This paper continues my work of [9], which showed there was a broad family of many valued logics that have a strict/tolerant counterpart. Here we consider a generalization of weak Kleene three valued logic, instead of the strong version that was background for that earlier work. We explain the intuition behind that generalization, then determine a subclass of strict/tolerant structures in which a generalization of weak Kleene logic produces the same results that the strong Kleene (...) generalization did. This paper provides much background, but is not self-contained. Some results from [9] are called on, and are not reproved here. [9] Melvin C. Fitting. “A Family of Strict/Tolerant Logics”. In: Journal of Philosophical Logic. Online. Print publication forthcoming. (shrink)

Strict/tolerant logic, ST, evaluates the premises and the consequences of its consequence relation differently, with the premises held to stricter standards while consequences are treated more tolerantly. More specifically, ST is a three-valued logic with left sides of sequents understood as if in Kleene’s Strong Three Valued Logic, and right sides as if in Priest’s Logic of Paradox. Surprisingly, this hybrid validates the same sequents that classical logic does. A version of this result has been extended to meta, (...) metameta, … consequence levels in Barrio et al.. In my earlier paper Fitting I showed that the original ideas behind ST are, in fact, much more general than first appeared, and an infinite family of many valued logics have Strict/Tolerant counterparts. This family includes both Kleene’s and Priest’s logic individually, as well as first degree entailment. For instance, for both the Kleene and the Priest logic, the corresponding strict/tolerant logic is six-valued, but with differing sets of strictly and tolerantly designated truth values. The present paper extends that generalization in two directions. We examine a reverse notion, of Tolerant/Strict logics, which exist for the same structures that were investigated in Fitting. And we show that the generalization extends through the meta, metameta, … consequence levels for the same infinite family of many valued logics. Finally we close with remarks on the status of cut and related rules, which can actually be rather nuanced. Throughout, the aim is not the philosophical applications of the Strict/Tolerant idea, but the determination of how general a phenomenon it is. (shrink)

Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egré, Ripley and van Rooij have put forward an approach based on a non-transitive notion of consequence which is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some interesting issues concerning the set of metainferences validated by this (...) logic. In this paper, we show that this logic, once it is adequately understood, is weaker than classical logic. Moreover, the logic is in a way similar to the paraconsistent logic LP. (shrink)

Strict/tolerant logic is a formally defined logic that has the same consequence relation as classical logic, though it differs from classical logic at the metaconsequence level. Specifically, it does not satisfy a cut rule. It has been proposed for use in work on theories of truth because it avoids some objectionable features arising from the use of classical logic. Here we are not interested in applications, but in the formal details themselves. We show that a wide range of (...) logics have strict/tolerant counterparts, with the same consequence relations but differing at the metaconsequence level. Among these logics are Kleene’s K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {K_3}$$\end{document}, Priest’s LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {LP}}$$\end{document}, and first-degree entailment, FDE. The primary tool we use is the bilattice. But it is more than a tool, it seems to be the natural home for this kind of investigation. (shrink)

In their recent article “A Hierarchy of Classical and Paraconsistent Logics”, Eduardo Barrio, Federico Pailos and Damien Szmuc present novel and striking results about meta-inferential validity in various three valued logics. In the process, they have thrown open the door to a hitherto unrecognized domain of non-classical logics with surprising intrinsic properties, as well as subtle and interesting relations to various familiar logics, including classical logic. One such result is that, for each natural number n, there (...) is a logic which agrees with classical logic on tautologies, inferences, meta-inferences, meta-meta-inferences, meta-meta-...-meta-inferences, but that disagrees with classical logic on n + 1-meta-inferences. They suggest that this shows that classical logic can only be characterized by defining its valid inferences at all orders. In this article, I invoke some simple symmetric generalizations of BPS’s results to show that the problem is worse than they suggest, since in fact there are logics that agree with classical logic on inferential validity to all orders but still intuitively differ from it. I then discuss the relevance of these results for truth theory and the classification problem. (shrink)

The paper presents a truth-maker semantics for Strict/Tolerant Logic (ST), which is the currently most popular logic among advocates of the non-transitive approach to paradoxes. Besides being interesting in itself, the truth-maker presentation of ST offers a new perspective on the recently discovered hierarchy of meta-inferences that, according to some, generalizes the idea behind ST. While fascinating from a mathematical perspective, there is no agreement on the philosophical significance of this hierarchy. I aim to show that there is (...) no clear philosophical significance of meta-inferences above the first level. (shrink)

We discuss four distinct semantic consequence relations which are based on Strong Kleene theories of truth and which generalize the notion of classical consequence to 3-valued logics. Then we set up a uniform signed tableau calculus, which we show to be sound and complete with respect to each of the four semantic consequence relations. The signs employed by our calculus are,, and, which indicate a strict assertion, strict denial, tolerant assertion and tolerant denial respectively. Recently, (...) Ripley applied the strict–tolerant account of assertion and denial to develop a new approach to truth and alethic paradox, which we call the Strict–Tolerant Conceptionof Truth. The paper aims to contribute to our understanding of STCT in at least three ways. First, by developing the strict–tolerant calculus.Second, by developing a semantic version of the strict–tolerant calculus, which informs us about the assertoric possibilities relative to a fixed ground model. Third, by showing that the strict–tolerant calculus and assertoric semantics jointly suggest that STCT's claim that the strict and tolerant can be understood in terms of one another has to be reconsidered. (shrink)

In a recent paper, Barrio, Pailos and Szmuc (BPS) show that there are logics that have exactly the validities of classical logic up to arbitrarily high levels of inference. They suggest that a logic therefore must be identified by its valid inferences at every inferential level. However, Scambler shows that there are logics with all the validities of classical logic at every inferential level, but with no antivalidities at any inferential level. Scambler concludes that in order to identify (...) a logic, we at least need to look at the validities and the antivalidities of every inferential level. In this paper, I argue that this is still not enough to identify a logic. I apply BPS’s techniques in a super/sub-valuationist setting to construct a logic that has exactly the validities and antivalidities of classical logic at every inferential level. I argue that the resulting logic is nevertheless distinct from classical logic. (shrink)

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of (...) it, which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases. (shrink)

