In his well known paper ‘Wittgenstein’s Builders’ Professor Rush Rhees has rightly criticized some appeals that Wittgenstein made to certain so-called ‘primitive languages’ while developing the early sections of the Philosophical Investigations. These appeals are taken by Wittgenstein to expose the shortcomings of an account given by Augustine at Confessions I, 8 of meaning and of learning language. I shall try first in this discussion to make it clear that at least some of Rhees’ criticisms and complaints are (...) made for very dubious reasons. Then I shall try to show that there are much better reasons for dissatisfaction, reasons which may tie up with recurrent muddles in Wittgenstein’s later thought. It will be contended that there is real point in construing several Wittgensteinian appeals to primitive languages as being somewhat vitiated by a systematic mistake. For, if I am right, a curious ambivalence arises in the text which results from confusion about what is largely false, because it is so inadequate, and what is wholly in error, because it is so fundamentally incoherent as to invite the label ‘Nonsense’. (shrink)

In the ideological construction of colonialism and, more widely, of any hierarchy of human communities, a crucial role is played by discourse on language. English nationalism and imperialism, in particular, developed extensive argumentations on language as an interpretation of the encounter with the other, on the basis of internal cultural developments that assigned to language the role of social discriminator. The paper investigates a strand of such argumentations during the period from the seventeenth to the nineteenth century: (...) the concept of “primitive” languages, described in a positive or, more often, in a negative light. The former arguments employ tones related to the idea of the “good savage” and stand in connection with narratives on the “language of Adam” and of the “Welsh Indians”, the latter uses a rhetoric extolling “progress” and “civilization” against the “immaturity” and “backwardness” of primitive languages, a perspective that was later to influence Darwinism. (shrink)

A theory of conceptual development must provide an account of the innate representational repertoire, must characterize how these initial representations differ from the adult state, and must provide an account of the processes that transform the initial into mature representations. In Carey, 2009 (The Origin of Concepts), I defend three theses: 1) the initial state includes rich conceptual representations, 2) nonetheless, there are radical discontinuities between early and later developing conceptual systems, 3) Quinean bootstrapping is one learning mechanism that underlies (...) the creation of new representational resources, enabling such discontinuity. I also claim that the theory of conceptual development developed in The Origin of Concepts addresses two of Fodor's challenges to cognitive science; namely, to show how learning could possibly lead to an increase in expressive power and to defeat Mad Dog Nativism, the thesis that all concepts lexicalized as mono-morphemic words are innate. A recent article by Georges Rey (Mind & Language, 29.2, 2014) argues that my responses to Fodor's challenges fail, because, he says, I fail to distinguish concept possession from manifestation and I do not confront Goodman's new riddle of induction. My response is to show that, and how, new primitives in a language of thought can be learned, that there are easy routes and hard ones to doing so, and that characterizing the learning mechanisms involved is the key to understanding both concept possession and constraints on induction. (shrink)

The purpose of this paper is to explore three alternative frameworks for understanding the nature of language and mentality, which accent syntactical, semantical, and pragmatical aspects of the phenomena with which they are concerned, respectively. Although the computational conception currently exerts considerable appeal, its defensibility appears to hinge upon an extremely implausible theory of the relation of form to content. Similarly, while the representational approach has much to recommend it, its range is essentially restricted to those units of (...) class='Hi'>language that can be understood in terms of undefined units. Thus, the only alternative among these three that can account for the meaning of primitive units of language is one emphasizing the basic role of skills, habits, and tendencies in relating signs and dispositions. (shrink)

Although Peano arithmetic is necessarily incomplete, Isaacson argued that it is in a sense conceptually complete: proving a statement of the language of PA that is independent of PA will require conceptual resources beyond those needed to understand PA. This paper gives a test of Isaacon’s thesis. Understanding PA requires understanding the functions of addition and multiplication. It is argued that grasping these primitive recursive functions involves grasping the double ancestral, a generalized version of the ancestral operator. Thus, (...) we can test Isaacon’s thesis by seeing whether when we phrase arithmetic in a context with the double ancestral operator, the result is conservative over PA. This is a stronger version of the test given by Smith, who argued that understanding the predicate “natural number” requires understanding the ancestral operator, but did not investigate what is required to understand the arithmetic functions. (shrink)

