Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

The Riddle of Gravitation

There is no doubt that both the special and general theories of relativity capture the imagination. The anti-intuitive properties of the special theory of relativity and its deep philosophical implications, the bizzare and dazzling predictions of the general theory of relativity: the curvature of spacetime, the exotic characteristics of black holes, the bewildering prospects of gravitational waves, the discovery of astronomical objects as quasers and pulsers, the expansion and the (possible) recontraction of the universe..., are all breathtaking phenomena. In this paper, we give a philosophical non-technical treatment of both the special and the general theory of relativity together with an exposition of some of the latest physical theories. We then give an outline of an axiomatic approach to relativity theories due to Andreka and Nemeti that throws light on the logical structure of both theories. This is followed by an exposition of some of the bewildering results established by Andreka and Nemeti concerning the foundations of mathematics using the notion of relativistic computers. We next give a survey on the meaning and philosophical implications of the the quantum theory and end the paper by an imaginary debate between Einstein and Neils Bohr reflecting both Einstein's and Bohr's philosophical views on the quantum world. The paper is written in a somewhat untraditional manner; there are too many footnotes. In order not to burden the reader with all the details, we have collected the more advanced material the footnotes. We think that this makes the paper easier to read and simpler to follow. The paper in full is adressed more to experts.Comment: 40 pages, LaTeX-fil

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

Palaeo-Philosophy: Archaic Ideas about Space and Time

This paper argues that efforts to understand historically remote patterns of thought are driven away from their original meaning if the investigation focuses on reconstruction of concepts, instead of cognitive complexes. My paper draws on research by Jan Assmann, Jean-Jacques Glassner, Keimpe Algra, Alex Purves, Nicholas Wyatt, and others on the cultures of Ancient Greece, Israel, Egypt, Mesopotamia, and Etruria through comparative analyses of the semantic fields of spatial and temporal terms, and how these terms are shaped by their relation to the sphere of the sacred. It shows that there are three super-ordinate timeframes - the cyclical, the linear and the static - each of which is composed of lower-order cycles (days, lunar months, and seasons). These timeframes reflect their cultures ideas about the nature, scope and power of the gods, and structure the common point-of-view about the present, the past and eternity. There are also super-ordinate spatial frames which reflect their cultures ideas about the heavens and which structure both the sacred precinct and the profane field of action and exchange. Close analysis of texts that use words such as eternity, forever, past, present, and future, for example, do not reveal that there is anything like a general abstract concept of time in virtue of which some thing or event can be said to be in time or to have its own time. Archaic patterns of thought do not differ from our modern patterns in having different concepts, but in not having anything like concepts at all

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.

Computably Based Locally Compact Spaces

ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambda-calculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth's effectively given domains and Jung's strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the way-below relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott's domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.Comment: 70pp, LaTeX2e, uses diagrams.sty; Accepted for "Logical Methods in Computer Science" LMCS-2004-19; see http://www.cs.man.ac.uk/~pt/ASD for related papers. ACM-class: F.4.

Do Your Thoughts Matter? Because They Are Made Of Matter? An Exploration On The Mind Body Problem Through The Lenses Of Philosophy, Neuropsychology, and Quantum Physics

In the late 21th century, the concept of Quantum Consciousness arose, joining theories and concepts from the disciplines of Philosophy, Neuropsychology, and Quantum Physics. The Preface of Mind, Matter, and Quantum Mechanics (Henry P. Stapp. 1993) states, “Nature appears to be composed of two completely different kinds of things; rock-like things and idea-like things. The first is epitomized by an enduring rock, the second by a fleeting thought. A rock can be experienced by many of us together, while a thought seems to be long to one of us alone” (Stapp, 1993 p. vii). Stapp refers to the mind-body problem, not as a problem, but as a connection; the mind-body connection. Descartes believed the mind and body could be separate and act separately. Stapp embellishes Descartes philosophy of “I think, therefore I am” through Quantum Physics by connecting the mind to the body; calling it the mind-body connection. Stapp states that nature is composed of two different things, rock-like things and idea-like things. I believe that rock-like things and idea-like things do not differ, but are one and the same. This paper explores the philosophies of Descartes, the contents of a thought and the ways in which neuropsychology and Quantum Physics help illuminate the question: do thoughts contain matter

Synthetic Philosophy of Mathematics and Natural Sciences Conceptual analyses from a Grothendieckian Perspective

ISBN-13: 978-0692593974. Giuseppe Longo. Synthetic Philosophy of Mathematics and Natural Sciences, Conceptual analyses from a Grothendieckian Perspective, Reflections on “Synthetic Philosophy of Contemporary Mathematics” by F. Zalamea, Urbanomic (UK) and Sequence Press (USA), 2012. Invited Paper, in Speculations: Journal of Speculative Realism, Published: 12/12/2015, followed by an answer by F. Zalamea.International audienceZalamea’s book is as original as it is belated. It is indeed surprising, if we give it a moment’s thought, just how greatly behind schedule philosophical reflection on contemporary mathematics lags, especially considering the momentous changes that took place in the second half of the twentieth century. Zalamea compares this situation with that of the philosophy of physics: he mentions D’Espagnat’s work on quantum mechanics, but we could add several others who, in the last few decades, have elaborated an extremely timely philosophy of contemporary physics (see for example Bitbol 2000; Bitbol et al. 2009). As was the case in biology, philosophy – since Kant’s crucial observations in the Critique of Judgment, at least – has often “run ahead” of life sciences, exploring and opening up a space for reflections that are not derived from or integrated with its contemporary scientific practice. Some of these reflections are still very much auspicious today. And indeed, some philosophers today are saying something truly new about biology..