Can an AGI create a more intelligent AGI? Under idealized assumptions, for a certain theoretical type of intelligence, our answer is: “Not without outside help”. This is a paper on the mathematical structure of AGI populations when parent AGIs create child AGIs. We argue that such populations satisfy a certain biological law. Motivated by observations of sexual reproduction in seemingly-asexual species, the Knight-Darwin Law states that it is impossible for one organism to asexually produce another, which asexually produces another, and (...) so on forever: that any sequence of organisms (each one a child of the previous) must contain occasional multi-parent organisms, or must terminate. By proving that a certain measure (arguably an intelligence measure) decreases when an idealized parent AGI single-handedly creates a child AGI, we argue that a similar Law holds for AGIs. (shrink)

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1-CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.

Despre necesitatea şi utilitatea dezvoltării unei filosofii specifice tehnologiei blockchain, accentuând pe aspectele ontologice. După o Introducere în care evidenţiez principalele direcţii filosofice pentru această tehnologie emergentă, în Tehnologia blockchain explicitez modul de funcţionare al blockchain, punând în discuţie direcţiile ontologice de dezvoltare în Proiectarea şi Modelarea acestei tehnologii. Următoarea secţiune este dedicată principalei aplicaţii a tehnologiei blockchain, Bitcoin, cu implicaţiile sociale ale acestei criptovalute. Urmează o secţiune de Filosofie în care identific tehnologia blockchain cu conceptul de heterotopie dezvoltat de (...) Michel Foucault şi o interpretez în lumina tehnologiei notaţionale dezvoltată de Nelson Goodman ca sistem notaţional. În secţiunea Ontologii dezvolt două direcţii de dezvoltare considerate de mine ca importante: Ontologia narativă, bazată pe ideea de ordine şi structură a istoriei transmise prin naraţia istoriei a lui Paul Ricoeur, şi un sistem ontologic bazat pe Ontologiile de intreprindere, bazat pe conceptele şi modelele de intreprindere specifice webului semantic, şi pe care eu îl consider cel mai bine dezvoltat şi cel care va deveni, probabil, sistemul ontologic formal, cel puţin în ceea ce priveşte aspectele economice şi juridice ale tehnologiei blockchain. În Concluzii vorbesc despre direcţiile viitoare de dezvoltare a filosofiei tehnologiei blockchain în general ca o teorie explicativă şi robustă din punct de vedere fenomenologic, consistentă şi care să permită testabilitatea, şi a ontologiilor în special, argumentând pentru necesitatea unei adopţii globale a unui sistem ontologic pentru a dezvolta soluţii transversale şi a rentabiliza tehnologia. -/- Cuvinte cheie: filosofie, blockchain, ontologie, bitcoin, proiectare, modele -/- CUPRINS -/- Abstract Introducere Tehnologia blockchain - - - Proiectare - - - Modele Bitcoin Filosofia Ontologii - - - Ontologii narative - - - Ontologii de intreprindere Concluzii Note Bibliografie -/- DOI: 10.13140/RG.2.2.25492.35204. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood (...) as a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

It is widely held that under ordinal utility, utility differences are ill-defined. Allegedly, for these to be well-defined (without turning to choice under risk or the like), one should adopt as a new kind of primitive quaternary relations, instead of the traditional binary relations underlying ordinal utility functions. Correlatively, it is also widely held that the key structural properties of quaternary relations are entirely arbitrary from an ordinal point of view. These properties would be, in a nutshell, (...) the hallmark of cardinal utility. While much is obviously true in these two tenets, this note explains why, as stated, they should be abandoned. Any ordinal utility function induces a rich quaternary relation. There is such a thing as ordinal utility differences. Furthermore, this induced quaternary relation respects, apart from completeness, the most standard structural properties of quaternary relations. These properties are, from an ordinal point of view, anything but arbitrary; from a quaternary perspective only completeness should be considered the hallmark—if any—of cardinal utility. These facts are explained to be especially relevant to the critical appreciation of the ordinalist methodology. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. (...) First, we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

