Since its existence, art objects have always been different from other objects in terms of perception and interpretation and have preserved their mystery for both the artist and the audience. This mystery was tried to be supported by various theories by the artist and the audience, and defined and defined with concepts such as spiritual development, spirituality and intuition. There is an ambiguity especially regarding intuition. The concept of intuition seems to be trapped in a bridge between the physical (...) world and the superphysical world. The concept of intuition has been interpreted together with the teachings of Theosophy and Buddhism and turned into an educational theory in today's art-based studies. In our age, interpretations beyond reality are increasing and unfortunately faced with the danger of losing the ground of reality. Art is a type of knowledge and knowing. Due to this feature, the compatibility of surreal discourses with this type of knowledge is discussed in this article, and on the other hand, it is tried to show how concepts such as synesthesia and intuition become possible on the basis of reality, based on the artistic experiences of world-famous artists. (shrink)

The paper is a critical discussion of the rich and insightful final chapter of Mitchell Green’s Self-Expression . There, Green seeks to elucidate the compelling, but inchoate intuition that when we’re fully and most expertly expressing ourselves, we can ‘push out’ from within not just our inner representations, but also the ways that we feel. I question, first, whether this type of ‘qualitative expression’ is really distinct from the other expressive forms that Green explores, and also whether it’s genuinely (...) ‘expressive’. I then scrutinize the nature of the ‘qualitative congruences’ that lie at the heart of Green’s theory; and I wonder whether they can play the role Green claims they can in providing a novel account of artistic expression. (shrink)

With the sustained, coherent perspective of an authored text and the diverse, authoritative views typical of an anthology, Philosophy of Art: Aesthetic Theory and Practice by David Boersema provides the context and commentary students need to comprehend the various issues in philosophy of art. Throughout the book, issues are examined using the lenses of the three broad areas of philosophy: metaphysics, epistemology, and value theory. That is, concerns are raised about what is expressed, how it is expressed, and why it (...) is expressed. Chapters on the artist, the audience, and the artwork further break down the discipline and are applied to the final chapters on the specific types of art. The differences between art and science as well as the relationship of art and society provide a refreshing discussion of overlooked areas in philosophy of art. (shrink)

Noël Carroll proposes a generalist theory of art criticism, which essentially involves evaluations of artworks on the basis of their success value, at the cost of rendering evaluations of reception value irrelevant to criticism. In this article, I argue for a hybrid account of art criticism, which incorporates Carroll's objective model but puts Carroll-type evaluations in the service of evaluations of reception value. I argue that this hybrid model is supported by Kant's theory of taste. Hence, I not only (...) present an alternative theory of metacriticism, which has the merit of reinstating the centrality of reception value in art critics’ evaluations, but also show that, contrary to a common conception, Kant's aesthetic theory can house a fruitful account of art criticism. The benefit of this hybrid account is that, despite being essentially particularist, it should be appealing even to generalists, including Carroll. (shrink)

I propose a distinction between design intentions, activities and products, as opposed to artistic intentions, activities and artworks. Examples of design products would include a specific type of car (or any other invention or device) as well as closer relatives of art such as decorative wall designs. In order to distinguish artistic from design intentions, I present an example in which two sculptors independently work on a single object to produce two sculptures, which are distinct just because (...) the artistic intentions of the sculptors are distinct. This case is then contrasted with an attempted parallel example for design intentions, which fails to produce two correspondingly distinct design products in spite of the different design intentions of its designers. I argue that this failure occurs because designs are types, for which any single token of a given type could not simultaneously be a token of some other type of the same general kind; whereas the possibility of my sculptural 'double artwork' example shows that such artworks cannot be types. I then further investigate the concept of a design, and conclude by arguing that an extension of my sculptural example shows that literary artworks also cannot be types. (shrink)

Issues concerning the spatial orientation of pictures play an important, though previously neglected, role in an adequate understanding of the nature and identity of visual artworks and other pictures. Using a previous contrast ('Artworks Versus Designs', BJA Vol. 41, No. 4, October 2001), I show that differing orientations of a design naturally give rise to distinct pictures, which may be appropriated as distinct artworks by a discerning artist--which also shows that such artworks cannot be types, since they share a common (...) token. The investigation also raises some significant issues concerning artistic printmaking, and strongly suggests in addition that two new concepts of interpretation--of identifying and constitutive interpretation--are required to adequately explain the artistic phenomena that are uncovered. (shrink)

This article proposes complementing actor-network theory (ANT) with Niklas Luhmann’s communication theory, in order to overcome one of ANT’s major shortcomings, namely, the lack of a conceptual repertoire to describe virtual processes such as sense-making. A highly problematic consequence of ANT’s actualism is that it cannot explain the differentiation of economic, legal, scientific, touristic, religious, medical, artistic, political and other qualities of actual entities, assemblages and relationships. By recasting Luhmann’s theory of functionally differentiated communication forms and sense-making as dealing (...) with different types of virtual attractors calling for actualizations in concrete assemblages, I propose a symmetrical understanding of societal differentiation processes as based on the co-production of virtual attractors and actual assemblages. (shrink)

Cet article traite du concept de « dégénérescence », importé dans la psychiatrie française par Benedict-Auguste Morel dans les années 1850, et largement diffusé par la suite, dans ce champ ainsi que dans celui de la criminologie. On tente d’analyser la reconfiguration qu’impose ce concept au savoir psychiatrique en dégageant la manière dont il permet d’intégrer en un ensemble cohérent plusieurs modèles théoriques: un paradigme neurologique, une théorie de l’automatisme morbide, un certain évolutionnisme. Sur ces bases, on essaie d’établir les (...) similitudes de fonds existant entre les conceptions psychiatrique et criminologique de la dégénérescence, en montrant qu’elles mobilisent le même réseau « structurant » d’analogies entre les figures de l’anormal (le fou, le sauvage, l’enfant, la femme, l’animal); et l’on s’efforce de montrer que ces théories promeuvent une logique dans laquelle les infractions aux différents types de normes (biologiques, sociales, morales, juridiques, psychologiques, économiques) sont susceptibles de faire systématiquement référence les unes aux autres, de se traduire les unes dans les autres. Puis l’on caractérise les transformations que ce nouveau paradigme impose aux principes de la thérapeutique, de la pédagogie et de la prise en charge des malades mentaux et des déviants. Pour finir, on présente les éléments latents qui dans ce paradigme préparent et indiquent déjà sa remise en cause et son renversement à venir. (shrink)