The strict-tolerant approach to paradox promises to erect theories of naïve truth and tolerant vagueness on the firm bedrock of classical logic. We assess the extent to which this claim is founded. Building on some results by Girard we show that the usual proof-theoretic formulation of propositional ST in terms of the classical sequent calculus without primitive Cut is incomplete with respect to ST-valid metainferences, and exhibit a complete calculus for the same class of metainferences. We also (...) argue that the latter calculus, far from coinciding with classical logic, is a close kin of Priest’s LP. (shrink)

Strict-Tolerant Logic ($$\textrm{ST}$$ ST ) underpins naïve theories of truth and vagueness (respectively including a fully disquotational truth predicate and an unrestricted tolerance principle) without jettisoning any classically valid laws. The classical sequent calculus without Cut is sometimes advocated as an appropriate proof-theoretic presentation of $$\textrm{ST}$$ ST. Unfortunately, there is only a partial correspondence between its derivability relation and the relation of local metainferential $$\textrm{ST}$$ ST -validity – these relations coincide only upon the addition of elimination rules and (...) only within the propositional fragment of the calculus, due to the non-invertibility of the quantifier rules. In this paper, we present two calculi for first-order $$\textrm{ST}$$ ST with an eye to recapturing this correspondence in full. The first calculus is close in spirit to the Epsilon calculus. The other calculus includes rules for the discharge of sequent-assumptions; moreover, it is normalisable and admits interpolation. (shrink)

Strictly speaking the law cannot admit toleration: It cannot tolerate ideas or behaviour which are contrary to its requirements. This logic explains why for centuries civilizations found no place for toleration. Then in the seventeenth and eighteenth centuries, philosophers and thinkers such as Spinoza, Locke, Bayle and later Voltaire or Malesherbes advocated tolerance, certain aspects of which were to be introduced into the legislation of many countries: freedom of opinion, the free movement of persons, freedom of assembly and of religion.

In a previous paper (see ‘Tolerant, Classical, Strict’, henceforth TCS) we investigated a semantic framework to deal with the idea that vague predicates are tolerant, namely that small changes do not affect the applicability of a vague predicate even if large changes do. Our approach there rests on two main ideas. First, given a classical extension of a predicate, we can define a strict and a tolerant extension depending on an indifference relation associated to that (...) predicate. Second, we can use these notions of satisfaction to define mixed consequence relations that capture non-transitive tolerant reasoning. Although we gave some empirical motivation for the use of strict and tolerant extensions, making use of them commits us to the view that sentences of the form ‘ p∨¬p ’ and ‘ p∧¬p ’ are not automatically valid or unsatisfiable, respectively. Some philosophers might take this commitment as a negative outcome of our previous proposal. We think, however, that the general ideas underlying our previous approach to vagueness can be implemented in a variety of ways. This paper explores the possibility of defining mixed notions of consequence in the more classical super/sub-valuationist setting and examines to what extent any of these notions captures non-transitive tolerant reasoning. (shrink)

Most nonmonotonic logics are assumed to be cumulative, which is often regarded as the minimum requirement for a logic. We argue that cumulativity, in particular, cumulative transitivity can be abandoned, in order to better characterize reasoning in uncertainty. But giving up cumulative transitivity makes it hard to obtain representation results for these logics. Borrowing the idea from strict-tolerant logics, we give some representation results for nonmonotonic logics that are not cumulatively transitive.

This article bears on four topics: observational predicates and phenomenal properties, vagueness, strict finitism as a philosophy of mathematics, and the analysis of feasible computability. It is argued that reactions to strict finitism point towards a semantics for vague predicates in the form of nonstandard models of weak arithmetical theories of the sort originally introduced to characterize the notion of feasibility as understood in computational complexity theory. The approach described eschews the use of nonclassical logic and related devices (...) like degrees of truth or supervaluation. Like epistemic approaches to vagueness, it may thus be smoothly integrated with the use of classical model theory as widely employed in natural language semantics. But unlike epistemicism, the described approach fails to imply either the existence of sharp boundaries or the failure of tolerance for soritical predicates. Applications ofmeasurement theory(in the sense of Krantz, Luce, Suppes, & Tversky (1971)) to vagueness in the nonstandard setting are also explored. (shrink)

Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations $st$, $ss$, $tt$, $ss\cap tt$, and $ts$, when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ the answer is trivial (no scheme works), and for $ss$ and $tt$ it is straightforward (they are the collapsible schemes, in which (...) the middle value acts like one of the classical values). For $st$, the schemes in question are the Boolean normal schemes that are either monotonic or collapsible. (shrink)

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that (...) a logic is to be identified with an infinite sequence of consequence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting consequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences. (shrink)

The present note investigates the connection between nonreflexive and nontransitive logics from a bounds-theoretic perspective. What will emerge is one way in which, if we focus on the ways in which strict and tolerant acts constrain one another, nonreflexive and nontransitive notions of consequence can be seen as simply reflecting different aspects of the same underlying reality.

Consider those many-valued logic models in which the truth values are a lattice that supplies interpretations for the logical connectives of conjunction and disjunction, and which has a De Morgan involution supplying an interpretation for negation. Assume that the set of designated truth values is a prime filter in the lattice. Each of these structures determines a simple many-valued logic. We show that there is a single Smullyan-style signed tableau system appropriate for all of the logics these structures determine. (...) Differences between the logics are confined entirely to tableau branch closure rules. Completeness, soundness, and interpolation can be proved in a uniform way for all cases. Since branch closure rules have a limited number of variations, in fact all the semantic structures determine just four different logics, all well-known ones. Asymmetric logics such as strict/tolerant, ST, also share all the same tableau rules, but differ in what constitutes an initial tableau. It is also possible to capture the notion of antivalidity using the same set of tableau rules. Thus a simple set of tableau rules serves as a unifying and classifying device for a natural and simple family of many-valued logics. (shrink)

