“Weak relevant model structures” (wr-ms) are defined on “weak relevant matrices” by generalizing Brady’s model structure ${\mathcal{M}_{\rm CL}}$ built upon Meyer’s Crystal matrix CL. It is shown how to falsify in any wr-ms the Generalized Modus Ponens axiom and similar schemes used to derive Curry’s Paradox. In the last section of the paper we discuss how to extend this method of falsification to more general schemes that could also be used in deriving Curry’s Paradox.

As it is well known, in the forties of the past century, Curry proved that in any logic S closed under Modus Ponens, uniform substitution of propositional variables and the Contraction Law, the naïve Comprehension axiom trivializes S in the sense that all propositions are derivable in S plus CA. Not less known is the fact that, ever since Curry published his proof, theses and rules weaker than W have been shown to cause the same effect as W causes. (...) Among these, the Contraction rule or the Modus Ponens axiom, for example, are to be noted. But, moreover, as Brady has proved, even the Generalized Modus Ponens axiom or the Generalized Contraction rule give rise to “Curry’s Paradox” under the same circumstances as W does. In some previous work by us, “weak relevant model structures” are defined on “weak relevant matrices” by generalizing Brady’s model structure MCL built upon Meyer’s Crystal matrix CL. We have proved that wr-ms only verify logics with the “depth relevance condition”. The aim of this paper is to show how to falsify gMPa and gRW in certain wr-ms. In particular, it will be shown that gMPa is falisfied in any wr-ms and gRW in any wr-ms verifying Routley and Meyer’s basic positive logic B+. (shrink)

Curry claims that the positive paradox principle, ` A ⊃ in his elementary statement presentation, ‘is valid in any normal interpretation’ . By previous definition, ‘an interpretation of a system S is a normal interpretation just when the proposition A is true when and only when ` A’ . But his argument to normal validity is interestingly, and relevantly, invalid.

Two semantic paradoxes, the Liar and Curry’s paradox, are analysed using a newly developed conception of procedural semantics (semantics according to which the truth of propositions is determined algorithmically), whose main characteristic is its departure from methodological realism. Rather than determining pre-existing facts, procedures are constitutive of them. Of this semantics, two versions are considered: closed (where the halting of procedures is presumed) and open (without this presumption). To this end, a procedural approach to deductive reasoning is developed, based on (...) the idea of simulation. As is shown, closed semantics supports classical logic, but cannot in any straightforward way accommodate the concept of truth. In open semantics, where paradoxical propositions naturally ‘belong’, they cease to be paradoxical; yet, it is concluded that the natural choice—for logicians and common people alike—is to stick to closed semantics, pragmatically circumventing problematic utterances. (shrink)

We propose contractionless constructive logic which is obtained from Nelson's constructive logic by deleting contractions. We discuss the consistency of a naive set theory based on the proposed logic in relation to Curry's paradox. The philosophical significance of contractionless constructive logic is also argued in comparison with Fitch's and Prawitz's systems.

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic (...) which individually, but not jointly, lack the problematic feature. (shrink)

This paper presents a new puzzle for certain positions in the theory of truth. The relevant positions can be stated in a language including a truth predicate T and an operation that takes sentences to names of those sentences; they are positions that take the T-schema A ↔ T to hold without restriction, for every sentence A in the language. As such, they must be based on a nonclassical logic, since paradoxes that cannot be handled classically will arise. The (...) bestknown of these paradoxes is probably the liar paradox – a sentence that says of itself that it is not true – but our concern here is not with the liar. Instead, our focus is a variant of Curry ’s paradox – a sentence that says of itself that if it is true, everything is [3, 5, 7, 11]. In §1, we present the standard version of Curry ’s paradox and the strain of response to it we wish to focus on. This strain of response crucially invokes non-normal worlds: worlds at which the laws of logic differ from the laws that actually hold. In §2, we go on to argue that, in light of temporal curry paradox, this strain of response ought also to accept non-normal times: times at which the laws of logic in the actual world differ from the laws that hold now. We then consider, in §3, what this would mean for the theorists in question. (shrink)

Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.