Chapter 1: An Introduction: The ‘Semantic Sting’ Argument Describes Dworkin’s theory as concerning the conditions of legal validity. “A legal system is a system of norms. Validity is a logical property of norms in a way akin to that in which truth is a logical property of propositions. A statement about the law is true if and only if the norm it purports to describe is a valid legal norm…It follows that there must be certain conditions which render certain norms, (...) but not others, legally valid. Hence it also follows that there can be two types of disagreement over the truth or falsehood of propositions about the law. People can disagree over the question ‘What are the conditions of legal validity?’, in which case their disagreement is a theoretical one. Or they can agree on the conditions of validity, and disagree as to whether or not those conditions actually obtain in a given case or not.” Argues that the best way to interpret Dworkin’s ‘semantic-sting argument’ is as a general argument against conventionalism in legal positivism. Summarizes the outcome of Dworkin’s argument as follows: “[I]f Dworkin is right about the legal reasoning of lawyers and judges, conventionalism would be self-defeating: if lawyers and judges recognize as legally binding not only those norms which are uncontroversially identifiable under the Rule of Recognition, that is, if what they recognize as binding is not only source-based law, then conventionalism turns out to be false on its own terms. In other words, either law is not what lawyers and judges think that it is, in which case law is not a matter of conventions, or – if it is what lawyers and judges think – conventionalism is false, as they do not see the law as purely a matter of conventions.” Dworkin’s theory, then, is a new conception of jurisprudence aiming to present itself as a rival to conventionalism. The book will be concerned with re-examining positivism in light of the interpretative challenge. Chapter 2: Meaning and Interpretation “[R]oughly, interpretation can be defined as an understanding or explanation of the meaning of an object.” Interpretation in this narrow sense is distinct from a less formal and broader use of interpretation that makes it equivalent to explanation. “Hence also, only those objects which are capable of bearing some meaning qualify as objects of interpretation.” [Why shouldn’t we think of interpretation as object-identification? This seems to be closer to what Dworkin has in mind.] Thus, a consideration of the philosophical work on meaning is appropriate. First, a consideration of Davidson on interpretation. The thesis is that Davidson’s “radical interpretation” can only account for sentence meaning and not for linguistic interpretation in general. Davidson cannot account for the non-rule-governed aspects of communication. “Semantics, as opposed to interpretation, concerns those aspects of communication which are rule or convention governed…but such rules are normally unavailable as reasons or justifications for an interpretation. On the contrary, interpretation is usually required because the issue is not determined by rules or conventions.” Interpretation, rather, is parasitic with understanding the meaning of an expression and not equivalent with it. As Dummett notes, any reflection on the meanings of words assumes prior knowledge of ordinary meanings. “This leads to the conclusion that understanding or explaining the meaning of an expression and interpreting it, are two conceptually separate things. It also indicates that seman ics can only be employed, if at all, to elucidate the concept of interpretation by way of contrast: interpretation concerns those aspects of communication which are under-determined by semantic rules or conventions.” Pragmatics, the study of problems posed by discrepancies between utterer’s meaning and sentence meaning, would seem to be closer to inquiry into interpretation. Both are concerned with understanding how meaning is possible where conventional rules leave off. Contemporary pragmatics, moreover, recognizes the logically indeterminate character of communicative inferences beyond conventional meaning and that the contextual conditions that account for the possibility cannot be realized in the semantic structure of the sentence. However, different criteria of success may be at work in pragmatics than interpretation. Pragmatics is concerned with how communication is achieved and thus the criteria of success would be grasping a speaker’s intentions. Many think, however, that communication is unimportant in certain spheres of communication, e.g. artistic interpretation, and that in these cases the aim of interpretation is not to uncover the actual artist’s intentions. What kind of meaning is interpretation aimed at then? It cannot be the meaning the object has for the interpreter. “Interpretation purports to be a statement about the object interpreted, not about the subject who offers the interpretation…it is crucial to remember that not everything we say about a work of art, or any other text, amounts to an interpretation of it. Only those aspects of a text which can shed light on its meaning form part of the text’s interpretation. What the text means for someone rarely entails anything about what the text means.” [This is puzzling to some extent for it remains somewhat ambiguous what it means to talk of subject-independent meaning.] Interpretive meaning attribution is, rather, counterfactual. “In general, I will suggest that the answer to this question consists in the fact that meaning is assigned by a counterfactual statement. Given that x is the meaning attributed to, for instance a text T, and x is not the literal meaning of T, nor is it the meaning of T intended by its author, then the attribution of meaning x to T can only be understood as the contention that on the basis of certain assumptions a certain fictitious or stipulated speaker would have meant x by expressing T.” Thus, on this account, interpretation, when it does not concern an actual speaker, turns out to be a kind of counterfactual intention attribution. Twofold point: “First, that interpretation is essentially a matter of attributing intentions, that is, in the pragmatics sense of ‘meaning’, namely, meaning that such-and-such by an act or expression. At the same time, interpretations need not be based on the intentions of actual authors; the meaning of an act or expression is understandable in terms of counterfactual intentions.” [Are we really restricted to understanding the coherence and integrity of meaning in such a way? Presumably the author is a kind of device for organizing meaning, much like Dworkin’s integrity, but need it take that format?] The logic of interpretation is typically reducible to intention attribution. [What role does the counterfactual author play in structuring meaning?] The criteria of interpretive success will depend upon the object, whether it is the real author or the counterfactual author. If the former, success will depend upon the retrieval of the author’s actual intentions. “Likewise, if the author is characterized in terms of some ideal representative of a certain genre, for instance, the presumptions which are taken to determine this c aracterization would provide the criteria of success for the particular interpretation offered.” [The criteria of success would seem, then, to be the principles, ideals, etc. that we attribute to the counterfactual author. These attributed qualities would be doing all the work – is the attribution of intention to the hypothetical author, then, really necessary at all?] [Does this account presuppose a weird dichotomy between what an object is and what it means? This dichotomy is different, if present at all, in Dworkin.] Chapter 3: Dworkin’s Theory of Interpretation and the Nature of Jurisprudence Three main insights of Dworkin’s theory of interpretation: “First, that interpretation strives to present its object in its best possible light. Second, that interpretation is essentially genre-dependent. And third, that there are certain constraints that determine the limits of possible interpretations of a given object.” Why the best light? Dworkin doesn’t really address this question in depth but claims that otherwise we are left with no way of justifying and choosing an interpretation or we are left with the author’s intention model, which we should reject. [Marmor doesn’t seem clear on what precisely it means to present something in its best light – seems to conflate it with most appealing light which, I take it, is different than what Dworkin has in mind.] “Dworkin is quite right to