Quentin Smith has argued that the new tenseless theory of time is faced with insurmountable problems and should be abandoned in favour of the tensed theory of time. Smith;s main argument attacks the fundamental premise of the tenseless theory: that tenseless truth conditions for tokens of tensed sentences adequately capture the meaning of tensed sentences. His position is that tenseless truth conditions cannot explain the logical relations between tensed sentences, thus the tensed theory must be accepted. Against Smith, (...) this paper adopts an alternative approach to the explanation of the entailment relations between sentences which contain indexicals. The approach drops the reliance upon tokens and instead relies on the evaluation of sentence types with respect to a context rather than upon actual or possible utterances of tokens of the types. This (new) version of the tenseless theory of time can adequately explain the relevant entailment relations between tensed sentences. (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)

We study, from a classical point of view, how the truth of a statement about higher type functionals depends on the underlying model. The models considered are the classical set-theoretic finite type hierarchy and the constructively more meaningful models of continuous functionals, hereditarily effective operations, as well as the closed term model of Gödel's system T. The main results are characterisations of prenex classes for which truth in the full set-theoretic model transfers to truth in the other (...) models. As a corollary we obtain that the axiom of choice is not conservative over Gödel's system T with classical logic for closed ∃2-formulas. We hope that this study will contribute to a clearer delineation and perception of constructive mathematics from a classical perspective. (shrink)

We present an overview of typed and untyped disquotational truth theories with the emphasis on their (non)conservativity over the base theory of syntax. Two types of conservativity are discussed: syntactic and semantic. We observe in particular that TB—one of the most basic disquotational theories—is not semantically conservative over its base; we show also that an untyped disquotational theory PTB is a syntactically conservative extension of Peano Arithmetic.

This book is a contribution to the flourishing field of formal and philosophical work on truth and the semantic paradoxes. Our aim is to present several theories of truth, to investigate some of their model-theoretic, recursion-theoretic and proof-theoretic aspects, and to evaluate their philosophical significance. In Part I we first outline some motivations for studying formal theories of truth, fix some terminology, provide some background on Tarski’s and Kripke’s theories of truth, and then discuss the prospects (...) of classical type-free truth. In Chapter 4 we discuss some minimal adequacy conditions on a satisfactory theory of truth based on the function that the truth predicate is intended to fulfil on the deflationist account. We cast doubt on the adequacy of some non-classical theories of truth and argue in favor of classical theories of truth. Part II is devoted to grounded truth. In chapter 5 we introduce a game-theoretic semantics for Kripke’s theory of truth. Strategies in these games can be interpreted as reference-graphs of the sentences in question. Using that framework, we give a graph-theoretic analysis of the Kripke-paradoxical sentences. In chapter 6 we provide simultaneous axiomatizations of groundedness and truth, and analyze the proof-theoretic strength of the resulting theories. These range from conservative extensions of Peano arithmetic to theories that have the full strength of the impredicative system ID1. Part III investigates the relationship between truth and set-theoretic comprehen- sion. In chapter 7 we canonically associate extensions of the truth predicate with Henkin-models of second-order arithmetic. This relationship will be employed to determine the recursion-theoretic complexity of several theories of grounded truth and to show the consistency of the latter with principles of generalized induction. In chapter 8 it is shown that the sets definable over the standard model of the Tarskian hierarchy are exactly the hyperarithmetical sets. Finally, we try to apply a certain solution to the set-theoretic paradoxes to the case of truth, namely Quine’s idea of stratification. This will yield classical disquotational theories that interpret full second-order arithmetic without set parameters, Z2- . We also indicate a method to recover the parameters. An appendix provides some background on ordinal notations, recursion theory and graph theory. (shrink)

Using focus groups, this small-scale, qualitative study investigated the way that students tend to think about religious truth-claims as compared to other types of truth-claim. All the student participants conceived of religious truth-claims as ‘opinions’, to be contrasted with the certain, indisputable ‘facts’ of science. For many students, it was the lack of empirical verification, as well as the existence of disagreement, which meant religious beliefs were relegated to this position. If these findings are generalisable, then (...) there are implications for the ongoing theoretical dispute over the extent to which truth should be a focus in religious education. The tendency for students to see religious claims as subjective lends support for a critical pedagogy that places evaluation of truth centre-stage. The findings also suggest a need for religious education and science teachers to include more reflection on the nature of the scientific method in their schemes of work in order to dispel the myths of unanimity and certainty in science. (shrink)

A statement is said to be true when the historic state of affairs to which it is correlated by the demonstrative conventions (the ones to which it ‘refers’) is of a type with which the sentence used in making it is correlated by the descriptive conventions.

