One way to study and understand the notion of truth is to examine principles that we are willing to associate with truth, often because they conform to a pre-theoretical or to a semi-formal characterization of this concept. In comparing different collections of such principles, one requires formally precise notions of inter-theoretic reduction that are also adequate to compare these conceptual aspects. In this work I study possible ways to make precise the relation of conceptual equivalence between notions of truth associated (...) with collections of principles of truth. In doing so, I will consider refinements and strengthenings of the notion of relative truth-definability proposed by Fujimoto (2010): in particular I employ suitable variants of notions of equivalence of theories considered in Visser (2006) and Friedman & Visser (2014) to show that there are better candidates than mutual truth-definability for the role of sufficient condition for conceptual equivalence between the semantic notions associated with the theories. In the concluding part of the paper, I extend the techniques introduced in the first and show that there is a precise sense in which ramified truth (either disquotational or compositional) does not correspond to iterations of comprehension. (shrink)

To get to grips with what Shapiro does and can say about higher-order vagueness, it is first necessary to thoroughly review and evaluate his conception of (first-order) vagueness, a conception which is both rich and suggestive but, as it turns out, not so easy to stabilise. In Sections I–IV, his basic position on vagueness (see Shapiro [2003]) is outlined and assessed. As we go along, I offer some suggestions for improvement. In Sections V–VI, I review two key paradoxes of higher-order (...) vagueness, while in Section VII, I explore a possible line of response to such paradoxes given by Keefe [2000]. In Section VIII, I assess whether which Shapiro might adapt Keefe's response to combat both paradoxes. (shrink)

It is a truism that semantic concepts are relative to language. I distinguish two kinds of them in accordance to their relativity to language L; the relativity is either explicit , or implicit . If language is explicated, the concepts of the former kind can be easily explicated in a plausible way and we resist semantic paradoxes. In the case of the latter concepts, the explication is also accessible and paradox-free. One can find then new interesting facts concerning famous Tarski’s (...) theorem. (shrink)

Van Heijenoort’s main contribution to history and philosophy of modern logic was his distinction between two basic views of logic, first, the absolutist, or universalist, view of the founding fathers, Frege, Peano, and Russell, which dominated the first, classical period of history of modern logic, and, second, the relativist, or model-theoretic, view, inherited from Boole, Schröder, and Löwenheim, which has dominated the second, contemporary period of that history. In my paper, I present the man Jean van Heijenoort (Sect. 1); then (...) I describe his way of arguing for the second view (Sect. 2); and finally I come down in favor of the first view (Sect. 3). There, I specify the version of universalism for which I am prepared to argue (Sect. 3, introduction). Choosing ZFC to play the part of universal, logical (in a nowadays forgotten sense) system, I show, through an example, how the usual model theory can be naturally given its proper place, from the universalist point of view, in the logical framework of ZFC; I outline another, not rival but complementary, semantics for admissible extensions of ZFC in the very same logical framework; I propose a way to get universalism out of the predicaments in which universalists themselves believed it to be (Sect. 3.1). Thus, if universalists of the classical period did not, in fact, construct these semantics, it was not that their universalism forbade them, in principle, to do so. The historical defeat of universalism was not technical in character. Neither was it philosophical. Indeed, it was hardly more than the victory of technicism over the very possibility of a philosophical dispute (Sect. 3.2). (shrink)

What are the objects of knowledge, belief, probability, apriority or analyticity? For at least some of these properties, it seems plausible that the objects are sentences, or sentence-like entities. However, results from mathematical logic indicate that sentential properties are subject to severe formal limitations. After surveying these results, I argue that they are more problematic than often assumed, that they can be avoided by taking the objects of the relevant property to be coarse-grained (“sets of worlds”) propositions, and that all (...) this has little to do with the choice between operators and predicates. (shrink)

It is shown that two formally consistent type-free second-order systems, due to Cocchiarella, and based on the notion of homogeneous stratification, are subject to a contingent version of Russell's paradox.