This book presents a new theory of the relationship between vagueness, context-sensitivity, gradability, and scale structure in natural language. Heather Burnett argues that it is possible to distinguish between particular subclasses of adjectival predicatesDLrelative adjectives like tall, total adjectives like dry, partial adjectives like wet, and non-scalar adjectives like hexagonalDLon the basis of how their criteria of application vary depending on the context; how they display the characteristic properties of vague language; and what the properties of their associated orders are. (...) It has been known for a long time that there exist empirical connections between context-sensitivity, vagueness, and scale structure; however, a formal system that expresses these connections had yet to be developed. This volume sets out a new logical system, called DelTCS, that brings together insights from the Delineation Semantics framework and from the Tolerant, Classical, Strict non-classical framework, to arrive at a full theory of gradability and scale structure in the adjectival domain. The analysis is further extended to examine vagueness and gradability associated with particular classes of determiner phrases, showing that the correspondences that exist between the major adjectival scale structure classes and subclasses of determiner phrases can also be captured within the DelTCS system. (shrink)

Rejecting the Cut rule has been proposed as a strategy to avoid both the usual semantic paradoxes and the so-called v-Curry paradox. In this paper we consider if a Cut-free theory is capable of accurately representing its own notion of validity. We claim that the standard rules governing the validity predicate are too weak for this purpose and we show that although it is possible to strengthen these rules, the most obvious way of doing so brings with it a serious (...) problem: an internalized version of Cut can be proved for a Curry-like sentence. We also evaluate a number of possible ways of escaping this difficulty. (shrink)

In a recent paper, Barrio, Tajer and Rosenblatt establish a correspondence between metainferences holding in the strict-tolerant logic of transparent truth ST+ and inferences holding in the logic of paradox LP+. They argue that LP+ is ST+’s external logic and they question whether ST+’s solution to the semantic paradoxes is fundamentally different from LP+’s. Here we establish that by parity of reasoning, ST+ can be related to LP+’s dual logic K3+. We clarify the distinction between internal and external (...) logic and argue that while ST+’s nonclassicality can be granted, its self-dual character does not tie it to LP+ more closely than to K3+. (shrink)

We study a probabilistic logic based on the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence). We examine probabilistic conditional knowledge bases associated with imprecise probability assessments defined on arbitrary families of conditional events. We introduce a notion of conditional interpretation defined directly in terms of precise probability assessments. We also examine a property of strong satisfiability which is related to the notion of toleration well known in default reasoning. In our framework we give more (...) general definitions of the notions of probabilistic consistency and probabilistic entailment of Adams. We also recall a notion of strict p-consistency and some related results. Moreover, we give new proofs of some results obtained in probabilistic default reasoning. Finally, we examine the relationships between conditional probability rankings and the notions of g-coherence and g-coherent entailment. (shrink)

This paper proposes an experimental investigation of the use of vague predicates in dynamic sorites. We present the results of two studies in which subjects had to categorize colored squares at the borderline between two color categories (Green vs. Blue, Yellow vs. Orange). Our main aim was to probe for hysteresis in the ordered transitions between the respective colors, namely for the longer persistence of the initial category. Our main finding is a reverse phenomenon of enhanced contrast (i.e. negative hysteresis), (...) present in two different tasks, a comparative task involving two color names, and a yes/no task involving a single color name, but not found in a corresponding color matching task. We propose an optimality-theoretic explanation of this effect in terms of the strict-tolerant framework of Cobreros et al. (J Philos Log 1–39, 2012), in which borderline cases are characterized in a dual manner in terms of overlap between tolerant extensions, and underlap between strict extensions. (shrink)

The cut-free validity theory $$\textsf{STV}$$ proposed by Barrio, Rosenblatt, and Tajer suffers from incompleteness with respect to its object language validity predicate. The validity predicate of $$\textsf{STV}$$ fails in validating some valid inferences of its underlying logic, the Strict Tolerant logic $$\textsf{ST}$$. In this paper, we will present the non-normal modal logic $$\textsf{ST}^{\Box \Diamond }$$ whose modalities $$\Box $$ and $$\Diamond $$ capture the tautologies/valid inferences and the consistent formulas of the logic $$\textsf{ST}$$, respectively. We show that $$\textsf{ST}^{\Box (...) \Diamond }$$ does not trivialize when extended with self-referential devices. We also show that such a solution poses a dilemma. If we extend $$\textsf{ST}^{\Box \Diamond }$$ in such a way that it allows iterated modal formulas among its theorems, then the resulting interpretation of $$\Box $$ as validity implies that metametainferences of $$\textsf{ST}$$ behave like classical logic. On the other hand, if we allow these modalities to receive intermediate truth values, we obtain formulas incompatible with the proposed reading of $$\Box $$. (shrink)

Many analyses of notion of _metainferences_ in the non-transitive logic ST have tackled the question of whether ST can be identified with classical logic. In this paper, we argue that the primary analyses are overly restrictive of the notion of metainference. We offer a more elegant and tractable semantics for the strict-tolerant hierarchy based on the three-valued function for the LP material conditional. This semantics can be shown to easily handle the introduction of _mixed_ inferences, _i.e._, inferences involving (...) objects belonging to more than one (meta)inferential level and solves several other limitations of the ST hierarchies introduced by Barrio, Pailos, and Szmuc. (shrink)

This paper explores the idea that vague predicates like “tall”, “loud” or “expensive” are applied based on a process of analog magnitude representation, whereby magnitudes are represented with noise. I present a probabilistic account of vague judgment, inspired by early remarks from E. Borel on vagueness, and use it to model judgments about borderline cases. The model involves two main components: probabilistic magnitude representation on the one hand, and a notion of subjective criterion. The framework is used to represent judgments (...) of the form “x is clearly tall” versus “x is tall”, as involving a shift of one’s criterion, and then to derive observed patterns of acceptance for borderline contradictions, namely sentences of the form “x is tall and not tall”, relative to the acceptance of their conjuncts. (shrink)

This paper presents and defends a way to add a transparent truth predicate to classical logic, such that and A are everywhere intersubstitutable, where all T-biconditionals hold, and where truth can be made compositional. A key feature of our framework, called STTT (for Strict-Tolerant Transparent Truth), is that it supports a non-transitive relation of consequence. At the same time, it can be seen that the only failures of transitivity STTT allows for arise in paradoxical cases.

