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Pablo Cobreros

Universidad de Navarra, Filosofía, Faculty Member

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I am professor of Philosophy in the University of Navarra where I took my PhD back in 2007 under the supervision of Prof. María Cerezo. My thesis is about vagueness and supervaluationism – the best bit was published in Studia Logica 2008. I spent one year, 2008, in UCL and the Institute of Philosophy London as a postdoc. I've been also visiting other places, recently the MCMP in LMU Munich with the support of the Humboldt Foundation. I am currently the coordinator of PLEXUS https://cordis.europa.eu/project/id/101086295

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Papers by Pablo Cobreros

Review of Richard Dietz, Sebastiano Moruzzi (eds.), Cuts and Clouds: Vagueness, its Nature, and its Logic

Notre Dame philosophical reviews, 2010

Supervaluationism and Necessarily Borderline Sentences

Disputatio, 2008

The supervaluationist theory of vagueness is committed to a particular notion of logical conseque... more The supervaluationist theory of vagueness is committed to a particular notion of logical consequence known as global validity. According to a recent objection, this notion of consequence is more problematic than is usually thought since i) it bears a commitment to some sort of bizarre inferences, ii) this commitment threatens the internal coherence of the theory and iii) we might find counterexamples to classically valid patterns of inference even in the absence of a definitely-operator (or similar device). As a consequence, the supervaluationist theory itself is in trouble. This paper discusses the objection.

Supervaluationism, Subvaluationism and the Sorites Paradox

The Sorites Paradox

Paraconsistent vagueness: a positive argument

Synthese, 2010

Paraconsistent approaches have received little attention in the literature on vagueness (at least... more Paraconsistent approaches have received little attention in the literature on vagueness (at least compared to other proposals). The reason seems to be that many philosophers have found the idea that a contradiction might be true (or that a sentence and its negation might both be true) hard to swallow. Even advocates of paraconsistency on vagueness do not look very convinced when they consider this fact; since they seem to have spent more time arguing that paraconsistent theories are at least as good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal. But it sometimes happens that the weakness of a theory turns out to be its mayor ally, and this is what (I claim) happens in a particular paraconsistent proposal known as subvaluationism. In order to make room for truth-value gluts subvaluationism needs to endorse a notion of logical consequence that is, in some sense, weaker than standard notions of consequence. But this weakness allows the subvaluationist theory to accommodate higher-order vagueness in a way that it is not available to other theories of vagueness (such as, for example, its paracomplete counterpart, supervaluationism).

Supervaluationism and Logical Consequence: A Third Way

Studia Logica, 2008

It is often assumed that the supervaluationist theory of vagueness is committed to a global notio... more It is often assumed that the supervaluationist theory of vagueness is committed to a global notion of logical consequence, in contrast with the local notion characteristic of modal logics. There are, at least, two problems related to the global notion of consequence. First, it brings some counterexamples to classically valid patterns of inference. Second, it is subject to an objection related to higher-order vagueness. This paper explores a third notion of logical consequence, and discusses its adequacy for the supervaluationist theory. The paper proceeds in two steps. In the first step, the paper provides a deductive notion of consequence for global validity using the tableaux method. In the second step, the paper provides a notion of logical consequence which is an alternative to global validity, and discusses i) whether it is acceptable to the supervaluationist and ii) whether it plays a better role in a theory of vagueness in the face of the problems related to the global notion.

Varzi on Supervaluationism and Logical Consequence

Mind, 2010

Though it is standardly assumed that supervaluationism applied to vagueness is committed to globa... more Though it is standardly assumed that supervaluationism applied to vagueness is committed to global validity, Achille Varzi (2007) argues that the supervaluationist should take seriously the idea of adopting local validity instead. Varzi's motivation for the adoption of local validity is largely based on two objections against the global notion: that it brings some counterexamples to classically valid rules of inference and that it is inconsistent with unrestricted higher-order vagueness. In this discussion I review these objections and point out ways to address them not considered in Varzi's paper.

Supervaluationism and Fara’s paradox of higher-order vagueness

Vagueness and Language Use. Palgrave McMillan, Oxford, 2010

This paper discusses Fara&amp;amp;amp;#x27;s so-called &amp;amp;amp;#x27;Paradox of Highe... more This paper discusses Fara&amp;amp;amp;#x27;s so-called &amp;amp;amp;#x27;Paradox of Higher-Order Vagueness&amp;amp;amp;#x27; concerning supervaluationism. In the paper I argue that supervaluationism is not committed to global validity, as it is largely assumed in the literature, but to a weaker notion of logical consequence I call &amp;amp;amp;#x27;regional validity&amp;amp;amp;#x27;. Then I show that the supervaluationist might solve Fara&amp;amp;amp;#x27;s paradox making use of this weaker notion of logical consequence. The paper is discussed by Delia Fara in the same volume.