In his treatise on sophisms, the medieval logician and philosopher J. Buridan expounded a theory on what we have come to call semantic paradoxes. His theory has not yet been fully understood. The present paper aims at showing that Barwise's and Etchemendy's considerations on paradoxes (founded upon Aczel's non-well-founded sets) provide the framework for an improved understanding. Barwise's and Etchemendy's account is contrasted with Kripke's. Finally, a recent analysis of Buridan's position by Epstein is criticized.

In diesem Aufsatz wird ein neues Paradoxon vorgestellt, der Super-Lügner. Er ist stärker als alle bekannten Lügner-Sätze, nicht mehr eindeutig selbstreferentiell und läßt sich darüber hinaus in eindeutig in die Tarski-Hierarchie einordnen. Eine unendlich große Familie von Super-Lügnern auf Metaebenen ist konstruierbar. Schließlich widersetzt sich der Super-Lügner der Auflösung durch die neue vielversprechende Reflexionslogik LR von U. Blau.

In diesem Aufsatz wird ein neues Paradoxon vorgestellt, der Super-Lügner. Er ist stärker als alle bekannten Lügner-Sätze, nicht mehr eindeutig selbstreferentiell und läßt sich darüber hinaus in eindeutig in die Tarski-Hierarchie einordnen. Eine unendlich große Familie von Super-Lügnern auf Metaebenen ist konstruierbar. Schließlich widersetzt sich der Super-Lügner der Auflösung durch die neue vielversprechende Reflexionslogik LR von U. Blau.

Ramsey était‐il “deflationniste”? C'est douteux, à lire attentivement le manuscrit “On Truth”. La position de Ramsey a nénmoins quelque chose de curieux, comme Austin l'a fait remarquer: quel est l'intérêt d'éliminer le prédicat de vérité sile problème de la vérité n'est pas en même temps éliminé? En poursuivant ces remarques, je suggére, à titre d'expérience de pensée, de lire autrement les fameuses “équivalencesT”.

Although the notion of common or mutual belief plays a crucial role in game theory, economics and social philosophy, no thoroughly representational account of it has yet been developed. In this paper, I propose two desiderata for such an account, namely, that it take into account the possibility of inconsistent data without portraying the human mind as logically and mathematically omniscient. I then propose a definition of mutual belief which meets these criteria. This account takes seriously the existence of computational (...) limitations. Finally, I point out that the epistemic logic (or theory) needed to make the definition work is subject to the Kaplan/Montague Paradox of the Knower. I argue that this is not a defect of the account, and I discuss briefly the bearing of recent work on the paradox of the Liar upon this problem. (shrink)

This paper argues for the thesis that, roughly put, it is impossible to talk about absolutely everything. To put the thesis more precisely, there is a particular sense in which, as a matter of semantics, quantifiers always range over domains that are in principle extensible, and so cannot count as really being ‘absolutely everything’. The paper presents an argument for this thesis, and considers some important objections to the argument and to the formulation of the thesis. The paper also offers (...) an assessment of just how implausible the thesis really is. It argues that the intuitions against the thesis come down to a few special cases, which can be given special treatment. Finally, the paper considers some metaphysical ideas that might surround the thesis. Particularly, it might be maintained that an important variety of realism is incompatible with the thesis. The paper argues that this is not the case. (shrink)

We present a strategy to dissolve semantic paradoxes which proceeds from an explanation of why paradoxical sentences or their definitions are semantically defective. This explanation is compatible with the acceptability of impredicative definitions, self-referential sentences and semantically closed languages and leaves the status of the so-called truth-teller sentence unaffected. It is based on platitudes which encode innocuous constraints on successful definition and successful expression of propositional content. We show that the construction of liar paradoxes and of certain versions of Curry's (...) paradox rests on presuppositions that violate these innocuous constraints. Other versions of Curry's paradox are shown not to be paradoxical at all once their presuppositions are made explicit. Part of what we say rehearses a proposal originally made by Laurence Goldstein in 1985. Like Goldstein we dispose of certain paradoxes by rejecting some of the premises from which they must be taken to proceed. However, we disagree with his more recent view that the premises to be rejected are neither true nor false. (shrink)