I discuss the problem of whether true contradictions of the form “x is P and not P” might be the expression of an implicit relativization to distinct respects of application of one and the same predicate P. Priest rightly claims that one should not mistake true contradictions for an expression of lexical ambiguity. However, he primarily targets cases of homophony for which lexical meanings do not overlap. There exist more subtle forms of equivocation, such as the relation of privative opposition (...) singled out by Zwicky and Sadock in their study of ambiguity. I argue that this relation, which is basically a relation of general to more specific, underlies the logical form of true contradictions. The generalization appears to be that all true contradictions really mean “x is P in some respects/to some extent, but not in all respects/not to all extent”. I relate this to the strict-tolerant account of vague predicates and outline a variant of the account to cover one-dimensional and multidimensional predicates. (shrink)

Most literature on vagueness deals with the phenomenon as applied to predicates. On the contrary, even the idea of vague connectives seems to be taken as an oxymoron. The goal of this article is to propose an understanding of vague logical connectives based on vague quantifiers. The main idea is that the phenomenon of vagueness translates to connectives in terms of the property of Abnormality. I also argue that Prior’s Tonk can, according to this approach, be considered a vague connective. (...) In order to do so, I provide a sound and complete interpretation for it, based on Strict-Tolerant semantics, in which Tonk has a non-normal truth-table. (shrink)

This paper revisits Buridan’s Bridge paradox (Sophismata, chapter 8, Sophism 17), itself close kin to the Liar paradox, a version of which also appears in Bradwardine’s Insolubilia. Prompted by the occurrence of the paradox in Cervantes’s Don Quixote, I discuss and compare four distinct solutions to the problem, namely Bradwardine’s “just false” conception, Buridan’s “contingently true/false” theory, Cervantes’s “both true and false” view, and then the “neither true simpliciter nor false simpliciter” account proposed more recently by Jacquette. All have in (...) common to accept that the Bridge expresses a truth-apt proposition, but only the latter three endorse the transparency of truth. Against some previous commentaries I first show that Buridan’s solution is fully compliant with an account of the paradox within classical logic. I then argue that Cervantes’s insights, as well as Jacquette’s treatment, are both supportive of a dialetheist account, and Jacquette’s in particular of the strict-tolerant account of truth. I defend dialetheist intuitions (whether in LP or ST guise) against two objections: one concerning the future, the other concerning the alleged simplicity of the Bridge compared to the Liar. (shrink)

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances (...) of Cut involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities. (shrink)

We expand first order models with a tolerance relation on thedomain. Intuitively, two elements stand in this relation if they arecognitively close for the agent who holds the model. This simplenotion turns out to be very powerful. It leads to a semanticcharacterization of the guarded fragment of Andréka, van Benthemand Németi, and highlights the strong analogies between modallogic and this fragment. Viewing the resulting logic – tolerance logic– dynamically it is a resource-conscious information processingalternative to classical first order logic. The (...) differences areindicated by several examples. (shrink)

Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in his strict finitistic metatheory, of equations between numerals on which concrete arithmetical sentences are evaluated. As a first step towards classical formalisation of strict finitism, we propose their counterparts in the classical metatheory with one additional assumption, and then extract the propositional (...) part of ‘strict finitistic logic’ from it and investigate. We will provide a sound and complete pair of a Kripke-style semantics and a sequent calculus, and compare with other logics. The logic lacks the law of excluded middle and Modus Ponens and is weaker than classical logic, but stronger than any proper intermediate logics in terms of theoremhood. In fact, all the other well-known classical theorems are found to be theorems. Finally, we will make an observation that models of this semantics can be seen as nodes of an intuitionistic model. (shrink)

Noniterative approaches to the sorites paradox accept single steps of soritical reasoning, but deny that these can be combined into valid chains of soritical reasoning. The distributed sorites is a puzzle designed to undermine noniterative approaches to the sorites paradox, by deriving an inconsistent conclusion using only single steps, but not chains, of soritical reasoning. This paper shows how a dialetheist version of the noniterative approach, the strict-tolerant approach, also solves the distributed sorites paradox, at no further cost, (...) by accepting the inconsistent conclusion. (shrink)

In order to reason in a non-trivializing way with contradictions, para- consistent logics reject some classically valid inferences. As a way of re- covering some of these inferences, Graham Priest ([Priest, 1991]) proposed to nonmonotonically strengthen the Logic of Paradox by allowing the se- lection of “less inconsistent” models via a comparison of their respective inconsistent parts. This move recaptures a good portion of classical logic in that it does not block, e.g., disjunctive syllogism, unless it is applied to (...) contradictory assumptions. In Priest’s approach the inconsistent parts of models are compared in an extensional way by considering their inconsis- tent objects. This distinguishes his system from the standard format of (inconsistency-)adaptive logics pioneered by Diderik Batens, according to which (atomic) contradictions validated in models form the basis of their comparison. A well-known problem for Priest’s extensional approach is its lack of the Strong Reassurance property, i.e., for specific settings there may be infinitely descending chains of less and less inconsistent models, thus never reaching a minimally inconsistent model. In this paper, we demonstrate that Strong Reassurance holds for the extensional approach under a cardinality-based comparison of the incon- sistent parts of models. Furthermore, we introduce and investigate the metatheory of the class of first-order nonmonotonic inconsistency-tolerant construct over the extensional or quantitative comparisons of their respec- tive models. Core model-theoretic properties for these logics, such as the Löwenheim-Skolem theorems, along with other nonmonotonic properties, are further studied. (shrink)

The paper presents both new data and a new analysis of the semantic and pragmatic properties of the class of absolute scalar adjectives within an extension of a well-known logical framework for the analysis of gradable predicates: the delineation semantics framework . It has been long observed that the context-sensitivity, vagueness and gradability features of absolute scalar predicates give rise to certain puzzles for their analysis within most, if not all, modern formal semantic frameworks. While there exist proposals for solving (...) these puzzles within other major frameworks , it has been argued that some of their aspects are particularly challenging for the analysis of absolute scalar predicates within the delineation approach. By combining insights into the relationship between context-sensitivity and scalarity from the DelS framework with insights into the relationship between tolerance/similarity relations and the Sorites paradox from Cobreros et al.’s Tolerant, Classical, Strict framework, I propose a new logical system, called Delineation TCS , in which to set analyses of four classes of adjectival predicates. I argue that this new framework allows for an analysis of absolute scalar adjectives that answers these challenges for delineation-based frameworks, while still preserving the heart of the Klein-ian approach. (shrink)