In the paper we investigate typed axiomatizations of the truth predicate in which the axioms of truth come with a built-in, minimal and self-sufficient machinery to talk about syntactic aspects of an arbitrary base theory. Expanding previous works of the author and building on recent works of Albert Visser and Richard Heck, we give a precise characterization of these systems by investigating the strict relationships occurring between them, arithmetized model constructions in weak arithmetical systems and suitable set existence (...) axioms. The framework considered will give rise to some methodological remarks on the construction of truth theories and provide us with a privileged point of view to analyze the notion of truth arising from compositional principles in a typed setting. (shrink)

Freedom interpreted as absence of restraints, as such, is a vacuous ideal. It requires reference to some setting and course of action in order to distinguish those restraints that block human effort from those, say, that support it. More broadly, this notion of freedom has little meaning and less value without some external criterion for evaluating human action and its conditions. Many thinkers have argued that the. criterion must be the absolute Truth ? truth that is unconditioned, indubitable, (...) and imperative. They have often argued that there is no freedom of error, and that the many persons who do not know such Truth are free only in obeying the few who do. This concept of freedom is often regarded as intolerable. A third possibility is to interpret freedom as a name for a range of significant alternatives for thought and action. Freedom, here, is compatible with truth interpreted in terms of the process of inquiry, rather than as substantive possession. It is a necessary condition for judging which restraints are real obstacles and which courses of action may be harmless, legitimate or good. The third meaning, especially, is illustrated with reference to the freedoms of utterance (expression) and speech (communication). (shrink)

Guy Rohrbaugh and Allan Hazlett have provided two arguments against the thesis that musical works are types. In short, they assume that, according to our modal talk and intuitions, musical works are modally flexible entities; since types are modally inflexible entities, musical works are not types. I argue that Rohrbaugh’s and Hazlett’s arguments fail and that the type/token theorist can preserve the truth of our modal claims and intuitions even if types are modally inflexible entities. (...) First, I consider two alternatives for the type/token theorist proposed in recent literature about the topic: the created types’ solution and the created abstract objects’ solution. I argue that none of them is attractive for the type/token theorist because they do not preserve the theoretical advantages of type/token theories to explain musical works’ repeatable nature. Then, I focus on the arguments’ common premise that musical works are modally flexible entities. A deeper analysis of musical practice will show that this premise is not true: our modal claims do not imply that musical works could have had different intrinsic but, instead, extrinsic properties. Finally, I show how the nested types theory may offer a satisfactory explanation of this fact and how it captures the truth of our modal talk about musical works. (shrink)

Biological taxonomists rely on the so-called ‘type method’ to regulate taxonomic nomenclature. For each newfound taxon, they lay down a ‘type specimen’ that carries with it the name of the taxon it belongs to. Even if a taxon’s circumscription is unknown and/or subject to change, it remains a necessary truth that the taxon’s type specimen falls within its boundaries. Philosophers have noted some time ago that this naming practice is in line with the causal theory of reference and its (...) central notion of rigid designation: a type specimen fixes the reference of a taxon name without defining it. Recently, however, this consensus has come under pressure in the pages of this journal. In a series of articles by Alex Levine, Joseph LaPorte, and Matthew Haber, it has been argued that type specimens belong only contingently to their species, and that this may bode problems for the relation between type method and causal theory. I will argue that this ‘contingency debate’ is a debate gone wrong, and that none of the arguments in defense of contingency withstand scrutiny. Taxonomic naming is not out of step with the causal theory, but conforms to it. However, I will also argue that this observation is itself in need of further explanation, since application of the type method in taxonomic practice is plagued by errors and ambiguities that threaten it with breaking down. Thus, the real question becomes why taxonomic naming conforms to the causal theory in the first place. I will show that the answer lies in the embedding of the type method into elaborate codes of nomenclature. (shrink)

Dynamic and proof-conditional approaches to discourse (exemplified by Discourse Representation Theory and Type-Theoretical Grammar, respectively) are related through translations and transitions labeled by first-order formulas with anaphoric twists. Type-theoretic contexts are defined relative to a signature and instantiated modeltheoretically, subject to change.