Here, I first prove that certain families of k-valued clones have the Gupta-Belnap fixed-point property. This essentially means that all propositional languages that are interpreted with operators belonging to those clones are such that any net of self-referential sentences in the language can be consistently evaluated. I then focus on two four-valued generalisations of the Kleene propositional operators that generalise the strong and weak Kleene operators: Belnap’s clone and Fitting’s clone, respectively. I apply the theorems from the initial part of (...) the paper to analyse the fixed-point property of Belnap’s and Fitting’s clones when some special operators that reflect the semantics are added. The conclusion of the paper is that Fitting’s clone is better suited than Belnap’s to provide self-referential languages with highly expressive resources. (shrink)

The present work, which was inspired by Kripke and McCarthy, is about a non-classical predicate logic system containing a truth predicate symbol. In this system, each sentence A is referred to not by a Gödel number but by its quotation name 'A'.

In Russell''s Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Church''s simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT. Despite the disappearance of orders which have a strong correlation with predicativity, predicative (...) logic still plays an influential role in Computer Science.An important example is the proof checker Nuprl, which is basedon Martin-Löf''s Type Theory which uses type universes. Those type universes,and also degrees of expressions in AUTOMATH, are closely related toorders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl.The novelty of our paper lies in: (1) a modest revival of Russell's orders, (2) the placing of the historical system RTT underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (Martin-Löf''s type theory and PTSs), and (3) the presentation of acomplex type system (Nuprl) as a simple and compact PTS. (shrink)

A case situation arising in a normal interaction among people is the baseline for discussing properties of the theory of viewpoints. In particular we consider how to ensure agreement on the meaning of certain utterances by agents who have different perspectives on the situation, while maintaning other knowledge as private. We argue that communication should be modeled as adding facts to the common knowledge of agents. We introduce the principle of ''referent sharing'' in communications and argue that common knowledge resulting (...) from communication should only use constants whose referent is manifest to the parties involved. (shrink)

I will use paradox as a guide to metaphysical grounding, a kind of non-causal explanation that has recently shown itself to play a pivotal role in philosophical inquiry. Specifically, I will analyze the grounding structure of the Predestination paradox, the regresses of Carroll and Bradley, Russell's paradox and the Liar, Yablo's paradox, Zeno's paradoxes, and a novel omega plus one variant of Yablo's paradox, and thus find reason for the following: We should continue to characterize grounding as asymmetrical and irreflexive. (...) We should change our understanding of the transitivity of grounding in a certain sense. We should require foundationality in a new, generalized sense, that has well-foundedness as its limit case. Meta-grounding is important. The polarity of grounding can be crucial. Thus we will learn a lot about structural properties of grounding from considering the various paradoxes. On the way, grounding will also turn out to be relevant to the diagnosis (if not the solution) of paradox. All the paradoxes under consideration will turn out to be breaches of some standard requirement on grounding, which makes uniform solutions of large groups of these paradoxes more desirable. In sum, bringing together paradox and grounding will be shown to be of considerable value to philosophy.1. (shrink)

I survey some important semantical and axiomatic theories of self-referential truth. Kripke's fixed-point theory, the revision theory of truth and appraoches involving fuzzy logic are the main examples of semantical theories. I look at axiomatic theories devised by Cantini, Feferman, Freidman and Sheard. Finally some applications of the theory of self-referential truth are considered.

Kripke's theory of partial truth offers a natural solution of the Liar paradox and an appealing explanation of why the Liar sentence seems to lack definite content. It seems vulnerable, however, to...

El objetivo de este artículo es investigar diversos resultados limitativos acerca del concepto de validez. En particular, argumento que ninguna teoría lógica de orden superior con semántica estándar puede tener recursos expresivos suficientes como para capturar su propio concepto de validez. Además, muestro que la lógica de la verdad transparente que Hartry Field desarrolló recientemente conduce a resultados limitativos similares.

