Propositional Functions Research Articles

The Axiom of Reducibility

The axiom of reducibility plays an important role in the logic of Principia Mathematica, but has generally been condemned as an ad hoc non-logical axiom which was added simply because the ramified type theory without it would not yield all the required theorems. In this paper I examine the status of the axiom of reducibility. Whether the axiom can plausibly be included as a logical axiom will depend in no small part on the understanding of propositional functions. If we understand propositional functions as constructions of the mind, it is clear that the axiom is clearly not a logical axiom and in fact makes an implausible claim. I look at two other ways of understanding propositional functions, a nominalist interpretation along the lines of Landini and a realist interpretation along the lines of Linsky and Mares. I argue that while on either of these interpretations it is not easy to see the axiom as a non-logical claim about the world, there are also appear to be difficulties in accepting it as a purely logical axiom.

The Axiom of Reducibility

The axiom of reducibility plays an important role in the logic of Principia Mathematica, but has generally been condemned as an ad hoc non-logical axiom which was added simply because the ramified type theory without it would not yield all the required theorems. In this paper I examine the status of the axiom of reducibility. Whether the axiom can plausibly be included as a logical axiom will depend in no small part on the understanding of propositional functions. If we understand propositional functions as constructions of the mind, it is clear that the axiom is clearly not a logical axiom and in fact makes an implausible claim. I look at two other ways of understanding propositional functions, a nominalist interpretation along the lines of Landini and a realist interpretation along the lines of Linsky and Mares. I argue that while on either of these interpretations it is not easy to see the axiom as a non-logical claim about the world, there are also appear to be difficulties in accepting it as a purely logical axiom.

“Predication, Things, and Kinds in Aristotle’s Metaphysics”

AbstractWhat in Aristotle corresponds, in whole or (more likely) in part, to our contemporary notion of predication? This paper sketches counterparts in Aristotle’s text to our theories of expression and of truth, and on this basis inquires into his treatment of sentences assigning an individual to its kinds. In some recent accounts, the Metaphysics offers a fresh look at such sentences in terms of matter and form, in contrast to the simpler theory on offer in the Categories. I argue that the Metaphysics initiates no change in this regard over the Categories. The point that form is (metaphysically) predicated of matter is a contribution, not to the account of statement predication, but to the analysis of compound material substances. Otherwise put, in our terms Aristotelian form is not - in particular, is not also - a propositional function, but a function from matter to compound material substances.

Towards many-valued/fuzzy interpretation of quantum mechanics

In this paper, we propose a new interpretation of quantum mechanics according to which Born probabilities are interpreted as many-valued truth values of statements concerning results of not-yet-performed experiments on quantum objects, or equivalently [0,1]-valued degrees to which quantum objects possess all their properties prior to measurements. The interpretation is based on mathematical results already published by the author, according to which any Birkhoff–von Neumann quantum logic can be isomorphically represented as a family of fuzzy sets or, equivalently, many-valued propositional functions.

Functions or Propositional Functions?

February 19, 2011 (11:48 am) E:\CPBR\RUSSJOUR\TYPE3002\russell 30,2 040 red.wpd Reviews 161 7 In, respectively, PaciWsm in Britain and Semi-Detached Idealists: the British Peace Movement and International Relations (Oxford: Oxford U. P., 2000). 8 See Monk 2: Chap. 13. FUNCTIONS OR PROPOSITIONAL FUNCTIONS? Alexander Paul Bozzo Philosophy / Marquette U. Milwaukee, wi 53233, usa alexander.bozzo@marquette.edu Michael Potter and Tom Ricketts, eds. The Cambridge Companion to Frege. Cambridge, uk: Cambridge U. P., 2010. Pp. vii, 630. isbn 978-0-521-62479-4 (pb). £57.00; £21.99 (pb). us$38.99 (pb). February 19, 2011 (11:48 am) E:\CPBR\RUSSJOUR\TYPE3002\russell 30,2 040 red.wpd 162 Reviews 1 Frege: Philosophy of Mathematics (Cambridge, ma: Harvard U. P., 1991), p. 111. The paragraph contains an expression of Frege’s Context Principle and the need for Hume’s Principle as a criterion of identity in the deWnition of number. 2 There are in fact Wve stated goals. The others are: (iii) to serve as a reference work for students and non-specialists, (iv) to provide a route into the study of Frege historically and to any relevant contemporary concerns in the philosophies of logic, language , mathematics, and mind, and (v) to provide, for new readers, the most convenient detailed guide to Frege currently available. Gottlob Frege is widely recognized as one of the chief progenitors of mathematical logic and philosophy of languagez—zindeed, the importance attributed to Frege’s innovations is such that Michael Dummett, when reXecting on§62 of Frege’s Die Grundlagen der Arithmetik, feels compelled to remark that the paragraph therein is “arguably the most pregnant philosophical paragraph ever written.”1 Whether or not one agrees with Dummett on this point, the magnitude of Frege’s inXuence on the current philosophical landscape is incontestable. Frege revolutionized the then dominant Aristotelian conception of logic, introducing a formal language now recognized as the predicate calculus. Central to this end were Frege’s insights on quantiWcation, the notation that expressed it, the logicist project, and the extension of mathematical notions like function and argument to natural language. The long-awaited Cambridge Companion to Frege is a compendium of Fregean scholarship that rigorously explores these and similar topics; editors Thomas Ricketts and Michael Potter have compiled a comprehensive collection of fourteen essays that individually provide focused appraisals of a number of Frege’s most substantial insights. On the advice of this journal’s editor, I have limited my review to the connections (or rather dissimilarities ) that exist between Frege and Russell, relying exclusively on Peter Hylton ’s contribution to the collection, “Frege and Russell”. Following a brief consideration of the volume as a whole, I move immediately into a detailed explication of Hylton’s analysis. The volume includes among its goals the following two aims: (i) to provide, for advanced students and specialists, a conspectus of recent interpretation of Frege, and (ii) to dispel the intimidation felt by readers who confront a diUcult thinker.2 I single out these ambitions because they most clearly represent the volume’s strengths and shortcomings. Enthusiasts of Frege will not be disappointed by the multitude of rigorously and technically written essays, essays that contribute substantially and originally to the collective understanding of Frege and his work. For instance, Thomas Ricketts provides a thoughtful and thorough analysis—an analysis that moves well beyond mere explication—of Frege’s concept/object distinction, in particular its relevance for Frege’s Context Principle and logicism more generally. Similarly, Richard Heck addresses a number of central yet largely unexplored issues pertaining to the overall signiWcance of February 19, 2011 (11:48 am) E:\CPBR\RUSSJOUR\TYPE3002\russell 30,2 040 red.wpd Reviews 163 3 I strongly encourage that non-specialists augment The Cambridge Companion to Frege with Joan Weiner’s Frege Explained: from Arithmetic to Analytic Philosophy (Chicago : Open Court, 2004), one of the best available introductory texts on Frege. 4 The essay in fact was available as early as 2005 in Hylton’s Propositions, Functions, and Analysis: Selected Essays on Russell’s Philosophy (New York, ny: Oxford U. P., 2005). The editors of The Cambridge Companion to Frege partially...