_ Source: _Volume 48, Issue 2, pp 244 - 264 Cassirer is important in 20th Century philosophy for the attention he gives to the fundamental relationship between myth and language. For Cassirer, myth is a non-subjective form of discourse wherein the origin of language coincides with both the human-divine encounter and the event of being itself. In this article, I trace the disagreement between Cassirer and Heidegger on the nature of the magical sign, which is at the heart (...) of mythical discourse. While Heidegger initially argues that this form of sign is structurally impossible on the basis of his accounts of signs and language in _Being and Time_, he later comes to recognize that he had not properly accounted for its possibility within his phenomenological deduction. I conclude by describing how Heidegger’s notion of poetry eventually comes into alignment with the way Cassirer describes the function of myth and the magical sign. (shrink)

A first-order language with a defined identity predicate is proposed whose apparatus for atomic predication is sensitive to grammatical categories of natural language. Subatomic natural deduction systems are defined for this naturalistic first-order language. These systems contain subatomic systems which govern the inferential relations which obtain between naturalistic atomic sentences and between their possibly composite components. As a main result it is shown that normal derivations in the defined systems enjoy the subexpression property which subsumes the subformula (...) property with respect to atomic and identity formulae as a special case. The systems admit a proof-theoretic semantics which does not only apply to logically compound but also to atomic and identity formulae—as well as to their components. The potential of the defined systems for a meticulous first-order analysis of natural inferences whose validity crucially depends on expressions of some of the aforementioned categories is demonstrated. (shrink)

Language Turned on Itself examines what happens when language becomes self-reflexive; when language is used to talk about language. Those who think, talk, and write about language are habitual users of various metalinguistic devices, but reliance on these devices begins early: kids are told, 'That's called a "rabbit"'. It's not implausible that a primitive capacity for the meta-linguistic kicks in at the beginning stages of language acquisition. But no matter when or how frequently (...) these devices are invoked, one thing is clear: they present theorists of language with a complex data pattern. Herman Cappelen and Ernest Lepore show that the study of these devices and patterns not only represents an interesting and neglected project in the philosophy of language, but also carries important consequences for other parts of philosophy. Part I is devoted to presenting data about various aspects of our metalinguistic practices. In Part II, the authors examine and reject the four leading metalinguistic theories, and offer a new account of our use of quotation in a variety of different contexts. But the primary goal of this book is not to promote one theory over another. Rather, it is to present a deeply puzzling set of problems and explain their significance. (shrink)

This paper rigorously examines the primitive foundations of economic reasoning through an original framework based on symbolic logic. Extending previous work, it formalizes economic conceptions (\(\mathbb{C}\)), symbols (\(s_i\)), and introduces a structured language (\(\mathcal{L_{\mathbb{C}}}\)) to define their formation and interpretation. Organized as a continuous chain of declarations and illustrations, the paper offers a concise, systematic approach to understanding the philosophy of economic reasoning through formal representations.

A short time ago, D. D. Lee wrote in Philosophy of Science: “Grammar contains in crystallized form the accumulated and accumulating experience, the Weltanschauung of a people.“ He thus called our attention once more to a theme which was much discussed during the 19th century but which has been in disrepute for some time in philosophical circles. It is a theme, however, which is not without merit. More than seven years of intensive study of primitive languages have convinced me (...) that the work started by Wilhelm von Humboldt deserves serious consideration even at present, and from the ethnological as well as from the philosophical point of view. (shrink)

In recent times Wittgenstein's work in logic has had an influence on other branches of philosophy. I am thinking, in particular, of social philosophy and the philosophy of religion. In these branches, Wittgenstein's followers have made much use of his notion of a language game. It has been argued, for example, that religion forms a language game of its own, having its own standards of reason, and is therefore not subject to criticism from outside. This argument has given (...) rise to controversy, some seeing it as a subtle attempt by the religious to evade criticism. I have come myself to feel that the notion of a language game has been put to uses with which Wittgenstein himself might not have agreed, or, if he had, would have been wrong to do so. In order to explain what I mean I should like to look closely at the opening section of Peter Winch's article ‘Understanding a Primitive Society’. (shrink)

Amodality is the thesis that concepts are not constituted by modal-specific representations. In this paper I assess the prospects for uncovering support for this claim in language by two different means. First, I examine the question of the amodal character of abstract concepts, but find it to be inconclusive pending a clearer account of the role of sensorimotor representations in language processing. Second, I evaluate the possibility of there being amodal primitive concepts in the context of Carey’s (...) account of representational primitives in core cognition. Despite their alleged iconicity, which seems to favor a modal view, I contend that there are grounds for regarding them as amodal in nature. I also challenge the discontinuity thesis that regards early primitives as being unrelated to mature, newly linguistically created primitives. (shrink)

