The inversion principle for logical rules expresses a relationship between introduction and elimination rules for logical constants. Hallnäs & Schroeder-Heister proposed the principle of definitional reflection, which embodies basic ideas of inversion in the more general context of clausal definitions. For the context of admissibility statements, this has been further elaborated by Schroeder-Heister. Using the framework of definitional reflection and its admissibility interpretation, we show that, in the sequent calculus of minimal propositional logic, the left introduction rules are admissible when (...) the right introduction rules are taken as the definitions of the logical constants and vice versa. This generalizes the well-known relationship between introduction and elimination rules in natural deduction to the framework of the sequent calculus. (shrink)

Au cours des dernières décennies, les travaux portant sur les pratiques humaines réelles ont pris de l'importance dans différents domaines de la philosophie, sans pour autant atteindre une position dominante. À ce jour, ce type de tournant pratique n'a cependant pas encore pénétré la philosophie de la logique. En première partie, j'esquisse ce que serait (ou pourrait être) une philosophie de la logique centrée sur l'étude des pratiques, en insistant en particulier sur sa pertinence et sur la manière de la (...) conduire. En deuxième partie, j'illustre cette approche centrée sur les pratiques au moyen d une étude de cas : le rôle joué par les langages formels en logique, en particulier dans les pratiques des logiciens. Ma thèse est que les langages formels jouent un rôle opératoire fondamental dans le travail des logiciens en tant que technologie pratique du crayon et du papier, génératrice de processus cognitifs - et qui plus spécifiquement vient contrebalancer certains de nos schémas cognitifs « spontanés » peu adéquats à la recherche en logique (ainsi que dans d'autres domaines). Cette thèse sera appuyée sur des données empiriques venant de la recherche en psychologie du raisonnement. Avec cette analyse j'espère montrer qu'une philosophie de la logique centrée sur l'étude des pratiques peut être fructueuse, en particulier si elle est complétée par les réflexions méthodologiques nécessaires. (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.

Logic isn’t special. Its theories are continuous with science; its method continuous with scientific method. Logic isn’t a priori, nor are its truths analytic truths. Logical theories are revisable, and if they are revised, they are revised on the same grounds as scientific theories. These are the tenets of anti-exceptionalism about logic. The position is most famously defended by Quine, but has more recent advocates in Maddy, Priest, Russell, and Williamson. Although these authors agree on many methodological issues about logic, (...) they disagree about which logic anti-exceptionalism supports. Williamson uses an anti-exceptionalist argument to defend classical logic, while Priest claims that his anti-exceptionalism supports nonclassical logic. This paper argues that the disagreement is due to a difference in how the parties understand logical theories. Once we reject Williamson’s deflationary account of logical theories, the argument for classical logic is undercut. Instead an alternative account of logical theories is offered, on which logical pluralism is a plausible supplement to anti-exceptionalism. (shrink)

We investigate effective categoricity for polymodal algebras. We prove that the class of polymodal algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. In particular, this implies that every categoricity spectrum is the categoricity spectrum of a polymodal algebra.

In the theory of meaning, it is common to contrast truth-conditional theories of meaning with theories which identify the meaning of an expression with its use. One rather exact version of the somewhat vague use-theoretic picture is the view that the standard rules of inference determine the meanings of logical constants. Often this idea also functions as a paradigm for more general use-theoretic approaches to meaning. In particular, the idea plays a key role in the anti-realist program of Dummett and (...) his followers. In the theory of truth, a key distinction now is made between substantial theories and minimalist or deflationist views. According to the former, truth is a genuine substantial property of the truth-bearers, whereas according to the latter, truth does not have any deeper essence, but all that can be said about truth is contained in T-sentences (sentences having the form: ‘P’ is true if and only if P). There is no necessary analytic connection between the above theories of meaning and truth, but they have nevertheless some connections. Realists often favour some kind of truth-conditional theory of meaning and a substantial theory of truth (in particular, the correspondence theory). Minimalists and deflationists on truth characteristically advocate the use theory of meaning (e.g. Horwich). Semantical anti-realism (e.g. Dummett, Prawitz) forms an interesting middle case: its starting point is the use theory of meaning, but it usually accepts a substantial view on truth, namely that truth is to be equated with verifiability or warranted assertability. When truth is so understood, it is also possible to accept the idea that meaning is closely related to truth-conditions, and hence the conflict between use theories and truth-conditional theories in a sense disappears in this view. (shrink)