Negation, questions, and structure building in a homesign system

Deaf children whose hearing losses are so severe that they cannot acquire spoken language, and whose hearing parents have not exposed them to sign language, use gestures called homesigns to communicate. Homesigns have been shown to contain many of the properties of natural languages. Here we ask whether homesign has structure building devices for negation and questions. We identify two meanings (negation, question) that correspond semantically to propositional functions, that is, to functions that apply to a sentence (whose semantic value is a proposition, ϕ) and yield another proposition that is more complex ( ⌝ ϕ for negation; ?ϕ for question). Combining ϕ with ⌝ or ? thus involves sentence modification. We propose that these negative and question functions are structure building operators, and we support this claim with data from an American homesigner. We show that: (a) each meaning is marked by a particular form in the child’s gesture system (side-to-side headshake for negation, manual flip for question); (b) the two markers occupy systematic, and different, positions at the periphery of the gesture sentences (headshake at the beginning, flip at the end); and (c) the flip is extended from questions to other uses associated with the wh-form (exclamatives, referential expressions of location) and thus functions like a category in natural languages. If what we see in homesign is a language creation process ( Goldin-Meadow, 2003), and if negation and question formation involve sentential modification, then our analysis implies that homesign has at least this minimal sentential syntax. Our findings thus contribute to ongoing debates about properties that are fundamental to language and language learning.

THE FUNCTIONS OF RUSSELL’S NO CLASS THEORY

Certain commentators on Russell’s “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions.” These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to meet the demands placed on classes by mathematics. I show that Russell did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extralinguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language.

Functional Monadic Bounded Algebras

The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator $${\exists}$$ reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions. We show that FMBA is characterised by the disjunction of the equations $${\exists}E = 1$$ and $${{\exists}E = 0}$$ . We also define a weaker notion of algebra, and show that every member of MBA is isomorphic to a relatively functional one.

Monadic Bounded Algebras

We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of MBA's as powerset algebras of certain directed graphs with a set of points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly $${{2^{{{3.2}^r}.2^{2^r - 1}}}}$$ elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r.

COMO NEGAR UM NOME

In the Tractatus Logico-Philosophicus, a name is always a propositional function. Wittgenstein makes a radical shift in the Fregean opposition between saturated and unsaturated entities. Any sentential component which is not itself a sentence is unsaturated. The proposition is therefore a synthesis of propositional functions. The name is just a limit case of propositional function, and as such it can be negated.

Word Order, Marking, and a Two-Dimensional Classification of Parts-of-Speech System Types

Within the framework of Hengeveld's approach to parts-of-speech systems, this article shows that Vulanović's two-dimensional classification of main parts-of-speech system types applies both to the types presented by Hengeveld, Rijkhoff, and Siewierska in their 2004 paper and to the new types recently reported by Hengeveld and van Lier. Hengeveld and van Lier's principles for explaining the attested main parts-of-speech system types are also discussed and a set of simpler and more accurate principles is proposed. Furthermore, the two-dimensional classification is extended to intermediate parts-of-speech system types, which enables a linguistic data analysis of the kind done in Vulanović and Köhler's 2009 article. This analysis shows that the proportion of languages that use fixed word order and grammatical markers to facilitate disambiguation between different propositional functions depends on the type of the parts-of-speech system. The dependence is expressed by a three-dimensional generalization of the sigmoid.

