The Embracing Revenge account of semantic paradox avoids the expressive limitations of previous approaches based on the Kripkean fixed point construction by replacing a single language with an indefinitely extensible sequence of languages, each of which contains the resources to fully characterize the semantics of the previous languages. In this paper we extend the account developed in Cook, Cook, Schlenker, and Tourville and Cook via the addition of intensional operators such as ``is paradoxical''. In this extended framework we are (...) able to characterize the difference between sentences, such as the Liar and the Truth-teller, that receive the same semantic value in minimal fixed points yet seem to involve distinct semantic phenomena. (shrink)

We identify a class of paradoxes that is neither set-theoretical nor semantical, but that seems to depend on intensionality. In particular, these paradoxes arise out of plausible properties of propositional attitudes and their objects. We try to explain why logicians have neglected these paradoxes, and to show that, like the Russell Paradox and the direct discourse Liar Paradox, these intensional paradoxes are recalcitrant and challenge logical analysis. Indeed, when we take these paradoxes seriously, we (...) may need to rethink the commonly accepted methods for dealing with the logical paradoxes. (shrink)

In , Frank Ramsey separates paradoxes into two groups, now taken to be the logical and the semantical. But he also revises the logical system developed in Whitehead and Russellthe intensional paradoxess interest in these problems seriously, then the intensional paradoxes deserve more widespread attention than they have historically received.

John Barker, in two recent essays, raises a variety of intriguing criticisms to challenge my interpretation of the liar paradox and the type of solution I proposein ‘Denying the Liar’ and ‘Denying the Liar Reaffirmed.’ Barker continues to believe that I have misunderstood the logical structure of the liar sentence and itsexpression, and that as a result my solution misfires. I shall try to show that on the contrary my analysis is correct, and that Barker does not properly grasp what (...) mysolution to the liar paradox involves. Additionally, I argue that Barker makes fundamental errors in the explanation of liar sentence formulations in intensional contexts and in the classical metatheory he invokes to support his criticisms. (shrink)

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other (...) axioms of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions. (shrink)

Andrew Bacon, John Hawthorne, and Gabriel Uzquiano have recently argued that free logics—logics that reject or restrict Universal Instantiation—are ultimately not promising approaches to resolving a family of intensional paradoxes due to Arthur Prior. These logics encompass ramified and contextualist approaches to paradoxes, and broadly speaking, there are two kinds of criticism they face. First, they fail to address every version of the Priorean paradoxes. Second, the theoretical considerations behind the logics make absolutely general statements about (...) all propositions, properties of propositions, etc., but because this sort of intensional quantification is always restricted in the logics, they cannot even express those considerations. I present a novel sort of free logic, which rejects the standard Universal Instantiation but validates a restricted form of the rule, and which addresses both kinds of criticism by allowing some propositions to be unrestricted in their quantification. (shrink)

Saul Kripke’s theory of truth suffers from expressive limitations – in particular, there are no extensional operators within that framework that allow one to characterize those sentences that fail to receive a truth value within the framework. Especially worrisome is the fact that there is no operator that outputs true on exactly the paradoxical sentences. In this paper I extend Kripke’s approach via the addition of extensional operators, which allows us to characterize many (but not all) such sentences, including the (...) paradoxical ones. (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)

Classical logic, of first or higher order, is extended with sentential operators and quantifiers, interpreted substitutionally over unrestricted substitution class. Operators mark a single layered, consistent metalanguage. Self-reference, arising from substitutional quantification over sentences, allows to express paradoxes which, unlike contradictions, do not lead to explosion. Semantics of the resulting language, using semi-kernels of digraphs, is non-explosive yet two-valued and has classical semantics as a special case for clasically consistent theories. A complete reasoning is obtained by extending LK with (...) two rules for sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. (shrink)

This note is an analysis of the paradox given by Myers. It is shown, assuming that the resources available in paraconsistent logic may be applied, how the conclusion of the paradox may be perfectly acceptable, but that the argument is, nonetheless, invalid. This provides a dialethic solution to the paradox.