Ontological pluralism is the view that there are different ways to exist. It is a position with deep roots in the history of philosophy, and in which there has been a recent resurgence of interest. In contemporary presentations, it is stated in terms of fundamental languages: as the view that such languages contain more than one quantifier. For example, one ranging over abstract objects, and another over concrete ones. A natural worry, however, is that the languages proposed by the pluralist (...) are mere notational variants of those proposed by the monist, in which case the debate between the two positions would not seem to be substantive. Jason Turner has given an ingenious response to this worry, in terms of a principle that he calls ‘logical realism’. This paper offers a counter-response on behalf of the ‘notationalist’. I argue that, properly applied, the principle of logical realism is no threat to the claim that the languages in question are notational variants. Indeed, there seems to be every reason to think that they are. (shrink)

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

Transfinite ordinal numbers enter mathematical practice mainly via the method of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition in works recasting proofs by transfinite recursion in other terms, mostly with the intention of eliminating the ordinals from the proofs. Leaving aside the different motivations which lead each specific case, we investigate the mathematics of this action of proof transforming and we address the problem of formalising the philosophical notion of elimination which (...) characterises this move. (shrink)

This paper argues that many so-called digital technologies can be construed as notational technologies, explored through the example of Monegraph, an art and digital asset management platform built on top of the blockchain system originally developed for the cryptocurrency bitcoin. As the paper characterizes it, a notational technology is the performance of syntactic notation within a field of reference, a technologized version of what Nelson Goodman called a “notational system.” Notational technologies produce abstracted entities through positive and reliable, or constitutive, (...) tests of socially acceptable meaning. Accordingly, this account deviates from typical narratives of blockchains, instead demonstrating that blockchain technologies are effective at managing digital assets because they produce abstracted identities through the performance of notation. Since notational technologies rely on configurations of socially acceptable meaning, this paper also provides a philosophical account of how blockchain technologies are socially embedded. (shrink)

The pragmatic notion of assertion has an important inferential role in logic. There are also many notational forms to express assertions in logical systems. This paper reviews, compares and analyses languages with signs for assertions, including explicit signs such as Frege’s and Dalla Pozza’s logical systems and implicit signs with no specific sign for assertion, such as Peirce’s algebraic and graphical logics and the recent modification of the latter termed Assertive Graphs. We identify and discuss the main ‘points’ of these (...) notations on the logical representation of assertions, and evaluate their systems from the perspective of the philosophy of logical notations. Pragmatic assertions turn out to be useful in providing intended interpretations of a variety of logical systems. (shrink)

This paper presents a diagrammatic notation for visualizing epistemic entities and relations. The notation was created during the Visualizing Worldviews project funded by the University of Toronto’s Jackman Humanities Institute and has been further developed by the scholars participating in the university’s Research Opportunity Program. Since any systematic diagrammatic notation should be based on a solid ontology of the respective domain, we first outline the current state of the scientonomic ontology. We then proceed to providing diagrammatic tools for visualizing the (...) epistemic entities and relations of this ontology. These basic diagramming techniques allow us to construct diagrams of various types for both synchronic and diachronic visualizations. The paper concludes by highlighting some future research directions. As the notation presented here is de facto accepted and used in scientonomy, the paper suggests no modifications. (shrink)

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's choice function. (...) This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these (...) accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic. (shrink)

We provide an intuitive motivation for the hyperreal numbers via electoral axioms. We do so in the form of a Socratic dialogue, in which Protagoras suggests replacing big-oh complexity classes by real numbers, and Socrates asks some troubling questions about what would happen if one tried to do that. The dialogue is followed by an appendix containing additional commentary and a more formal proof.

ABSTRACT (ENG) One of the concerns of Greek philosophy centred on the question of how a manifold and ordered universe arose out of the primitive state of things. From the mythical accounts dating around the seventh century B.C. to the cosmologies of the Classical period in Ancient Greece, many theories have been proposed in order to answer to this question. How these theories differ in positing a “something” that pre-existed the ordered cosmos has been widely discussed. However, scholars have rarely (...) made explicit how they differ in style of thought. In the span of four centuries the first deductive arguments of the Eleatic philosophers culminated in the emergence of logical proof and a form of explanation of natural phenomena, which consisted of searching for the simplest and fewest premises and deducting implications. In this paper it will be discussed how, at distinct stages of its development, the deductive thinking informed the solutions proposed to solve the chaos-order problem, that of how an ordered universe has been possible. -/- ABSTRACT (ITA) Una delle più importanti questioni della filosofia greca è stata quella di comprendere come sia stato possibile un universo ordinato a partire da uno stato primordiale. Dalle teogonie del VII secolo a.C. fino alle cosmologie dei filosofi dell’età classica, sono state proposte diverse teorie per dare risposta a questa domanda. Come esse differiscano nel postulare l’esistenza di un “qualcosa” di primordiale che preesisteva all’ordine del cosmo è stato molto discusso. Pochi studiosi, però, le hanno esaminate sullo sfondo della lenta evoluzione del pensiero deduttivo, culminata nella dimostrazione in geometria e in una forma di spiegazione dei fenomeni che consisteva nel cercare semplici premesse e inferire conclusioni. In questo articolo si mostrerà come il lento affermarsi della spiegazione razionale prima, dell’argomento deduttivo e della dimostrazione in geometria poi, abbiano dato forma alle diverse risposte al problema caos-ordine, e in particolare alla domanda su come sia sorto un universo ordinato. (shrink)