We introduce a two-valued and a three-valued truth-valuational substitutional semantics for the Quantified Argument Calculus (Quarc). We then prove that the 2-valid arguments are identical to the 3-valid ones with strict-to-tolerant validity. Next, we introduce a Lemmon-style Natural Deduction system and prove the completeness of Quarc on both two- and three-valued versions, adapting Lindenbaum’s Lemma to truth-valuational semantics. We proceed to investigate the relations of three-valued Quarc and the Predicate Calculus (PC). Adding a logical predicate T to Quarc, (...) true of all singular arguments, allows us to represent PC quantification in Quarc and translate PC into Quarc, preserving validity. Introducing a weak existential quantifier into PC allows us to translate Quarc into PC, also preserving validity. However, unlike the translated systems, neither extended system can have a sound and complete proof system with Cut, supporting the claim that these are basically different calculi. (shrink)

The first theory of confirmation that Carnap developed in detail is to be found in "Testability and Meaning". In this paper, he addressed the issue of a definition of empiricism, several years after abandoning the quest for a unique and universal logical framework supposed to be the basis of a clear distinction between the meaningful sentences of science and the pseudo-sentences of metaphysics. The principle of tolerance (according to which everyone is free to build up his own form of language (...) as he wishes) had been adopted near the end of 1932, at a time when it was already obvious to Carnap that a strictly verificationist criterion of meaning was inadequate. He therefore considered a variety of empiricisms and a variety of choices for the language of science. As Carnap put it, "there are many different possibilities in framing an empiricist language" and, correspondingly, several degrees of liberalization of the criterion of meaning. It was in this context that Carnap provided both a logical (syntactical) and an empirical analysis of confirmation (and of testing), before distinguishing requirements of different strengths which served the purpose of defining several versions of empiricism. (shrink)