Supervaluationism and Classical Logic

The supervaluationist theory of vagueness provides a notion of logical consequence that is akin t... more The supervaluationist theory of vagueness provides a notion of logical consequence that is akin to classical consequence. In the absence of a definitely operator, supervaluationist consequence coincides with classical consequence. In the presence of 'definitely', we might find counterexamples to classically valid patterns of inference within supervaluationist reasoning. Foes of supervaluationism emphasize the last result to argue against the supervaluationist theory, concluding that 'supervaluations invalidate our natural mode of deductive thinking' (Williamson, 1994, 152). This and related objections, however, do not consider particular ways in which we might provide systems of deduction for supervaluationist consequence. The present talk considers two ways in which we can carry out this task.

Vagueness: Subvaluationism

Philosophy Compass, 2013

Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theor... more Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theory of vagueness. But these theories cannot be taken apart, for they are in a relation of duality that can be made precise. This paper provides an introduction to the subvaluationist theory of vagueness in connection to its dual, supervaluationism. A survey on the supervaluationist theory can be found in the Compass paper of Keefe (2008); our presentation of the theory in this paper will be short to get rapidly into the logical issues. This paper is relatively self-contained. A modest background on propositional modal logic is, though not strictly necessary, advisable. The reader might find useful the Compass papers Kracht (2011) and Negri (2011) (though these papers cover issues of more complexity than what is demanded to follow this paper).

Tolerance and degrees of truth

Cornell University - arXiv, Jul 26, 2022

Inferences and Metainferences in ST

Journal of Philosophical Logic, 2020

In a recent paper, Barrio, Tajer and Rosenblatt establish a correspondence between metainferences... more 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 K+. 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 K+.

Tolerant reasoning: nontransitive or nonmonotonic?

Synthese, 2017

The principle of tolerance characteristic of vague predicates is sometimes presented as a soft ru... more The principle of tolerance characteristic of vague predicates is sometimes presented as a soft rule, namely as a default which we can use in ordinary reasoning, but which requires care in order to avoid paradoxes. We focus on two ways in which the tolerance principle can be modeled in that spirit, using special consequence relations. The first approach relates tolerant reasoning to nontransitive reasoning; the second relates tolerant reasoning to nonmonotonic reasoning. We compare the two approaches and examine three specific consequence relations in relation to those, which we call: strict-to-tolerant entailment, pragmatic-to-tolerant entailment, and pragmatic-to-pragmatic entailment. The first two are nontransitive, whereas the latter two are nonmonotonic.

Priest’s Motorbike and Tolerant Identity

Trends in Logic, 2014

In his chapter ‘Non-transitive identity’ [8], Graham Priest develops a notion of non-transitive i... more In his chapter ‘Non-transitive identity’ [8], Graham Priest develops a notion of non-transitive identity based on a second-order version of \(LP\). Though we are sympathetic to Priest’s general approach to identity we think that the account can be refined in different ways. Here we present two such ways and discuss their appropriateness for a metaphysical reading of indefiniteness in connection to Evans’ argument.

Pragmatic Interpretations of Vague Expressions: Strongest Meaning and Nonmonotonic Consequence

Journal of Philosophical Logic, 2014

Recent experiments have shown that naive speakers find borderline contradictions involving vague ... more Recent experiments have shown that naive speakers find borderline contradictions involving vague predicates acceptable. In Cobreros et al. (Journal of Philosophical Logic, 41, 347-385, 2012a) we proposed a pragmatic explanation of the acceptability of borderline contradictions, building on a three-valued semantics. In a reply, Alxatib et al. (Journal of Philosophical Logic, 42, 619-634, 2013) show, however, that the pragmatic account predicts the wrong interpretations for some examples involving disjunction, and propose as a remedy a semantic analysis instead, based on fuzzy logic. In this paper we provide an explicit global pragmatic interpretation rule, based on a somewhat richer semantics, and show that with its help the problem can be overcome in pragmatics after all. Furthermore, we use this pragmatic interpretation rule to define a new (nonmonotonic) consequence-relation and discuss some of its properties.