In this paper, we distinguish two versions of Curry's paradox: c-Curry, the standard conditional-Curry paradox, and v-Curry, a validity-involving version of Curry's paradox that isn’t automatically solved by solving c-curry. A unified treatment of curry paradox thus calls for a unified treatment of both c-Curry and v-Curry. If, as is often thought, c-Curry paradox is to be solved via non-classical logic, then v-Curry may require a lesson about the structure—indeed, the substructure—of the validity relation itself.

In this paper we study the systemsP andP * (see Arruda and da Costa,O paradoxo de Curry-Moh Shaw-Kwei, Boletim da Sociedade Matemtica de São Paulo 18 (1966)) and some related systems. In the last section, we prove that certain set theories havingP andP * as their underlying logics are non-trivial.

In approaching Ch. 4 of Saving Truth from Paradox, it might be helpful first to revisit Curry’s original paper, and to revisit Lukasiewicz too, to provide more of the scenesetting that Field doesn’t himself fill in. So in §1 I’ll say something about Curry, in §2 we’ll look at what Lukasiewicz was up to in his original three-valued logic, and in §3 we’ll look at the move from a three-valued to a many-valued Lukasiewicz logic. In §4, I move (...) on to announce a theorem by H´. (shrink)

Theories where truth is a naive concept fall under the following dilemma: either the theory is subject to Curry’s Paradox, which engenders triviality, or the theory is not trivial but the resulting conditional is too weak. In this paper we explore a number of theories which arguably do not fall under this dilemma. In these theories the conditional is characterized in terms of non-deterministic matrices. These non-deterministic theories are similar to infinitely-valued Łukasiewicz logic in that they are consistent and (...) their conditionals are quite strong. The difference is the following: while Łukasiewicz logic is \-inconsistent, the non-deterministic theories might turn out to be \-consistent. (shrink)

Although formal analysis provides us with interesting tools for treating Curry’s paradox, it certainly does not exhaust every possible reading of it. Thus, we suggest that this paradox should be analysed with non-formal tools coming from pragmatics. In this way, using Grice’s logic of conversation, we will see that Curry’s sentence can be reinterpreted as a peculiar conditional sentence implying its own consequent.

This paper presents a range of new triviality proofs pertaining to naïve truth theory formulated in paraconsistent relevant logics. It is shown that excluded middle together with various permutation principles such as A → (B → C)⊩B → (A → C) trivialize naïve truth theory. The paper also provides some new triviality proofs which utilize the axioms ((A → B)∧ (B → C)) → (A → C) and (A → ¬A) → ¬A, the fusion connective and the Ackermann constant. An (...) overview over various ways to formulate Leibniz’s law in non-classical logics and two new triviality proofs for naïve set theory are also provided. (shrink)

We argue that narratives are central to the success of historical reconstruction. Narrative explanation involves tracing causal trajectories across time. The construction of narrative, then, often involves postulating relatively speculative causal connections between comparatively well-established events. But speculation is not always idle or harmful: it also aids in overcoming local underdetermination by forming scaffolds from which new evidence becomes relevant. Moreover, as our understanding of the past’s causal milieus become richer, the constraints on narrative plausibility become increasingly strict: a narrative’s (...) admissibility does not turn on mere logical consistency with background data. Finally, narrative explanation and explanation generated by simple, formal models complement one another. Where models often achieve isolation and precision at the cost of simplification and abstraction, narratives can track complex changes in a trajectory over time at the cost of simplicity and precision. In combination both allow us to understand and explain highly complex historical sequences. (shrink)

We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail (...) Abelian ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”. (shrink)

