There are two paradoxes of satisfaction, and they are of different kinds. The classic satisfaction paradox is a version of Grelling’s: does ‘does not satisfy itself’ satisfy itself? The Unsatisfied paradox finds a predicate, P, such that Px if and only if x does not satisfy that predicate: paradox results for any x. The two are intuitively different as their predicates have different paradoxical extensions. Analysis reduces each paradoxical argument to differing rule sets, wherein their respective pathologies lie. Having different (...) pathologies, they are paradoxes of different kinds. Furthermore, each of these satisfaction paradoxes has an analogue with the same pathology in set theory. Therefore, these analogues are respectively of the same two kinds. This level of abstraction is significant in that it tracks two related but different pathologies. Thus, not all paradoxes of semantics and set theory share the same pathology: there are at least two kinds of paradox cutting across the semantic and set-theoretic distinction. (shrink)

The Pinocchio paradox, devised by Veronique Eldridge-Smith in February 2001, is a counter-example to solutions to the Liar that restrict the use or definition of semantic predicates. Pinocchio’s nose grows if and only if what he is stating is false, and Pinocchio says ‘My nose is growing’. In this statement, ‘is growing’ has its normal meaning and is not a semantic predicate. If Pinocchio’s nose is growing it is because he is saying something false; otherwise, it is not growing. ‘Because’ (...) stands here for a non-semantic relation; it might be supposed to be causal or of some other nature, but it is not semantic. The paradox is discussed in relation to Tarski’s and Kripke’s theories of truth. Although the paradox is not necessarily a counter-example to a theory of a truth predicate, it is a problem for a theory of truth of the kind preserved by validity. (shrink)

Semantic dialetheists astutely dodge Explosion, the logical contagion of everything being true if a single contradiction is true. A dialetheia is contained in their semantics, and sustained by a paraconsistent logic. Graham Priest has shown that this is a solution to the Liar paradox. I use the Pinocchio paradox, devised by Veronique Eldridge-Smith, as a counter-example. The Pinocchio paradox turns on the truth of Pinocchio, whose nose grows if and only if what he is saying is not true, saying ‘My (...) nose is growing’. It is not just a matter of interpretation whether Pinocchio’s nose is and is not growing. (shrink)

The Pinocchio paradox poses one dialetheia too many for semantic dialetheists (Eldridge-Smith 2011). However, Beall (2011) thinks that the Pinocchio scenario is merely an impossible story, like that of the village barber who shaves just those villagers who do not shave themselves. Meanwhile, Beall maintains that Liar paradoxes generate dialetheia. The Barber scenario is self-contradictory, yet the Pinocchio scenario requires a principle of truth for a contradiction. In this and other respects the Pinocchio paradox is a version of the Liar, (...) unlike the Barber. One wonders why some Liars would be impossible if others generate dialetheias. (shrink)

The main thesis of this paper is that we sometimes are disposed to accept false and even jointly inconsistent claims by virtue of our semantic competence, and that this comes to light in the sorites and liar paradoxes. Among the subsidiary theses are that this is an important source of indeterminacy in truth conditions, that we must revise basic assumptions about semantic competence, and that classical logic and bivalence can be upheld in the face of the sorites paradox.

Think of the color spectrum, spread out before you. You can identify the different colors with ease. But if you are asked to indicate the point at which one color ends and the next begins, you are at a loss. "There is no such point", is a natural thought: one color just shades gradually into the next.

The aim of this paper is to propose a formal approach to reasoning about desires, understood as logical propositions which we would be pleased to make true, also acknowledging the fact that desire is a matter of degree. It is first shown that, at the static level, desires should satisfy certain principles that differ from those to which beliefs obey. In this sense, from a static perspective, the logic of desires is different from the logic of beliefs. While the accumulation (...) of beliefs tend to reduce the remaining possible worlds they point at, the accumulation of desires tends to increase the set of states of affairs tentatively considered as satisfactory. Indeed beliefs are expected to be closed under conjunctions, while, in the positive view of desires developed here, one can argue that endorsing \ as a desire means to desire \ and to desire \. However, desiring \ and \ at the same time is not usually regarded as rational, since it does not make much sense to desire one thing and its contrary at the same time. Thus when a new desire is added to the set of desires of an agent, a revision process may be necessary. Just as belief revision relies on an epistemic entrenchment relation, desire revision is based on a hedonic entrenchment relation satisfying other properties, due to the different natures of belief and desire. While epistemic entrenchment relations are known to be qualitative necessity relations, hedonic relations obeying a set of reasonable postulates correspond to another set-function in possibility theory, called guaranteed possibility, that drive well-behaved desire revision operations. Then the general framework of possibilistic logic provides a syntactic setting for encoding desire change. The paper also insists that desires should be carefully distinguished from goals. (shrink)