Tolerance and Mixed Consequence in the S’valuationist Setting

Studia Logica, 2012

In a previous paper (see 'Tolerant, Classical, Strict', henceforth TCS) we investigated a semanti... more 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.

Reaching Transparent Truth

Mind, 2013

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

Identity, Leibniz's Law and Non-transitive Reasoning

Metaphysica, 2013

Arguments based on Leibniz's Law seem to show that there is no room for either indefinite or cont... more Arguments based on Leibniz's Law seem to show that there is no room for either indefinite or contingent identity. The arguments seem to prove too much, but their conclusion is hard to resist if we want to keep Leibniz's Law. We present a novel approach to this issue, based on an appropriate modification of the notion of logical consequence.

Foreword: Three-valued logics and their applications

Journal of Applied Non-Classical Logics, 2014

Three-valued logics belong to a family of nonclassical logics that started to flourish in the 192... more Three-valued logics belong to a family of nonclassical logics that started to flourish in the 1920s and 1930s, following the work of (Lukasiewicz, 1920), and earlier insights coming from Frege and Peirce (see (Frege, 1879), (Frege, 1892), (Fisch and Turquette, 1966)). All of them were moved by the idea that not all sentences need be True or False, but that some sentences can be indeterminate in truth value. In his pioneering paper, Lukasiewicz writes: "Three-valued logic is a system of non-Aristotelian logic, since it assumes that in addition to true and false propositions there also are propositions that are neither true nor false, and hence, that there exists a third logical value."

Tolerant, Classical, Strict

Journal of Philosophical Logic, 2010

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooi... more 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.

Tolerance and degrees of truth

This paper explores the relations between two logical approaches to vagueness: on the one hand th... more This paper explores the relations between two logical approaches to vagueness: on the one hand the fuzzy approach defended by Smith (2008), and on the other the strict-tolerant approach defended by Cobreros, Egré, Ripley and van Rooij (2012). Although the former approach uses continuum many values and the latter implicitly four, we show that both approaches can be subsumed under a common three-valued framework. In particular, we defend the claim that Smith's continuum many values are not needed to solve what Smith calls `the jolt problem', and we show that they are not needed for his account of logical consequence either. Not only are three values enough to satisfy Smith's central desiderata, but they also allow us to internalize Smith's closeness principle in the form of a tolerance principle at the object-language. The reduction, we argue, matters for the justification of many-valuedness in an adequate theory of vague language.

Review of Richard Dietz, Sebastiano Moruzzi (eds.), Cuts and Clouds: Vagueness, its Nature, and its Logic

Notre Dame philosophical reviews, 2010

Supervaluationism and Necessarily Borderline Sentences

Disputatio, 2008

The supervaluationist theory of vagueness is committed to a particular notion of logical conseque... more The supervaluationist theory of vagueness is committed to a particular notion of logical consequence known as global validity. According to a recent objection, this notion of consequence is more problematic than is usually thought since i) it bears a commitment to some sort of bizarre inferences, ii) this commitment threatens the internal coherence of the theory and iii) we might find counterexamples to classically valid patterns of inference even in the absence of a definitely-operator (or similar device). As a consequence, the supervaluationist theory itself is in trouble. This paper discusses the objection.

Supervaluationism, Subvaluationism and the Sorites Paradox

The Sorites Paradox

Paraconsistent vagueness: a positive argument

Synthese, 2010

Paraconsistent approaches have received little attention in the literature on vagueness (at least... more Paraconsistent approaches have received little attention in the literature on vagueness (at least compared to other proposals). The reason seems to be that many philosophers have found the idea that a contradiction might be true (or that a sentence and its negation might both be true) hard to swallow. Even advocates of paraconsistency on vagueness do not look very convinced when they consider this fact; since they seem to have spent more time arguing that paraconsistent theories are at least as good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal. But it sometimes happens that the weakness of a theory turns out to be its mayor ally, and this is what (I claim) happens in a particular paraconsistent proposal known as subvaluationism. In order to make room for truth-value gluts subvaluationism needs to endorse a notion of logical consequence that is, in some sense, weaker than standard notions of consequence. But this weakness allows the subvaluationist theory to accommodate higher-order vagueness in a way that it is not available to other theories of vagueness (such as, for example, its paracomplete counterpart, supervaluationism).

