Although logical settings are typically concerned with tracking alethic considerations, frameworks exist in which topic-theoretic considerations—e.g., tracking subject-matter or topic—are given equal importance. Intuitions about extending topic through a propositional language are generally straightforward for extensional cases. For a number of reasons, arriving at a compelling account of the subject-matter of intensional operators—such as intensional conditionals—is a more difficult task. In particular, the framework of topic-sensitive intentional modals (TSIMs) championed by Francesco Berto and his collaborators leave the topics (...) of intensional formulae undefined, which artificially constricts the expressivity of the theory. This paper proposes an approach to fill in this lacuna, emphasizing an analogous problem in Parry-style containment logics. In this setting, the approach receives a proof-of-concept through the introduction of a natural and general family of subsystems of Parry’s $${\textsf{PAI}}$$ PAI —with sound and complete axiomatizations—that allow a fine degree of control over the topics of intensional conditionals. (shrink)

The purpose of this article is to provide a non‐contradictory interpretation of sentences such as “Smith's murderer might not have murdered Smith”. An anti‐actualist, two‐dimensional framework including partial functions provides the basis for my solution. I argue for two claims. The modal profile of the proposition expressed by “The F might not have been an F” is complex: at any world where there is a unique F the proposition is true; at any world without a unique F the proposition has (...) no truth‐value; hence, at no world is it false. It remains an open semantic and epistemological question which of the first two kinds of world the actual world is. The semantic method should be based on explicit intensionalization in lieu of actualism. Actualism accords a privileged role to the actual world. Explicit intensionalization places all possible worlds, including the actual one, on an equal footing. Syntactically, a lambda‐bound world variable replaces the actual‐world constant or operator, while the other world variable is existentially bound. (shrink)

Consistency with the formal Church’s thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by Maietti and Sambin From sets and types to topology and analysis: practicable foundations for constructive mathematics, Oxford University Press, Oxford, 2005). Here we show that this is the case for the intensional level of the two-level Minimalist Foundation, for (...) short MF, completed in 2009 by the second author. The intensional level of MF consists of an intensional type theory à la Martin-Löf, called mTT. The consistency of mTT with CT and AC is obtained by showing the consistency with the formal Church’s thesis of a fragment of intensional Martin-Löf’s type theory, called \, where mTT can be easily interpreted. Then to show the consistency of \ with CT we interpret it within Feferman’s predicative theory of non-iterative fixpoints \ by extending the well known Kleene’s realizability semantics of intuitionistic arithmetics so that CT is trivially validated. More in detail the fragment \ we interpret consists of first order intensional Martin-Löf’s type theory with one universe and with explicit substitution rules in place of usual equality rules preserving type constructors -rule which is not valid in our realizability semantics). A key difficulty encountered in our interpretation was to use the right interpretation of lambda abstraction in the applicative structure of natural numbers in order to model all the equality rules of \ correctly. In particular the universe of \ is modelled by means of \-fixpoints following a technique due first to Aczel and used by Feferman and Beeson. (shrink)

Attempting to deflate ontological debates, the proponent of Quantifier Variance (QV) claims that there are multiple quantifier meanings of equal metaphysical merit. According to Hirsch—the main proponent of QV—metaphysical merit should be understood intensionally: two languages have equal merit if they allow us to express the same possibilities. I examine the notion of metaphysical merit and its purported link to intensionality. That link, I argue, should not be supported by adopting an intensional theory of semantic content. Rather, I give (...) a general strategy for supporting claims about metaphysical merit and examine whether that strategy can be used to link merit and intensionality. Though I don’t deliver a definitive verdict, the discussion provides a clearer framework for articulating and evaluating claims about metaphysical merit. (shrink)

Franz Brentano upheld the medieval Aristotelian doctrine of intentionality as a mark of the mental, distinguishing physical from psychological phenomena by the intentionality of the psychological and nonintentionality of the physical. But to implement, even to fully understand and appreciate Brentano’s thesis, it is necessary to know when intentionality does or does not obtain. The task of formulating satisfactory criteria of intentionality has proved elusive. The magnitude of difficulty is indicated by the number and variety of criteria proposed and defeated (...) by counter examples. Intentionality may be a primitive irreducible property incapable of reductive conceptual analysis. But nonreductive criteriological characterizations of intentionality demonstrate an equally stubborn resistance to philosophical methods. (shrink)

We present a version of arithmetic in all finite types based on a systematic use of an internally definable notion of observational equivalence for dealing with equalities at higher types. For this system both intensional and extensional models are possible, the deduction theorem holds and the soundness of the Dialectica interpretation is provable inside the system itself.