maintain that without having some views about the values inherent in the genre to which the text is taken to belong, no interpretation can take off the ground. The values we associate with the genre partly, but crucially, determine what would make sense to say about the text, what are the kinds of meaning we could ascribe to it.” There is no necessity for an interpretation to present an object in its best light with regards to the genre it is taken to be part of. Interpretations “could simply strive to present it in a certain light, perhaps better than some, worse than others, but in a way which highlights an aspect of the meaning of the text which may be worth paying attention to for some reason or another.” [What kind of reasons would count in choosing to present it in one light rather than another?] There is simply zero grounds for Dworkin’s assertion that we have no reason to pay attention to an interpretation that does not present it in its best light. Take a psychoanalytic interpretation of Hamlet, for example – not the best light, but a legitimate one. [This seems related to the next criticism – clearly Dworkin would have to loosen his insistence on the best, though not to detriment.] Moreover, especially with regards to art, it doesn’t appear possible to make an all things considered judgment about the best interpretation given the incommensurability of different values that are taken to inhere in a given genre. [Dworkin can respond here that we may have different available interpretations that could be justified with respect to the relevant value. Moreover, a lack of common denominator does not prevent any comparison whatsoever.] Constructive interpretation concerns social practices constituted by norms – it imposes a purpose or value rendering intelligible the normative character of the practice. There is a separate aspect of constructive interpretation that maintains that elements of the practice are sensitive to the point of the practice. It is important to note that different social practices will institutionalize themselves in different ways – thus, different practices will establish varying conditions on how the practice can be altered. The position of the positivists is that evaluative judgments concerning what the law should be are insufficient for determining what the law is given the nature of law’s institutionalization.&nb p; [This would seem to reside on a certain conception of law, i.e. an interpretation about where the value of law lies.] “[T]his being one of the main points of dispute between Dworkin and his positivist opponents, Dworkin cannot at this initial state presume law to be sensitive to its value in the manner that other, non-institutional practices might be, without incurring the charge of having assumed the very point at issue.” The upshot for Dworkin is that any explanation of a social practice such as law requires the same kind of reasoning as participation in that practice. This will be called the hermeneutic thesis. We can try to reconstruct a justification for this view based on the necessity of shared background assumptions – but this will not do since Dworkin’s claim is stronger: it contends that participants’ and theorists’ must “adopt one and the same normative point of view.” [The sense of normative here is unclear. It is true, though, that shared background assumptions do not imply viewing the law as valuable in any particular way, or at least it does not imply this in any strong sense.] Dworkin makes two distinct points with the hermeneutic thesis: “First, that the explanation of a social practice, like law or the arts, is essentially interpretative, and as such, necessarily value laden. Second, that the interpretation of such social practices, which Dworkin calls ‘argumentative’, is unique in the sense that the practice itself is an evaluative enterprise, and that therefore the interpreter of such a practice must form an evaluative judgment of her own about those values which are inherent in the practice that she purports to interpret.” [As stated, the first thesis is terribly ambiguous – for Dworkin, it clear implies a commitment to constructive interpretation. For Marmor, who accepts it, it is totally unclear what commitment he intends to make. He seems to mean simply that we have criteria for success concerning what counts as a successful explanation.] Mamor denies the second thesis. “What Dworkin seems to ignore here is that there is a crucial difference between forming a view about the values which are manifest in a social practice, like law, and actually having evaluative judgments about them.” [The problem, however, it that it is a contentious matter what values are part of the law – a theorist’s account of what values are constitutive of a particular legal system must rely on the kind of arguments which participants rely on. Of course, a theorist might simply note a debate about values and attempt to remain agnostic, but this will fail to be a full theoretical account of what law is!] Dworkin’s thesis seems to depend exclusively on his account of constructive interpretation. [This seems exactly right.] Chapter 4: Coherence, Holism, and Interpretation: The Epistemic Foundation of Dworkin’s Legal Theory Begins with discussion of Rawls’ reflective equilibrium. Sums up dual criticism as follows: “[T]he assumption that intuitions are independently true and the converse one, that their truth depends on fitting a coherent scheme, both seem to yield paradoxical results.” Dworkin, in his article “The Original Position”, claims that Rawls cannot commit himself to ethical realism. “[T]he natural model presupposes some form of ethical realism, while the constructive model does not. One important difficulty arising from an attempt to apply the natural model to Rawls is, that under the natural model, any theory which dopes not account for an intuition, at least for one which is held firmly, cannot be wholly satisfactory, just as scientific theory which does not account for certain observational data it is supposed to cover, would not be satisfactory in a familiar way.” Rawls, rather, is committed to a constructivist view of morality. Moreover, the reason we value coherence epistemically in moral theory is that it itself is a value of political morality – fairness requires consistency and publicity in the application of moral standards. Marmor notes the following problem with such a view: “[I]f coherence is justified, as it is here, with reference to certain moral values, that is, a specific conception of fairness, then we face the following problem: the presupposed values of fairness must themselves be based upon intuitive convictions, in which case the question of their truth cannot be ignored. If they are taken to be true…we are driven back to the perplexities of the natural model.” Discusses, helpfully, the Fish/Dworkin debate. The upshot is that Dworkin’s interpretive model is best off if it commits itself to a Quineian holism. However, this undermines his distinction between internal and external skepticism since holism must deny that moral judgments constitute a closed system. [This analysis seems exactly right.] Endorses Simmonds view that Dworkin’s theory does not meet the requirements of complexity he identifies as necessary to avoid vicious circularity in interpretive judgments of fit and identity. Coherence is doing the work through and through and therefore the interpretive theory of law is circular. [Will require further investigation, though it is unclear how coherence is doing any work in the pre-interpretive stage. At this stage, the judgments, if interpretive, seem to be derived from different sources. In any case, using coherence as a value seems to be highly suspect.] Why doesn’t Dworkin just reject Fish’s assumptions: because of his jurisprudence. “If legal texts can have a meaning that is not entirely dependent on a process of interpretation, then it is at least sometimes the case that the law can simply be understood, and applied, without the mediation of interpretation. And if that is the case, then the argument from interpretation against legal positivism collapses. It is no longer the case that every conclusion about what the law is, depends on evaluative considerations about what it ought to be.” Chapter 5: Semantics, Realism, and Natural Law Assessment of Michael Moore’s legal realism and its implications. [The final criticisms of Moore are not decisive.] Chapter 6: Constructive Identification and Razian Authority A consideration of Dworkin’s denial that the communication model of interpretation is appropriate for legal interpretation. Marmor argues that