This paper discusses certain problems arising within the treatment of the senses of functions in Alonzo Church's Logic of Sense and Denotation. Church understands such senses themselves to be "sense-functions," functions from sense to sense. However, the conditions he lays out under which a sense-function is to be regarded as a sense presenting another function as denotation allow for certain undesirable results given certain unusual or "deviant" sense-functions. Certain absurdities (...) result, e.g., an argument can be found for equating any two senses of the same type. An alternative treatment of the senses of functions is discussed, and is thought to do better justice to Frege's original theory. (shrink)

This book aims to develop certain aspects of Gottlob Frege’s theory of meaning, especially those relevant to intensional logic. It offers a new interpretation of the nature of senses, and attempts to devise a logical calculus for the theory of sense and reference that captures as closely as possible the views of the historical Frege. (The approach is contrasted with the less historically-minded Logic of Sense and Denotation of Alonzo Church.) Comparisons of Frege’s theory with (...) those of Russell and others are given. It is in the end shown that developing Frege’s theory in these ways reveals serious problems hitherto largely unnoticed, including those possibly rendering a Fregean intensional logic inconsistent even if his naïve class theory is excluded. (shrink)

In the Grundlagen Frege says that "line a is parallel to line b" differs from "the direction of a = the direction of b" in that "we carve up the content in a way different from the original way". It seems that such recarving is crucial to Frege's logicist program of defining numbers, but it also seems incompatible with his later theory of sense and reference. I formulate a restriction on recarving, in particular, that no names may be introduced (...) that introduce new possibilities of reference failure, which is observed by Frege's examples. This restriction discriminates between various relatives of the "slingshot" argument which rely on a step of recarving. I offer an argument for the restriction based on Fregean principles, which I formalize in Church's "Logic of Sense and Denotation", and briefly discuss various axioms of his "Alternative (0)" which are incompatible with recarving. (shrink)

There is an inconsistency among claims made (or apparently made) in separate articles by Alonzo Church concerning Frege's distinction between sense and denotation taken together with plausible assertions by Frege concerning his notion of ungerade Sinn-i.e., the sense that an expression allegedly takes on in positions in which it has ungerade Bedeutung, denoting its own customary sense. As with any inconsistency, the difficulty can be avoided by relinquishing one of the joint assumptions from which contradiction may (...) be derived. Yet what seems the most plausible resolution, in light of Church's own arguments, involves abandoning the theory of sense and denotation, in the form in which it has been staunchly advocated by both Frege and Church. That theory has come under sustained criticism from several quarters. Whereas the present difficulty supplements the case against the theory, it is unlike more familiar criticisms, since it is not framed from the perspective of an alternative theory (or picture) but is based instead on an internal conflict, and therefore has special force. (shrink)

This volume began as a remembrance of Alonzo Church while he was still with us and is now finally complete. It contains papers by many well-known scholars, most of whom have been directly influenced by Church's own work. Often the emphasis is on foundational issues in logic, mathematics, computation, and philosophy - as was the case with Church's contributions, now universally recognized as having been of profound fundamental significance in those areas. The volume will be of interest (...) to logicians, computer scientists, philosophers, and linguists. The contributions concern classical first-order logic, higher-order logic, non-classical theories of implication, set theories with universal sets, the logical and semantical paradoxes, the lambda-calculus, especially as it is used in computation, philosophical issues about meaning and ontology in the abstract sciences and in natural language, and much else. The material will be accessible to specialists in these areas and to advanced graduate students in the respective fields. (shrink)