Translated by Drew S. Burk and Anthony Paul Smith. Excerpted from Struggle and Utopia at the End Times of Philosophy , (Minneapolis: Univocal Publishing, 2012). THE END TIMES OF PHILOSOPHY The phrase “end times of philosophy” is not a new version of the “end of philosophy” or the “end of history,” themes which have become quite vulgar and nourish all hopes of revenge and powerlessness. Moreover, philosophy itself does not stop proclaiming its own death, admitting itself to be half dead (...) and doing nothing but providing ammunition for its adversaries. With our sights set on clearing up this nuance, we differentiate philosophy as an institutional entity, and the philosophizability of the World and History, of “thought-world,” which universalizes the narrow concept of philosophy and that of “capital.” We also give an eschatological and apocalyptical cause to this end, of “times” or “ages” rather than those of philosophical practice. Last but not least, it is the Future itself in the performativity of its ultimatum that determines this end times, reversing these times from the Identity the Future accorded to it, withdrawing the thought-world from the lie of its death. Why this style of axioms and oracles, these more or less subtle distinctions, old and debased, with an appeal to the ultimata , to end times and last words? We fight to give, parallel to the concept of Hell, its new theoretical position, for its philosophical return and its non-philosophical transformation. No more so than any other word, Hell is not a metaphor here, just the Principle of Sufficient World. Every man, no doubt, has his “hell” readily available, connivance, control, conformism, domestication, schooling, alienation, extermination, exploitation, oppression, anxiety, etc. We have our little list that the Contemporaries established in the previous century in the same way one used to construct lists of virtues and vices or honors and wealth. They invented it for us without knowing it, for us-the-Futures who have as our responsibility to invent its use. In the Christian and Gnostic tradition, the struggle of the End Times takes place “on earth.” The most sophisticated of believers have it taking place in Heaven as well, above all in Heaven. The various kinds of gnosis imagine infinite falls and vertiginous highs, the vertigo of salvation. On Earth as in Heaven, a hell is available. The Marxists have the law of profit and the control of production, the class struggle Capital imposes on man. The Nietzscheans, the dull grumble of the struggle in the foundations of World and History, the domestication of man, the society of control. The phenomenologists, the capture of being, the most superficial amongst them, the age of suspicion. But all of these “hells” are taken from World, History, Society, and Religion. What we call Hell is no longer of the order of these specific and total intra-worldly generalities, it is both more singular and more universal, no longer being at all of the same order, it is the determinant Identity of these small hells strung out through history but unified here in the name of the last Humaneity. It is even found within the French idiom for hell [ enfer ], en-fer literally means “in-irons.” We are as much “in-irons” as we are “alive” [ en-Vie ]. We believe in Hell but as non-philosophers and it is even because we are non-philosophers that we can believe in it outside of any sort of religion. Hell is less mythological than ideological, it combines philosophizability with universal capital. Under what form? A single term could work for them without being a metaphor or something they would participate in by analogy, a more innovative and conjectural term than control, more universal than profit. It would denote the growing and permanent extortion of a surplus-value of communication, of speed and of urgency in change, in productivity and in work, in the pressure of images and slogans. It would be worse than solicitation, more tenacious than capture, more active and persecutory than control, softer and more insidious than a frontal attack, just as perverse as questioning and accusation, less brutal and offensive than extermination, less ritualized than inquisition, it would be soft and dispersed, instantaneous and vicious, it would be a crime without declared violence. Collusion and conformism, it would be afraid to show itself. Related to rumor, from which it borrows its infinite and tortuous ways. It is harassment. As a modernized form of Hell, perhaps harassment has a long future in front of it, of innocent torture, slow assassination, in short the fall, but radical with no way of recovering and which tolerates only salvation. THE PHILOSOPHICAL PAST OF NON-PHILOSOPHY Non-philosophy is thus Man as the utopian identity of the philosophical form of the World, a utopia destined to transform it. We still have to understand these equations, in particular that of the being-uni-versed from Man, and this book adheres to this by re-exposing non-philosophy in a different way via one of its new possibilities. It uses this opportunity to once again take up formulations that lead to objections answering certain external critics, as well as revisiting diverging interpretations specific to non-philosophers. A portion of this book is devoted to going through these theories via a strict or “lengthy” presentation of non-philosophy, and its defense against more expeditious solutions. This work of rectification is the occasion, merely the occasion, for refocusing non-philosophy on Man (the “Man-in-person,” “Humaneity”), and in a more innovative manner, on its utopian vocation established since the book Future Christ . As for this “occasion,” it is quite obvious. A school of posture, not to say a school of thought, supposes a minimum of closure from the most liberating of knowledge, a heritage, its utilization, and its no less certain dispersal. Within its development, a variety of interpretations will no doubt appear, deviations that are as much normalizations, and a struggle against this multiplication of divergent tendencies. These are perhaps not inevitable evils, especially here, merely a normal development according to the twisting paths of history. But the problem is made worse by the fact that this school of non-philosophers is that of utopia. Not the former attempts devoted to commenting on the worst authoritarian and criminal forms of the past and the present, but utopia as the determinant principle for human life, or to put it another way, of the Future as an irreducible presupposition of (for) thinking the World and History. Non-philosophers are engaged in an aleatory navigation between the respect for the most rigorous utopia, whose rules are not that of the reproductive imagination but those from the Future determining imagination itself, as well as the temptations, diversions, and remorse of history. Little by little, we will begin to understand that the Future as we understand it no longer has any temporal consistency or positive content, without being an empty form or a nothingness, but that it is foreclosed to past and present History, just as it is foreclosed to the place of places, the World, and that it is the only method for establishing the practice of thinking in a non-imaginary instance. Because it is the World and History that are imaginary and have a terrible materiality, it is not necessarily utopia. We will overlap two objectives here: the defense of non-philosophy against the (non-) philosophers that we are, only occasionally, and the introduction of philosophy to a rigorous future. Together they set out to definitively render, without any possible return to philosophical conformism or towards the facilities of the past and present, the non-philosophical enterprise understood as utopia or uchronia. Imagination and speculation, left to themselves and thus undistinguished, are quite good for participating in the grand game of History but have little value or worse for the Future which is unimaginable and unintelligible and must be maintained as such. MAN-IN-PERSON AS SUSPENSION OF THE PHILOSOPHICAL CHORA The point where philosophical resistance is concentrated is without a doubt the invention of the Name-of-Man, first name, oracular as much as axiomatic, of the determining cause for the non-philosophical posture. And that which concentrates the differends is the style of non-philosophy as identity that possesses the dual aspect, of discipline and of the oeuvre, of the theorem and of the oracle. But the real difficulty in understanding the simplicity of non-philosophy is profoundly hidden in the depths of philosophy itself. Because philosophy, from Parmenides to Derrida, even Levinas, continues to be a divided gesture, without a veritable immanence, transforming its thematic contents of transcendence in also forgetting to transform the operative transcendence in the element from which the ontology of surface is established which we will call, in memory of Plato, the chorismos. The general effect of the chora literally gives place to philosophy, demanding binding and sutures to which we will once and for all “oppose” the Man-in-person, his power of cloning, and his future being. All philosophy contains a hinter-philosophy in which it deploys its operations and weaves its tradition like an understudy of a topographical nature and in the best of cases being itself topological. Philosophy as well consists of two levels, its pre-ontological operative conditions on the one hand, and its superficial theme on the other, it too has its presupposed, but is not aware of it or erases it within the unity of appearance named Logos. Rightly, the Logos, and its flash or lightning nature, possesses a “dark precursor,” the chora, which is as much a virtual image, and philosophy, dazzled by its own lightning flash, seems to completely forget about it at the same time it sets itself up within it. Non-philosophy risks taking this same path, of confusing what it believes to be the real with its phantom double, contenting itself to working on the thematic level of philosophy, not its surface objects and its idle chatter (we stopped talking about this a long time ago and in any case they are merely simple materials for inducing a work of transformation), but the transcendence-form of its objects. In the end it risks, through precipitation, taking back up the heritage of philosophy, a heritage of a misunderstood presupposed, even more profound than the play of transcendences. This is what the imperative of the radicality of immanence meant, to treat immanence in an immanent manner, not to make a new object out of it. And from here we get non- (philosophy) and its refusal of the Platonic chorismos , symbol of all abstraction, and thus all transcendental appearance. There are no illusions. The message will leave a heritage in tattered pieces and interpretations. But it was difficult not to dispute the differend to its core. There will be complete confusion of the multiple, possible, and necessary effectuations of non-philosophy with its interpretations. The non-philosophical or human freedom of philosophical effectuation and the philosophical freedom of interpretation. Effectuations demand non-philosophy to return to zero from the point of view of its philosophical material and thus also but within these limits the formulation of its axioms , but in no way providing from the outset divergent interpretations of the aforementioned axioms. They are divergent because they do not take into account the material from which these axioms are derived within non-philosophy, and because they do not see themselves as symptoms of another vision of the World. The utterances of non-philosophy are not mathematical theorems and pure axioms, they merely have a mathematical aspect . They are, by their extraction or origin, mathematical and transcendental. And by their determined function in-Real, within non-philosophy, they are identically in-the-last-Humaneity entities which have an aspect of an axiom and an aspect of interpretation (or an oracular aspect as we say) that attempts (sometimes it is ourselves who provide the occasion) to isolate and transform, in complete freedom of interpretation. There will be an opportunity to complain about