The logic BN4 was defined by R.T. Brady as a four-valued extension of Routley and Meyer’s basic logic B. The system EF4 is defined as a companion to BN4 to represent the four-valued system of (relevant) implication. The system Ł was defined by J. Łukasiewicz and it is a four-valued modal logic that validates what is known as strong Łukasiewicz-type modal paradoxes. The systems EF4-M and EF4-Ł are defined as alternatives to Ł without modal paradoxes. This paper (...) aims to define a Belnap-Dunn semantics for EF4, EF4-M and EF4-Ł. It is shown that EF4, EF4-M and EF4-Ł are strongly sound and complete w.r.t. their respective semantics and that EF4-M and EF4-Ł are free from strong Łukasiewicz-type modal paradoxes. (shrink)

In this paper we study the systems P and $P^{\ast}$ (see Arruda and da Costa, O paradoxo de Curry-Moh Shaw-Kwei, Boletim da Sociedade Matemātica de São Paulo 18 (1966)) and some related systems. In the last section, we prove that certain set theories having P and $P^{\ast}$ as their underlying logics are non-trivial.

The logic B+ is Routley and Meyer’s basic positive logic. Wedefine the logics BK+ and BK′+ by adding to B+ the K rule and to BK+the characteristic S4 axiom, respectively. These logics are endowed witha relatively strong non-constructive negation. We prove that all the logicsdefined lack the K axiom and the standard paradoxes of consistency.

In lieu of an abstract, here is a brief excerpt of the content:Royce, Racism, and the Colonial IdealWhite Supremacy and the Illusion of Civilization in Josiah Royce's Account of the White Man's Burden1Tommy J. CurryNo colony can be made by a theory of Imperialism, it can only be made by people who want to colonize and are capable of maintaining themselves as colonists.—Sir Sydney OlivierIntroductionAs with most historic white figures in philosophy, their repopularization and reintroduction into contemporary circles commits their (...) work, regardless of its initial silence, to speak to the problem of anti-Black racism in America. Josiah Royce is no different in this regard. In recent years, white scholars like Jacquelyn Kegley and Shannon Sullivan have introduced revisions to Royce's thought that make it appear multicultural and antiracist, despite the anachronism of such a contention. According to Kegley, "Royce believes, as others in the contemporary scene, that 'race' as a concept cannot be eliminated, [because] it plays too crucial a role, both positive and negative, in self- and social identification" (216). While Kegley is correct that contemporary scholars like Lucius Outlaw and Cornel West have adamantly defended the importance of race as a social and cultural entity, her claim is in stark contradiction to the racial eliminativism of other scholars like Anthony Appiah and Alain Locke, whom she also contends that Royce agrees with simultaneously.2 While it makes perfect sense for Kegley to try to put Royce in conversation with contemporary Black thinkers in race theory, her work is indicative of a trend aiming to gain currency for historic white thinkers in current race debates at the expense of historical accuracy. Black scholars who studied under Royce and applied his theories to the race problem from the mid-1900s onward, like William H. Ferris3 and William T. Fontaine,4 are ignored by white scholars because the first application of Roycean philosophy supported assimilationism. It is only today with the political and ideological [End Page 10] ethicizations of racial diversity and multiculturalism that Royce's work is interpreted as critically conserving race.5In contrast to contemporary works on Royce and race, this article seeks to evaluate the current claims for the relevance of Royce's views in contemporary debates on race. I maintain that only by taking Royce out of his historical context (late nineteenth-and early twentieth-century thought) do his ideas on race acquire a level of importance that cannot be supported by the actual evidence he himself advances. Conversely, by actually placing Royce in his historical context, this article will demonstrate that Royce's 1905 article entitled "Race Questions" (later published as "Race Questions and Prejudices" in 1906) is an unapologetic extension of his 1900 article entitled "Characteristic Tendencies of American Civilization," where Royce adamantly champions British colonialism and assimilation as the remedy to the burgeoning race problem both at home and abroad. I begin with a critical analysis that places Josiah Royce's attitudes firmly within the boundaries of his times. In what follows, I will attempt to contextualize Royce's seemingly progressive stance on Japan as part of a more general reaction whites had during the early 1900s to Russia's defeat by the Japanese. Instead of reading Royce's stance on Japan as a moment of exceptionalism, I contend that Royce's attitude was in fact a popular disposition that signaled the inevitable shift of race science away from biological determinism toward more environmentalist explanations. Secondly, and perhaps more importantly, I will show that Royce's disposition toward Blacks was firmly rooted in a colonial and assimilationist logic that ultimately sought the cultural destruction of African-descended people. A closer look at Royce's prefatory remarks on British colonialism from 1900 and his valorization of English colonial administration in Jamaica actually reveals an insidious racial dynamic in Royce's thought that is anything but antiracist. Finally, having established Royce's philosophical stake in American assimilation and Western domination, I argue that the "provincial revision" currently in vogue amongst American philosophers working with Royce is simply not enough to overturn Royce's clearly racist investments in dealing with the race question.If I Can Think It... Then I Can Do It—Reading Royce... (shrink)