In an article entitled "Is BAD a semantic primitive?", John Myhill suggested that the concept 'bad' should be removed from the list of semantic primitives put forward by Anna Wierzbicka and Cliff Goddard. Myhill argued that 'bad' is semantically decomposable, that there is no word in Biblical Hebrew that corresponds to the English word bad and, thus, no linguistic form that represents the primitive BAD in this language, and that 'bad' is dispensable in the semantic analysis and (...) can be replaced with other components without any loss or change of meaning. Discussing and illustrating some fundamental questions in the search for universal semantic primitives, the present author reconsiders these findings and finds a different answer to John Myhill's question. (shrink)

This paper explores the prospects for using the notion of a primitive normative attitude in responding to the sceptical argument about meaning developed in chapter 2 of Saul Kripke’s Wittgenstein on Rules and Private Language. It takes as its stalking-horse the response to Kripke’s Wittgenstein developed in a recent series of important works by Hannah Ginsborg. The paper concludes that Ginsborg’s attempted solution fails for a number of reasons: it depends on an inadequate response to Kripke’s Wittgenstein’s ‘finitude’ (...) objection to reductive dispositionalism; it erroneously rejects the idea that a speaker’s understanding of an expression guides her use; it threatens to collapse into either full-blown non-reductionism or reductive dispositionalism; and there is no motive for accepting it over forms of non-reductionism such as those developed by Barry Stroud and John McDowell. (shrink)

In setting up a formal system to specify a grammar formalism, the conventional (mathematical) wisdom is to start with primitives (basic primitive structures) as simple as possible, and then introduce various operations for constructing more complex structures. An alternate approach is to start with complex (more complicated) primitives, which directly capture some crucial linguistic properties and then introduce some general operations for composing these complex structures. These two approaches provide different domains of locality, i.e., domains over which various types (...) of linguistic dependencies can be specified. The latter approach, characterized as complicate locally, simplify globally (CLSG), pushes non‐local dependencies to become local, i.e., they arise in the basic primitive structures to start with.The CLSG approach has led to some new insights into syntactic description, semantic composition, language generation, statistical processing, and psycholinguistic phenomena, all these with possible relevance to the cognitive architecture of language. In this paper, we will describe these results in an introductory manner making use of the framework of lexicalized tree‐adjoining grammar (LTAG), a key example of the CLSG approach, thereby describing the interplay between formal analysis on the one hand and linguistic and processing issues on the other hand. (shrink)

Theories of collective intentions must distinguish genuinely collective intentions from coincidentally harmonized ones. Two apparently equally apt ways of doing so are the ‘neo-reductionism’ of Bacharach (2006) and Gold and Sugden (2007a) and the ‘non-reductionism’ of Searle (1990, 1995). Here, we present findings from theoretical linguistics that show that we is not a cognitive primitive, but is composed of notions of I and grouphood. The ramifications of this finding on the structure both of grammatical and lexical systems suggests that (...) an understanding of collective intentionality does not require a primitive we-intention, but the notion of grouphood implicit in team reasoning, coupled with the individual concept I. This, we argue, supports neo-reductionism but poses difficulties for non-reductionism. (shrink)

Starting from six kinds of periodicity of words we define six sets of words which are primitive in different senses and we investigate their relationships. We show that only three of the sets are external Marcus contextual languages with choice but none of them is an external contextual language without choice or an internal contextual language. For the time complexity of deciding any of our sets by one-tape Turing machines, n2 is a lower bound and this is (...) optimal in two cases. The notions of roots and degrees of words and languages, which are strongly connected to periodicity and primitivity, are also considered, and we show that there can be an arbitrarily large gap between the complexity of a language and that of its roots. (shrink)

Mathematics is a critical part of much scientific research. Physics in particular weaves math extensively into its instruction beginning in high school. Despite much research on the learning of both physics and math, the problem of how to effectively include math in physics in a way that reaches most students remains unsolved. In this paper, we suggest that a fundamental issue has received insufficient exploration: the fact that in science, we don’t just use math, we make meaning with it in (...) a different way than mathematicians do. In this reflective essay, we explore math as a language and consider the language of math in physics through the lens of cognitive linguistics. We begin by offering a number of examples that show how the use of math in physics differs from the use of math as typically found in math classes. We then explore basic concepts in cognitive semantics to show how humans make meaning with language in general. The critical elements are the roles of embodied cognition and interpretation in context. Then, we show how a theoretical framework commonly used in physics education research, resources, is coherent with and extends the ideas of cognitive semantics by connecting embodiment to phenomenological primitives and contextual interpretation to the dynamics of meaning-making with conceptual resources, epistemological resources, and affect. We present these ideas with illustrative case studies of students working on physics problems with math and demonstrate the dynamical nature of student reasoning with math in physics. We conclude with some thoughts about the implications for instruction. (shrink)