The present work outlines a logical and philosophical conception of propositions in relation to a group of puzzles that arise by quantifying over them: the Russell-Myhill paradox, the Prior-Kaplan paradox, and Prior's Theorem. I begin by motivating an interpretation of Russell-Myhill as depending on aboutness, which constrains the notion of propositional identity. I discuss two formalizations of of the paradox, showing that it does not depend on the syntax of propositional variables. I then extend to propositions a modal predicative response (...) to the paradoxes articulated by an abstraction principle for propositions. On this conception, propositions are “shadows” of the sentences that express them. Modal operators are used to uncover the implicit relation of dependence that characterizes propositions that are about propositions. The benefits of this approach are shown by application to other intensional puzzles. The resulting view is an alternative to the plenitudinous metaphysics of impredicative comprehension principles. (shrink)

This paper is about the standard Reflection Principle :235–256, 1984) and the Group Reflection Principle :478–502, 2007; Bovens and Rabinowicz in Episteme 8:281–300, 2011; Titelbaum in Quitting certainties: a Bayesian framework modeling degrees of belief, OUP, Oxford, 2012; Hedden in Mind 124:449–491, 2015). I argue that these principles are incomplete as they stand. The key point is that deference is an intensional relation, and so whether you are rationally required to defer to a person at a time can depend (...) on how that person and that time are designated. In this paper I suggest a way of completing the Reflection Principle and Group Reflection Principle, and I argue that so completed these principles are plausible. In particular, they do not fall foul of the Sleeping Beauty case :143–147, 2000), the Cable Guy Paradox :112–119, 2005), Arntzenius’ prisoner cases :356–370, 2003), or the Puzzle of the Hats :281–300, 2011). (shrink)

We expand on the fixed point semantic approach of Kripke via the addition of two unary intensional operators: a paradoxicality operator Π where Π(Φ) is true at a fixed point if and only if Φ is paradoxical (i.e., if and only if Φ receives the third, non-classical value on all fixed points that extend the current fixed point), and an unbounded truth operator Υ⊤ where Υ⊤(Φ) is true at a fixed point if and only if any fixed point extending (...) the current fixed point can be extended to one on which Φ receives the value true. We prove a generalized version of Kripke’s fixed point theorem guaranteeing the existence of models of this new language, as well as an expressive completeness result. We conclude with an exploration of the significant improvements in expressive power that result from the addition of these new operators, and we precisely identify what still cannot be said on this intensional extension of the Kripkean framework. (shrink)

I propose to downgrade the Liar paradox from what Quine called an antinomy to a much weaker veridical paradox. I then apply Quine’s strategy of rejecting veridical paradoxes by exposing an unacceptable premise to the Liar. I argue that upon analysis the intension of a standard Liar sentence presupposes the existence of a non-empty empty set; and that since such an object is impossible this presupposition may be rejected, downgrading the Liar. I then briefly argue that Dialetheism can be (...) motivated without supposing the existence of a non-empty empty set. (shrink)

The principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox and Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to (...) higher-order logic. Our assessment of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior’s and Kaplan’s derivations at face value. (shrink)

– The paradoxes of analysis and identity each consist of a pair of statements sharing the same referents, but differing in their informativeness properties. Carnap employs a different solution for each of these paradoxes. Church, Davidson, and others have maintained that the two paradoxes can, and should, be resolved by a single method, viz. one based on the Fregean distinction between sense and reference.The present paper argues that Carnap's solution for the paradox of analysis is unsatisfactory on (...) several counts, but primary among these is that Carnap is driven to hold that no semantic distinction can be drawn between the informativeness properties occurring in this paradox. Nevertheless, the paper shows how the required distinction can be formulated within the general framework of Carnap's extensional‐intensional system of semantics, and that such a distinction provides a single solution for both paradoxes. (shrink)

In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy (...) due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. (shrink)

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) (...) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. (shrink)

Lasersohn has argued that the use of Russell's analysis of the definite determiner in Montague Grammar, which is responsible for giving the correct prediction in the case of the Temperature Paradox, is also responsible for giving the wrong prediction in the case of the Gupta Syllogism. In this paper I argue against Lasersohn, and show that the problem of the Gupta Syllogism can be solved by making a minor addition to Intensional Montague Grammar. This solution is one that Lasersohn (...) discusses but rejects. I will show that his critique of it is ill-founded. (shrink)