We prove a set-theoretic version of Hájek, Paris and Shepherdson's theorem [HPS00] as follows: The set ω of natural numbers must contain a non-standard natural number in any natural Tarskian semantics of CŁ0, the set theory with comprehension principle within Lukasiewicz's infinite-valued predicate logic. The key idea of the proof is a generalization of the derivation of Moh Shaw-Kwei's paradox, which is a Russell-like paradox for many-valued logic.

The logic B+ is Routley and Meyer's basic positive logic. We show how to introduce a minimal intuitionistic negation and an intuitionistic negation in B+. The two types of negation are introduced in a wide spectrum of relevance logics built up from B+. It is proved that although all these logics have the characteristic paradoxes of consistency, they lack the K rule (and so, the K axioms).

The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical inferences. If one (...) or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes. (shrink)

IntroductionCurry’s paradox is well known.See, e.g., Priest , ch. 6. It comes in both set theoretic and semantic versions. Here we will concentrate on the semantic versions. Historically, these have deployed the notion of truth. Those who wish to endorse an unrestricted T-schema have mainly endorsed a logic which rejects the principle of Absorption, \\models A\rightarrow B\). High profile logics of this kind are certain relevant logics; these have semantics which show how and why this principle is not valid. (...) Of more recent times, paradoxes which are clearly in the same family have been appearing; but these concern the notion of validity itself. The standard semantics of relevant logics seem powerless to address these. But they can. This note shows how. The upshot can be seen as a return to the roots of relevant logic, in a sense to become clear.BackgroundThe Usual Curry ParadoxesLet us start with a couple of standard forms of the paradox. I .. (shrink)

In this paper, we aim to devise a logic that can deal with both the paradoxes that motivate dialetheism and the paradoxes related to the irrelevance of material implication. We propose the semantics and the sequent calculus of a relevant logic inspired by difference-making accounts of causation and arguably true to Graham Priest’s Logic of Paradox \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document}: a relevant logic that validates those and only those \documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document}-consequences that are not irrelevant. The new logic’s relevant implication is then added to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document} to obtain a logic with the logical vocabulary required to cope with the mentioned paradoxes. (shrink)

An analysis of Moore's paradox is given in doxastic logic. Logics arising from formalizations of various introspective principles are compared; one logic, K5c, emerges as privileged in the sense that it is the weakest to avoid Moorean belief. Moreover it has other attractive properties, one of which is that it can be justified solely in terms of avoiding false belief. Introspection is therefore revealed as less relevant to the Moorean problem than first appears.

After an introduction to set the stage, we consider some variations on the reasoning behind Curry's Paradox arising against the background of classical propositional logic and of BCI logic and one of its extensions, in the latter case treating the "paradoxicality" as a matter of nonconservative extension rather than outright inconsistency. A question about the relation of this extension and a differently described (though possibly identical) logic intermediate between BCI and BCK is raised in a final (...) section, which closes with a handful of questions left unanswered by our discussion. (shrink)

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large (...) range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management. (shrink)

In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz’s law are formulated and each is shown to cause problems when combined with naive truth theories.