There is a widely held view on measurement inferences, that goes back to Stevens’s ([1946]) theory of measurement scales and ‘permissible statistics’. This view defends the following prohibition: you should not make inferences from averages taken with ordinal scales (versus quantitative scales: interval or ratio). This prohibition is general—it applies to all ordinal scales—and it is sometimes endorsed without qualification. Adhering to it dramatically limits the research that the social and biomedical sciences can conduct. I provide a Bayesian (...) analysis of this inferential problem, determining when measurements from ordinal scales can be used to confirm hypotheses about relative group averages. The prohibition, I conclude, cannot be upheld, even in a qualified sense. The beliefs needed to make average comparisons are less demanding than those appropriate for quantitative scales. I illustrate with the alleged paradigm ordinal scale, Mohs’ scale of mineral hardness, arguing that the literature has mischaracterized it. (shrink)

The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter.

In contemporary natural languages semantics one will often see the use of special brackets to enclose a linguistic expression, e.g. ⟦carrot⟧. These brackets---so-called denotation brackets or semantic evaluation brackets---stand for a function that maps a linguistic expression to its "denotation" or semantic value (perhaps relative to a model or other parameters). Even though this notation has been used in one form or another since the early development of natural language semantics in the 1960s and 1970s, Montague himself didn't make use (...) of this notation in his series of groundbreaking papers on semantics. That raises the question: When was the ⟦.⟧-notation introduced to semantics? This note answers that question. (shrink)

Gauthier and Hobbes reduce Prisoners Dilemmas to co-ordination problems (CPs). Many think rational, face-to-face agents can solve any CP by agreed fiat. But though an agent can rationally use a symmetry-breaking technique (ST) to decide between equal options, groups cannot unless their members' STs luckily converge. Failing this, the CP is escapable only by one agent's non-rational stubbornness, or by the group's "conquest" by an outside force. Implications: one's strategic rationality is group-relative; there are some optimums groups in principle cannot (...) rationally choose; thus justice cannot always be a rationally contracted optimum. Howard Sobel provides the point of departure. (shrink)

Wittgenstein composed five original musical fragments during his transitional middle period, in which he employs musical notation as a means by which to convey his philosophical thoughts. This is an overlooked aspect of the importance of aesthetics, and musical thinking in particular, in the development of Wittgenstein’s philosophy. We explain and evaluate the way the music interlinks with Wittgenstein’s philosophical thoughts. We show the direct relation of these musical examples as precursors to some of Wittgenstein’s most celebrated ideas (the push (...) beyond the inner/outer picture, the problem of rule-following, and aspect perception). We also show how important broad aesthetic notions such as style and composition concerning the very possibility of philosophizing were to Wittgenstein. (shrink)