Propositions are more than the bearers of truth and the meanings of sentences: they are also the objects of an array of attitudes including belief, desire, hope, and fear. This variety of roles leads to a variety of paradoxes, most of which have been sorely neglected. Arguing that existing work on these paradoxes is either too heavy-handed or too specific in its focus to be fully satisfactory, I develop a basic intensional logic and pursue and compare three (...) strategies for addressing the paradoxes, one employing truth-value gaps, one restricting propositional quantification, and one restricting our ability to have attitudes like belief and desire. This results in four distinct resolutions of the paradoxes, all but one of which are novel and all of which receive novel and general implementations. While resolving the paradoxes is of course the ultimate goal, I do not here argue that any one of the resolutions is superior. These paradoxes have been so little studied that my primary goal is only to identify the most fundamental costs and benefits of the various approaches one can take to addressing them. Each resolution I develop has significant drawbacks, which I argue highlight tensions between the different roles propositions play. Past researchers have skirted these tensions, and the issues raised by these paradoxes more generally, by focusing on non-propositional paradoxes, such as the most familiar forms of the Liar paradox. At the least, then, I hope this dissertation establishes that the propositional paradoxes deserve attention not only because of their consequences for intensional logic, but also because of their consequences for our understanding of content, truth, quantification, and a host of mental attitudes. (shrink)

The paper focuses on two apparent paradoxes arising from our use of intensional verbs: first, their object can be something which does not exist, i.e. something which is nothing; second, the fact that entailment from a qualified to a non-qualified object is not guaranteed. In this paper, I suggest that the problems share a solution, insofar as they arise in connection with intensional verbs that ascribe mental states. The solution turns on (I) a properly intensional or (...) nonrelational notion of representation and (II) a notion of “putting a representation on display”. (shrink)

In ‘The Unsatisfied Paradox’ (The Reasoner 6(12), p.184-5), Peter Eldridge-Smith has argued that no unique solution for the logical paradoxes is likely to exist in the presence of the following two kinds of paradox: 1. The Unsatisfied kind. 2. The Satisfiable kind. We argue that both kinds of paradoxes typically contain some kind of self-reference used for an attempt of self-diagonalization, and that consequently they may solvable in the same way, namely, by the acknowledgement that no intensional (...) object is available to itself for reference or quantification. (shrink)

Russell's "new contradiction" about "the totality of propositions" has been connected with a number of modal paradoxes. M. Oksanen has recently shown how these modal paradoxes are resolved in the set theory NFU. Russell's paradox of the totality of propositions was left unexplained, however. We reconstruct Russell's argument and explain how it is resolved in two intensional logics that are equiconsistent with NFU. We also show how different notions of possible worlds are represented in these intensional (...) logics. (shrink)

This is a critical discussion of Nino B. Cocchiarella’s book “Formal Ontology and Conceptual Realism.” It focuses on paradoxes of hyperintensionality that may arise in formal systems of intensional logic.

Montague’s analysis of the well-known temperature paradox poses a problem for Gupta’s syllogism, whose surface syntax differs from the temperature syllogism in the addition of the intensional adverb necessarily. Lasersohn (2005) argues that the puzzle arising from these syllogisms can be solved if one adopts the Fregean presuppositional treatment of definite descriptions, and concludes that the temperature-Gupta puzzle provides an argument in favor of such treatment. This paper shows that the analysis of definite descriptions is in fact orthogonal to (...) the puzzle. Instead, it will be shown that the differences between the two syllogisms stem from the temporal interpretation of their premises. (shrink)

Evidence is drawn together to connect sources of inconsistency that Frege discerned in his foundations for arithmetic with the origins of the paradox derived by Russell in "Basic Laws" I and then with antinomies, paradoxes, contradictions, riddles associated with modal and intensional logics. Examined are: Frege's efforts to grasp logical objects; the philosophical arguments that compelled Russell to adopt a description theory of names and a eliminative theory of descriptions; the resurfacing of issues surrounding reference, descriptions, identity, substitutivity, (...) paradox in the debates concerning modal and intensional logics; the development of the New Theory of Reference. I consider this to be the philosophical ground upon which the debates regarding that theory should take place. (shrink)

Some issues raised by the notion of surveyability and how it is represented mathematically are explored. Wright considers the sense in which the positive integers are surveyable and suggests that their structure will be a weakly finite, but weakly infinite, totality. One way to expose the incoherence of this account is by applying Wittgenstein's distinction between intensional and extensional to it. Criticism of the idea of a surveyable proof shows the notion's lack of clarity. It is suggested that this (...) concept should be replaced by that of a feasible operation, as strict finitism aims to understand the boundaries of legitimate mathematical knowledge. (shrink)