intentions do play a crucial role in the identification of legal norms in a way that is incompatible with Dworkin’s “coherence thesis”. Marmor takes Dworkin to be committed to the constructive identification thesis: that the identification of something as part of the legal, artistic, etc. genre can be done sufficiently by evaluative considerations. “Here, one must maintain that evaluative considerations are sufficient to determine whether something is a legal norm or a work of art”. [Fit, however, constrains possible evaluative considerations – thus evaluative considerations are not sufficient in themselves.] “Now the crucial point here is this: if you maintain the possibility of constructive identification in art, you must assume that works of art can be identified as such on the basis of certain features they happen to possess, features which contain no reference to any particular intention to create a work of art…unless we take intentions into account, how can we discriminate between the concept of an aesthetic artifact and the con ept of a work of art?” [It is unclear, and I think false, that constructive interpretation of anything requires identifying something irrespective of an artificers intentions. Those intentions may be relevant, dependent upon the genre under which we are trying to classify the object. Nothing, seemingly, about constructive interpretation in general excludes this possibility.] Marmor’s point is that it turns out to be impossible to identify art as art without reference to an artist’s intentions – especially given the state of contemporary art. [It is notable that Marmor’s response to counterexample implicitly depends upon a fully developed theory of aesthetics. His cursory comments on the matter of identifying art seem insufficient to say the least.] The rest of the chapter considers the incompatibility of Raz’s account of authority with constructive interpretation. The incompatibilities mentioned do not move beyond what Raz identifies in his article “Authority, Law and Morality”, but a defense of this notion of authority is defended in several [inconclusive] respects. [Raz and Marmor want to separate identification from content. This is not clearly a defensible separation.] Chapter 7: No Easy Cases? Chapter defends notion that there is a distinction between easy and hard cases in the positivist sense. Takes a Hartian approach and ultimately defends the distinction by associating it Wittgenstein’s conception of rule following. The point is that one need to look the purpose of rule in order to understand what the rule requires. Chapter 8: Legislative Intent and the Authority of Law Examines the following doctrine concerning the role of legislative intent in adjudication: “[F]irst, it would hold that laws, at least in certain cases, are enacted with relatively specific intentions, and that this is a matter of fact which is discernible through an ordinary fact-finding procedure. Second, that in certain cases the presence of such a fact, namely, that the law was enacted with a certain intention, provides judges with a reason to decide the legal dispute in accordance with the relevant legislative intent.” [It is important that for Marmor, interpretive strategies are only appropriate in hard cases. However, we should wonder what grounds positivism can offer for deciding a hard case in any manner which is not the morally best manner according to the judgment of the judge. There are no legal grounds, by definition, in hard cases and so what justificatory strategy would endorse any judicial decision except for the morally best one?] We can ascribe intentions to a legislative body, when this is possible at all, by taking the shared intentions of the majority, rather than some group intention, to be the significant intentions. Moreover, we should expect a good deal of consensus about intentions for otherwise it is hard to explain how legislative bodies are able to produce so much legislation. Discussion of kinds of intentions concludes as follows: “I have distinguished between three main types of intention that are potentially relevant from the legal point of view. Apart from the intentions that are manifest in the language of the law itself, legislators typically have further intentions in enacting a given law, and sometimes they would have certain intentions bearing on its proper application. I have also suggested that some of these further intentions may be essentially non-avowable, in which case they are rendered initially irrelevant. Finally, I have pointed out that considerations of consistency require that the legislator’s application intentions be taken into account only if, and to the extent that, they are in accord with his further intentions.” The manner in which intentionalism is to be justif ed relies on the distinction, Marmor maintains, between expert and mere collective action authority. “The point I wanted to make is strictly conditional: if, and only if, a certain law is justified on the basis of the expertise branch of the normal justification thesis, would it make sense to defer to the legislature’s intentions in the interpretation of the law, that is, to the extent that there is, in fact,; such an intention and it can clarify something that needs clarification.” Insofar as a legislature can be considered an expert on the matter requiring adjudication and interpretation should the body’s intention be taken into account. Chapter 9: Constitutional Interpretation Begins by laying out general necessary features of a constitution. Then Marmor moves on to consider general questions of constitutional legitimacy. First, he considers what grounds might be offered for the legitimacy of a constitution at all. He notes that the central question is what permits a single generation to bind future generations. He then examines four arguments to avoid this problem. The first is based on the moral legitimacy of the constitution; Marmor rejects this solution for the [truly dubious] reason that the constitution would make no practical difference because it would not supply reasons in addition to those provided by morality [but, presumably, the point is that these legally valid rules would be institutionally enshrined]. The second is based on the moral authority of the framers and Marmor rightly rejects this on the basis of the idealization of the framers it requires and less rightly on the basis that there cannot be moral authorities. Third is the “argument from interpretation” which claims that “as long as the particular content of the constitution is determined by its interpretation, and the authoritative interpretation at any time correctly instantiates the values which ought to be upheld in the community, the constitution would be morally legitimate”. The fourth is Raz’s which argues that a constitution is valid so long as its constraints are morally permissible because conventions of this type might aid the continuity of the legal system, i.e. it is important to have a convention of this type and the constitution is a kind of this type. Marmor appears to endorse these two approaches and concludes: “The conditions for the legitimacy of a constitution must comprise the following conditions. First, the values and principles enshrined in it must be morally permissible…Second, when certain choices are made in particular cases, they would be legitimate if they are either morally underdetermined, or else, morally correct…It follows from this that both arguments must assume that at least in those areas in which the constitution would make a moral difference, it can be interpreted to make the difference that it should, that is, according to the true moral principles that should apply to the particular case...the moral legitimacy of constitutions very much dependent on the practices of their interpretation. In other words, a great deal of the burden of moral legitimacy is shifted by these arguments to the application of the constitution, thus assuming that the constitution is legitimate only if the courts are likely to apply the constitution in a morally desirable way.” [This implies that the only significant value constitutions have is in providing a conventional practice where some such practice is required.] Moves on to argue that courts typically rely, and rightly so, on moral considerations when deciding constitutional issues because typically, in cases of the kind that reach the supreme or constitutional court, there is no law on the case. Much of this relies on his earlier discussion about the nature of interpretation in the law – that legal interpretation is always a matter of cha ging the law. (shrink)