Some Carrollian posthumous manuscripts reveal, in addition to his famous ‘logical diagrams’, two mysterious ‘logical charts’. The first chart, a strange network making out of fourteen logical sentences a large 2D ‘triangle’ containing three smaller ones, has been shown equivalent—modulo the rediscovery of a fourth smaller triangle implicit in Carroll's global picture—to a 3D tetrahedron, the four triangular faces of which are the 3+1 Carrollian complex triangles. As it happens, such an until now very mysterious 3D logical shape—slightly deformed—has been (...) rediscovered, independently from Carroll and much later, by a logician , a mathematician and a linguist studying the geometry of the ‘opposition relations’, that is, the mathematical generalisations of the ‘logical square’. We show that inside what is called equivalently ‘n-opposition theory’, ‘oppositional geometry’ or ‘logical geometry’, Carroll's first chart corresponds exactly, duly reshap.. (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)

In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator ?. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a translation of mHC into (...) K4.Grz and prove that this translation is exact, i. e. theorem-preserving and deducibility-invariant. We have established (however, in this note we do not present a proof of this) that the lattice of all extensions of mHC is isomorphic to the lattice of normal extensions of K4.Grz (a generalization of the Kuznetsov and Muravitsky theorem). (shrink)

Given a 4-tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators ( \({a \wedge b}\) and \({\overline{a} \wedge \overline{b}}\) ), or dissimilarity indicators ( \({a \wedge \overline{b}}\) and \({\overline{a} \wedge b}\) ) pertaining to the pair (a, b), to the ones associated with the pair (c, d). There are 120 semantically distinct logical proportions. One of them models the analogical proportion which corresponds to a statement of the form “a (...) is to b as c is to d”. The paper inventories the whole set of logical proportions by dividing it into five subfamilies according to what they express, and then identifies the proportions that satisfy noticeable properties such as full identity (the pair of equivalences defining the proportion hold as true for the 4-tuple (a, a, a, a)), symmetry (if the proportion holds for (a, b, c, d), it also holds for (c, d, a, b)), or code independency (if the proportion holds for (a, b, c, d), it also holds for their negations \({{(\overline{a},\overline{b}, \overline{c}, \overline{d})}}\) ). It appears that only four proportions (including analogical proportion) are homogeneous in the sense that they use only one type of indicator (either similarity or dissimilarity) in their definition. Due to their specific patterns, they have a particular cognitive appeal, and as such are studied in greater details. Finally, the paper provides a discussion of the other existing works on analogical proportions. (shrink)

The traditional view of lying holds that this phenomenon involves two central components: stating what one does not believe oneself and doing so with the intention to deceive. This view remained the generally accepted view of the nature of lying until very recently, with the intention-to-deceive requirement now coming under repeated attack. In this article, I argue that the tides have turned too quickly in the literature on lying. For while it is indeed true that there can be lies where (...) there is no intention on the part of the speaker to deceive the hearer, this does not warrant severing the connection between lying and deception altogether. Thus, I defend the following account of lying: A lies to B if and only if (1) A states that p to B, (2) A believes that p is false, and (3) A intends to be deceptive to B in stating that p. (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 view that lying is morally worse than merely misleading is a very natural one, which has had many prominent defenders. Nonetheless, here I will argue that it is misguided: holding all else fixed, acts of mere misleading are not morally preferable to acts of lying, and successful lying is not morally worse than merely deliberately misleading. In fact, except in certain very special contexts, I will suggest that – when faced with a felt need to deceive – we might (...) as well just go ahead and lie. (shrink)

Abstract This paper discusses the semantic theory presented in Robert Brandom’s Making It Explicit . I argue that it is best understood as a special version of dynamic semantics, so that these semantics by themselves offer an interesting theoretical alternative to more standard truth-conditional theories. This reorientation also has implications for more foundational issues. I argue that it gives us the resources for a renewed argument for the normativity of meaning. The paper ends by critically assessing the view in both (...) its development and motivations. Content Type Journal Article Pages 1-22 DOI 10.1007/s11098-011-9768-4 Authors Bernhard Nickel, Harvard University, Cambridge, MA, USA Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116. (shrink)

Islamic building codes require mosques to face Mecca. The further Islam spreads, the more apt are believers to fall into a quandary. X faces Y only when the front of X is closer to Y than any other side of X. So the front of the mosque should be oriented along a shortest path to Mecca. Which way is that? Does the path to Mecca tunnel through the earth? Or does the path follow the surface of the earth?