Visual imagery is a form of sensory imagination, involving subjective experiences typically described as similar to perception, but which occur in the absence of corresponding external stimuli. We used the Activation Likelihood Estimation algorithm (ALE) to identify regions consistently activated by visual imagery across 40 neuroimaging studies, the first such meta-analysis. We also employed a recently developed multi-modal parcellation of the human brain to attribute stereotactic co-ordinates to one of 180 anatomical regions, the first time this approach has been combined (...) with the ALE algorithm. We identified a total 634 foci, based on measurements from 464 participants. Our overall comparison identified activation in the superior parietal lobule, particularly in the left hemisphere, consistent with the proposed ‘top-down’ role for this brain region in imagery. Inferior premotor areas and the inferior frontal sulcus were reliably activated, a finding consistent with the prominent semantic demands made by many visual imagery tasks. We observed bilateral activation in several areas associated with the integration of eye movements and visual information, including the supplementary and cingulate eye fields (SCEFs) and the frontal eye fields (FEFs), suggesting that enactive processes are important in visual imagery. V1 was typically activated during visual imagery, even when participants have their eyes closed, consistent with influential depictive theories of visual imagery. Temporal lobe activation was restricted to area PH and regions of the fusiform gyrus, adjacent to the fusiform face complex (FFC). These results provide a secure foundation for future work to characterise in greater detail the functional contributions of specific areas to visual imagery. (shrink)

The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter. Implications to time-dependent Schrodinger wave equation and uncertainty (...) principle are hinted. (shrink)

The main objective of this PhD Thesis is to present a method of obtaining strong normalization via natural ordinal, which is applicable to natural deduction systems and typed lambda calculus. The method includes (a) the definition of a numerical assignment that associates each derivation (or lambda term) to a natural number and (b) the proof that this assignment decreases with reductions of maximal formulas (or redex). Besides, because the numerical assignment used coincide with the length of a specific sequence (...) of reduction - the worst reduction sequence - it is the lowest upper bound on the length of reduction sequences. The main commitment of the introduced method is that it is constructive and elementary, produced only through analyzing structural and combinatorial properties of derivations and lambda terms, without appeal to any sophisticated mathematical tool. Together with the exposition of the method, it is presented a comparative study of some articles in the literature that also get strong normalization by means we can identify with the natural ordinal methods. Among them we highlight Howard[1968], which performs an ordinal analysis of Godel’s Dialectica interpretation for intuitionistic first order arithmetic. We reveal a fact about this article not noted by the author himself: a syntactic proof of strong normalization theorem for the system of typified lambda calculus λ⊃ is a consequence of its results. This would be the first strong normalization proof in the literature. (written in Portuguese). (shrink)

Transformed RAVAL NOTATION solves Syllogism problems very quickly and accurately. This method solves any categorical syllogism problem with same ease and is as simple as ABC… In Transformed RAVAL NOTATION, each premise and conclusion is written in abbreviated form, and then conclusion is reached simply by connecting abbreviated premises.NOTATION: Statements (both premises and conclusions) are represented as follows: Statement Notation a) All S are P, SS-P b) Some S are P, S-P c) Some S are not P, S / PP (...) d) No S is P, SS / PP (- implies are and / implies are not) All is represented by double letters; Some is represented by single letter. No S is P implies No P is S so its notation contains double letters on both sides. -/- RULES: (1) Conclusions are reached by connecting Notations. Two notations can be linked only through common linking terms. When the common linking term multiplies (becomes double from single), divides (becomes single from double) or remains double then conclusion is arrived between terminal terms. (Aristotle’s rule: the middle term must be distributed at least once) -/- (2)If both statements linked are having – signs, resulting conclusion carries – sign (Aristotle’s rule: two affirmatives imply an affirmative) -/- (3) Whenever statements having – and / signs are linked, resulting conclusion carries / sign. (Aristotle’s rule: if one premise is negative, then the conclusion must be negative) -/- (4)Statement having / sign cannot be linked with another statement having / sign to derive any conclusion. (Aristotle’s rule: Two negative premises imply no valid conclusion) Syllogism conclusion by Tranformed Raval’s Notation is in accordance with Aristotle’s rules for the same. It is visually very transparent and conclusions can be deduced at a glance, moreover it solves syllogism problems with any number of statements and it is quickest of all available methods. By new Raval method for solving categorical syllogism, solving categorical syllogism is as simple as pronouncing ABC and it is just continuance of Aristotle work on categorical syllogism. It’s believed that Boole's system could handle multi-term propositions and arguments, whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, it’s claimed that Aristotle's system could not deduce: "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle". Above conclusion is reached at a glance with Raval's Notations (Symbolic Aristotle’s syllogism rules). Premise: "No (square that is a quadrangle) is a (rhombus that is a rectangle)" Raval's Representations: S – Q, S – Q / Rh – Re, Rh – Re Premise: "No (rhombus that is a rectangle) is a (square that is a quadrangle)". Raval's Representations: Rh – Re, Rh – Re / S – Q, S - Q Conclusion: "No (quadrangle that is a square) is a (rectangle that is a rhombus)" Raval’s Representations: Q – S, Q – S / Re – Rh, Re – Rh As “ Q – S” follows from “S – Q” and “Re – Rh” from “Rh – Re”. Given conclusion follows from the given premises. Author disregards existential fallacy, as subset of a null set has to be a null set. -/- . (shrink)