Supervaluationism and Logical Consequence: A Third Way

Studia Logica, 2008

It is often assumed that the supervaluationist theory of vagueness is committed to a global notio... more It is often assumed that the supervaluationist theory of vagueness is committed to a global notion of logical consequence, in contrast with the local notion characteristic of modal logics. There are, at least, two problems related to the global notion of consequence. First, it brings some counterexamples to classically valid patterns of inference. Second, it is subject to an objection related to higher-order vagueness. This paper explores a third notion of logical consequence, and discusses its adequacy for the supervaluationist theory. The paper proceeds in two steps. In the first step, the paper provides a deductive notion of consequence for global validity using the tableaux method. In the second step, the paper provides a notion of logical consequence which is an alternative to global validity, and discusses i) whether it is acceptable to the supervaluationist and ii) whether it plays a better role in a theory of vagueness in the face of the problems related to the global notion.

Varzi on Supervaluationism and Logical Consequence

Mind, 2010

Though it is standardly assumed that supervaluationism applied to vagueness is committed to globa... more Though it is standardly assumed that supervaluationism applied to vagueness is committed to global validity, Achille Varzi (2007) argues that the supervaluationist should take seriously the idea of adopting local validity instead. Varzi's motivation for the adoption of local validity is largely based on two objections against the global notion: that it brings some counterexamples to classically valid rules of inference and that it is inconsistent with unrestricted higher-order vagueness. In this discussion I review these objections and point out ways to address them not considered in Varzi's paper.

Supervaluationism and Fara’s paradox of higher-order vagueness

Vagueness and Language Use. Palgrave McMillan, Oxford, 2010

This paper discusses Fara&amp;amp;amp;#x27;s so-called &amp;amp;amp;#x27;Paradox of Highe... more This paper discusses Fara&amp;amp;amp;#x27;s so-called &amp;amp;amp;#x27;Paradox of Higher-Order Vagueness&amp;amp;amp;#x27; concerning supervaluationism. In the paper I argue that supervaluationism is not committed to global validity, as it is largely assumed in the literature, but to a weaker notion of logical consequence I call &amp;amp;amp;#x27;regional validity&amp;amp;amp;#x27;. Then I show that the supervaluationist might solve Fara&amp;amp;amp;#x27;s paradox making use of this weaker notion of logical consequence. The paper is discussed by Delia Fara in the same volume.

Supervaluationism and Classical Logic

The supervaluationist theory of vagueness provides a notion of logical consequence that is akin t... more The supervaluationist theory of vagueness provides a notion of logical consequence that is akin to classical consequence. In the absence of a definitely operator, supervaluationist consequence coincides with classical consequence. In the presence of 'definitely', we might find counterexamples to classically valid patterns of inference within supervaluationist reasoning. Foes of supervaluationism emphasize the last result to argue against the supervaluationist theory, concluding that 'supervaluations invalidate our natural mode of deductive thinking' (Williamson, 1994, 152). This and related objections, however, do not consider particular ways in which we might provide systems of deduction for supervaluationist consequence. The present talk considers two ways in which we can carry out this task.

Vagueness: Subvaluationism

Philosophy Compass, 2013

Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theor... more Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theory of vagueness. But these theories cannot be taken apart, for they are in a relation of duality that can be made precise. This paper provides an introduction to the subvaluationist theory of vagueness in connection to its dual, supervaluationism. A survey on the supervaluationist theory can be found in the Compass paper of Keefe (2008); our presentation of the theory in this paper will be short to get rapidly into the logical issues. This paper is relatively self-contained. A modest background on propositional modal logic is, though not strictly necessary, advisable. The reader might find useful the Compass papers Kracht (2011) and Negri (2011) (though these papers cover issues of more complexity than what is demanded to follow this paper).

Tolerance and degrees of truth

Cornell University - arXiv, Jul 26, 2022

Inferences and Metainferences in ST

Journal of Philosophical Logic, 2020

In a recent paper, Barrio, Tajer and Rosenblatt establish a correspondence between metainferences... more 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 K+. 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 K+.

Tolerant reasoning: nontransitive or nonmonotonic?

Synthese, 2017

The principle of tolerance characteristic of vague predicates is sometimes presented as a soft ru... more The principle of tolerance characteristic of vague predicates is sometimes presented as a soft rule, namely as a default which we can use in ordinary reasoning, but which requires care in order to avoid paradoxes. We focus on two ways in which the tolerance principle can be modeled in that spirit, using special consequence relations. The first approach relates tolerant reasoning to nontransitive reasoning; the second relates tolerant reasoning to nonmonotonic reasoning. We compare the two approaches and examine three specific consequence relations in relation to those, which we call: strict-to-tolerant entailment, pragmatic-to-tolerant entailment, and pragmatic-to-pragmatic entailment. The first two are nontransitive, whereas the latter two are nonmonotonic.