Preference, according to many theories of human behaviour, is a very important phenomenon. It is therefore some what surprising that philosophers of mind pay so little attention to it. One question about preference concerns its variety. Is preference always preference for one option or state of affairs rather than another? Or is there also, as ordinary language suggests, object-preference – preferences for one person rather than another, for one country rather than another, for one value rather than another? Another (...) question or rather group of questions concerns the nature of preference. Is it a mental state, disposition, act or episode, a theoretical construct, a purely behavioural phenomenon? If it is a mental state or act, is it an intellectual, affective or a conative phenomenon? If it is an affective phenomenon, does it enjoy a positive or negative “valence”? Is preference to be understood as a relation between a person’s attitudes or is it a primitive phenomenon? (shrink)

Critique of Norman Malcolm’s ‘Wittgenstein: The Relation of Language to Instinctive Behaviour’. Rhees points out the danger of thinking of instinctive reactions as the foundations of language. The reactions are primitive, Rhees argues, in relation to primitive means of communication, ie, in relation to people who already speak a language. What we need to emphasise is the way in which primitive reactions are taken up in our ways of thinking and forms of life. That (...) cannot be reduced to something ‘instinctive’. (shrink)

We correct a mistake in the paper “Generalized periodicity and primitivity for words” [4] and justify the existence of regular languages all of whose roots are not even context-sensitive.

This book argues that a primitive society is formed on the basis of kinship ties while a civilized society is formed on the basis of language. Yu presents a new theory about the importance of language in the growth of the state.

The square of a language L is the set of all words pp where p ∈ L. The square of a regular language may be regular too or context-free or none of both. We give characterizations for each of these cases and show that it is decidable whether a regular language has one of these properties.

As computers became multi-component systems in the 1950s, handling the speed differentials efficiently was identified as a major challenge. The desire for better understanding and control of ‘concurrency’ spread into hardware, software, and formalism. This paper examines the way in which the problem emerged and was handled across various computing cultures from 1955 to 1985. In the machinic culture of the late 1950s, system programs called ‘monitors’ were used for directly managing synchronisation. Attempts to reframe synchronisation in the subsequent algorithmic (...) culture pushed the problem to a higher level of abstraction; Dijkstra’s semaphores were a reaction to the algorithms’ complexity. Towards the end of the 1960s, the culture of ‘structured programming’ created a milieu in which Dijkstra, Hoare, and Brinch Hansen (among others) aimed for a concurrency primitive which embodied the new view of programming. Via conditional critical regions and Dijkstra’s ‘secretaries’, the co-produced ‘monitor’ appeared to provide the desired encapsulation. The construct received embodiment in a few programming languages; this paper ends by considering Modula and Concurrent Pascal. (shrink)

The logics of the modal operators and of the quantifiers show striking analogies. The analogies are so extensive that, when a special class of entities (possible worlds) is postulated, natural and non-arbitrary translation procedures can be defined from the language with the modal operators into a purely quantificational one, under which the necessity and possibility operators translate into universal and existential quantifiers. In view of this I would be willing to classify the modal operators as ‘disguised’ quantifiers, and I (...) think that wholehearted acceptance of modal language should be considered to carry ontological commitment to something like possible worldsConsidered as two languages for describing the same subject matter, modal and purely quantificational languages show interesting differences. The operator variables of the purely quantificational languages give them more power than the modal languages, but at least some of the functions performed by the apparatus of operator variables are also performed, in a more primitive and less versatile way, by actuality operators in modal languages.A final note. Quine has written much on the inter-relations of quantifiers, identity, and the concept of existence. These, he holds, form a tightly knit conceptual system which has been evolved to a high point of perfection, but which might conceivably change yet further.29 He has also dropped hints about the possibility of a simpler, primitive or defective version of the system, in which the quantifiers are not backed up in their accustomed way by the concept of identity. He has dubbed the resulting concept a ‘pre-individuative’ concept of existence, or a concept of ‘entity without identity.’ What would a pre-individuative concept of existence be like? Quine has sometimes suggested that one might be embodied in the use of mass nouns, but the identity concept is used in connection with stuff as well as with things: “is that the same coffee that was in the cup last night?” I would submit that modality provides a better case. In view of the comparative weakness of modal languages, compared to the explicitly quantificational ones Quine takes as canonical, there is surely a sense in which the concept of existence embodied in that disguised existential quantifier, the possibility operator, is a defective one. And as we have seen, one of the differences between modal operators and explicit quantifiers is that modal operators cannot be joined with the identity predicate in the way quantifiers with operator variables can. Surely, then, there is a sense in which ordinary speech, as opposed to the metaphysical theorizing of a Leibniz or a David Lewis, conceives of possible worlds as entities without identity. (shrink)