The presence of a basic and general form of creativity in people is investigated through experiments with music. The results indicate that “generative” creativity—‐the ability to spontaneously generate a music by varying a basic set of musical elements—is a basic human endowment, unlike “constructive” creativity—‐the type of creativity exhibited by composers and other artists—‐which is the result of training and the spehal development of faculties. Generative creativity's coming to the fore in contemporary people would contribute to the development of (...) the personality and help bring about more fulfilled, better balanced people and societies. (shrink)

There are several reasons to explore the role theater played in the life of Søren Kierkegaard and in the inspiration for his works. There are probably more reasons to analyze the role Kierkegaard played for theater, both as a source of inspiration and as a thinker reflecting on different facets of drama, performance, and acting. In the present study I focus on the diversity and complexity of Kierkegaard’s views on theater to elaborate on the possible connections and types of influence (...) he exercised on stage artists and theorists, shaping our contemporary theater theories. Approaches include literature, philosophy, theology, staging, acting, audience, and the history of theater, both in Kierkegaard’s times and later. (shrink)

The papers comprising Zur Farbenlehre, best known portion of Goethe's writings on color and optics, appeared between 1808 and 1810. Portions of Zur Farbenlehre, translated by the painter Charles Lock Eastlake and frequently reprinted under the title Theory of Colours, achieved immediate notoriety because of Goethe's insistent questioning of Newton's methodology. Acknowledging no mentors except Theophrastus and the physicist Robert Boyle, Goethe compared the Newtonian theory of colors--indelicately, some think--to a once proud castle still revered long after it has fallen (...) into decay. The furor aroused by this intemperate analogy has not fully subsided yet. During the twentieth century Max Planck answered the impertinence with a vigorous ad hominem attack and Rudolph Carnap followed suit. We still lack a serious analysis of Goethe's color theories or his arguments against Newton. And the early vituperation seems quaint. Questioning Newton's methodology--or even his epistemology--is a legitimate activity and has even become commonplace. The 1840 edition of Eastlake's translation enjoys a perennial popularity among artists and a few perceptual psychologists, who have been allowed to monopolize for too long a book which is primarily a highly provocative inquiry in the philosophy of science. Someday an imaginative publisher may be inspired to make available more extensive translations of Goethe's copious writings on color, complemented by the type of careful editing, annotation and analysis which I. Bernard Cohen and Alexander Koyré provide for the writings of Newton. Until then one is grateful for Eastlake.--P. S. (shrink)

Dust is a distinctive material that, in addition to its physical properties, reveals anthropological and cultural dimensions, particularly within aesthetic contexts. In a collaborative project focused on “dust,” a theoretical-systematic approach is combined with an artistic-practical-participatory one. Philosophical reflections and artistic concepts related to the material “dust,” specific artworks involving dust, and the relationship between artwork and theory are interwoven. Thus, the text discusses various types of dust, the role of the artist, different modes of perception, cultural context, (...) forms of dust and materiality, media-specific characteristics, and questions of mediation. Throughout this process, common practices of artistic and theoretical engagement are transformed and developed. (shrink)

What is the relationship between place and cultural production? How do we account for the interaction between the domain of spatiality and that of artistic expression? In particular, how might we conceptualize the connections between space and literature? Here, I attempt to map the principal ways in which the central thematic issues we associate with literary expression are related to questions about space and place. By elucidating these matters, I hope to arrive at a rationale for an approach to (...) setting out the links between symbolic expression and spatiality. Literature is envisaged here as one specific type of cultural expression, and the suggestion is that much of what applies to literature can be taken as... (shrink)