the complex character of the language of non-philosophy, an idiom saturated with classical references, sophisticated in a contemporary way. Its freedom of decision up against the whole of philosophy demands these effects of “complication” and “privatization,” as the saying goes. But it also demands fighting against the drift [ dérive ] of the pedagogical-all and the mediatic-all that leads philosophy into the shallow depths of opinion, which is the site of its impossible death. The noble idealism of “pop-philosophy” has been consumed into a “philo-reality;” against this we propose philo-fiction. Parricide, which is at the bottom of these interpretations and which we can judge as being quite fertile, although it has informed tradition, only takes place once or within one lone meaning. In regards to Parmenides, it was possible; Plato introduced the Other as non-being and language, bringing into existence the philosophical system of the World, but is it possible to repeat it again with the same fecundity in regards to non-philosophy, this time in introducing (non) religion or (non) art, still mixing them without taking into account this mixture, alternatively as a philosophical or religious resentment? If philosophy begins via a crime, it is no doubt obliged to continue down similar pathways, to the effect that the crimes of philosophy, once the founding crime has been committed, are a reaction of self-defense. It is undoubtedly from this that we get Marx’s declaration that history begins by tragedy and repeats itself or ends in farce. The preservation of rigor and fecundity is, in every respect, a psychologically difficult task within a theory such as non-philosophy. Having posited an essential objective of liberation in regards to philosophy and its services, one has often understood this objective as an authorization of providing particular interpretations of its axioms and ends up obliterating their scope. This ends up confusing, on one hand, two kinds of freedoms in regards to non-philosophy, the freedom of its interpretations and the freedom of its effectuations. On the other hand, any defense of “principles” against precipitated interpretations is immediately taxed with a will to orthodoxy, a prohibitive objection when we are dealing with, as is the case here, a heretical theory of thought. Nevertheless, it is time to stop confusing heresy as the cause of thought with an ideology of heresies, which is certainly not at all our object, but rather a form of normalization. As for the “disciplinary” aspect, which is not the only aspect, it demands something other than philosophical “answers to objections,” a precision in the definition and use of its procedures in the formation of utterances, since non-philosophy is neither a supplementary doctrine interior to philosophy nor a vision of the world but one whose priority is a “vision of Man,” or rather Man as “vision” that implies a theory and a practice of philosophy. In the end, struggle is only one aspect of non-philosophy, not its whole or telos, struggle coming only from its materiality. In particular, if the discipline of non-philosophy is inseparable from struggle, it is not a question of reducing the monomaniacal obsession of its “marching orders.” This would reduce its complexity and kill its indivisibility, deploying it in a “long march” and a form of Maoization whose philosophical presuppositions no longer have any pertinence here, a case of the One and the Two, which are now cloned and no longer tied together. More generally, non-philosophy is a complex thought composed of a multitude of aspects, which is to say, unilateral interpretations, of a philosophical origin but reduced by their determination in-the-last-instance . The “liberalism” of non-philosophy is merely one of the aspects of which it is capable, not an essence. Similarly, it is only capable of having a “Maoist” aspect. Let us generalize. The weakness of non-philosophy is due to a specific cause, the determination-in-the-last-Humaneity of a subject for the World. Everything that has a right to the philosophical city can be said about it in turn and in a retaliatory mode since Man contributes nothing of himself that Man takes from the World. We can consider non-philosophy as being pretentious, absurd, idealistic, empty, materialistic, formalistic, contradictory, modern, post-modern, Zen, Buddhist, Marxist; it endures or tolerates, perhaps “appeals” to, or at least renders possible, sarcasms, ironies, and insults without even talking about the misunderstandings, partly for the same reason as psychoanalysis. All of this goes beyond simple “deviations.” They are its aspects, which is to say, its “unilateral” philosophical interpretations in both senses of the word, being either sufficient coming from the mouth of philosophers, or reduced to their absolute dimension of sufficiency and totality in the mouths of non-philosophers, and both times due to the weakness and strength of Man-in-person as their determination only in-the-last-instance. The non-philosopher is certainly not a Saint Paul fantasizing about a new Church. The non-philosopher is either a (Saint) Sebastian whose flesh is pierced with as many arrows as there are Churches, or a Christ persecuted by a Saint Paul. What is engaged in here is the practice of retaliation. A negative rule of the non-philosophical ethics of outlawed discussion by way of argumentation (the sufficient is you, the orthodox one is always you, you are the fashionable one, and when a master you are someone else) that is founded on the confusion of effectuations of non-philosophy and of its overall interpretations. Retaliation is the law but as with any too-human law, it must acquire a dimension that displaces it, or rather emplaces it and takes away its authority but not all of its effectiveness. If the non-philosopher is only authorized by himself, which is to say by philosophy but limited by the Real-of-the-last-instance, its critique of other non-philosophers can merely be retaliatory under the same conditions, only by the Real limited in-the last-instance. THE TREE OF PHILOSOPHICAL SAINTLINESS The thematic horizon or material of these debates is in the relationships between philosophy, religion as gnosis, and non-philosophy. It is inevitable, regarding non-philosophers in general (whether they are non-philosophers by name or simply its neighbors) that we often end up evoking Marx’s Holy Family and imagining, arranged on the neighboring branches of the tree of philosophy and annexed, sometimes abusively, to non-philosophy, authors who would quite evidently and quite rightly refuse this label. So it is that we find, for example, a Saint Michel, a Saint Alain, a Saint Gilles 1 without even mentioning the youngest who aspire as well to the freedom of “saintliness” and who make their muted voices heard here. If there is a Holy Family of non-philosophers, it extends completely beyond these three, provided that the sectarian spirit can save us. This book is organized in the following manner: To begin, in order to recall the essential part of the problematic, we have organized a Summary of Non-Philosophy , a vade-mecum of notions and basic problems, in a classical style. Secondly, there is Clarifications On the Three Axioms of Non-Philosophy , designed to posit their proper use as much as to elucidate their meaning. Thirdly, an analysis of Philosophizability and Practicity , both being extreme constituents of philosophical material or the contents of the third axiom. Fourthly, the heart of this work: Let us Make a Tabula Rasa of the Future or of Utopia as Method . Fifthly, we have a theoretical outline of a non-institutional utopia, The International Organization of Non-Philosophy, L’Organisation Non-Philosophique Internationale, (ONPHI) already created in practice but under the conditions of possibility and functioning from which here we put into question “de jure,” thus not without a perplexity concerning “facts,” in any case, without the capability of “getting to the bottom of things.” Sixthly, an essay characterizing The Right and the Left of Non-Philosophy , a brief topology of several philosophizing and normalizing positions of proponents or tenants of this problematic. Seventhly, Rebel in the Soul: A Theory of Future Struggle , a systematic discussion starting from a confrontation of non-philosophical gnosis and non-religious gnosis to the extent that they pose, posed or perhaps still will pose themselves as rivals to non-philosophy in a mixture of fidelity and infidelity. Despite the fact that it can also be read as putting non-philosophy into perspective: it pits against a standard Platonism two contemporary appropriations of Gnosticism. On the basis of the paradigm of Man who never ceases to come as the Future-in-person, each one of these moments strives to reestablish not the “true” non-philosophy and its orthodoxy, but the minimal conditions to respect in order to allow for its maximum fecundity. And in order to bring about one of the last possibilities of its development, making explicit Humaneity as a utopia-for-the-World. In introducing these considerations in the form of a “testament” and “ultimatum,” we want to indicate two things: First, that this is the last time we will intervene in order to caution non-philosophers against the temptation of returning and looking backwards towards philosophy. Only a disillusioned nostalgia for the former World and its traditions barely remain permissible to us.… Secondly, that non-philosophy is also a sort of ultimatum for considering one’s life and transforming one’s thought from the perspective of a uni-version rather than a conversion. Man as future is this ultimatum in action, not an impatient self-proclaimed genius, and philosophy is his testament. It is obviously the ultimatum that determines this testament as “old” with a view towards a life that is, itself, non-testamentary. In and of itself, the “old” can never bear a veritable eschaton. Thus, this book intersects according to the logic of this paradigm, under the sign of the ultimate or “last” as future, philosophy as testament and cautionary note for maintaining the non-philosophical oeuvre as “future” or “utopian.” We will see that between these two dimensions it cradles a theory of struggle. In the end, this book envisions non-philosophers in multiple ways. It inevitably sees them as subjects of knowledge, most often academics insofar as life in the world demands, but above all as close relatives of three great human types. The analyst and political militant are quite obvious, for non-philosophy is close to psychoanalysis and Marxism insofar as it transforms the subject in transforming philosophy. Here again, one must have a sense not of certain nuances but of aspects (of the interpretations, albeit unilateralized) and not in order to construct a simple proletarization or militarization of thought as theory. To be rigorous, rather than authoritarian or spiteful, is its task. And lastly, non-philosophy is a close relative of the spiritual but definitely not the spiritualist. Those who are spiritual are not at all spiritualists, for the spiritual oscillate between fury and tranquil rage, they are great destroyers of the forces of Philosophy and the State, which are united under the name of Conformism. They haunt the margins of philosophy, gnosis, mysticism, science fiction and even religions. Spiritual types are not only abstract mystics and quietists; they are heretics for the World. The task is to bring their heresy to the capacity of utopia, and their utopia to the capacity of the paradigm. NOTES We will recognize allusions, and sometimes references, to closely related or distant themes, but which are related, in the work of Michel Henry, Alain Badiou and via the representative of “non-religious” gnosis of a Platonic origin, in Gilles Grelet. It goes without saying that these discussions are current and local, neither concerning the ensemble of doctrines nor prejudging the eventual evolution of certain amongst them. This concerns defining certain proximities with non-philosophy (rather than adversaries which in some sense they are) and typological and emblematic differends (rather than conflicts with a certain author). (shrink)