It has recently been suggested that the type/token theorist concerning musical works cannot come up with an adequate semantic theory of those sentences in which we purport to talk about such works. Specifically, it has been claimed that, since types are abstract entities, a type/token theorist can only account for the truth of sentences such as “The 1812 Overture is very loud” and “Bach's Two Part Invention in C has an F-sharp in its fourth measure” by adopting an (...) untenable semantic claim: namely, that the predicates in such sentences, once applied to musical works, undergo a systematic shift in their meanings. This article is a sustained explanation of why our talk about musical works in fact provides no problem for the type/token theorist. First, we demonstrate that the aforementioned “meaning shift” approach to the sentences’ predicates is well motivated and very credible. Second, we explain how the type/token theorist can adopt the best available version of an alternative, generic quantificational approach to such sentences. Third, we establish that other semantic theories, presented as undermining the type/token theory by giving us a reason for adopting eliminativism about types, are much less theoretically virtuous than the two theories that a type/token theorist can freely adopt. (shrink)

This paper presents a type-free property-theoretic system in the spirit of a framework proposed by Menzel and then supplements it with a theory of truth and exemplification. The notions of a truth-relevantly complex (simple) sentence and of a truth-relevant subsentence are introduced and then used in order to motivate the proposed theory. Finally, it is shown how the theory avoids Russell's paradox and similar problems. Some potential applications to the foundations of mathematics and to natural language semantics (...) are sketched in the introduction. (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)

An examination of conditionals in di¤erent languages leads to a distinction of three types of conditionals instead of the usual two (indicative and subjunctive). The three types can be explained by the degree of acceptance or as-if acceptance of the truth of the antecedent. The labels subjunctive and indicative are shown to be inadequate. So-called indicative conditionals comprise two classes, the very frequent uncertain-fact conditionals and the quite rare accepted-fact conditionals. Uncertain-fact conditionals may have a time shift (...) in contemporary English and the future subjunctive in Portuguese (though not all of them do). Moreover, paraphrases of if with in case or supposing are usually possible with approximately the same meaning. Accepted-fact conditionals never have these features. (shrink)

In my paper , I noted that Fitch's argument, which purports to show that if all truths are knowable then all truths are known, can be blocked by typing knowledge. If there is not one knowledge predicate, ‘ K’, but infinitely many, ‘ K 1’, ‘ K 2’, … , then the type rules prevent application of the predicate ‘ K i’ to sentences containing ‘ K i’ such as ‘ p ∧¬ K i⌜ p⌝’. This provides a motivated response (...) to Fitch's argument so long as knowledge typing is itself motivated. It was the burden of my paper to explore the case that knowledge typing is as motivated as truth typing by drawing on the parallels between epistemic paradoxes generated by sentences of the kind ‘this sentence is unknown’ and semantic paradoxes generated by sentences such as ‘this sentence is untrue’. Given that typing truth is one of the acknowledged options for solving semantic paradoxes, if the parity argument succeeds it follows that epistemic typing is as well-motivated as truth typing and that the typing response to Fitch's argument is correspondingly strong.Halbach presents an apparent problem for this argument. Let ‘ N’ and ‘ P’, respectively, denote the necessity and possibility predicates, ‘ K 1’ the knowledge predicate of first type, and let ‘γ’ be a K-free sentence such that γ ↔¬ P⌜ K 1⌜γ⌝⌝; we know such a ‘γ’ exists by a standard diagonalization argument. If we assume that γ is knowable at the next knowledge type, that is, at type 1, and that the possibility typing does not interfere with the knowledge typing, a contradiction quickly ensues. Formally: γ ↔¬ P⌜ …. (shrink)