Definite descriptions, I shall argue, have two possible functions. 1] They are used to refer to what a speaker wishes to talk about, but they are also used quite differently. Moreover, a definite description occurring in one and the same sentence may, on different occasions of its use, function in either way. The failure to deal with this duality of function obscures the genuine referring use of definite descriptions. The best known theories of definite descriptions, those of Russell and Strawson, (...) I shall suggest, are both guilty of this. Before discussing this distinction in use, I will mention some features of these theories to which it is especially relevant. (shrink)

In this paper, we argue for a particular informative and unified analysis of normative reasons. According to this analysis, a fact F is a reason to act in a certain way just in case it is evidence that one ought to act in that way. Similarly, F is a reason to believe a certain proposition just in case it is evidence for the truth of this proposition. Putting the relatively uncontroversial claim about reasons for belief to one side, we present (...) several arguments in favor of our analysis of reasons for action. We then turn to consider a series of objections to the analysis. We conclude that there are good reasons to accept the analysis and that the objections do not succeed. (shrink)

The aim of my contribution is to investigate the ground of habits and rituals; they are based on the same processes even though they have different functions depending on the context (personal or social). My discussion will mostly centered on the nature and function of rituals, as necessary practices in human social life (but also in animal life). After a brief introduction of different perspectives on the notions of “habit” and “ritual”, I propose an interpretation of rituals as collective activity, (...) which is based on the same mechanisms of habits formation but it is expressed in a “we-form”, from which is created and institutionalized. (shrink)

Aristotle’s Prior Analytics identifies deductions simpliciter with inferential necessity, so that a deduced conclusion is necessarily inferred from some premises. Modern logical reconstructions claim that inferential necessity in Aristotle corresponds to logical validity. However, this logical reconstruction fails on two accounts. First, logical validity does not highlight Aristotle’s distinction between inferential necessity and predicative necessity, meaning that the inferential necessity of a deduction is not of the same kind as the predicative necessity of a non‑deductive argument. Second, logical validity does (...) not explain the relevance of Aristotle’s distinction between complete and incomplete deductions. Logicians speak of complete deduction by adding the term “obvious” or “transparent” to logical validity, and then criticize Aristotle’s view for being unclear. However, Aristotle’s position is not confronted with this difficulty. There is nothing to add to inferential necessity, which already means complete deducibility, as opposed to incomplete deducibility, deemed to be potentially complete. Accordingly, the Prior Analytics reduces the incomplete deductions to the complete deductions in order to prove the potential, inferential necessity of the incomplete deductions. Logical validity would have been faithful to Aristotle’s text, if it had been possible to coin a notion of potential validity, distinct from both validity and invalidity. (shrink)

How can it be valuable to use vagueness in a normative text? The effect is to make a vague norm, and vagueness seems repugnant to the very idea of making a norm. It leaves conduct (to some extent) unregulated, when the very idea of making a norm is to regulate conduct. A vague norm leaves the persons for whom the norm is valid with no guide to their conduct in some cases - and the point of a norm is to (...) guide conduct. A vague norm in a system of norms does not control the officers or officials responsible for applying the norms or resolving disputes - and part of the value of a system of norms is to control the conduct of the persons to whom the system gives normative power. In this essay I will seek to resolve these puzzles, and to show that vagueness can be valuable to lawmakers (and valuable to them, because their use of it is valuable to the people to whom the law is addressed). Far from being repugnant to the idea of making a norm, vagueness is a central technique of normative texts: it is needed in order to pursue the purposes of formulating such texts. Not all norms are vague. But vagueness is of central importance to the very idea of guiding conduct by norms. (shrink)

This chapter provides a survey of issues about diagrams in traditional geometrical reasoning. After briefly refuting several common philosophical objections, and giving a sketch of diagram-based reasoning practice in Euclidean plane geometry, discussion focuses first on problems of diagram sensitivity, and then on the relationship between uniform treatment and geometrical generality. Here, one finds a balance between representationally enforced unresponsiveness (to differences among diagrams) and the intellectual agent's contribution to such unresponsiveness that is somewhat different from what one has come (...) to expect in modern logic. Finally, challenges and opportunities for further work are indicated. (shrink)