We give a formal interpretation of Martin-Löf's Constructive Theory of Types in Elementary Topos Theory which is presented as a formalised theory with intensional equality of objects. Types are interpreted as arrows and variables as sections of their types. This is necessary to model correctly the working of the assumption x ∈ A. Then intensional equality interprets equality of types. The normal form theorem which asserts that the interpretation of a type is intensional equal (...) to the pullback of its “alignment” along some “base” arrow relates this interpretation to categorical semantic of types. (shrink)

Dans cette note, nous étudions la structure logique de la notion d'égalité. Après avoir présenté divers concepts connexes à la notion d'égalité, nous suggérons que les notions de propriétés homotopiques et de propriétés hétérotopiques constituent le support logique et sémantique d'une théorie de l'égalité qui aille au-delà de la pure analyse syntaxique des concepts.In this note, we examine the logical structure of the notion of equality. After having introduced the various concepts which are traditionally linked with the notion of (...) equality, extensional and intensional equality, we suggest that the notion of homotopic versus heterotopic properties constitute the logico-semantical basis for a theory of equality beyond the mere syntactical analysis of the relevant concepts. (shrink)

The Transparent Intensional Logic explicates the meaning of a linguistic expression as a construction. The construction is a hyperintensional entity. It is characterised as instructions for a „calculation“ of a concrete value. In the terminology of Pavel Tichy a linguistic expression denotes its meaning , which construes the signification of the expression. If the signification is an intension, we can call it a reference of the expression. In several semantic conceptions the question of the denotation and of the reference (...) are not treated equally. The terminological chaos can be resolved in a specific way thanks to „fine-grained“ character of the logical analysis offered by TIL. (shrink)

The classical theory of semantic information (ESI), as formulated by Bar-Hillel and Carnap in 1952, does not give a satisfactory account of the problem of what information, if any, analytically and/or logically true sentences have to offer. According to ESI, analytically true sentences lack informational content, and any two analytically equivalent sentences convey the same piece of information. This problem is connected with Cohen and Nagel's paradox of inference: Since the conclusion of a valid argument is contained in the premises, (...) it fails to provide any novel information. Again, ESI does not give a satisfactory account of the paradox. In this paper I propose a solution based on the distinction between empirical information and analytic information. Declarative sentences are informative due to their meanings. I construe meanings as structured hyperintensions, modelled in Transparent Intensional Logic as so-called constructions. These are abstract, algorithmically structured procedures whose constituents are sub-procedures. My main thesis is that constructions are the vehicles of information. Hence, although analytically true sentences provide no empirical information about the state of the world, they convey analytic information, in the shape of constructions prescribing how to arrive at the truths in question. Moreover, even though analytically equivalent sentences have equal empirical content, their analytic content may be different. Finally, though the empirical content of the conclusion of a valid argument is contained in the premises, its analytic content may be different from the analytic content of the premises and thus convey a new piece of information. (shrink)

In this paper we analyze an extension of Martin-Löf s intensional set theory by means of a set contructor P such that the elements of P are the subsets of the set S. Since it seems natural to require some kind of extensionality on the equality among subsets, it turns out that such an extension cannot be constructive. In fact we will prove that this extension is classic, that is “ true holds for any proposition A.

It is argued herein that there are two distinct ways in which all observation vocabularies are prejudiced with respect to theory. An argument based on the demands of adequate translation is invoked to show that even the simplest of our observation predicates must display the first and more obvious grade of bias--intensional bias. It is also argued that any observation vocabulary whose predicates are corrigibly applicable must manifest a second and equally serious grade of bias--extensional bias--independently of whatever (...) class='Hi'>intensional bias its predicates may or may not have. (shrink)

The analysis of atomic sentences and their subatomic components poses a special problem for proof-theoretic approaches to natural language semantics, as it is far from clear how their semantics could be explained by means of proofs rather than denotations. The paper develops a proof-theoretic semantics for a fragment of English within a type-theoretical formalism that combines subatomic systems for natural deduction [20] with constructive (or Martin-Löf) type theory [8, 9] by stating rules for the formation, introduction, elimination and equality (...) of atomic propositions understood as types (or sets) of subatomic proof-objects. The formalism is extended with dependent types to admit an interpretation of non-atomic sentences. The paper concludes with applications to natural language including internally nested proper names, anaphoric pronouns, simple identity sentences, and intensional transitive verbs. (shrink)