In reading Wittgenstein one can, and for the most part perhaps should, treat the expression ‘language-game’ as a term of art, a more or less arbitrarily chosen item of terminology meaning something like ‘an actual or possible way of using words’. It would then be a fairly routine task to work out answers to such questions as what features of the ways a word is used are emphasized by this term of art, what philosophical purposes are served by the (...) description of primitive language-games or of variations on actual language-games, or in exactly what way those purposes are supposed to be served. (shrink)

We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain.

A topological constraint language is a formal language whose variables range over certain subsets of topological spaces, and whose nonlogical primitives are interpreted as topological relations and functions taking these subsets as arguments. Thus, topological constraint languages typically allow us to make assertions such as “region V1 touches the boundary of region V2”, “region V3 is connected” or “region V4 is a proper part of the closure of region V5”. A formula f in a topological constraint language (...) is said to be satisfiable if there exists an assignment to its variables of regions from some topological space under which φ is made true. This paper introduces a topological constraint language which, in addition to the usual mechanisms for expressing Boolean combinations of regions and their topological closures, includes primitives for bounding the number of components of a region. We call this language T CC, a rough acronym for “topological constraint language with component counting”. Our main result is that the problem of determining the satisfiability of a T CC-formula is NEXPTIME-complete. (shrink)

This paper concerns the formal semantic analysis of imperative sentences. It is argued that such an analysis cannot be deferred to the semantics of propositions, under any of the three commonly adopted strategies: the performative analysis, the sentence radical approach to propositions, and the (nondeclarative) mood-as-operator approach. Whereas the first two are conceptually problematic, the third faces empirical problems: various complex imperatives should be analysed in terms of semantic operators over simple imperatives. One particularly striking case is the Dutch pluperfect (...) imperative. It is argued that this construction should be analysed as a genuine counterfactual imperative. On the constructive side, in the last part of the paper a formal semantic analysis of imperatives is presented, in the framework of Update Semantics. On this analysis, imperatives are sui generis semantic entities, on a par with propositions. The analysis also includes an account of the counterfactual imperatives. (shrink)

Wittgenstein's anti‐psychologism had induced him not to investigate the concepts that informed the psychological presuppositions of the Tractatus; only the essence of any possible symbolism seemed relevant to his concerns. The private language arguments have shown the incoherence of the idea that the foundations of language lie in private mental objects that constitute, or explain, the meanings of primitive indefinables of language. For language is 'alive' for one only in so far as one thinks or (...) understands the senses attached to sentences. Thinking is a psychological process of coming into intellectual contact with, grasping, these lexical souls. One deep root of our troubles lies in a supposition which Wittgenstein criticized in the context of his discussion of the name/object model as misapplied to the concept of pain and its relation to pain‐behaviour. We must remind ourselves how specialized is the language‐game of saying what we think, of communicating our thoughts, of telling them to others. (shrink)

This book will attempt to achieve a constructive and positive correla tion between mythic-symbolic language and philosophical anthropolo gy. It is intended as a reflection on the philosophical accomplishment of Paul Ricoeur. The term mythic-symbolic language in this context means the language of the multivalent symbol given in the myth with its psychological and poetic counterparts. The term symbol is not con ceived as an abstract sign as it is used in symbolic logic, but rather as a (...) concrete phenomenon - religious, psychological, and poetic. The task inherent in this correlation is monumental when one considers the dual dilemma of problematic and possibility which is at its heart. The prob lematic arises out of the apparent difficulty presented by the so-called challenge of modernity which seems to require the elimination of my thic-symbolic language as an intelligible mode of communication. Mythic-symbolic language is sometimes eliminated because in a world molded by abstract conceptualizations conceptUalizations of science, such a language is thought to be unintelligible. The claim is that its "primitive" explana tions have been transcended by our modernity. Others believe that the problem of mythic-symbolic language is the problem of the myth. If the mythic forms of language could be eliminated, the truth of such language could be preserved through its translation into an intelligible mode of discourse. The problematic is heightened further by the relation of consider ations of language to philosophical anthropology. (shrink)

Quine’s way of dealing with the semantical paradoxes (Ways of Paradox, pp. 9-10) is criticized. The criticism is based on three premises: (1) no learnable language has infinitely many semantical primitives; (2) any language of which Quine’s theory is true has infinitely many semantical primitives; (3) English is a learnable language. The conclusion drawn is that Quine’s theory is not true of English.