Critical discussion of Alan Code's paper "Aristotle's Investigation of a Basic Logical Principle: Which Science Investigates the Principle of Non-Contradiction?".

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)

Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0.1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) ∨ ((B → A) ∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MTL and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are (...) provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1]. (shrink)

I argue that it may well be the case that space and time do not consist of points, indeed that they have no smallest parts. I examine two different approaches to such pointless spaces : a topological approach and a measure theoretic approach. I argue in favor of the measure theoretic approach.

An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers.

Following the speech act theory, we take hypotheses and assertions as linguistic acts with different illocutionary forces. We assume that a hypothesis is justified if there is at least a scintilla of evidence for the truth of its propositional content, while an assertion is justified when there is conclusive evidence that its propositional content is true. Here we extend the logical treatment for assertions given by Dalla Pozza and Garola (1995, Erkenntnis, 43, 81–109) by outlining a pragmatic logic for assertions (...) and hypotheses. On the basis of this extension we analyse the standard logical opposition relations for assertions and hypotheses. We formulate a pragmatic square of oppositions for assertions and a hexagon of oppositions for hypotheses. Finally, we give a mixed hexagon of oppositions to point out the opposition relations for assertions and hypotheses. (shrink)

This paper is about philosophical disputes where the literal content of what speakers communicate concerns such object-level issues as ground, supervenience, or real definition. It is tempting to think that such disputes straightforwardly express disagreements about these topics. In contrast to this, I suggest that, in many such cases, the disagreement that is expressed is actually one about which concepts should be employed. I make this case as follows. First, I look at non-philosophical, everyday disputes where a speaker employs a (...) metalinguistic usage of a term. This is where a speaker uses a term to express a view about the meaning of that term, or, relatedly, how to correctly use that term. A metalinguistic negotiation is a metalinguistic dispute that concerns a normative issue about what a word should mean, or, similarly, about how it should be used, rather than the descriptive issue about what it does mean. I argue that the same.. (shrink)

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)

For each integer n≥0, we provide an explicit functional characterization of the free n-generated BL-algebra, together with an explicit construction of the corresponding normal forms.

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)

The ontological argument fails because of an operator order switch between (1) “necessarily there is an perfect being” and (2) “there is a being which necessarily is perfect”. Here (1) is trivially true logically but (2) problematic. Since Kant's criticisms were directed at the notion of existence, not at the step from (1) to (2), they are misplaced. They are also wrong, because existence can be a predicate. Moreover, Kant did not anticipate Frege's claim that “is” is ambiguous between existence, (...) predication, identity, and class‐inclusion. To restore the ontological argument, an extra premise is needed to the effect that it is known who the existentially perfect being is. The question is raised whether Kant could have meant the failure of this extra premise by his thesis that existence is not a real predicate. (shrink)

Since the ground-breaking contributions of M. Dummett (Dummett 1978), it is widely recognized that anti-realist principles have a critical impact on the choice of logic. Dummett argued that classical logic does not satisfy the requirements of such principles but that intuitionistic logic does. Some philosophers have adopted a more radical stance and argued for a more important departure from classical logic on the basis of similar intuitions. In particular, J. Dubucs and M. Marion (?) and (Dubucs 2002) have recently argued (...) that a proper understanding of anti-realism should lead us to the so-called substructural logics (see (Restall 2000)) and especially linear logic. The aim of this paper is to scrutinize this proposal. We will raise two kinds of issues for the radical anti-realist. First, we will stress the fact that it is hard to live without structural rules. Second, we will argue that, from an anti- realist perspective, there is currently no satisfactory justi cation to the shift to substructural logics. (shrink)

In fact, law is necessarily very vague. So if vagueness is a problem for legal theory, it is a serious problem. The problem has to do with the ideal of the rule of law and with the very idea of law: if vague standards provide no guidance in some cases, how can the life of a community be ruled by law? The problem has long concerned philosophers of law; the papers at this symposium address it afresh by asking what legal (...) theory may have to learn from (or contribute to) work on vagueness in philosophy of language and philosophy of logic. Here I will not try to state the implications of vagueness for philosophy of law; I will try to set the stage by showing that vagueness is both an important and an unavoidable feature of law. (shrink)

In a recent paper, Sven Danielsson argued that the ‘original paradoxes' of deontic logic, in particular Ross's paradox and Prior's paradox of derived obligation, can be solved by restricting the modal inheritance rule. I argue that this does not solve the paradoxes.