Several authors have argued that a version of Curry's paradox involving validity motivates rejecting the structural rule of contraction. This paper criticizes two recently suggested alternative responses to “validity Curry.” There are three salient stages in a validity Curry derivation. Rejecting contraction blocks the first, while the alternative responses focus on the second and third. I show that a distinguishing feature of validity Curry, as contrasted with more familiar forms of Curry's paradox, is that paradox arises already at (...) the first stage. Accordingly, blocking the second or third stages won't suffice for resolving the paradox. (shrink)

Routley-Meyer ternary relational semantics was introduced in the early seventies of the past century. RM-semantics was intended to model classical relevant logics such as the logic of the relevant conditional R and the logic of Entailment E. But, ever since Routley and Meyer’s first papers on the topic, this essentially malleable semantics has been used for characterizing more general relevant logics or even non-relevant logics. The aim of this paper is to provide an RM-semantics with respect to which (...) Łukasiewicz 3-valued logic Ł3 is sound and complete. Ł3 is understood as the set of all valid formulas in Łukasiewicz 3-valued matrices MŁ3. In this sense, leaning on previous work by us, Ł3 is axiomatized as an extension of Routley and Meyer’s basic positive logic B+, labelled Ł3. And the RM-semantics for Ł3 is actually defined for this particular axiomatization of Ł3. The result presented in the paper is interesting from the Universal Logic perspective, in the sense that it connects Łukasiewicz many-valued logics and similar systems to relevant logics from the point of view of the latter, the 3-termed relational point of view, in particular. (shrink)

Curry's paradox for "if.. then.." concerns the paradoxical features of sentences of the form "If this very sentence is true, then 2+2=5". Standard inference principles lead us to the conclusion that such conditionals have true consequents: so, for example, 2+2=5 after all. There has been a lot of technical work done on formal options for blocking Curry paradoxes while only compromising a little on the various central principles of logic and meaning that are under threat. -/- Once we (...) have a sense of the technical options, though, a philosophical choice remains. When dealing with puzzles in the logic of conditionals, a natural place to turn is independently motivated semantic theories of the behaviour of "if... then...". This paper argues that the closest-worlds approach outlined in Nolan 1997 offers a philosophically satisfying reason to deny conditional proof and so block the paradoxical Curry reasoning, and can give the verdict that standard Curry conditionals are false, along with related "contraction conditionals". (shrink)

Beall and Cotnoir (2017) argue that theists may accept the claim that God's omnipotence is fully unrestricted if they also adopt a suitable nonclassical logic. Their primary focus is on the infamous Stone problem (i.e., whether God can create a stone too heavy for God to lift). We show how unrestricted omnipotence generates Curry‐like paradoxes. The upshot is that Beall and Cotnoir only provide a solution to one version of the Stone problem, but that unrestricted omnipotence generates other problems (...) which they do not adequately address. (shrink)

This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, (...) and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity". (shrink)

Deflationists about truth seek to undermine debates about the nature of truth by arguing that the truth predicate is merely a device that allows us to express a certain kind of generality. I argue that a parallel approach is available in the case of logical consequence. Just as deflationism about truth offers an alternative to accounts of truth's nature in terms of correspondence or justification, deflationism about consequence promises an alternative to model-theoretic or proof-theoretic accounts of consequence's nature. I then (...) argue, against considerations put forward by Field and Beall, that Curry's paradox no more rules out deflationism about consequence than the liar paradox rules out deflationism about truth. (shrink)

Curry’s paradox is well known. The original version employed a conditional connective, and is not forthcoming if the conditional does not satisfy contraction. A newer version uses a validity predicate, instead of a conditional, and is not forthcoming if validity does not satisfy structural contraction. But there is a variation of the paradox which uses “external validity”. And since external validity contracts, one might expect the appropriate version of the Curry paradox to be inescapable. In this paper we show that (...) this is not the case. We consider two ways of formalising the notion of external validity, and show that in both of these the paradox is not forthcoming without the appropriate forms of contraction. (shrink)