It is a fundamental intuition about truth that the conditions under which a sentence is true are given by what the sentence asserts. My aim in this paper is to show that this intuition captures the concept of truth completely and correctly. This is conceptual deflationism, for it does not go beyond what is asserted by a sentence in order to define the truth status of that sentence. This paper, hence, is a defense of deflationism as a (...) conceptual account of truth. This defense is developed in four stages. In the first stage I present a distinction between two types of deflationism, conceptual and metaphysical. This is the central stage of the argument and its main conclusion is that conceptual deflationism when joined with the principle of bivalence is inconsistent with metaphysical deflationism, that is, conceptual deflationism together with bivalence entails a non-deflationary metaphysical account of truth. In the second and third stages of the argument I argue that the totality of the Tarskian biconditionals, when interpreted as definitional biconditionals, offers a description of the nature of truth. In the fourth, and final, stage of the argument I advance a positive case for conceptual deflationism. I explain how the revision theory of truth provides this sort of deflationism with its best evidence: a clear demonstration of its consistency and a compelling argument for its material adequacy. (shrink)

The majority of physicalists are type-B physicalists – believing that the phenomenal-physical truths are only knowable a posteriori. This paper aims to show why this view is misguided. The strategy is to design an agent who (1) has full general physical knowledge, (2) has phenomenal concepts, and yet (3) is wired such that she would be in a position to immediately work out the phenomenal-physical truths. I argue that this derivation yields a priori knowledge. The possibility of such a creature (...) entails that – contrary to type-B physicalism – there is not an ideal epistemic gap between the phenomenal and the physical truths. Out of this argument against type-B physicalism emerges a positive result: a new and compelling version of type-A physicalism, roughly a type-A phenomenal concept strategy. (shrink)

Fitch's argument purports to show that if all truths are knowable then all truths are known. The argument exploits the fact that the knowledge predicate or operator is untyped and may thus apply to sentences containing itself. This article outlines a response to Fitch's argument based on the idea that knowledge is typed. The first part of the article outlines the philosophical motivation for the view, comparing it to the motivation behind typing truth. The second, formal part presents a (...) logic in which knowledge is typed and demonstrates that it allows nonlogical truths to be knowable yet unknown. (shrink)

This paper is a contribution to the development of model theory of fuzzy logic in narrow sense. We consider a formal system EvŁ of fuzzy logic that has evaluated syntax, i. e. axioms need not be fully convincing and so, they form a fuzzy set only. Consequently, formulas are provable in some general degree. A generalization of Gödel's completeness theorem does hold in EvŁ. The truth values form an MV-algebra that is either finite or Łukasiewicz algebra on [0, 1].The (...) classical omitting types theorem states that given a formal theory T and a set Σ of formulas with the same free variables, we can construct a model of T which omits Σ, i. e. there is always a formula from Σ not true in it. In this paper, we generalize this theorem for EvŁ, that is, we prove that if T is a fuzzy theory and Σ forms a fuzzy set , then a model omitting Σ also exists. We will prove this theorem for two essential cases of EvŁ: either EvŁ has logical constants for all truth values, or it has these constants for truth values from [0, 1] ∩ ℚ only. (shrink)

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our (...) theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to. (shrink)

This paper advances a version of physicalism which reconciles the “a priori entailment thesis” (APET) with the analytic independence of our phenomenal and physical vocabularies. The APET is the claim that, if physicalism is true, the complete truths of physics imply every other truth a priori. If so, “cosmic hermeneutics” is possible: a demon having only complete knowledge of physics could deduce every truth about the world. Analytic independence is a popular physicalist explanation for the apparent “epistemic gaps” (...) between phenomenal and physical truths. The two are generally seen as incompatible, since the demon’s deductions seem to presuppose analytic connections between physical and phenomenal terms. I begin by arguing, in support of the APET, that implications from the complete truths of physics to phenomenal truths cannot be a posteriori. Such implications are (according to the physicalist) necessarily true. But they cannot be Kripke-style a posteriori necessities, since (according to the physicalist) the complete truths of physics fix any relevant a posteriori facts about the reference of terms. I then show how the physicalist can turn the tables: the demon can exploit the physical fixing of reference to bridge the gap between the vocabularies, by deducing when phenomenal and physical terms co-refer. This opens the way for a “type-C” physicalism, which accepts in-principle deducibility while still appealing to analytic independence to explain why we (who are not demons) find it impossible to see phenomenal-physical connections a priori. (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)