In this dissertation, we construct a consistent, complete quotational logic G$\sb1$. We first develop a semantics, and then show the undecidability of circular quotation and anaphorism . Next, a complete axiom system is presented, and completeness theorems are shown for G$\sb1$. We show that definable truth exists in G$\sb1$. ;Later, we replace equality in G$\sb1$ with an equivalence relation. An axiom system and completeness theorems are provided for this equality-free version of G$\sb1$, which is useful in program verification. (...) ;Interpolation and definability are discussed in intensional and extensional contexts. We show that G$\sb1$ contains the complete first order theory of arithmetic by implicit definitional extension. ;The last chapter deals with representability and fixed points. Derived induction schemata are produced, and G$\sb1$ is shown to be a recursion theory. We give an example of how G$\sb1$ may be used to solve domain equations of Scott. (shrink)

In an earlier paper, [5], I gave semantics and tableau rules for a simple firstorder intensional logic called FOIL, in which both objects and intensions are explicitly present and can be quantified over. Intensions, being non-rigid, are represented in FOIL as (partial) functions from states to objects. Scoping machinery, predicate abstraction, is present to disambiguate sentences like that asserting the necessary identity of the morning and the evening star, which is true in one sense and not true in another.In (...) this paper I address the problem of axiomatizing FOIL. I begin with an interesting sublogic with predicate abstraction and equality but no quantifiers. In [2] this sublogic was shown to be undecidable if the underlying modal logic was at least K4, though it is decidable in other cases. The axiomatization given is shown to be complete for standard logics without a symmetry condition. The general situation is not known. After this an axiomatization for the full FOIL is given, which is straightforward after one makes a change in the point of view. (shrink)

I attempt to provide a general account of deontic context, equally applicable within and an intensional and a hyperintensional framework; I compare Rothschild's and Yablo's accounts of the semantics for deontic logic, deontic updating and denotic duality with my own accounts; and I conclude with some general remarks on negation.

This chapter is concerned with representing the semantics of natural language plurals and mass terms in property theory; a weak first-order theory of Truth, Propositions and Properties with fine-grained intensionality (Turner 1990, Turner 1992, Aczel 1980). The theory allows apparently coreferring items to corefer without inconsistency. This is achieved by using property modifiers which keep track of the property used to refer to a term, much like Landman’s roles (Landman 1989). We can thus predicate apparently contradictory properties of “the judge” (...) and “the cleaner,” for example, even if they turn out to be the same individual. The same device can also be used to control distribution into mereological terms: when we say “the dirty water is liquid,” we can infer that those parts that are dirty water are liquid without inferring that the dirt is liquid. The theory shows how we can formalise some aspects of natural language semantics without being forced to make certain ontological commitments. This is achieved in part by adopting an axiomatic methodology. Axioms are proposed that are just strong enough to support intuitively acceptable inferences, whilst being weak enough for some ontological choices to be avoided (such as whether or not the extensions of mass terms should be homogeneous or atomic). The axioms are deliberately incomplete, just as in basic PT, where incomplete axioms are used to avoid the logical paradoxes. The axioms presented are deliberately too weak to say much about ”non-denoting’ definite descriptors. For example, we cannot erroneously prove that they are all equal. Neither can we prove that predication of such definites results in a proposition. This means that we cannot question the truth of sentences such as “the present king of France is bald.”. (shrink)

If every concept ist not, in trivial terms, to have a content, then the content of a concept has to be understood as the set of all concepts that it contains , the concept itself and the tautological concept being excepted. Since a negated concept can only include, in terms of content, the negations of the special concepts, the negated individual concepts are consequently without content . The individual concept, whose content is the sufficient condition for establishing a possible individual (...) as extension, is indispensable if Leibniz's theses are to be upheld: equality of extent is also always equality of content. The intensional conceptual estimation can also be extensionally interpreted. Conceptual negations are however indispensable if a Spinozistic monism is to be avoided and a multitude of the most general elementary concepts obtained. (shrink)