The symbolic argument against women's ordination supposes that the theological significance of Christ's sex is his saving relationship to the Church, which takes the form of that of a bridegroom and his bride. It infers that a male priest alone is fit to represent Christ in his capacity as the Saviour of the Church, and thus that only men should be ordained. Since the emergence of the symbolic argument, however, scholars have rediscovered a long tradition of understanding Christ's saving relationship (...) to the Church in maternal terms. While remaining neutral on whether women ultimately ought to be ordained or not, I argue that the kind of reasoning in the symbolic argument, if updated with the Jesus as Mother tradition, would suggest that it is fitting for a female priest to represent Jesus as Mother, just as it is fitting for a male priest to represent Christ the Bridegroom. Other critics of the symbolic argument tend to contest what is seen as its ‘literalist’ or ‘essentialist’ or overly gendered-valanced assumptions about priestly representation. I show that even if we grant to the symbolic argument more gendered and ‘essentialist’ views on each of the major points of disagreement in these debates, women might still be fit to represent Christ because of the maternal ways that Christ has traditionally been thought to relate to the Church. As a result, the symbolic argument may be repurposed to support women's ordination rather than undermine it. (shrink)

This study examines problems related to the representation of music. It constructs the sender/message/perceiver/result model, a prototype broad enough to incorporate a large variety of music and other notation systems, including those having to do with computers. The work defines music notation itself, describes various models for studying the subject--including the binary types prescriptive/descriptive, and symbolic/iconic--and assesses music notation as a contemporary practice. It encompasses a review of the actions and intentions of composers, performers, and audiences, and a consideration of (...) the message and how it is affected by media , reference systems , and meanings. Finally, through an examination of the possibilities of the self acting as a notation system within models posited by feminist and queer theory, I attempt to demonstrate the power of notation to affect other systems and worlds, including the worlds of music, art, and politics. ;The consideration of these topics draws on theories from linguistics, semiotics, structuralism, post-structuralism, cybernetics, and music, as well as the philosophies of John Cage, Jean-Jacques Nattiez, Nelson Goodman, and others. The work examines various languages and applications used to program computers for musical purposes, though this examination does not incorporate programs dedicated exclusively to printing music. Nor does the study focus on non-Western notations, suggestions for reforms of notation, or encyclopedic lists of notation devices. ;The expository writing is augmented by excerpts from material I gathered in numerous interviews. There are separate sections containing interviews of Leonard Meyer, Allan Kaprow, John Cage, Earle Brown, Roger Reynolds, and Heidi von Gunden. There is also a large, partially-annotated bibliography. ;My hope is that an inclusive, interdisciplinary exploration of music notation will be useful in broadening the theoretical and creative boundaries of the field, as well as bridging gaps among musicians using various music technologies, and artists and theorists from diverse fields and perspectives. (shrink)

In Languages of Art, Nelson Goodman presents a general theory of symbolic notation. However, I show that his theory could not adequately explain possible cases of natural language notational uses, and argue that this outcome undermines, not only Goodman's own theory, but any broadly type versus token based account of notational structure.Given this failure, an alternative representational theory is proposed, in which different visual or perceptual aspects of a given physical inscription each represent a different letter, word, or other notational (...) item. Such a view is strongly supported by the completely conventional relation between inscriptions and notation, as shown by encryption techniques etc. (shrink)

The design process in Kashmiri carpet weaving is distributed over a number of actors and artifacts and is mediated by a weaving notation called talim. The script encodes entire design in practice-specific symbols. This encoded script is decoded and interpreted via design-specific conventions by weavers to weave the design embedded in it. The cognitive properties of this notational system are described in the paper employing cognitive dimensions (CDs) framework of Green (People and computers, Cambridge University Press, Cambridge, 1989) and Blackwell (...) et al. (Cognitive technology: instruments of mind—CT 2001, LNAI 2117, Springer, Berlin, 2001). After introduction to the practice, the design process is described in ‘The design process’ section which includes coding and decoding of talim. In ‘Cognitive dimensions of talim’ section, after briefly discussing CDs framework, the specific cognitive dimensions possessed by talim are described in detail. (shrink)

