoanalysis, ideology critique, and structural semiology as its major theoretic coordinates. Interest in these coordinates has declined against the ascent of the semiotics of Charles Peirce, the two approaches are sometimes construed as irreconcilable, but the dissertation seeks to integrate Peirce to the coordinates of detotalization. This integration requires that Peirce be read in the way that Jacques Derrida and Umberto Eco propose to read him, by moderating his realism. This is achieved through theorization of the notion of retroactivity. Chapters one through four restate the coordinates of detotalization in terms of retroactivity, and chapter five searches the domains of cognitive and biosemiotics for the Peircean equivalent of retroactivity. The black pyramid schema is a picture of the Peirce-Hjelmslev hybrid, where Peirce is integrated to detotalization. In the schema, semiotics is organized by the domains of sign function and sign production, and the Peircean trichotomy is reconciled to the Saussurean dichotomy by means of this division. The synthesis has two advantages. In one direction, the subjectivist relativism of detotalization is anchored by the empirical and logical applications of cognitive and biosemiotics. In the other direction, cognitive and biosemiotics are enhanced by the textual procedures of retroactivity, which account for the external without compromising the definition of the sign by importing a naïve referent. There is a scientific explanation for the profound articulatory matrix, but there is also a need within scientific semiotics for the textual experimentation characteristic of detotalization. Retroactivity as the bridge concept between the two divided camps of semiotics also restores its original ambition, to provide a unifying vocabulary for the fractured social sciences.https://www.ester.ee/record=b540146

A workshop on the gathering of information for problem formulation

Issued as Quarterly progress reports no. [1-5], Proceedings and Final contract report, Project no. G-36-651Papers presented at the Workshop/Symposium on Human Computer Interaction, March 26 and 27, 1981, Atlanta, G