I investigate the role of the subject in judgment in Kant, Frege, and Husserl, situating it in the broader and less-often-considered context of their accounts of presentation as well as judgment. Contemporary philosophical usage of “representation” tends to elide the question of what Kant called the constitution of content, because of a reluctance, traced to Frege’s anti-psychologism, to attend to subjectivity. But for Kant and Husserl, anti-psychologism allows for synthesis as the subjective act necessary for both “mere presentation” and judgment. (...) In Begriffshrift, Frege alludes to a significant logical role for the subjective act of judgment, and in later work, traces of this logical role remain in the intensional notions of grasping a thought and judging as acknowledging its truth. But Frege’s anti-psychologism blocks interpreting these subjective notions in term of synthesis. Although similar in certain ways to Frege and equally anti-psychologistic, Husserl’s theory of judgment in the Logical Investigations maintains a role for subjective syntheses for presentations and judgments, and goes beyond Kant in allowing for a kind of objectivity at the level of non-judgment presentations. These two great anti-psychologists at the dawn of the parallel heydays of linguistic and phenomenological analysis are thus differentiated by the fates they assign to the act of synthesis. (shrink)

The dominant conceptions of moral status in the English-speaking literature are either holist or individualist, neither of which accounts well for widespread judgments that: animals and humans both have moral status that is of the same kind but different in degree; even a severely mentally incapacitated human being has a greater moral status than an animal with identical internal properties; and a newborn infant has a greater moral status than a mid-to-late stage foetus. Holists accord no moral status to any (...) of these beings, assigning it only to groups to which they belong, while individualists such as welfarists grant an equal moral status to humans and many animals, and Kantians accord no moral status either to animals or severely mentally incapacitated humans. I argue that an underexplored, modal-relational perspective does a better job of accounting for degrees of moral status. According to modal-relationalism, something has moral status insofar as it capable of having a certain causal or intensional connection with another being. I articulate a novel instance of modal-relationalism grounded in salient sub-Saharan moral views, roughly according to which the greater a being's capacity to be part of a communal relationship with us, the greater its moral status. I then demonstrate that this new, African-based theory entails and plausibly explains the above judgments, among others, in a unified way. (shrink)

In this paper we define Martin-L¨of complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional Martin-L¨of type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on the category of 1-truncated Martin-L¨of complexes and that this category is Quillen equivalent to the category of groupoids. In particular, 1-truncated Martin-L¨of (...) complexes are a model of homotopy 1-types. In order to establish these facts we give a proof-theoretic analysis, using a modified version of Tait’s logical predicates argument, of the propositional equality classes of terms of identity type in the 1-truncated theory. (shrink)

When formalizing mathematics in constructive type theories, or more practically in proof assistants such as Coq or Agda, one is often using setoids. In this note we consider two categories of setoids with equality on objects and show, within intensional Martin-Löf type theory, that they are isomorphic. Both categories are constructed from a fixed proof-irrelevant family F of setoids. The objects of the categories form the index setoid I of the family, whereas the definition of arrows differs. The (...) first category has for arrows triples →F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,\rightarrow \,F)$$\end{document} where f is an extensional function. Two such arrows are identified if appropriate composition with transportation maps makes them equal. In the second category the arrows are triples 2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2)$$\end{document} where R is a total functional relation between the subobjects F,F↪Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F, F \hookrightarrow \Sigma $$\end{document} of the setoid sum of the family. This category is simpler to use as the transportation maps disappear. Moreover we also show that the full image of a category along an E-functor into an E-category is a category. (shrink)

I critically discuss both the particular doctrinal and general meta-philosophical or methodological tenets of Mark Johnston's paper "Human Beings", attending to several weaknesses in his argument. One of the most important amongst them is an apparent reliance on a substitution of identicals within an intensional context as he argues that continuity of functioning brain is essential to the persistence of "Human Beings" as allegedly singled out by his methodology; another equally important is a simple lacuna in place of an (...) argument that candidate entities for re-identification by means we take for granted in the case of persons cannot be what I call "mentalistically" individuated. (shrink)