Raphaël Van Daele L’expression chinoise ziran 自然 s’entend aussi bien à propos de la « nature », cette région de l’être dont l’émergence et les processus ne relèvent pas de la volonté ou de l’action humaine, qu’à propos d’un certain type d’agir humain. Le geste expert de l’artiste est l’exemple privilégié d’un tel agir. Loin de nuire à la cohérence de cette notion, le fait que celle-ci soit mobilisée dans le discours théorique sur l’art autant que dans le discours (...) sur la nature témoigne d’un lien organique qui, dans la pensée chinoise, unit différents domaines de la réflexion. Le présent article propose d’esquisser une lecture philosophique de la notion de ziran. Pour ce faire, nous explorerons le rôle joué par ziran dans l’oeuvre esthétique de Cai Yong 蔡邕 (133-192) et dans la pensée de Guo Xiang 郭象 († 312). (shrink)

Contextual type theories are largely explored in their applications to programming languages, but less investigated for knowledge representation purposes. The combination of a constructive language with a modal extension of contexts appears crucial to explore the attractive idea of a type-theoretical calculus of provability from refutable assumptions for non-monotonic reasoning. This paper introduces such a language: the modal operators are meant to internalize two different modes of correctness, respectively with necessity as the standard notion of constructive verification (...) and possibility as provability up to refutation of contextual conditions. (shrink)

Prolegomena It is fitting to begin this book on intuitionistic type theory by putting the subject matter into perspective. The purpose of this chapter is to ...

Neil Tennant’s core logic is a type of bilateralist natural deduction system based on proofs and refutations. We present a proof system for propositional core logic, explain its connections to bilateralism, and explore the possibility of using it as a type theory, in the same kind of way intuitionistic logic is often used as a type theory. Our proof system is not Tennant’s own, but it is very closely related, and determines the same consequence relation. The difference, (...) however, matters for our purposes, and we discuss this. We then turn to the question of strong normalization, showing that although Tennant’s proof system for core logic is not strongly normalizing, our modified system is. (shrink)

I explore the possibility of a structuralist interpretation of homotopy type theory (HoTT) as a foundation for mathematics. There are two main aspects to HoTT's structuralist credentials. First, it builds on categorical set theory (CST), of which the best-known variant is Lawvere's ETCS. I argue that CST has merit as a structuralist foundation, in that it ascribes only structural properties to typical mathematical objects. However, I also argue that this success depends on the adoption of a strict typing system (...) which undermines the metaphysical seriousness of this structuralism. Homotopy type theory adds to CST a distinctive theory of identity between sets, which arguably allows its objects to be seen as ante rem structures. I examine the prospects for such a view, and address many other interpretive problems as they arise. (shrink)

We introduce a predicative version of topos based on the notion of small maps in algebraic set theory, developed by Joyal and one of the authors. Examples of stratified pseudotoposes can be constructed in Martin-Löf type theory, which is a predicative theory. A stratified pseudotopos admits construction of the internal category of sheaves, which is again a stratified pseudotopos. We also show how to build models of Aczel-Myhill constructive set theory using this categorical structure.

Intuitively speaking, a multiple artwork is one that admits of multiple ‘instances’ which are capable of playing a particular role in the appreciation of the work. The ‘explananda’ in the title of this article are things that have been proposed as requiring explanation by any adequate ontology of multiple artworks so conceived. This assumes that the ontology of art is in the business of explaining certain things, an assumption I defend. At least nine purported explananda have been proposed in the (...) relevant literature. I begin by offering a preliminary sketch of these explananda, identifying how they are grounded in our ordinary artistic practice and discourse, and how they have structured recent debates in the ontology of art. I next argue that the notion of ‘instance’ must be understood in a particular way if instance multiplicity is to capture the standard distinction between singular and multiple art forms. I then assess the relative significance and implications of the nine explananda for an adjudication of the debates in the ontology of art. I identify problems for the historically dominant ‘type’ theory of multiples, and propose an alternative account that speaks to all nine explananda. I conclude by reflecting on where this leaves us and how we should proceed. (shrink)

Law as Art presents a radical new legal theory, the Law as Art Hypothesis, which conceives law, not as a system of rules, but as a distinctive kind of art work. Law is differentiated as art by the Law as Compound Artistic Type Hypothesis, which uses the heuristic metaphor of the Operatic Music Drama, the most elementally complex compound art form, to develop an idea of legal art as a distinctive empowered text, supported by the arts of drama, (...) painting, sculpture, dress-design, architecture, rhetoric and communication to form an elementally developed yet integrated unitary art work. Part I develops a new realist epistemology to support a contemporary action-type ontology of art, differentiated as art by virtue of its artistic value. Part II opens with a critical review of the arts in legal theory, before detailing the Law as Art and Law as Compound Artistic Type Hypothesis and locating them within contemporary scholarship. Legal philosophical implications are considered and there is an acronym key and glossary, bibliography and index. (shrink)

This volume presents a Type Theory of Law (TTL), claiming that this is a unique theory of law that stems from the philosophical understanding of Jung's psychological types applied to the phenomenon of law. Furthermore, the TTL claims to be a universal, general and descriptive account of law. To prove that, the book first presents the fundamentals of Jungian psychological types, as they had been invented by Jung and consequently developed further by his followers. The next part of the (...) book describes how the typological structure of an individual determines their understanding of law. It then addresses the way in which inclusive legal theory can be understood based on this typology. Finally, the book describes the TTL in general and descriptive terms and puts it into context. All in all, the book shows how the integral or inclusive approach to understanding the nature of law is not only in tune with our time, but also relevant for presenting a more persuasive picture of law than the older exclusivist or dualist approaches of strict natural law and rigid legal positivism did. (shrink)

Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s type-restrictions are unjustifiable, (...) the type-restrictions imposed by STT are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for CTT. We end by examining an alternative approach to cumulative types due to Florio and Jones; we argue that their theory is best seen as a misleadingly formulated version of STT. (shrink)