For wise men and academics, the order of the world had been an implicit truth for a long time. Nobody had doubted it. The purpose of philosophy and theological discourse has always been to record and contemplate the mundane and celestial world. Because it has a Creator, the universe (and the human being) are in complete order (id est they are in orderliness), in a rigorous hierarchy and, at the same time, they have a meaning, a purpose, they are part (...) of a meaningful whole. In an orderly world, life instantly acquires meaning. This venerable certainty has become, in time, a simple hypothesis. Philosophers are no longer sure that the world is orderly. The following essays comment on the decay of a certainty and, in the end, on its loss. Borges and Georges Perec dared to record (with bitter irony) this loss. And this tripartite essay will speak about them. (shrink)

Darwiche and Pearl’s seminal 1997 article outlined a number of baseline principles for a logic of iterated belief revision. These principles, the DP postulates, have been supplemented in a number of alternative ways. Most suggestions have resulted in a form of ‘reductionism’ that identifies belief states with orderings of worlds. However, this position has recently been criticised as being unacceptably strong. Other proposals, such as the popular principle (P), aka ‘Independence’, characteristic of ‘admissible’ operators, remain commendably more modest. In this (...) paper, we supplement the DP postulates and (P) with a number of novel conditions. While the DP postulates constrain the relation between a prior and a posterior conditional belief set, our new principles notably govern the relation between two posterior conditional belief sets obtained from a common prior by different revisions. We show that operators from the resulting family, which subsumes both lexicographic and restrained revision, can be represented as relating belief states associated with a ‘proper ordinal interval’ (POI) assignment, a structure more fine-grained than a simple ordering of worlds. We close the paper by noting that these operators satisfy iterated versions of many AGM era postulates, including Superexpansion, that are not sound for admissible operators in general. (shrink)

Basic abstraction principles are reached through ontology, which was traditionally conceived as a depiction of the world itself. Ontology is also described using conceptual modeling (CM) that defines fundamental concepts of reality. CM is one of the central activities in computer science, especially as it is mainly used in software engineering as an intermediate artifact for system construction. To achieve such a goal, we propose Stoic CM (SCM) as a description of what a system must do functionally with minimal ambiguity. (...) As a case study, we apply SCM to investigate the ontology of BPMN (business process modeling notation). Such an undertaking would demonstrate SCM notions and simultaneously may offer a viable ontological foundation for BPMN. SCM defines the being of things and actions in reality based on Stoic notions of existence and subsistence. It has two levels of specification: (1) a subsistence static model where things and actions subsist and (2) an existence dynamic model where things and actions exist in time. From the Stoic ontological point of view, while a thing existing has a clear denotation, subsistence indicates the thing is “being there,” but it is inactive (does not participate in an event). We apply SCM to BPMN processes that involve buying a new car with many notions, such as activity, task, event, and message. The result indicates that SCM produces a tighter representation of reality, thus providing the necessary description of the part in the application world to be used as requirements for developing the software system. (shrink)

The editor's introduction discusses Clarence I. Lewis's conceptual pragmatism when compared with post-empiricist epistemology and argues that several Cartesian assumptions play a major role in the work, not unlike those of Logical Positivism. The suggestion is made that the Cartesian legacy still hidden in Logical Positivism turns out to be a rather heavy ballast for Lewis’s project of restructuring epistemology in a pragmatist key. More in detail, the sore point is the nature of inter-subjectivity. For Lewis, no less than for (...) the Logical Positivists at the time of the Protocols Controversy and Husserl in the Cartesian Meditations, this is a problem without a solution. The reason is that all these philosophers are apparently unable to realize that the existence of a plurality of knowing subjects cannot be treated at once both as a speculative problem and a methodological one. Lewis, thanks to his pragmatist approach both comes closer to the right answer and offers an even more naïve unsatisfactory solution to the pseudo-problem under discussion. The fact that he has clear in mind that inter-subjectivity means not only a plurality of linguistic utterances but also a co-existence of different kinds of practical behaviour. Eventually, the very idea of mind, the key-idea in the book, suffers from the above mentioned tension. In fact, if inter-subjective communication and action is considered at a methodological level, the very idea of mind would not need an analysis, and no kind of ‘reflexive’ analysis. Methodology might be limited to a ‘naïve’ level where the existence of the world and a plurality of subjects be taken as a bedrock of uncritically accepted evidence. Philosophical reflection on ultimate evidence, instead, would take a different approach, maybe the one Wittgenstein was putting into practice in the same years when Mind and the world order was written, namely it would be bound to question the very meaning of the idea of ‘mind’ as an undue fiction – the same carried out by Descartes – when he assumed the Cogito to be at once a body of self-evident truths and a thing or substance, the familiar Platonic idea of psyche or soul. (shrink)