In this paper I undertake to resolve a main pragmatic puzzle triggered by Bank-type cases. After accepting ‘sanitized’ intuitions about Truth-Values, as reflected in x-phi experiments, the pragmatic puzzle about whether the husband is inconsistent remains, and if he isn’t, which intuitively is the case, how are we to explain it. The context in such cases is pragmatic, with awareness of high risks, and the treatment I propose is pragmatic as well, but not Gricean. I offer a new Pragmatics (...) whose main tools are Steering Thrust and Posting. It focuses on what I call Action-Directed Pragmatics. An overall view of Pragmatics centers on the primary resort to Sayability rather than to Truth-Values, and to the Pragmatic phenomenon of co-present operative norm-types, which can conflict in some cases vis-à-vis particular actions and about what subjects can say. The very satisfactory explanation of the above puzzle that I offer is in such terms. In particular, attention is focused on the phenomenon of Negative-Polarity Pragmatic Functors. (shrink)

Temporal Type Theory (TTT) has been recently introduced as a topos-theoretic approach to understanding the behaviour of systems over time. A truly innovative point of TTT is that it makes truth inherently dependent on time; this is to be contrasted with the classical approach in which past, present and future are related via logical operators. Further on this line of research, the notion of truth is substituted by the ‘time duration’ over which a proposition is true, giving rise (...) to the logic of temporal landscapes. In this paper, we embed the main notions of the Calendar and Time Type System (a framework that was built for reasoning about calendric notions in the context of the Web and is centred around the notion of time granularity) in Temporal Type Theory, attempting to address a common requirement in the practical use of temporal formalisms: the ability to speak about certain time moments and appropriate time windows in (past and future) time, in the form of hours, dates, holidays, etc. With a view to concrete applications, we provide examples on how calendar types can be combined with temporal operators of TTT to augment the expressive power of this framework. In that respect, the addition of calendar types genuinely adds to the range of Temporal Type Theory as a powerful specification language. (shrink)

The fact that the names of two philosophers who lived and worked in such different periods, under conditions of quite dissimilar cultures and civilizations, share the title of this article is in itself enough to require clarification. Nikolai Berdiaev, who belonged to a current of Russian philosophy that called itself mystical, hardly needs to be presented to the reader. Muhyiuddin ibn-'Arabi , the greatest mystic of the Arab Middle Ages, is known as the founder of a philosophical conception that was (...) later referred to as the "unity of being" and enjoyed broad popularity among Islamic thinkers, philosophers, and poets of the late Middle Ages. Moreover, motifs of this conception are not difficult to find in the works of contemporary literati enthralled by the ideas of Islamic mysticism, i.e., of Sufism. Hence these two outstanding figures can be compared on the common ground that they belonged to the current of philosophical thought of mankind that is called mysticism or mystical philosophy. Despite all the differences that doubtlessly exist among Russian, Arabic, and, let us say, European mysticism, there are also certain fundamental common features of this way of looking at the world that enable us to bring all these currents together under a single generic concept. (shrink)

In this paper I address the question 'How is knowledge of logical truths possible'. The sought-after explanation should be independent of what the true story about logical truth is. In particular, I try to account for the epistemic warrant that is conferred upon logical beliefs when they are neither inferred from other beliefs nor grounded on empirical evidence or testimony. The need for such an account is motivated by the apparent failure of the notions ofanalyticity on the one hand (...) and intuition on the other in addressing the relevant question. I end up defending an account according to which warranted logical beliefs can be grounded on pure reasoning: they can be inferentially formed on the basis of pieces of suppositional reasoning. (shrink)

Speaking of Professor Geach's belief that pronouns in natural language function like the bound variables in quantification theory, Gareth Evans, in ‘Pronouns, Quantifiers, and Relative Clauses - I’ says :I want to try to show that there are pronouns with quantifier antecedents that function in a quite different way. Such pronouns typically stand in a different grammatical relation to their antecedents, and; in contrast with bound pronouns, must be assigned a reference, so that their most immediate sentential contexts can always (...) be assigned a truth value. The relevant grammatical relation appears to be Klima's relation of ‘in construction with’. When the pronoun is in construction with its antecedent, as in [‘Some man loves his mother’] and [‘No man is happy when he is in love’] the result is a bound pronoun. (shrink)