Homotopy Type Theory is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics (...) might be expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory, and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics?2.1 A characterization of a foundation for mathematics2.2 Autonomy3 The Basic Features of Homotopy Type Theory3.1 The rules3.2 The basic ways to construct types3.3 Types as propositions and propositions as types3.4 Identity3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types5.1 Tokens as mathematical objects?5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy7.1 Framework7.2 Semantics7.3 Metaphysics7.4 Epistemology7.5 Methodology8 Possible Objections to this Account8.1 A constructive foundation for mathematics?8.2 What are concepts?8.3 Isn’t this just Brouwerian intuitionism?8.4 Duplicated objects8.5 Intensionality and substitution salva veritate9 Conclusion9.1 Advantages of this foundation. (shrink)

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

In his famous paper Der Wahrheitsbegriff in den formalisierten Sprachen (Polish edition: Nakładem/Prace Towarzystwa Naukowego Warszawskiego, wydzial, III, 1933), Alfred Tarski constructs a materially adequate and formally correct definition of the term “true sentence” for certain kinds of formalised languages. In the case of other formalised languages, he shows that such a construction is impossible but that the term “true sentence” can nevertheless be consistently postulated. In the Postscript that Tarski added to a later version of this paper (Studia Philosophica, (...) 1, 1935), he does not explicitly include limits for the kinds of language for which such a construction is possible. This absence of such limits has been interpreted as an implied claim that such a definition of the term “true sentence” can be constructed for every language. This has far-reaching consequences, not least for the widely held belief that Tarski changed from an universalistic to an anti-universalistic standpoint. We will claim that the consequence of anti-universalism is unwarranted, given that it can be argued that the Postscript is not in conflict with the existence of limits outside of which a definition of “true sentence” cannot be constructed. Moreover, by a discussion of transfinite type theory, we will also be able to accommodate other of the changes made in Tarski’s Postscript within a type-theoretical framework. The awareness of transfinite type theory afforded by this discussion will lead, in turn, to an account of Tarski’s Postscript that shows a gradual change in his logical work, rather than any of the more radical transitions which the Postscript has been claimed to reflect. (shrink)

Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems culminating in the well-known and powerful Calculus of Constructions. (...) The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalize mathematics. The only prerequisites are a good knowledge of undergraduate algebra and analysis. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarize themselves with the material. (shrink)

This paper is part of a broader project whose aim is to present a coherent unified approach to natural language dialogue semantics using tools from type theory. Here we explore aspects of our approach which relate to situation theory and situation semantics. We first point out a relationship between type theory and the Austinian notion of truth. We then consider how records in type theory might be used to represent situations and how dependent record types can be (...) used to model constraints on situations. We then sketch treatments of attitude phenomena for which Barwise and Perry proposed situation semantic analyses (perception complements, belief, the Pierre puzzle) as well as two other intensional phenomena (intensional verbs and intentional identity). Finally we give a characterisation of the type theory used and a small illustrative fragment of English. (shrink)

Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a foundation for (...) mathematics might be expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory (but is compatible with the homotopy interpretation), and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics? 2.1 A characterization of a foundation for mathematics 2.2 Autonomy3 The Basic Features of Homotopy Type Theory 3.1 The rules 3.2 The basic ways to construct types 3.3 Types as propositions and propositions as types 3.4 Identity 3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types 5.1 Tokens as mathematical objects? 5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy 7.1 Framework 7.2 Semantics 7.3 Metaphysics 7.4 Epistemology 7.5 Methodology8 Possible Objections to this Account 8.1 A constructive foundation for mathematics? 8.2 What are concepts? 8.3 Isn’t this just Brouwerian intuitionism? 8.4 Duplicated objects 8.5 Intensionality and substitution salva veritate9 Conclusion 9.1 Advantages of this foundation. (shrink)

Recent debates in metaphysics have highlighted the significance of type theories, such as Simple Type Theory (STT), for our philosophical analysis. In this chapter, I present the salient features of a constructive type theory in the style of Martin-Löf, termed CTT. My principal aim is to convey the flavour of this rich, flexible and sophisticated theory and compare it with STT. I especially focus on the forms of quantification which are available in CTT. A further aim (...) is to argue that a comparison between a plurality of theories is beneficial to the philosophical analysis. We may, for example, discover helpful features of one theory that we may want to import into another context, thus enriching our repertoire of formal tools. Or, through comparison with a less well-known theory, we may gain a better understanding of the characteristics of the theories we are familiar with. As argued in this chapter, CTT has much to offer in all of these respects. (shrink)

In the formal semantics based on modern type theories, common nouns are interpreted as types, rather than as predicates of entities as in Montague’s semantics. This brings about important advantages in linguistic interpretations but also leads to a limitation of expressive power because there are fewer operations on types as compared with those on predicates. The theory of coercive subtyping adequately extends the modern type theories and, as shown in this paper, plays a very useful role (...) in making type theories more expressive for formal semantics. It not only gives a satisfactory solution to the basic problem of ‘multiple categorisation’ caused by interpreting common nouns as types, but provides a powerful formal framework to model interesting linguistic phenomena such as copredication, whose formal treatment has been found difficult in a Montagovian setting. In particular, we show how to formally introduce dot-types in a type theory with coercive subtyping and study some type-theoretic constructs that provide useful representational tools for reference transfers and multiple word meanings in formal lexical semantics. (shrink)

In recent years, several formalisms for program construction have appeared. One such formalism is the type theory developed by Per Martin-L f. Well suited as a theory for program construction, it makes possible the expression of both specifications and programs within the same formalism. Furthermore, the proof rules can be used to derive a correct program from a specification as well as to verify that a given program has a certain property. This book contains a thorough introduction to (...) class='Hi'>type theory, with information on polymorphic sets, subsets, monomorphic sets, and a full set of helpful examples. (shrink)

Selection type theories solve adaptation problems. Natural selection, clonal selection for antibody production, and selective theories of higher brain function are examples. An abstract characterization of typical selection processes is generated by analyzing and extending previous work on the nature of natural selection. Once constructed, this abstraction provides a useful tool for analyzing the nature of other selection theories and may be of use in new instances of theory construction. This suggests the potential fruitfulness of research (...) to find other theory types and construct their abstractions. (shrink)

The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Per Martin-Löf into homotopy theory and higher-dimensional category theory.