This paper discusses conditional probability $$P$$ P , or the probability of A given B. When $$P>0$$ P > 0 , the ratio formula determines $$P$$ P . When $$P=0$$ P = 0 , the ratio formula breaks down. The Borel–Kolmogorov paradox suggests that conditional probabilities in such cases are indeterminate or ill-posed. To analyze the paradox, I explore the relation between probability and intensionality. I argue that the paradox is a Frege case, similar to those that arise in many (...) probabilistic and non-probabilistic contexts. The paradox vividly illustrates how an agent’s way of representing an entity can rationally influence her credal assignments. I deploy my analysis to defend Kolmogorov’s relativistic treatment of conditional probability. (shrink)

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting phrase’. (...) Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the ‘substitutional theory’, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process. (shrink)

This dissertation is a study of some recent theories of truth. The theories fall into three groups: The Revision Theories, the context-sensitive theories, and the "Chrysippian theories". ;The "Chrysippian theories" are based on the intuition that pathologicalities arising from the concept of truth can be recognized and acknowledged with the concept of truth itself. Thus, from the pathologicality of the Liar, for example, we can conclude that the Liar is not true. This leads to immediate difficulties since the Liar claims (...) just that--that it is not true--and it would thus seem to be true. The Chrysippian theories are designed to avoid these difficulties while preserving the motivating intuition. Haim Gaifman's Pointer Semantics, and Brian Skyrms' Intensional Fixed-Point Theory of Truth are examples of such theories. They are exposited and criticized. ;The context-sensitive theories dealt with are Tyler Burge's Indexical Theory of Truth and the theory of Jon Barwise and John Etchemendy. These theories claim that an ambiguity due to a contextual parameter is the reason for the difficulties we encounter. Burge offers a theory of Tarskian levels coupled with a pragmatic theory that explains the assignment of levels to occurrences of 'true'. Barwise and Etchemendy make explicit a situational parameter in their account of propositions. They consider the ambiguity with respect to the situation a propositions is about a main source of the paradoxes. ;Finally, the Revision Theories of Truth of Anil Gupta and Nuel Belnap and two variations of this theory are considered. One is due to Aladdin Yaqub, the other, the "fully-varied theory", is due to Gupta and Belnap but remains largely unexplored. Revision theories are put forward as the most promising account of the difficulties. They are based on a definitional reading of all Tarski biconditionals. The circularity implied by such a reading is handled by Gupta's and Belnap's general theory of definitions. ;Yaqub's objections raised against the standard theory are examined, and his proposed solution is shown to be inadequate. The fully-varied Revision Theory is shown to provide a better solution. (shrink)

The main objective of this paper is to examine how theories of truth and reference that are in a broad sense Fregean in character are threatened by antinomies; in particular by the Epimenides paradox and versions of the so-called Russell-Myhill antinomy, an intensional analogue of Russell’s more well-known paradox for extensions. Frege’s ontology of propositions and senses has recently received renewed interest in connection with minimalist theories that take propositions (thoughts) and senses (concepts) as the primary bearers of truth (...) and reference. In this paper, I will present a rigorous version of Frege’s theory of sense and denotation and show that it leads to antinomies. I am also going to discuss ways of modifying Frege’s semantical and ontological framework in order to avoid the paradoxes. In this connection, I explore the possibility of giving up the Fregean assumption of a universal domain of absolutely all objects, containing in addition to extensional objects also abstract intensional ones like propositions and singular concepts. I outline a cumulative hierarchy of Fregean propositions and senses, in analogy with Gödel’s hierarchy of constructible sets. In this hierarchy, there is no domain of all objects. Instead, every domain of objects is extendible with new objects that, on pain of contradiction, cannot belong to the given domain. According to this approach, there is no domain containing absolutely all propositions or absolutely all senses. (shrink)

We construct a a system PLRI which is the classical propositional logic supplied with a ternary construction , interpreted as the intensional identity of statements and in the context . PLRI is a refinement of Roman Suszko’s sentential calculus with identity (SCI) whose identity connective is a binary one. We provide a Hilbert-style axiomatization of this logic and prove its soundness and completeness with respect to some algebraic models. We also show that PLRI can be used to give a (...) partial solution to the paradox of analysis. (shrink)

This piece continues the tradition of arguments by John Lucas, Roger Penrose and others to the effect that the human mind is not a machine. Kurt Gödel thought that the intensional paradoxes stand in the way of proving that the mind is not a machine. According to Gödel, a successful proof that the mind is not a machine would require a solution to the intensional paradoxes. We provide what might seem to be a partial vindication of (...) Gödel and show that if a particular solution to the intensional paradoxes is adopted, one can indeed give an argument to the effect that the mind is not a machine. (shrink)