Priest’s Motorbike and Tolerant Identity

Trends in Logic, 2014

In his chapter ‘Non-transitive identity’ [8], Graham Priest develops a notion of non-transitive i... more In his chapter ‘Non-transitive identity’ [8], Graham Priest develops a notion of non-transitive identity based on a second-order version of \(LP\). Though we are sympathetic to Priest’s general approach to identity we think that the account can be refined in different ways. Here we present two such ways and discuss their appropriateness for a metaphysical reading of indefiniteness in connection to Evans’ argument.

Pragmatic Interpretations of Vague Expressions: Strongest Meaning and Nonmonotonic Consequence

Journal of Philosophical Logic, 2014

Recent experiments have shown that naive speakers find borderline contradictions involving vague ... more Recent experiments have shown that naive speakers find borderline contradictions involving vague predicates acceptable. In Cobreros et al. (Journal of Philosophical Logic, 41, 347-385, 2012a) we proposed a pragmatic explanation of the acceptability of borderline contradictions, building on a three-valued semantics. In a reply, Alxatib et al. (Journal of Philosophical Logic, 42, 619-634, 2013) show, however, that the pragmatic account predicts the wrong interpretations for some examples involving disjunction, and propose as a remedy a semantic analysis instead, based on fuzzy logic. In this paper we provide an explicit global pragmatic interpretation rule, based on a somewhat richer semantics, and show that with its help the problem can be overcome in pragmatics after all. Furthermore, we use this pragmatic interpretation rule to define a new (nonmonotonic) consequence-relation and discuss some of its properties.

Tolerance and Mixed Consequence in the S’valuationist Setting

Studia Logica, 2012

In a previous paper (see 'Tolerant, Classical, Strict', henceforth TCS) we investigated a semanti... more 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.

Reaching Transparent Truth

Mind, 2013

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

Identity, Leibniz's Law and Non-transitive Reasoning

Metaphysica, 2013

Arguments based on Leibniz's Law seem to show that there is no room for either indefinite or cont... more Arguments based on Leibniz's Law seem to show that there is no room for either indefinite or contingent identity. The arguments seem to prove too much, but their conclusion is hard to resist if we want to keep Leibniz's Law. We present a novel approach to this issue, based on an appropriate modification of the notion of logical consequence.

Foreword: Three-valued logics and their applications

Journal of Applied Non-Classical Logics, 2014

Three-valued logics belong to a family of nonclassical logics that started to flourish in the 192... more Three-valued logics belong to a family of nonclassical logics that started to flourish in the 1920s and 1930s, following the work of (Lukasiewicz, 1920), and earlier insights coming from Frege and Peirce (see (Frege, 1879), (Frege, 1892), (Fisch and Turquette, 1966)). All of them were moved by the idea that not all sentences need be True or False, but that some sentences can be indeterminate in truth value. In his pioneering paper, Lukasiewicz writes: "Three-valued logic is a system of non-Aristotelian logic, since it assumes that in addition to true and false propositions there also are propositions that are neither true nor false, and hence, that there exists a third logical value."

Tolerant, Classical, Strict

Journal of Philosophical Logic, 2010

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooi... more 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.

Tolerance and degrees of truth

This paper explores the relations between two logical approaches to vagueness: on the one hand th... more This paper explores the relations between two logical approaches to vagueness: on the one hand the fuzzy approach defended by Smith (2008), and on the other the strict-tolerant approach defended by Cobreros, Egré, Ripley and van Rooij (2012). Although the former approach uses continuum many values and the latter implicitly four, we show that both approaches can be subsumed under a common three-valued framework. In particular, we defend the claim that Smith's continuum many values are not needed to solve what Smith calls `the jolt problem', and we show that they are not needed for his account of logical consequence either. Not only are three values enough to satisfy Smith's central desiderata, but they also allow us to internalize Smith's closeness principle in the form of a tolerance principle at the object-language. The reduction, we argue, matters for the justification of many-valuedness in an adequate theory of vague language.