Rejecting structural contraction has been proposed as a strategy for escaping semantic paradoxes. The challenge for its advocates has been to make intuitive sense of how contraction might fail. I offer a way of doing so, based on a “naive” interpretation of the relation between structure and logical vocabulary in a sequent proof system. The naive interpretation of structure motivates the most common way of blaming Curry-style paradoxes on illicit contraction. By contrast, the naive interpretation will not as easily motivate (...) one recent noncontractive approach to the Liar paradox. (shrink)

The paper is concerned with the epistemological status of testimony and the question of what may confer justification on true testimonial beliefs and enable us to call such beliefs knowledge. In particular, it addresses certain anti-reductionist arguments in the epistemology of testimony and their incompatibility with the grammatical categories of egophoricity (conjunct/disjunct marking) and evidentiality (information source marking) present in the architecture of natural languages. First, the tradition of epistemological individualism and its rationale are discussed, as well as certain attempts (...) within this tradition to include testimony-based beliefs in the body of legitimate and rational beliefs. The next section is concerned with the anti-reductionist approach to testimonial beliefs and some arguments supporting ‘the credulity principle’ grounded in Thomas Reid's philosophy of common sense. The last two sections show how the anti-reductionist argumentation is threatened with an insoluble dilemma when considered in the context of languages with the grammatical categories of egophoricity (conjunct/disjunct marking) and evidentiality (information source marking). Special attention is paid to the justificatory role of egophoric and evidential marking; how these grammatical categories may influence the process of belief formation, and consequently, how this bears upon the nature of justification for testimonial beliefs. The final conclusion of the paper is that the anti-reductionist claim concerning the directness and non-inferential nature of testimonial knowledge is bound to meet objections grounded in the structure of many natural languages. This leads to a paradox: in discussion with reductionism, anti-reductionists do seek support for their arguments in common sense beliefs reflected in everyday language, and, on the other hand, their claims are incompatible with the linguistic data found in a number of natural languages. (shrink)

In “God of the Gaps: A Neglected Reply to God’s Stone Problem”, Jc Beall and A. J. Cotnoir offer a gappy solution to the paradox of (unrestricted) omnipotence that is typified by the classic stone problem. Andrew Tedder and Guillermo Badia, however, have recently argued that this solution could not be extended to a more serious Curry-like version of the paradox. In this paper, we show that such a gappy solution does extend to it.

An investigation into the assessment of the moral components which were developed by John Wilson, is reported. Tests fox the classroom measurement of two components were developed. The components were; PHIL(CC), the claiming of concern for other persons as an overriding, universal, and prescriptive principle in moral decision making; and; GIG, knowledge of factual information which is relevant in making moral decisions which subjects face. The test development exercise was undertaken at a time when public interest in moral education was (...) growing. The recent demand for moral education in Auckland is reviewed. Over the last fifteen years, since the Currie Commission Report, reports by committees investigating the purposes of schools have increasingly emphasised moral and social education as school objectives. The Department of Education appeared to be sympathetic towards the cause of moral education. The submissions made by the public during the Educational Development Conference suggested that, in general, parents and citizens were prepared to consider innovative programmes in social or moral education, although there was little agreement on what form such training or education should take. A number of teachers were supporters of moral education. The primary purpose in constructing tests for Wilson's components was to provide an instrument which would assist in the evaluation of moral education curriculum projects in Auckland secondary schools. Evidence concerning descriptive, content, domain selection, construct and concurrent validity is presented. Kuder-Richardson, retest and criterion-referenced reliability studies were undertaken. It is claimed that an instrument with sufficient validity and reliability has bean produced for the summative evaluation of curriculum projects, and the diagnostic investigation of class groups using the test as a criterion-referenced measure. Auckland intermediate and secondary school pupils were surveyed, using the tests produced and punch card recording in an attempt to identify significant variables. Over 1,100 children completed the tests under controlled conditions. Significant variables identified using the test for PHIL(CC) were socio-economic level for twelve-year-old children, and intelligence for sixteen-year-old children. The effect of schooling appeared to be significant at all levels. Age does not appear to markedly increase children's concern for others. Age was related to performance in the knowledge test. Older children knew more. Other significant variables for GIG were socio-economic level (middle levels performed better) and the effects of schooling. There was some evidence that females know more than males. In both tests it appears that there is considerable interaction between the variables. Suggestions for the further development of the tests are given. (shrink)