This book is in the Studies in Philosophy: Outstanding Dissertations series. Its central theme is that Frege’s concept-notation is inadequate because it does not formalize his semantic theory after the introduction of the sense-reference distinction in 1891. This failing, according to Klement, opens Frege up to a number of philosophical and logical challenges that can be met only by completing the project of showing “how Frege’s mature semantic views would be incorporated into his mature logical system”, a project which, (...) Klement plausibly argues, Frege would have endorsed. This involves: allowing the concept-notation to represent explicitly what is expressed in natural language with indirect constructions such as “said that...” or “believes that... ”; allowing principles to be formulated over this notion which capture inferences involving indirect constructions such as: The Morning Star is a planet, Gottlob believes the Morning Star is a planet, so, Gottlob believes something true; being able to respond to certain objections to Frege’s views, some first bruited by Russell and in later correspondence, others involving the identity conditions on intensional entities and “quantifying in” ; making the concept-notation reflect Frege’s ontology; and evaluating the consistency of Frege’s ontology. The chapters of this book are devoted to fulfilling this project “based upon a careful reading of Frege’s own works”, and their contents are mostly self-explanatory. Chapter 2 sets out the formal apparatus of the Grundgeseze, even the bits Frege left out. Chapter 3 is the most philosophical. Senses are construed as interpersonally accessible information uniquely true of the references they determine—senses are objects and functions in the “third realm.” Singular terms are associated with an impossibly rich hierarchy of senses as objects. Functors, including predicates, are associated with senses as functions. Sentences are associated with thoughts as objects, composed of constituent sense-functions, and in many cases, also sense-objects. Robust compositionality at the level of sense raises the question of the nature of the relation between constituents and the whole—is it functional, or not? Compositionality potentially conflicts with a number of other claims, and Klement’s conclusion is reasonable: “the context principle, the priority thesis, and the possibility of there existing multiple decompositions for the same Gedanke, do not pose any great difficulty for the compositionality principle or the understanding of Gedanke as having determinate inner structures”. Chapter 4 sets out Church’s original system and investigates reformulations suggested by Church himself, Kaplan, and others. Chapter 5 presents Klement’s formalization of an intensional logic of sense and reference, based on the results of chapters 2 and 4, and sufficient for indirect speech and quantifying-in, but containing a contradiction. Chapter 6 takes up arguments of early Russell, Searle, and Bergmann. Chapter 7 addresses strategies for repairing the inconsistencies of chapter 5, but in a Fregean spirit: dropping some realms of entities, restricting their number, or modifying our conception of their nature. (shrink)

This is a thesis in formal semantics. It consists of two parts corresponding to the distinction, due to Richard Montague, between universal grammar and specific semantic theories. The first part concerns universal grammar and is intended to provide a precise and unified conceptual framework within which different theories of formal semantics can be represented and compared. ;The second part of the thesis is concerned with intensional logic, i.e., with the logical analysis of discourse involving so called oblique contexts. These (...) contexts are characterized by the apparent failure of certain classically valid principles of substitutivity. Two kinds of reactions among logicians to the appearance of oblique contexts in natural languages are distinguished: (1) the intensional point of view, according to which oblique contexts are seen as genuine counterexamples to classical logic and to the related semantical principle of extensionality: the principle that the denotation of an expression is a function of the denotations of its parts; and (2) the extensional point of view, according to which the conflict between oblique contexts and classical logic is only apparent. The extensional point of view therefore calls for analyses of oblique contexts that preserve the laws of classical logic and the principle of extensionality. In this thesis, we are concerned with the precise formulation of and comparison between different programs for the formal semantic representation of oblique contexts. In Chapter 5, we are concerned with Church's program of representing oblique contexts within a formal language whose semantics is fully extensional and based on Frege's distinction between sense and denotation. The ambiguity between the "direct" and "indirect" denotation of an expression A is avoided by letting A be replaced in oblique contexts by an expression which denotes A's ordinary sense. We provide a possible-worlds semantics for a version of Church's formal language with the following features: (i) Objectual quantification into oblique constructions is accounted for. The treatment of quantification requires for each entity x an essence of x, by which we mean a sense which determines x in those worlds in which x exists and is undefined in all other worlds. (ii) The variables range over actually existing entities only. It is allowed that entities of certain types are contingent, i.e., the same entity may exist in one world and fail to exist in other worlds. (iii) In order to combine a proper treatment of the cross-world identification of functions with varying domains, partial functions are allowed. (iv) Senses are allowed to be partially defined. For example, the sense of a proper name is undefined in a world in which the denotation of the name does not exist. (v) Senses of functions from objects of type a to objects of type b are identified with functions from senses of objects of type a to senses of objects of type b. In Chapters 6 and 7, we pursue the Russellian Alternative in intensional semantics, where a sense/denotation distinction is avoided. Apparent failures of substitutivity principles for singular terms are explained along the lines of Russell's theory of definite descriptions. Failures of substitutivity of coextensional sentences and predicates are explained by Russell's doctrine that sentences and predicate denote propositions and propositional functions , respectively. Russellian semantics is fully extensional in the sense that the Russellian denotation of a designator is a function of the Russellian denotations of its parts. In Chapter 6, a theory of Russellian propositions is developed within set theory. This theory is then applied in Chapter 7 to the task of providing a set-theoretic semantics for Russellian type theory. (shrink)