Seguendo l’esposizione data in (Orsi 2012), riguardante una comparazione fra alcuni aspetti dell’opera di Carl Schmitt e di Jürgen Habermas in filosofia politica, centrata sulla nozione di ordine ed inquadrata, nelle sue basi, entro la sociologia delle religioni di Max Weber, sarà possibile, oltre l’individuazione in essa di un comune punto di convergenza fra il pensiero dei questi autori nella nozione di ordine, portare avanti, su un piano teoretico di livello superiore, un ulteriore raffronto più orientato verso la metodologia della (...) ricerca filosofica così come intesa da Martin Heidegger, la quale permetterà tra l’altro di vedere sia Schmitt che Habermas da un altro possibile punto di vista prospettico. (shrink)

Till just few decades ago, scholars used to use the label ‘Ockhamism’ to mark a turning-point in the history of mediaeval philosophy, above all in the history of natural philosophy. That turning-point was exemplified by the once so-called ‘Buridanian school’, today known simply as ‘Parisian school of natural philosophy’, whose leading representative was for sure John Buridan. But looking carefully at some crucial points of the Picard master’s idea of ‘nature’, concerning specifically the relationship between God and secondary causes on (...) one side, and the role played by free will, both the divine and the human one, on the other side, one will find out that Buridan’s ‘Ockhamism’ is at least questionable: Scotus’ modal logic is the conceptual framework, within which he redetermines the boundaries of natural laws, but the ‘pure ockhamist’ idea of the prevalence of the will on the intellect in God’s nature is the source of an image of the world in which voluntary behaviours and natural laws are intertwined. Therefore, Buridan’s natural philosophy results as a peculiar synthesis of two of the main divergent lines of thinking of late mediaeval philosophy. (shrink)

Dopo mezzo secolo di oblio, la natura della coscienza è ora l'argomento più caldo nelle scienze comportamentali e nella filosofia. A partire dal lavoro pionieristico di Ludwig Wittgenstein negli anni '30 (i libri blu e brown) e dagli anni '50 ad oggi dal suo logico successore John Searle, ho creato la seguente tabella come euristica per promuovere questo studio. Le righe mostrano vari aspetti o modi di studio e le colonne mostrano i processi involontari e i comportamenti volontari che comprendono (...) i due sistemi (processi duali) della struttura logica della coscienza (LSC), che possono anche essere considerati come la struttura logica della razionalità (LSR-Searle), del comportamento (LSB), della personalità (LSP), della realtà (LSOR), dell'intenzionalità (LSI) -il termine filosofico, filosofico la Psicologia Descrittiva della Coscienza (DPC) , la Psicologia descrittiva del Pensiero (DPT) –o meglio, il Linguaggio della Psicologia Descrittiva del Pensiero (LDPT), termini introdotti qui e nei miei altri scritti molto recenti. Coloro che desiderano un quadro aggiornato completo per il comportamento umano dalla moderna vista a due systems possono consultare il mio libro 'La struttura logica dellafilosofia, psicologia, Mind e il linguaggio in Ludwig Wittgenstein e John Searle' 2nd ed (2019). Coloro che sono interessati a più dei miei scritti possono vedere 'Talking Monkeys--Filosofia, Psicologia, Scienza, Religione e Politica su un Pianeta Condannato--Articoli e Recensioni 2006-2019 3rd ed (2019) e Suicidal Utopian Delusions nel 21st Century 4th ed (2019). (shrink)