PASSION AND SIMILARITY: THE CLINICAL APPLICATION OF MATTE BLANCO'S IDEAS

I will focus on Matte Blanco's hypothesis of emotion and the unconscious as being substantially the same. This concept allows us to apply the same instruments to understanding emotional reactions as we use with manifestations of the unconscious, such as dreams and symptoms. The tendency towards undifferentiation in the deepest strata of the mind is capable of mobilizing dramatic experiences of inappropriate amplification of the concept of identity. When a simple relation of similarity is translated into equivalence, a symmetrization arises and different people or situations on a basis of a single common characteristic are treated as if identical and interchangeable. In my clinical example, Franco had a panic attack with florid somatic symptoms on the same day he felt that Sara, a woman in whom he had become interested, was attracted to someone else. The intensity of feelings had grown exponentially from the moment the suspicion had arisen that he might have to ‘share her with another’. The infinite quality of his emotion suggested that his jealousy had constructed Sara as part of a broader class of women whom, for whatever reason, he felt to be indispensable to his survival. Only the understanding of this propositional function, through the decrease of the incandescence of emotions, allowed Sara, considered as a class, to be seen as an individual person.

Making Sense of Truth-Makers

This essay argues that propositions are made true by facts. A proposition is the sense expressed by a statement (sentence token used to make a truth claim). Facts are positive or negative constitutive properties of the domain of discourse (usually the actual world). The presence of horses is a positive constitutive property of the world; the absence of unicorns is a negative one. This notion of constitutive properties accords well with the Hume-Kant claim that existence is not a property of any individual said to exist. While Frege held existence to be a property of concepts and Russell held it to be a property of propositional functions, our view sees existence as a property of a domain of discourse. To say that Native Dancer exists is simply to say that the world is characterized by the presence of Native Dancer; to say that Pegasus does not exist is to say the world is characterized by the absence of Pegasus. Such properties of presence and absence are facts. Facts make true propositions true; nothing makes false propositions false (they simply fail to be made true). Facts are not items in the world; they are (constitutive) properties of the world.

Word Order, Marking, and Parts-of-Speech Systems

Parts-of-speech systems, as defined by Hengeveld et al. in their 2004 article on parts-of-speech systems and word order, form the framework for this article, in which we investigate how the distribution of languages with respect to certain linguistic features is related to the number of propositional functions and the number of lexeme classes. The linguistic features are the presence or absence of fixed word order and markers, which may disambiguate between different propositional functions. We show that the relation can be modelled by a three-dimensional generalization of the sigmoid, for which we provide theoretical justification.

The Complexity of Generalized Satisfiability for Linear Temporal Logic

In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post's lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P.

On Whitehead’s Recurrent Themes and Consistent Style

Abstract The way in which Stengers thinks "with" rather than "about" White-head is explained through an examination of the transformational function of speculative propositions. The article then investigates the significance o recurrent theme of "coherence" throughout Whitehead’s philosophical and socio-organizational writings. This theme guides and unifies his thought through various conceptual adventures.

Russell And Frege On The Logic of Functions

I compare Russell’s theory of mathematical functions, the “descriptive functions” from Principia Mathematica ∗30, with Frege’s well known account of functions as “unsaturated” entities. Russell analyses functional terms with propositional functions and the theory of definite descriptions. This is the primary technical role of the theory of descriptions in P M . In Principles of Mathematics and some unpublished writings from before 1905, Russell offered explicit criticisms of Frege’s account of functions. Consequenly, the theory of descriptions in “On Denoting” can be seen as a crucial part of Russell’s larger logicist reduction of mathematics,aswellasanexcursionintothetheoryof reference.

On Whitehead’s Recurrent Themes and Consistent Style

The way in which Stengers thinks “with” rather than “about” Whitehead is explained through an examination of the transformational function of speculative propositions. The article then investigates the significance of the recurrent theme of “coherence” throughout Whitehead’s philosophical and socio-organizational writings. This theme guides and unifies his thought through its various conceptual adventures.

The Early Life Of Russell’s Notion Of A Propositional Function

In this paper I describe the birth of Russell’s notion of a propositional function on 3 May 1902 (precise time of day unknown) and its immediate context and implications. In particular, I consider its significance in relation to the development of his views on analysis.

Entzweiung als Zeitalter der Bildung

Derrida’s discovery of „supplementary“ logic as the basis of Rousseau’s thinking is assessed in the historical contexts both of modern philosophy and metaphysics between Leibniz and Hegel. Rousseau’s supplementary proposition proves not to be identical with the modern propositional function, but with the „infinite proposition“ („das unendliche Urteil“) in Hegels Science of Logic. It is shown that this proposition structures Hegel’s concept of disunion („Entzweiung“) in the productive overcoming of Rousseau’s anthropological critique of representation as developed in the Phenomenology of Spirit.