The use of the primitive notion of mereological fusion (also known as composition and sum) has been considered by various philosophers and logicians, including Aristotle, G. Leibniz, S. Leśniewski, K. Fine, J. Ketland, T. Schindler, and S. Kleishmid. The problem of finding an axiomatization of Classical Mereology with primitive fusion, instead of the primitive notion of being a part, is quite old and was formally considered by C. Lejewski. Lejewski somehow axiomatized classical mereology using primitive fusion (...) (1962, and also later in 1984). However, his mereology is formulated in a very rich nonclassical language of Leśniewski’s ontology with the essential use of quantification over function symbols, including function symbols “part of” and “sum of.” Lejewski’s approach is not expressible in modern mereologies, and we do not use it in any way. We prove that classical mereology is axiomatizable using primitive mereological fusion within the framework of two-sorted logic. The theory presented here is the first contemporary axiomatization of classical mereology with primitive mereological fusion. (shrink)

Ludwig Wittgenstein imagines a variety of eccentric social practices—like a tribe trained “to give no expression of feeling of any kind”. But he also speaks of “the common behavior of mankind” that is rooted in “natural/primitive reactions”. This emphasis on the uniformities of human behavior raises questions about the plausibility of some of his imagined language games. Indeed, it suggests the claim of evolutionary psychologists that there are biologically based human universals that shape social practices. But in contrast (...) to E.O. Wilson's belief that “genes hold culture on a leash”, Wittgenstein sees culture as a mediator—rather than a conduit—of “natural reactions”. This suggests that social science can incorporate the claims of evolutionary psychology without scanting the centrality of culture in action and allows that nature can be overwhelmed by culture. (shrink)

We begin with the following quotation from Karp [1964]: My interest in infinitary logic dates back to a February day in 1956 when I remarked to my thesis supervisor, Professor Leon Henkin, that a particularly vexing problem would be so simple if only I could write a formula which would say x = 0 or x = 1 or x = 2 etc. To my surprise, he replied, "Well, go ahead." Traditionally, expressions in formal systems have been regarded as signifying (...) finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas— such as that in the above quotation—would be naturally identified as infinite sets. A "language" of this kind is called an infinitary language: in this article we discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion we shall see that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. In §1 we lay down the basic syntax and semantics of infinitary languages and demonstrate their expressive power by means of examples. §2 is devoted to those infinitary languages which permit only finite quantifier sequences: these languages turn out to be relatively well-behaved. In §3 we discuss the compactness problem for infinitary languages and its.. (shrink)

A notation for the language of physics is given, and a system of axioms constructed. It is argued that from the standpoint of a 'realistic' ontology our method is preferable to Carnap's 'coordinate languages.' The primitive ideas are the part-whole relation μ and the set H of coordinate systems. Only such statements are intended in the axioms as are non-controversial; i.e. no open cosmological questions are prejudged.

It has been assumed by some writers that Wittgenstein’s talk of primitive reactions amounts to a theory of concept formation out of instinctive behaviour. Others have argued that Wittgenstein is thinking of reactions within language‐games, which therefore belong to its structure not its origins. The author concurs, but argues that Wittgenstein also has in mind that it belongs to the grammars of certain concepts that the language‐games in which they lie are themselves supplementary to natural behavioural forms. (...) This provides a context in which the notion of pre‐linguistic behaviour may be employed legitimately in philosophy whilst avoiding unjustified theorising. (shrink)

In this paper we provide quantifier-free, constructive axiomatizations for several fragments of plane Euclidean geometry over Euclidean fields, such that each axiom contains at most 4 variables. The languages in which they are expressed contain only at most ternary operations. In some precisely defined sense these axiomatizations are the simplest possible.

In this paper we provide a quantifier-free constructive axiomatization for Euclidean planes in a first-order language with only ternary operation symbols and three constant symbols . We also determine the algorithmic theories of some ‘naturally occurring’ plane geometries.