Kremer presented three approaches of comparing fixed-point and revision theories of truth in Kremer, 363–403, 2009). Using these approaches, he established the relationships among ten fixed-point theories suggested by Kripke in, 690–716, 1975) and three revision theories presented by Gupta and Belnap in. This paper continues Kremer’s work. We add five other revision theories to the comparisons, including the theory proposed by Gupta in, 1–60, 1982), the theory proposed by Herzberger in, 61–102, 1982), the theory based on fully-varied revision sequences (...) proposed by Gupta and Belnap in, the theory proposed by Yaqūb in, and the theory based on weakly consistent revision sequences. We show that, the notion of Thomason model defined by Belnap’s limit rule is not equivalent to the one defined by Gupta’s limit rule, and that the theory based on fully-varied revision sequences is ≤2-equivalent to the one based on the greatest intrinsic fixed point of σ. (shrink)

There has been much debate recently as to whether the notion of truth, as applied to one's home language, is metaphysically neutral, the interesting metaphysical questions arising elsewhere (in relation to such notions as mind-independence or objectivity or existence). ' On one side, the minimalists, as they have come to be known, favour deflationary accounts of truth such as the redundancy or disquotational theories and conclude that the notion of truth is applicable to declarative sentences in general - at least (...) where there are fairly definite standards of appropriateness for the assertion or denial of such sentences - irrespective of the metaphysical commitments of the propositions the sentences express. Opponents have criticized the minimalist account of truth or denied that it follows from this account that truth is metaphysically neutral. My sympathies in this debate lie with the minimalist but I wish to focus on a serious problem for the minimalist - that posed by the Liar paradox. This problem has featured very little in the recent debate but seems to me to yield the strongest objection to minimalism. I develop the objection in the next section, in §2 sketching what I feel to be the minimalist's best response, namely developing the position not in terms of Tarskian classical biconditionals but rather in terms of the naive rules for truth. It emerges, then, that the minimalist will be forced to repudiate classical logic. The last section, §3, deals with objections to the strategy of §2. (shrink)

Approaches to truth and the Liar paradox seem invariably to face a dilemma: either appeal to some sort of hierarchy, or declare apparently perfectly coherent concepts incoherent. But since both options lead to severe expressive restrictions, neither seems satisfactory. The aim of this paper is a new approach, which avoids the dilemma and the resulting expressive restrictions. Previous approaches tend to appeal to some new sort of semantic value for the truth predicate to take. I argue that such approaches inevitably (...) lead to the dilemma in question. In contrast, the present proposal sticks with classical semantic values, but allows the compositional rules associated with these to admit of exceptions. I show how such an approach can be developed systematically. (shrink)

We introduce an epistemic theory of truth according to which the same rational degree of belief is assigned to Tr(. It is shown that if epistemic probability measures are only demanded to be finitely additive (but not necessarily σ-additive), then such a theory is consistent even for object languages that contain their own truth predicate. As the proof of this result indicates, the theory can also be interpreted as deriving from a quantitative version of the Revision Theory of Truth.

In Zwicker (1987) the hypergame paradox is introduced and studied. In this paper we continue this investigation, comparing the hypergame argument with the diagonal one, in order to find a proof schema. In particular, in Theorems 9 and 10 we discuss the complexity of the set of founded elements in a recursively enumerable relation on the set N of natural numbers, in the framework of reduction between relations. We also find an application in the theory of diagonalizable algebras and construct (...) an undecidable formula. (shrink)

Is there a key for ‘translating' some set-theoretical paradoxes into counterpart semantical paradoxes and vice-versa? There is, and this encourages the hope of a unified solution. The solution turns not on inventing new axioms that do not entail contradiction, but on imposing a completely intuitive restriction on the comprehension axiom of naive set theory in order to avoid illegitimate (circular) stipulation.

This paper contributes to the study of paracompleteness and paraconsistency. We present two logics that address the following questions in novel ways. How can the paracomplete theorist characterize the formulas that defy excluded middle while maintaining that not all formulas are of this kind? How can the paraconsistent theorist characterize the formulas that obey explosion while still maintaining that there are some formulas not of this kind?