First we have individual variables, as usual in first-order logics. (We do not have individual constants, but this is a minor point.) The propositional logic LP has justification constants, but in FOLP these are generalized to allow individual variables as arguments. Thus we have as justification constants c, c(x), c(x, y), . . . . Similarly LP has justification variables, but in FOLP these can be parametrized with individual variables p, p(x), p(x, y), . . . . To keep terminology (...) in line with past papers, we will still refer to things as justification constants and justification variables, even though they have structure to them. As in LP, justification terms are built up from justification constants and justification variables using ·, +, ! as usual. In addition there is a new constructor, genx, introduced by Artemov, and there is one further new constructor, exsx, introduced in this paper. If t is a justification term and x is an individual variable, genxt and exsxt are justification terms. An individual variable x is free in a justification term unless it is bound by genx or exsx. More specifically, the free variables of p(x, y, . . .) and of c(x, y, . . .) are {x, y, . . .}, the free variables of s · t and of s + t are the free variables of s together with the free variables of t, the free variables of !s are the free variables of s, and the free variables of genxt and of exsxt are the free variables of t except for x. Formulas are built up from atomic formulas, including ⊥, in the way standard in first-order logic, together with the additional formation rule: t:X is a formula provided t is a justification term, X is a formula, and all free variables of X occur in t. We assume ⊃, ⊥, and ∀ are basic, with other connectives and quantifier defined. The axiomatization used here is a combination of an LP axiomatization and a standard axiomatization of first-order logic, together with a version of the Barcan formula, and one additional axiom that corresponds to the converse Barcan formula.. (shrink)

The main result of this paper is Theorem 3.1 which is a criterion for weak o-minimality of a linearly ordered structure in terms of realizations of 1-types. Here we also prove some other properties of weakly o-minimal structures. In particular, we characterize all weakly o-minimal linear orderings in the signature $\{<, =\}$. Moreover, we present a criterion for density of isolated types of a weakly o-minimal theory. Lastly, at the end of the paper we present some remarks on the Exchange (...) Principle for algebraic closure in a weakly o-minimal structure. (shrink)

We present a basis for the admissible rules of intuitionistic propositional logic. Thereby a conjecture by de Jongh and Visser is proved. We also present a proof system for the admissible rules, and give semantic criteria for admissibility.

In this paper, the notion of degree of inconsistency is introduced as a tool to evaluate the sensitivity of the Full Bayesian Significance Test (FBST) value of evidence with respect to changes in the prior or reference density. For that, both the definition of the FBST, a possibilistic approach to hypothesis testing based on Bayesian probability procedures, and the use of bilattice structures, as introduced by Ginsberg and Fitting, in paraconsistent logics, are reviewed. The computational and theoretical advantages of using (...) the proposed degree of inconsistency based sensitivity evaluation as an alternative to traditional statistical power analysis is also discussed. (shrink)

Pluralism has many meanings. An assessment of the need for logical pluralism with respect to scientific knowledge requires insights in its domain of application. So first a specific form of epistemic pluralism will be defended. Knowledge turns out a patchwork of knowledge chunks. These serve descriptive as well as evaluative functions, may have competitors within the knowledge system, interact with each other, and display a characteristic dynamics caused by new information as well as by mutual readjustment. Logics play a role (...) in the organization of the chunks, in their applications and in the exchange of information between them. Epistemic pluralism causes a specific form of logical pluralism. Against this background, the occurrence of inconsistencies will be discussed together with required reactions and systematic ways to explicate them. Finally, the place of inconsistencies in the sciences will be considered. Seven theses will be proposed and argued for. The implications of each of these for pluralism will be considered. The general tenet is that paraconsistency plays an important role, bound to become more explicit in the future, but that the occurrence of inconsistencies does not basically affect the need for pluralism. (shrink)

Scientific pluralism is the view according to which there is a plurality of scientific domains and of scientific theories, and these theories are empirically adequate relative to their own respective domains. Scientific monism is the view according to which there is a single domain to which all scientific theories apply. How are these views impacted by the presence of inconsistent scientific theories? There are consistency-preservation strategies and inconsistency-toleration strategies. Among the former, two prominent strategies can be articulated: Compartmentalization and Information (...) restriction. Among the inconsistency-toleration strategies, we have: Paraconsistent compartmentalization and Dialetheism. In this paper, I critically assess the adequacy of each of these four views. (shrink)