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. Yanofsky (...) describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

Recent debates over the role of recognition by the community for one’s development and flourishing generally discuss community in a univocal sense: the way that recognition functions in particular communities is not fundamentally different from the way it functions in the larger community. They also tend to logically prioritize a fundamental human identity over particular religious, ethnic, or societal identities, which are understood to be secondary to, and derivative of, this basic identity. In his depiction of how communal recognition (...) contributes to individual selfhood, Søren Kierkegaard challenges these accounts by (1) differentiating between how communal recognition occurs in society in general and in one particular community, the Christian church and (2) questioning the idea that the self created in the former is more fundamental than that created in the latter. For Kierkegaard, one’s identities as both a Christian and a fully-developed self depend essentially on recognition by the Other: initially and fundamentally on recognition by God alone, and secondarily and derivatively on recognition by the human Other. (shrink)

This paper examines and discusses an argument for the collapse of modal distincions offered by Quine in "Reference and Modality" and in Word and Object that relies exclusively on a version of the Principle of Substitution. It is argued that the argument does not affect its historical targets: Carnap's treatment of modality, presented in Meaning and Necessity, and Church's Logic of Sense and Denotation, developed by Kaplan; nor does it affect a treatment of modality inspired in (...) Frege's treatment of oblique contexts. It is argued, nevertheless, that the immunity of those systems to Quine's argument depends on the success of their rejection of the Principle of Substitution presupposed by Quine. (shrink)

In traditional societies, knowledge is organized in hierarchical chains through which authority is legitimated by custom. Because the majority of the population is illiterate, sacred knowledge is conveyed orally and ritualistically, but the ultimate source of religious authority is typically invested in the Book. The hadith are a good example of traditional practice. These chains of Islamic knowledge were also characteristically local, consensual and lay, unlike in Christianity, with its emergent ecclesiastical bureaucracies, episcopal structures and ordained priests. In one (...) class='Hi'>sense, Islam has no church. While there are important institutional differences between the world religions, network society opens up significant challenges to traditional authority, rapidly increasing the flow of religious knowledge and commodities. With global flows of knowledge on the Internet, power is no longer embodied and the person is simply a switchpoint in the information flow. The logic of networking is that control cannot be concentrated for long at any single point in the system; knowledge, which is by definition only temporary, is democratically produced at an infinite number of sites. In this Andy Warhol world, every human can, in principle, have their own site. While the Chinese Communist Party and several Middle Eastern states attempt to control this flow, their efforts are only partially successful. The result is that traditional forms of religious authority are constantly disrupted and challenged, but at the same time the Internet creates new opportunities for evangelism, religious instruction and piety. The outcome of these processes is, however, unknown and unknowable. There is a need, therefore, to invent a new theory of authority that is post-Weberian in reconstructing the conventional format of charisma, tradition and legal rationalism. (shrink)

This is the first critical study of The Logic of Sense, Gilles Deleuze's most important work on language and ethics, as well as the main source of his vital philosophy of the event.James Williams explains the originality of Deleuze's work with careful definitions of all his innovative terms and a detailed description of the complex structure he constructs. This reading makes connections to his ground-breaking work on literature, to his critical but also progressive relation to the sciences, and (...) to his controversial denial of the priority of standard logics, human values and 'meaning' in thinking.This book will open new debates and develop current ones around Deleuze's work in philosophy, politics, literature, linguistics, cultural studies and sociology. (shrink)