The aim of the paper is to sketch some solutions that arose along the work on Logic of Strict Processes . Three main topics are discussed: negation based on implication constructed in intuitionistic fashion; satisfiability in multimodal contexts and a proposal of a first order semantics for Dynamic Logic of Strict Processes . The system of DLSP differs from the original LSP in using the set of contexts, which are treated as ordered sets of formulas. The interpretation of (...) a context is a transition system, which is constructed solely of simple processes. After the set of non-allowed processes is constructed, the negation of molecular formula can be understood as a set o processes that, when combined with processes associated with a non-negated formula, produce non-allowed processes. Satisfiablity in a transition system is defined by a special modal operator [φ]φ which is true only when the relation of metaimplication between φ and φ holds. In conclusion the author briefly reviews first order system based on DLSP and points to several open problems waiting for further investigation. (shrink)

Standard deontic logic (SDL) is defined on the basis of possible world semantics and is a logic of alethic-deontic modalities rather than deontic modalities alone. The interpretation of the concepts of obligation and permission comes down exclusively to the logical value that a sentence adopts for the accessible deontic alternatives. Here, we set forth a different approach, this being a logic which additionally takes into consideration whether sentences stand in relation to the normative system or to the system of values (...) under which we predicate the deontic qualifications. By taking this aspect into account, we arrive at a logical system which preserves laws proper to a deontic logic but where the standard paradoxes of deontic logic do not arise. It is a logic of strictly-deontic modalities DR. (shrink)

In this paper we study essential hereditary undecidability. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation essential tolerance, or, in (...) the converse direction, lax interpretability that interacts in a good way with essential hereditary undecidability. We introduce the class of $$\Sigma ^0_1$$ Σ 1 0 -friendly theories and show that $$\Sigma ^0_1$$ Σ 1 0 -friendliness is sufficient but not necessary for essential hereditary undecidability. Finally, we adapt an argument due to Pakhomov, Murwanashyaka and Visser to show that there is no interpretability minimal essentially hereditarily undecidable theory. (shrink)

Rudolf Carnap's 'Principle of Tolerance' was undoubtedly one of the most infl uential precepts in 20th Century philosophy. Introduced in The Logical Syntax of Language, Carnap's Principle suffered from ambiguities which aroused important philosophical questions from Moritz Schlick and Alberto Coffa. Specifi cally, their questions arise from the application of the Principle to the matter of including extra-logical transformation rules in the defi nition of a language, which Carnap regarded as an important difference between his own conventionalist philosophy of logic (...) and the 'absolutist' conception of Wittgenstein's Tractatus. Yet both Schlick and Coffa suggested that applying the Tolerance precept to this matter exceeds the limits of its legitimate deployment. Upon considering their objections, as well as the criticism of Wittgenstein which evoked them, it is argued that the philosophical signifi cance of the Principle of Tolerance requires its strict limitation to matters pertaining to the determination of meaning. (shrink)

The goal of this paper is a comprehensive analysis of basic reasoning patterns that are characteristic of vague predicates. The analysis leads to rigorous reconstructions of the phenomena within formal systems. Two basic features are dealt with. One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-increment of a walking distance is a walking distance). The other is the existence of borderline cases. The paper shows why these should be treated as different, though (...) related phenomena. Tolerance is formally reconstructed within a proposed framework of contextual logic, leading to a solution of the Sorites paradox. Borderline-vagueness is reconstructed using certain modality operators; the set-up provides an analysis of higher order vagueness and a derivation of scales of degrees for the property in question. (shrink)