An overcomplete logic is a logic that ‘ceases to make the difference’: According to such a logic, all inferences hold independently of the nature of the statements involved. A negation-inconsistent logic is a logic having at least one model that satisfies both some statement and its negation. A negation-incomplete logic has at least one model according to which neither some statement nor its negation are satisfied. Paraconsistent logics are negation-inconsistent yet non-overcomplete; paracomplete logics are negation-incomplete yet non-overcomplete. A paranormal logic (...) is simply a logic that is both paraconsistent and paracomplete. -/- Despite being perfectly consistent and complete with respect to classical negation, nearly every normal modal logic, in its ordinary language and interpretation, admits to some latent paranormality: It is paracomplete with respect to a negation defined as an impossibility operator, and paraconsistent with respect to a negation defined as non-necessity. In fact, as it will be shown here, even in languages without a primitive classical negation, normal modal logics can often be alternatively characterized directly by way of their paranormal negations and related operators. So, instead of talking about ‘necessity’, ‘possibility’, and so on, modal logics could be seen just as devices tailored for the study of (modal) negation. This paper shows how and to what extent this alternative characterization of modal logics can be realized. (shrink)

Both in formal and computational natural language semantics, the classical correspondence view of meaning – and, more specifically, the view that the meaning of a declarative sentence coincides with its truth conditions – is widely held. Truth (in the world or a situation) plays the role of the given, and meaning is analysed in terms of it. Both language and the world feature in this perspective on meaning, but language users are conspicuously absent. In contrast, the inferentialist semantics that Robert (...) Brandom proposes in his magisterial book ‘Making It Explicit’ puts the language user centre stage. According to his theory of meaning, the utterance of a sentence is meaningful in as far as it is a move by a language user in a game of giving and asking for reasons (with reasons underwritten by a notion of good inferences). In this paper, I propose a proof-theoretic formalisation of the game of giving and asking for reasons that lends itself to computer implementation. In the current proposal, I flesh out an account of defeasible inferences, a variety of inferences which play a pivotal role in ordinary (and scientific) language use. (shrink)

ABSTRACT It is well-known that Artificial Intelligence requires Logic. But its Classical version shows too many insufficiencies. So, it is very necessary to introduce more sophisticated tools, as may be Fuzzy Logic, Modal Logic, Non-Monotonic Logic, and so on. When you are searching the possible precedent of such new ideas, we may found that they are not totally new, because some ancient thinkers have suggested many centuries ago similar concepts, certainly without adequate mathematical formulation, but in the same line: against (...) the dogmatism and the dualistic vision of the world: absolutely true vs. absolutely false, black vs. white, good or bad by nature, 0 vs.1, etc. We attempt to analyze here some of these greatly unexplored, and very interesting early origins. (shrink)

In Logical Self-Defense , Johnson and I introduced the criteria of acceptability, relevance and sufficiency as appropriate for the evaluation of arguments in the sense of reasons offered in support of a claim. These three criteria have been widely adopted, but each has been subjected to a number of criticisms; and also 30 years of research have intervened. How do these criteria stand up today? In this paper I argue that they still have a place in argument analysis and evaluation, (...) but in much-modified roles. (shrink)

The style of arithmetic in the treatises the Neo-Pythagorean authors is strikingly different from that of the "Elements". Namely, it is characterised by the absence of proof in the Euclidean sense and a specific genetic approach to the construction of arithmetic that we are going to describe in our paper. Lack of mathematical sophistication has led certain historians to consider this type of mathematics as a feature of decadence of mathematics in this period [Tannery 1887; Heath 1921]. The alleged absence (...) of originality in these works has also given grounds to believe that “the arithmetic presented in these works derives substantially from an ancient, primitive stage of Pythagorean arithmetic” and to use them as “an index of the character of arithmetic science in the 5th century” [Knorr 1975]. In this paper, we take Nicomachus’ Introduction to Arithmetic for point of departure, because it is the richest and most well organised treatise representing this tradition. However, we also take into account the works of other Neo-Pythagorean authors. We are going to show that the Neo-Pythagorean arithmetic might have been developed in a natural, self-contained manner as a simple theory of counting over a domain of concrete initial objects, designated by fixed signs. This approach can be comfortably realised without appealing to assumptions of axiomatic character, but relying upon some ‘genetic’ constructions intented to be carried out by means of the designated entities. (shrink)