A version of intuitionistic type theory is presented here in which all logical symbols are defined in terms of equality. This language is used to construct the so-called free topos with natural number object. It is argued that the free topos may be regarded as the universe of mathematics from an intuitionist's point of view.

We suggest a way of bringing together type theory and unification-based grammar formalisms by using records in type theory. The work is part of a broader project whose aim is to present a coherent unified approach to natural language dialogue semantics using tools from type theory.

The paradox of propositions, presented in Appendix B of Russell's The Principles of Mathematics (1903), is usually taken as Russell's principal motive, at the time, for moving from a simple to a ramified theory of types. I argue that this view is mistaken. A closer study of Russell's correspondence with Frege reveals that Russell carne to adopt a very different resolution of the paradox, calling into question not the simplicity of his early type theory but the simplicity of his (...) early theory of propositions. (shrink)

The article deals with an approach to the analysis of propositional attitudes based on the type-theoretical semantics proposed by A. Ranta and originating from the type theory of P. Martin-Löf. Type-theoretical semantics contains the notion of context and tools of extracting information from it in an explicit form. This allows us to correctly formalize the dependence on contexts typical of propositional attitudes. In the article the context is presented as a dependent sum type (Record type (...) in the proof assistant Coq). Ranta’s approach is refined and applied to the analysis of Quine’s phrase “Ralph believes that someone is a spy”. Three variants of formalization for this phrase are described which differ in the content of contextual knowledge and the way the truth values of the phrase are derived. Contexts are connected through the function of conversion, making it possible to relate truth values. As a result, it is shown that the instruments for working with contexts provided by type-theoretical semantics allow us to avoid the problem of opacity described by Quine. Provided formalization along with proofs is coded in Coq and made freely available. (shrink)

The aim of this paper is to define a partial Propositional Type Theory. Our system is partial in a double sense: the hierarchy of (propositional) types contains partial functions and some expressions of the language, including formulas, may be undefined. The specific interpretation we give to the undefined value is that of Kleene’s strong logic of indeterminacy. We present a semantics for the new system and prove that every element of any domain of the hierarchy has a name in (...) the object language. Finally, we provide a proof system and a (constructive) proof of completeness. (shrink)

In this article we define a logical system called Hybrid Partial Type Theory ( $\mathcal {HPTT}$ ). The system is obtained by combining William Farmer’s partial type theory with a strong form of hybrid logic. William Farmer’s system is a version of Church’s theory of types which allows terms to be non-denoting; hybrid logic is a version of modal logic in which it is possible to name worlds and evaluate expressions with respect to particular worlds. We motivate this (...) combination of ideas in the introduction, and devote the rest of the article to defining, axiomatising, and proving a completeness result for $\mathcal {HPTT}$. (shrink)

The proofs of universally quantified statements, in mathematics, are given as “schemata” or as “prototypes” which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of λ-calculus and act as “proof-schemata”, as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i. (...) e. Girard's system F, where type-quantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context. (shrink)

Pure type systems arise as a generalisation of simply typed lambda calculus. The contemporary development of mathematics has renewed the interest in type theories, as they are not just the object of mere historical research, but have an active role in the development of computational science and core mathematics. It is worth exploring some of them in depth, particularly predicative Martin-Löf’s intuitionistic type theory and impredicative Coquand’s calculus of constructions. The logical and philosophical differences and similarities (...) between them will be studied, showing the relationship between these type theories and other fields of logic. (shrink)

This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory, an important form (...) of type theory based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined. (shrink)

This article proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, and Licata as the (...) implicit foundation, confining our presentation to elementary results such as function extensionality, the derivation of weak connections and path induction, the groupoid structure of types, and the Eckmman–Hilton duality. (shrink)

A logic-enriched type theory is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named and , which we claim correspond closely to the classical predicative systems of second order arithmetic and . We justify this claim by translating each second order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise (...) different approaches to the foundations of mathematics.The two LTTs we construct are subsystems of the logic-enriched type theory , which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system has also been claimed to correspond to Weyl’s foundation. By casting and as LTTs, we are able to compare them with . It is a consequence of the work in this paper that is strictly stronger than .The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work. (shrink)

The logic of identity contains riches not seen through the coarse lens of predicate logic. This is one of several lessons to draw from the subtle treatment of identity in Martin‐Löf type theory, to which the reader will be introduced in this article. After a brief general introduction we shall mainly be concerned with the distinction between identity propositions and identity judgements. These differ from each other both in logical form and in logical strength. Along the way, connections to (...) philosophical debates concerning identity are noted. Some use of logical symbolism is inevitable in any serious discussion of type theory, but the emphasis here is on basic ideas rather than technicalities. (shrink)

In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B prop [x : A], i. e. to require that the predicate ∨ ¬ B) is provable, is equivalent, when working within the framework of Martin-Löf's Intuitionistic Type Theory, to require that there exists a decision function ψ: A → Boole such that = Booletrue) ↔ B). Since we will also show that the proposition x = Booletrue [x: Boole] is decidable, we (...) can alternatively say that the main result of this paper is a proof that the decidability of the predicate Bprop [x : A] can be effectively reduced by a function ψ A → Boole to the decidability of the predicate ψ = Booletrue [x : A]. All the proofs are carried out within the Intuitionistic Type Theory and hence the decision function ψ, together with a proof of its correctness, is effectively constructed as a function of the proof of ∨ ¬ B). (shrink)