This article consists of three parts, two introductory, in which the limits and the methods of analysis of dialogues are expounded, and the major part, in which the main features of a philosophical theory of disputation are outlined.It was an essential aim of the philosophical analysis of argumentative dialogues to develop tools of substantiation for cases in which logic doesn't help any more. In the first part of this paper I show that such tools can and will be developed only (...) by analyzing argumentations (“argumentation” in the sense of a monologue in which arguments for a thesis are brought forward), and that the analysis of argumentative dialogues doesn't contribute anything to the development of such tools.The systematically first task of the philosophical analysis of dialogues consists in understanding the general practical aims of philosophically interesting types of dialogue. Only subsequently the rules of the dialogue can be reconstructed as good means for reaching these aims. Dialogical games constructed without referring to such a purpose are externally senseless and useless.The third part is an outline of a philosophical theory of disputation (“disputation” here will mean: (learned) dialogue in which the participants cooperatively though perhaps controversially attempt to find out by means of arguments and mutual criticism whether a thesis is true or false). Disputations contain argumentations, and many functions of a disputation can also be fulfilled by argumentations alone. Certifying the truth of convictions is the specific aim of disputation. This is accomplished by eliminating errors of substantiation and foundation as effectively as possible, in revising false convictions and their foundations, thereby making the remaining convictions more certain. Based on this analysis of the aim of disputation, the basic rules of disputations will be critically reconstructed: possible moves, rules of sequence, and the internal aim and ends of disputations. (shrink)

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

This paper explores the relation between the philosophical idea that logic is a science studying logical forms, and a mathematical feature of logical systems called the principle of uniform substitution, which is often regarded as a technical counterpart of the philosophical idea. We argue that at least in one interesting sense the principle of uniform substitution does not capture adequately the requirement that logic is a matter of form and that logical truths are formal truths. We show that some specific (...) logical expressions can produce propositions of different kinds and the resulting diversity of informational types can lead to a justified failure of uniform substitution without undermining the view that logic is a purely formal discipline. (shrink)

Martin-Löf’s type theory stems simultaneously from Frege and Russell’s logic-ontological ideas and Husserl’s phenomenology. The article examines this intermediate status of type theory using as examples Martin-Löf ’s syntactical-semantic method and the role of evidence and canonical objects in his approach. Martin-Löf borrows the syntactical-semantic method from Frege and extends it drawing on Husserl’s theory of meaning. In type theory this method leads to the identity (isomorphism) of syntax and semantics (formal logic and formal ontology). Unlike traditional formal logic the (...) type theory is an interpreted system from the very beginning. Being intuitionistic, Martin-Löf ’s theory is based on the notion of proof, not truth. From the meaning theory point of view, it is a variant of proof-theoretic semantics (Gentzen, Prawitz, Dummett) which understands meaning as an object constructed according to certain rules. So understood, the proof is based on evidence, which allows us to associate it with the theory of intentionality by Husserl. The article compares Martin-Löf ’s type theory with Husserl’s intentionality theory, especially with the latter’s noematic component. We may consider type-theoretical rules for constructing objects and operating with them as a concretization and formalization of the phenomenological notion of noema. Both are explications of the more general concept of meaning. The article discusses the interrelation between notions of sense and meaning (Sinn, Bedeutung) in Frege, Husserl and Martin-Löf. This reveals the uncertainty of Martin-Löf ’s position in relation to meaning theories of Frege and Husserl. (shrink)

Granting that there could be true subjunctive conditionals of libertarian freedom (SCLs), I argue (roughly) that there could be such conditionals only in connection with individual "possible creatures" (in contrast to types). This implies that Molinism depends on the view that, prior to creation, God grasps possible creatures in their individuality. In making my case, I explore the notions of counterfactual implication (that relationship between antecedent and consequent of an SCL which consists in its truth) and counterfactual relevance (...) (that feature of an antecedent in virtue of which it counterfactually implies something or other). (shrink)