The study examines Russian philosopher Andrei Smirnov’s conception of the logic of sense as a way of providing exposition of the reasons for the Russian-Ukrainian war. The logic of sense is simultaneously a theory of rules of sense-setting and the very rules of sense-setting created by a culture and the ruling culture. Smirnov thinks that the reasons lie in the clash between common-human European culture and its logic of sense and all-human Russian (...) culture and its logic of sense. Smirnov sees no difference between Russians and Ukrainians and considers the Ukrainian culture an essential and indispensable part of Russian culture. Therefore, the expansion of the common-human European culture to Ukraine challenges the all-human Russian culture. (shrink)

The purpose of this article is to integrate two outstanding problems within the philosophy of science. The first concerns what role aesthetics plays in scientific thinking. The second is the problem of how logically testable ideas are generated (the so-called "psychology of research" versus "logic of (dis)proof" problem). I argue that aesthetic sensibility is the basis for what scientists often call intuition, and that intuition in turn embodies (in a literal physiological sense) ways of thinking that have their (...) own meta-logic. Thus, aesthetics is a form of cognition. Scientists think not in equations or words or other logical abstractions, but emotionally and sensually, using visual and aural images, kinesthetic and other proprioceptive feelings, sensations, patterns, and analogies. These aesthetic forms of thinking have their own logics that I call "synosia", from the root words synaesthesia (a combining of senses) and gnosis, "to know". Synosia denotes understanding that integrates feeling that one knows with feeling what one knows. Eminent scientists universally describe an explicitly secondary process in which such personal knowledge must be "translated" into a formal language, such as words or equations, in order to be communicated to other people. Many of the unsolved problems that philosophers of science (as well as psychologists and artificial intelligence researchers) have had in making sense of scientific thinking have arisen from confusing the form and content of the final translations with the hidden means by which scientific insights are actually achieved. (shrink)

An infinite binary sequence X is Kolmogorov–Loveland random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence.One of the major open problems in the field of effective randomness is whether Martin-Löf randomness is the same as KL-randomness. Our (...) first main result states that KL-random sequences are close to Martin-Löf random sequences in so far as every KL-random sequence has arbitrarily dense subsequences that are Martin-Löf random. A key lemma in the proof of this result is that for every effective split of a KL-random sequence at least one of the halves is Martin-Löf random. However, this splitting property does not characterize KL-randomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2-random. Furthermore, we show for any KL-random sequence A that is computable in the halting problem that, first, for any effective split of A both halves are Martin-Löf random and, second, for any computable, nondecreasing, and unbounded function g and almost all n, the prefix of A of length n has prefix-free Kolmogorov complexity at least n−g. Again, the latter property does not characterize KL-randomness, even when restricted to left-r.e. sequences; we construct a left-r.e. sequence that has this property but is not KL-stochastic and, in fact, is not even Mises–Wald–Church stochastic.Turning our attention to KL-stochasticity, we construct a non-empty class of KL-stochastic sequences that are not weakly 1-random; by the usual basis theorems we obtain such sequences that in addition are left-r.e., are low, or are of hyperimmune-free degree.Our second main result asserts that every KL-stochastic sequence has effective dimension 1, or equivalently, a sequence cannot be KL-stochastic if it has infinitely many prefixes that can be compressed by a factor of α<1. This improves on a result by Muchnik, who has shown that were they to exist, such compressible prefixes could not be found effectively. (shrink)

Writings, including articles, letters, and unpublished work, by one of the twentieth century's most influential figures in mathematical logic and philosophy. Alonzo Church's long and distinguished career in mathematics and philosophy can be traced through his influential and wide-ranging writings. Church published his first article as an undergraduate at Princeton in 1924 and his last shortly before his death in 1995. This volume collects all of his published articles, many of his reviews, his monograph The Calculi of Lambda-Conversion, (...) the introduction to his important and authoritative textbook Introduction to Mathematical Logic, a substantial amount of previously unpublished work (including chapters for the unfinished second volume of Introduction to Mathematical Logic), and a selection of letters to such correspondents as Rudolf Carnap and W. V. O. Quine. With the exception of the reviews, letters, and unpublished work, these appear in chronological order, for the most part in the format in which they were originally published. Church's work in calculability, especially the monograph on the lambda-calculus, helped lay the foundation for theoretical computer science; it attracted the interest of Alan Turing, who later completed his PhD under Church's supervision. (Church coined the term “Turing machine” in a review.) Church's influential textbook, still in print, defined the field of mathematical logic for a generation of logicians. In addition, his close connection with the Association for Symbolic Logic and his many years as review editor for the Journal of Symbolic Logic are documented in the reviews included here. (shrink)

First published in 1931, A Study in the Philosophy of Malebranche examines the theories which constitute the philosophical system of Malebranche. Church specifically analyses theories pertaining to Malebranche's vision in god; knowledge; occasionalism; and imagination and sense.