The question of the independence of the axioms of the theory of sets has been dealt with in a number of works, although not in a final manner. The writer will be concerned solely with the axiomatic system of Zermelo and Fraenkel, and only with that feature of the system whereby all the objects of the underlying domain are sets (so that there is no difference between objects in general and sets in particular).A special place among the axioms is occupied (...) by a purely relational one, an axiom of definiteness, which establishes the character of equality within the system.Zermelo introduces equality intensionally—if two symbolsxandyrepresent the same object, we write = (x, y). According to this there is no necessity for axioms to assure the interchangeability of equal objects as arguments of the primitive relationϵ(, ). For under the intensional interpretation it is clear that:Similarly it is clear that there is no necessity for a separate axiom that will assure to equality the properties of an equivalence relation. Therefore Zermelo introduces only the followingaxiom of extensionality:(if one set is a subset of another set, and the second set is also a subset of the first, then the two sets are equal). (shrink)

What kind of equality should we value and why? Current debate centres around whether distributive equality is valuable. However, it is not the only (potentially) morally significant form of equality. David Miller and T. M. Scanlon have emphasised the importance of social equality—a strongly egalitarian notion distinct from distributive equality, and which cannot be reduced to a concern for overall welfare or the welfare of the worst-off. However, as debate tends to focus on distribution, social (...) equality has been neglected and we do not have a clear understanding of what it is and why it might be valuable. This paper aims to address this gap. (shrink)

As part of recent epistemic challenges to democracy, some have endorsed the implementation of epistemic constraints on voting, institutional mechanisms that bar incompetent voters from participating in public decision-making procedures. This proposal is often considered incompatible with a commitment to political equality. In this paper, I aim to dispute the strength of this latter claim by offering a theoretical justification for epistemic constraints on voting that does not rest on antiegalitarian commitments. Call this the civic accountability justification for epistemic (...) constraints on voting. On this view, voters stand to one another in a normative relation of reciprocal accountability that requires them to uphold an epistemically responsible conduct. Modest epistemic constraints on voting are justifiable because they secure that participation in voting is conforming to this normative requirement. Thanks to its distinctive features, the civic accountability justification can overcome two problems that egalitarians commonly associate with the idea of epistemic constraints on voting: the disrespect problem and the hierarchy problem. (shrink)

Many writers claim that democratic government rests on a principled commitment to the ideal of political equality. The ideal of political equality holds that political institutions ought to be arranged so that they distribute political standing equally to all citizens. I reject this common view. I argue that the ideal of political equality, under its most plausible characterizations, lacks independent justificatory force. By casting doubt on the ideal of political equality, I provide indirect support for the (...) claim that democratic government is only instrumentally justified. (shrink)

This is a concise and profound book from one of the world's leading political and legal philosophers about a major theme, equality, and the proposition that humans are all one another's equals. Jeremy Waldron explores the implications of this fundamental tenet for law, politics, society and economy in the company of John Locke, whose work Waldron regards 'as well-worked-out a theory of basic equality as we have in the canon of political philosophy'. Throughout the text, which is based (...) on the Carlyle Lectures given in Oxford in 1999, Jeremy Waldron discusses contemporary approaches to equality and rival interpretations of Locke, and this dual agenda gives the whole an unusual degree of accessibility and intellectual excitement, of interest to philosophers, political theorists, lawyers and theologians around the world. (shrink)

In recent years, a distinction between two concepts of equality has been much discussed: 'distributive’ equality involves people having equal amounts of a good such as welfare or resources, and ‘social’ or ‘relational’ equality involves the absence of social hierarchy and the presence of equal social relations. This contrast is commonly thought to have important implications for our understanding of the relationship between equality and justice. But the nature and significance of the distinction is far from (...) clear. I examine several accounts of this issue and argue none are entirely satisfactory. In turn, I offer an alternative proposal. Relational equality, on my account, involves a concern with each person having an equal 'civic status'. I characterize this concern and show it has distinctive and normatively significant positional and temporal aspects. (shrink)

Beginning with liberalism's foundational idea of moral equality as the basis for treating people with equal concern and respect, Christine Koggel offers a modified account of what makes human beings equal and what is needed to achieve equality. Koggel utilizes insights from care ethics but switches the focus from care as a moral response within personal relationships to the broader network of relationships within which care is given or withheld. The result is an account of moral personhood and (...) agency that is richer than the view propounded by liberal theory and care ethics. This exciting and original work challenges theoretical resistance to the idea that relationships are relevant to an understanding of equality, and it provides an opening to a critical analysis of relationships and care that informs and transforms our understanding of what is needed to treat all people with equal concern and respect. (shrink)