The notion of cognitive act is of importance for an epistemology that is apt for constructive type theory, and for epistemology in general. Instead of taking knowledge attributions as the primary use of the verb 'to know' that needs to be given an account of, and understanding a first-person knowledge claim as a special case of knowledge attribution, the account of knowledge that is given here understands first-person knowledge claims as the primary use of the verb 'to know'. This means (...) that a cognitive act is an act that counts as cognitive from a first-person point of view. The method of linguistic phenomenology is used to explain or elucidate our epistemic notions. One of the advantages of the theory is that an answer can be given to some of the problems in modern epistemology, such as the Gettier problem. (shrink)

According to act theories, propositions are structured cognitive act‐types. Act theories appear to make propositions inherently representational and truth‐evaluable, and to provide solutions to familiar problems with alternative theories, including Frege’s and Russell’s problems, and the third‐realm and unity problems. Act theories have critical problems of their own, though: acts as opposed to their objects are not truth evaluable, not structured in the right way, not expressed by sentences, and not the objects of propositional attitudes. I show (...) how identifying propositions with other cognitive event‐types, namely thoughts, has the perceived virtues of act theories without the defects. (shrink)

This paper argues that there are two camps in narrative theology, "pure" (e.g., Hans Frei) and "impure" (e.g., Paul Ricoeur) narrative theologians. Narrative theology, reflection on religious claims embedded in stories, is one of the most significant currents of late twentieth century thought. H. Richard Niebuhr initiated the conversation when he wrote in 1941 of "The Story of Our Lives." If his theme lay undeveloped for several decades, it burst onto the theological scene in the early 1970s. Demurrers followed, and (...) we now recognize the limits of the metaphor" narrative." Still, scholarly interest has not waned. Books and articles keep coming: on narrative in the Bible, narrative and theology, narrative and morality, narrative tradition, narrative practice, even narrative truth. The conversation, it appears, is as lively as ever. And if it has produced lasting contributions, it is because of the quality of the minds it has managed to attract: Paul Ricoeur, Hans Frei, David Tracy, George Lindbeck, Stanley Hauerwas, Julian Hartt, Sallie McFague, Johann Baptist Metz. (shrink)

Studies the threefold hierarchy of certainty, from its origins in Mahāyāna Buddhism, through Islam, to 17th century China. This tripartite scheme may be traced back to the ancient Buddhist scheme of the threefold wisdom as systematized by Vasubandhu of Gandhāra in the 4th-5th centuries CE. Following the advent of Islam in the 8th century, it was combined with Qur'anic notions of certainty. Initially taken up by early Islamic mystics such as Sahl al-Tustarī and al-Ḥākim al-Tirmiḏī, the notion of yaqīn was (...) gradually systematized into the three-level hierarchy of “knowledge or science of certainly”, “essence” of certainty, and “truth or reality of certainty”, a hierarchy that bears a distinct resemblance to the Buddhist threefold path of wisdom as discussed by Marc-Henri Deroche. Half a millennium later, this threefold hierarchy of levels of certainty, remotely inspired by Buddhism and integrated into the philosophical Sufism of Ibn ʿArabī and his Persian disciple Jāmī, this complex of ideas may have resurfaced in 17th century China. (shrink)

I investigate why Frege rejected the theory of types, as Russell presented it to him in their correspondence. Frege claims that it commits one to violations of the law of excluded middle, but this complaint seems to rest on a dogmatic refusal to take Russell’s proposal seriously on its own terms. What is at stake is not so much the truth of a law of logic, but the structure of the hierarchy of the logical categories, something Frege seems (...) to neglect. To come to a better understanding of Frege’s response, I proceed to investigate his conception of the nature of the logical categories, and how it differs from Russell’s. I argue that, for Frege, our grasp of the logical categories cannot be severed from our grasp of the Begriffsschrift notation itself. Russell, on the other hand, attaches no such importance to notation. From Frege’s point of view, Russell has not succeeded in presenting an alternative conception of the logical hierarchy, since such a conception must go in tandem with the development of a notation. Moreover, Frege has good reasons to think that Russell’s proposal does not admit of a suitable notation. (shrink)