There are three distinct questions associated with Simpson's paradox, (i) Why or in what sense is Simpson's paradox a paradox? (ii) What is the proper analysis of the paradox? (iii) How one should proceed when confronted with a typical case of the paradox? We propose a "formar" answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson's paradox. Our account contrasts sharply with Pearl's causal (and questionable) account of the first two (...) questions. We argue that the "how to proceed question?" does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth's account of the paradox. Our research on the paradox suggests that the "how to proceed question" needs to be divorced from what makes Simpson's paradox "paradoxical.". (shrink)

There are three distinct questions associated with Simpson’s paradox. Why or in what sense is Simpson’s paradox a paradox? What is the proper analysis of the paradox? How one should proceed when confronted with a typical case of the paradox? We propose a “formal” answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson’s paradox. Our account contrasts sharply with Pearl’s causal account of the first two questions. We argue that the (...) “how to proceed question?” does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth’s account of the paradox. Our research on the paradox suggests that the “how to proceed question” needs to be divorced from what makes Simpson’s paradox “paradoxical.”. (shrink)

Most of us experience the world through competing perspectives. A job or a religion seems important and fulfilling when looked at in one way; but from a different angle they seem tedious or ridiculous. A friend is obtuse from one point of view, wise from another. Continuing to hold both views at once can be unsettling, highlighting conflicts between our own judgments and values, and undermining our ability to live purposefully and effectively. Yet, as Jennifer Church argues in this book, (...) inner conflict can be a good thing, and not just as a temporary road bump on the road to resolution. This book describes several desirable types of "double consciousness"--or being of two minds--and explains why and how they should be maintained. Church looks critically at some common ideas about identity, including a popular belief about narratives that suggests our lives should "make sense" as a story. She also examines how empathy can helpfully cause us to be of two minds, and how various forms of irony and laughter enable us to benefit from holding onto opposing views. Finally, Church shows the merit of acknowledging reality while sometimes being guided by fantasy. Why It's OK to Be of Two Minds is for anyone who's held two opposing views simultaneously, which is to say it's for everyone. Key Features: Argues against a longstanding philosophical idea: that it is important to resolve inner conflicts that result from competing systems of beliefs Examines the role of empathy and friendship in maintaining a valuable form of double consciousness Considers how irony and laughter allow us to dedicate ourselves to our particular projects while acknowledging their ultimate insignificance Shows how fantasies that conflict with our beliefs can make a positive contribution to the way we live our lives. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 12.1 (2005) 59-61 [Access article in PDF] Space and Normativity Jennifer Church Keywords space, normativity, reasons, unconscious I appreciate the thoughtful criticisms and helpful suggestions of my commentators. In this brief reply, I can only begin to address the many interesting issues that they raise.I am not sure whether R.D. Hinshelwood views my paper as operating within the constraints of analytic philosophy, which he equates (...) with "rationalist" philosophy and traces back to Aristotle, or as moving beyond those boundaries. Three initial remarks may help to situate my position. (1) My paper is in the analytic tradition insofar as it is an attempt to articulate the constitutive structure of concepts that are especially interesting and fundamental, where such articulation may or may not lead to conceptual revision. (2) My particular focus is on the concept of reason, with the aim of thinking through its appropriate use with respect to unconscious thought and the emotions; implicit in this project is the assumption that there are some things that should not be explained by appeal to reasons. (3) I wish to show how some peculiarities surrounding the attribution of unconscious reasons and emotional reasons can be understood by reference to spatial reasoning rather than reasoning that is more linguistic in character.Hinshelwood sympathizes with my attempt to describe a kind of reasoning that is not "rationalist" insofar as it does not rely on the manipulation of symbols and, indeed, tends to ignore the distinction between symbol and thing. He agrees that there is something spatial about such ways of reasoning. But Hinshelwood argues that more needs to be said to establish a connection between the spatial maneuverings of animals and the nonpropositional thinking of humans. He thinks that "psychoanalysis can in effect fill in the gap in Church's argument with a developmental argument" (2005, 47). The developmental account he has in mind is based on Freud's description of how an infant who automatically tries to ingest what feels good and excrete what feels bad will also come to introject (i.e., to imagine ingesting) what seems good and eject (i.e., imagine excreting) what seems bad. In the words of Hinshelwood, this is a hypothesis about how "bodily space becomes psychic space, and emerges in the sense of having a mind" (2005, 47). I am sympathetic to this account of how we acquire a sense of having a mind, but I do not think that we need to have a sense of having a mind in order to have a mind. A cat can have beliefs and desires (and can have the normative experience of having taken the wrong turn) without ever imagining an inner space with mental contents. Likewise, a cat can experience anger or fear in virtue of experiencing certain changes within its body despite having no experience of its mind as such. In short, I view an account of introjection as an important contribution to developmental psychology, but I think it comes [End Page 59] after, not before, an account of unconscious and emotional reasons.Gerrit Glas maintains that my attempt to ground the normativity of unconscious reasoning (or spatial reasoning) in the feeling of 'getting it right' versus getting it wrong' runs afoul of this dilemma: either (a) the normativity of unconscious reasoning ends up very much like the normativity of conscious reasoning, and the needed normativity is already built into the feeling that I appeal to, or (b) the normativity of unconscious reasoning is reduced to physical sensations and is no longer normative at all. I do not view this as a dilemma because I embrace option (a). I agree that the feel of being on the right track is much the same, whether or not we can say what that track is, let alone why it is right rather than wrong. Likewise, it feels the same to realize that I have made a wrong turn, whether or not I know what the wrong turn was. Also, I agree that normativity is implicit in the feel of getting it right. I am not appealing to such a feeling... (shrink)