We reexamine some of the classic problems connected with the use of cardinal utility functions in decision theory, and discuss Patrick Suppes's contributions to this field in light of a reinterpretation we propose for these problems. We analytically decompose the doctrine of ordinalism, which only accepts ordinal utility functions, and distinguish between several doctrines of cardinalism, depending on what components of ordinalism they specifically reject. We identify Suppes's doctrine with the major deviation from ordinalism that conceives of utility functions (...) as representing preference differences, while being non- etheless empirically related to choices. We highlight the originality, promises and limits of this choice-based cardinalism. (shrink)

We define a notion of the intelligence level of an idealized mechanical knowing agent. This is motivated by efforts within artificial intelligence research to define real-number intelligence levels of compli- cated intelligent systems. Our agents are more idealized, which allows us to define a much simpler measure of intelligence level for them. In short, we define the intelligence level of a mechanical knowing agent to be the supremum of the computable ordinals that have codes the agent knows to be codes (...) of computable ordinals. We prove that if one agent knows certain things about another agent, then the former necessarily has a higher intelligence level than the latter. This allows our intelligence no- tion to serve as a stepping stone to obtain results which, by themselves, are not stated in terms of our intelligence notion (results of potential in- terest even to readers totally skeptical that our notion correctly captures intelligence). As an application, we argue that these results comprise evidence against the possibility of intelligence explosion (that is, the no- tion that sufficiently intelligent machines will eventually be capable of designing even more intelligent machines, which can then design even more intelligent machines, and so on). (shrink)

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential (...) to broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...) of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...) diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear. (shrink)

There are many different ways to talk about the world. Some ways of talking are more expressive than others—that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power, translation, and notational variance. In doing so, I show how these investigations (...) help to clarify the role that expressive power plays within debates in metaphysics, logic, and the philosophy of language. (shrink)

Traditionally, the investigation of truth has been anchored in a priori reasoning. Cognitive science deviates from this tradition by adding empirical data on how people understand and use concepts. Building on psychophysics and machine learning methods, we introduce conceptual scaling, an approach to map people's understanding of abstract concepts. This approach, allows computing participant-specific conceptual maps from obtained ordinal comparison data, thereby quantifying perceived similarities among abstract concepts. Using this approach, we investigated individual's alignment with philosophical theories on truth (...) and the predictive capacity of conceptual maps. The obtained results indicated that, while people's understanding of truth is multifaceted and encapsulates notions of coherence and authenticity, alignment is best for the correspondence theory of truth. Furthermore, conceptual maps allowed predicting individual outcomes with an accuracy of roughly 70%. This research demonstrates that conceptual scaling offers accurate descriptions of individual's understanding of abstract concepts, behavioral predictions, and quantification of alignment with theoretical perspectives. (shrink)

The objective of this article is to take into account the functioning of representational cognitive tools, and in particular of notations and visualizations in mathematics. In order to explain their functioning, formulas in algebra and logic and diagrams in topology will be presented as case studies and the notion of manipulative imagination as proposed in previous work will be discussed. To better characterize the analysis, the notions of material anchor and representational affordance will be introduced.

Usual narratives among prehistoric archaeologists consider typological approaches as part of a past and outdated episode in the history of research, subsequently replaced by technological, functional, chemical, and cognitive approaches. From a historical and conceptual perspective, this paper addresses several limits of these narratives, which (1) assume a linear, exclusive, and additive conception of scientific change, neglecting the persistence of typological problems; (2) reduce collective developments to personal work (e.g. the “Bordes’” and “Laplace’s” methods in France); and (3) presuppose the (...) coherence and identity of these “methods” over time. It explores the case of the “Structural and analytical typology” method, developed in France, Spain, and Italy from the 1950s to the 2000s by Georges Laplace and his collaborators for lithic implements. This paper (1) provides a detailed historical account of the evolving content of this collective endeavour over five decades; (2) it addresses the epistemological question of what makes the identity and unity of a scientific method, demonstrating that the core component of the “analytical typology” lies in its particular way to represent real-world phenomena through its notation system; and (3) it reveals how this little known but significant episode of advances in the methods and theory of archaeology, contemporary but independent of the “New Archaeology” trend in English-speaking archaeology, was instrumental in the continuation of evolutionary perspectives in France and in the development of quantitative and formal methods in archaeology in southwestern Europe, foreseeing crucial knowledge representation issues raised today by digital methods in archaeology and data curation. (shrink)