If properties are to play a useful role in semantics, it is hard to avoid assuming the naïve theory of properties: for any predicate Θ(x), there is a property such that an object o has it if and only if Θ(o). Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic is weakened appropriately; the main difficulty is finding a semantics that can handle a conditional obeying reasonable laws without engendering paradox. (...) I employ a semantics which is infinite-valued, with the values only partially ordered. Can the solution be adapted to naïve set theory? Probably not, but limiting naïve comprehension in set theory is perfectly satisfactory, whereas this is not so in a property theory used for semantics. (shrink)

I first focus on Gomez-Torrente’s defense of a neo-Kripkean treatment of theoretical identification statements like “Water is H2O” against objections that ‘water’ is vague and that ‘H2O’ does not designate a genuine explanatory kind. On my reading, Gomez-Torrente accepts and while interpreting the identity statement as correctly attributing vague identity to the pair consisting of the vague natural kind water and the related kind H2O. My discussion sketches a framework for making sense talk of vague objects, properties, and identities that (...) seems to fit his discussion. Next I focus on his treatment of a problem posed by seemingly conflicting perceptual reports of secondary qualities, like being hot, or red, made by equally competent observers. Since the observers are equally competent, it is hard to explain the apparent inconsistency. Whereas Gomez-Torrente treats the reports as so fine-grained as not to conflict, I suggest dispensing with ultrafine-grained semantic contextual parameters in favor of richer pragmatic parameters determining negotiated asserted contents. Next I question whether his plausible individually sufficient but not disjunctively necessary rules for uses of names and demonstratives should be understood as providing speaker referents versus contributions to illocutionary contents of uses of sentences containing those terms. Finally, I explain why I take the crowning jewel of the book to be the account in chapter 4 of natural numbers as plural cardinality properties, and of verbal, and Arabic, numerals as Millian names of those properties, the referents of which are fixed by genuinely semantic reference-fixing expressions. (shrink)

I justify a non-logician speaking of the liar antinomy by Peano view that semantic antinomies are linguistic issues. The remarks I formulate below follow Alfred Gawroński’s idea, namely, that the liar antinomy is an apparent antinomy.

This paper is concerned with the interaction between formal semantics and the foundations of mathematics. We introduce a formal theory of truth, TLR, which extends the classical first order theory of pure combinators with a primitive truth predicate and a family of truth approximations, indexed by a directed partial ordering. TLR naturally works as a theory of partial classifications, in which type-free comprehension coexists with functional abstraction. TLR provides an inner model for a well known subsystem of second order arithmetic; (...) indeed, TLR is proof-theoretically equivalent to Predicative Analysis. (shrink)

§1. Introduction. When truth-theoretic paradoxes are generated, two factors seem to be at play: the behaviour that truth intuitively has; and the facts about which singular terms refer to which sentences, and so on. For example, paradoxicality might be partially attributed to the contingent fact that the singular term, "the italicized sentence on page one", refers to the sentence, The italicized sentence on page one is not true. Factors of this second kind might be represented by a ground model: an (...) interpretation of all the names, function symbols, and predicates in the potentially self-referential language under study, with the exception of the predicate "x is true". Formally, suppose that L is an uninterpreted first order language. M = 〈D, I〉 is a classical model for L if D is a nonempty set; and I is a function assigning to each name of L a member of D, to each n-place function symbol of L a function from Dn to D, and to each nonlogical n-place predicate of L a function from Dn to {t, f}. Suppose furthermore that L+ is obtained by adding a new one-place predicate T to L, and that L has a quote name ‘A’ for each sentence A of L+. We follow Gupta and Belnap [5] in defining S =df {A: A is a sentences of L+}. A classical model M = 〈D, I〉 for L is a ground model for L iff both S ⊆ D, and I(‘A’) = A for each A ∈ S. A classical ground model for L is a representation of the supposedly unproblematic fragment of L+: a representation of which terms refer to which objects, and of which objects have which nonsemantic properties. (shrink)