In Hume's dialogue with the Hobbesian-Mandevillian "selfish system" of morals, Hume seems to reject its conclusions in morals, but accept them in politics. No skeptic of moral claims like Mandeville, Hume sought to ground objective moral standards in his moral sentiment philosophy, yet, like Mandeville, Hume argued that in political life human beings act based largely on self-interest and a limited generosity. I argue that Hume, however, is ultimately ambivalent about the selfish system's conclusions in politics. He puts forth both (...) a nonmoral and a moral solution to the problem of cohesion in modern liberal states. First, he agrees with the selfish system's nonmoral tactic of channeling the self-interest of citizens through well-constructed institutions toward salutary ends. Second, arguing that the first solution is insufficient for the health of a political regime, Hume seeks to expand the limited moral sense of citizens through moral and aesthetic education and through an empowerment of local politics. Hume's second solution is a means within liberalism to combat its own tendencies toward the dissolution of communal ties and the creation of conditions ripe for the emergence of "sensible knaves.". (shrink)

Imagine an equilateral triangle “pointing upward”—its horizontal base under its apex angle. A semiotic triangle has the following three “vertexes”: (apex) an expression, (lower-left) one of the expression’s conceptual meanings or senses, and (lower-right) the referent or denotation determined by the sense [1, pp. 88ff]. One example: the eight-letter string ‘coleslaw’ (apex), the concept “coleslaw” (lower-left), and the salad coleslaw (lower-right) [1, p. 84f]. Using Church’s terminology [2, pp. 6, 41]—modifying Frege’s—the word ‘coleslaw’ expresses the concept “coleslaw”, the (...) word ‘coleslaw’ denotes or names the salad coleslaw, and the concept “coleslaw” determines the salad coleslaw—recalling Frege’s principle that sense determines denotation. Church [2, p. 6] wrote: -/- We shall say that a name denotes or names its denotation and expresses its sense. […] Of the sense we say that it determines its denotation, or is a concept of the denotation. -/- Aristotle seems cognizant of distinctions going beyond those in semiotic triangles. The expression Aristotle’s semiotic pyramids seem warranted by Aristotle’s Categories, 1a1: -/- When [two] things have a name (onoma) in common and the concept (logos) of being (ousia) which corresponds to the name in each case is different, they are called same-named (homonuma). Thus, for example, both a man and a picture [of an animal] are called animals. These have only a name in common. In each case the name’s concept of being [an animal] is different; for if one says what being an animal is for each of them, one will give two distinct concepts. -/- Semiotic triangles and pyramids in Aristotle’s logic are compared to those in Church’s [2]. [1] JOHN CORCORAN, Sentence, proposition, judgment, statement, and fact, Many Sides of Logic, College Publications, 2009. [2] ALONZO CHURCH, Introduction to Mathematical Logic, Princeton, 1956. -/- The semiotic pyramid in Categories, 1a1 has a square base under the vertex ‘animal’. On the corners of the square are: the concept “animal”, the concept “animal picture”, the animals, and the animal pictures. The animals are homonymous with the animal pictures. People find Aristotle’s example far-fetched or inept even if the experience of pointing to a picture while saying “That is Tarski” is familiar. Imagine looking at a painting while thinking “That is an animal”. Without putting too fine a point on this, notice that in Aristotle’s sense it is individual things that are homonymous, not words. It would be natural to say