Any theory of truth must find a way around Curry’s paradox, and there are well-known ways to do so. This paper concerns an apparently analogous paradox, about validity rather than truth, which JC Beall and Julien Murzi call the v-Curry. They argue that there are reasons to want a common solution to it and the standard Curry paradox, and that this rules out the solutions to the latter offered by most “naive truth theorists.” To this end they recommend a radical (...) solution to both paradoxes, involving a substructural logic, in particular, one without structural contraction. In this paper I argue that substructuralism is unnecessary. Diagnosing the “v-Curry” is complicated because of a multiplicity of readings of the principles it relies on. But these principles are not analogous to the principles of naive truth, and taken together, there is no reading of them that should have much appeal to anyone who has absorbed the morals of both the ordinary Curry paradox and the second incompleteness theorem. (shrink)

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical (...) means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. A comparison to Mach’s doctrine is used to be revealed the fundamental and philosophical reductionism of Husserl’s phenomenology leading to a kind of Pythagoreanism in the final analysis. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem. (shrink)

Starting with Jaakko Hintikka’s seminal work, epistemic logic has now grown up to a huge academic industry. When we look back to its history, however, there is a practical paradox. While Hintikka’s original purpose was to facilitate epistemological discussion, most subsequent work of epistemic logic has been done outside of philosophy, and has been ignored by most mainstream epistemologists. At the current point, one might naturally wonder, does epistemic logic still have something to do with epistemology? Along (...) with Hintikka and a growing number of formal epistemologists, we believe the answer is “yes”. Contemporary epistemology has largely focused on the goal of providing an analysis of knowledge and an answer to skepticism. To avoid skepticism, many epistemologists prefer to define knowledge in terms of evidence and relevant alternatives. A typical saying is that “in order to know that p, one only needs evidence to eliminate all relevant alternatives, rather than all alternatives”. We see such a project as perfectly congenial to the spirit of Hintikka’s epistemic logic. In this paper, we propose an epistemic logic to deal with evidence and relevant alternatives explicitly, and extend EREL to its dynamic version to investigate how different notions of knowledge behavior with the decrease and increase of relevant alternatives. To illustrate its philosophical application, we provide a new formalization of the so called factivity problem of epistemic contextualism. We conclude with some remarks on further developments of our framework. (shrink)

Based on his Inclosure Schema and the Principle of Uniform Solution (PUS), Priest has argued that Curry’s paradox belongs to a different family of paradoxes than the Liar. Pleitz (2015, The Logica Yearbook 2014, pp. 233–248) argued that Curry’s paradox shares the same structure as the other paradoxes and proposed a scheme of which the Inclosure Schema is a particular case and he criticizes Priest’s position by pointing out that applying the PUS implies the use of a paraconsistent logic (...) that does not validate Contraction, but that this can hardly seen as uniform. In this paper, we will develop some further reasons to defend Pleitz’ thesis that Curry’s paradox belongs to the same family as the rest of the self-referential paradoxes & using the idea that conditionals are generalized negations. However, we will not follow Pleitz in considering doubtful that there is a uniform solution for the paradoxes in a paraconsistent spirit. We will argue that the paraconsistent strategies can be seen as special cases of the strategy of restricting Detachment and that the latter uniformly blocks all the connective-involving self-referential paradoxes, including Curry’s. (shrink)