also in his sense that two things are homonyms if one is homonymous with the other. In contrast, we use the words homonym and homonymous to relate words that are spelled the same and pronounced the same but have different meaning. Consider the noun ‘center’ and the verb ‘center’. Consider the noun ‘smell’ and the verb ‘smell’. The spelling of two homonyms is an ambiguity, or an ambiguous spelling. We need appropriate adjectives to distinguish the Categorical senses of ‘homonym’ and ‘homonymous’ from the current English sense just mentioned. I propose ‘ontological’ for the sense relating things and ‘linguistic’ for that relating words. Given that all words are things but not all things are words, we ask are words that are linguistically homonymous also ontologically homonymous? END OF POST ABSTRACT. (shrink)

This essay reviews two installments in the Heidelberg Academyʼs Historical and Critical Commentary series on the works of Friedrich Nietzsche. While Sarah Scheibenberger’s volume focuses on Truth and Lie in an Extra-Moral Sense, highlighting the sources and influence of Nietzsche’s text, Jochen Schmidt and Sebastian Kaufmann provide a detailed and extremely useful contextualization of Daybreak and of Nietzsche’s poetry.

Church’s type theory, aka simple type theory, is a formal logical language which includes classical first-order and propositional logic, but is more expressive in a practical sense. It is used, with some modifications and enhancements, in most modern applications of type theory. It is particularly well suited to the formalization of mathematics and other disciplines and to specifying and verifying hardware and software. It also plays an important role in the study of the formal semantics of natural language. (...) When utilizing it as a meta-logic to semantically embed expressive non-classical logics further topical applications are enabled in artificial intelligence and philosophy. A great wealth of technical knowledge can be expressed very naturally in it. With possible enhancements, Church’s type theory constitutes an excellent formal language for representing the knowledge in automated information systems, sophisticated automated reasoning systems, systems for verifying the correctness of mathematical proofs, and a range of projects involving logic and artificial intelligence. Some examples and further references are given in Sections 1.2.2 and 5 below. Type theories are also called higher-order logics, since they allow quantification not only over individual variables, but also over function, predicate, and even higher order variables. Type theories characteristically assign types to entities, distinguishing, for example, between numbers, sets of numbers, functions from numbers to sets of numbers, and sets of such functions. As illustrated in Section 1.2.2 below, these distinctions allow one to discuss the conceptually rich world of sets and functions without encountering the paradoxes of naive set theory. Church’s type theory is a formulation of type theory that was introduced by Alonzo Church in Church 1940. In certain respects, it is simpler and more general than the type theory introduced by Bertrand Russell in Russell 1908 and Whitehead & Russell 1927a. Since properties and relations can be regarded as functions from entities to truth values, the concept of a function is taken as primitive in Church’s type theory, and the λ-notation which Church introduced in Church 1932 and Church 1941 is incorporated into the formal language. Moreover, quantifiers and description operators are introduced in a way so that additional binding mechanisms can be avoided, λ-notation is reused instead. λ-notation is thus the only binding mechanism employed in Church’s type theory. (shrink)