Non transitive identity

Sorites and the liar: towards a superclassical solution

Classical logic is often taken as a default assumption. This default assumption, however, seems t... more Classical logic is often taken as a default assumption. This default assumption, however, seems to be revoked in the case of paradoxes such as the sorites and the liar. A common strategy to solve these paradoxes is to weaken classical logic in one or another form, as in the case of paracomplete and paraconsistent logics. In a standard paracomplete solution to the sorites, the least number principle is not generally valid; in the case of the liar, excluded middle is not valid. In a standard paraconsistent solution to the sorites, on the other hand, modus ponens is not valid; in the case of the liar, the solution restricts the applicability of ex contradictione quodlibet. Our strategy, by contrast, consists in a strengthening of classical logic. In our proposal we make use of a logic that is at least as strong as classical logic: it coincides with classical logic for the classical vocabulary but is stronger for a suitably expanded language. In this sense, our approach can be appropriately called 'superclassical'.

Glosario de términos Lógica y Filosofía de las Matemáticas

Una parte importante dentro de cualquier disciplina científica es el manejo de un vocabulario téc... more Una parte importante dentro de cualquier disciplina científica es el manejo de un vocabulario técnico. La intención de este glosario es dar definiciones breves sobre nociones relativamente técnicas que puedan aparecer a lo largo de la asignatura. Este glosario no sustituye los apuntes ni la participación en clase y no tiene sentido estudiarlo de memoria. Se trata de venir aél cuando alguna noción no esté clara del todo. Iré completando este glosario poco a poco. Agradecería que además de los términos que ya he incluido, me fuerais diciendo cuáles os parece que deberían estar aquí. Otras sugerencias, sobre el modo de organizarlo, por ejemplo, también serán bienvenidas (de momento está todo un poco desordenado...)

Lógica II (Spanish)

Tema 1: Introducción P. Cobreros Lógica II: Tema 1 El objeto de la lógica La lógica proposicional... more Tema 1: Introducción P. Cobreros Lógica II: Tema 1 El objeto de la lógica La lógica proposicional Deducción Corrección y completud Resumen El objeto de la lógica Validez o consecuencia lógica La formalización La lógica proposicional El lenguaje proposicional Semántica Deducción Deducción y consecuencia lógica Tablas analíticas Contramodelos Corrección y completud Corrección Completud Resumen P. Cobreros Lógica II: Tema 1 El objeto de la lógica La lógica proposicional Deducción Corrección y completud Resumen Validez o consecuencia lógica Noción intuitiva de consecuencia lógica El objeto de la lógica es la validez de los argumentos o, desde otro punto de vista, la relación de consecuencia lógica. Un argumento es un conjunto de enunciados con cierto orden: premisas y conclusión. La lógica trata de determinar en qué consiste la validez de un argumento o, desde otro punto de vista, la relación de consecuencia lógica. Además trata de esclarecer qué argumentos son válidos y cuáles no; y aportar procedimientos para responder a esta cuestión. De modo intuitivo diremos que un argumento es válido ssi necesariamente preserva la verdad. O, existe una relación de consecuencia lógica entre unas premisas y una conclusión ssi necesariamente, si las premisas son verdaderas, la conclusión es verdadera. P. Cobreros Lógica II: Tema 1 El objeto de la lógica La lógica proposicional Deducción Corrección y completud Resumen Validez o consecuencia lógica Válido vs. 'probativo' Que un argumento sea válido no significa que prueba la verdad de la conclusión. Un argumento es probativo ssi es válido y sus premisas son verdaderas. Pr1: Si llueve en Cascante, todos los animales irán al cielo después de la muerte Pr2: llueve en Cascante C: Todos los animales irán al cielo después de la muerte P. Cobreros Lógica II: Tema 1 El objeto de la lógica La lógica proposicional Deducción Corrección y completud Resumen La formalización

History and Philosophy of Logic (Spanish)

Logic and Paradoxes

This is a three-week course on logic and paradoxes: classical logic, K3 and LP and also a bit of ... more

Logic and Paradoxes

VII Navarra Workshop on Vagueness

Inferences and metainferences in ST

The logic ST has been proposed to deal with paradoxes of vagueness and with the semantic paradoxe... more The logic ST has been proposed to deal with paradoxes of vagueness and with the semantic paradoxes. There is something very distinctive about ST: namely, it is classical logic for a classical language, but it provides ways of strengthening classical logic to deal with paradoxes. For example, the logic ST + (ST for a language with a transparent truth predicate and self-referential sentences) is a conservative extension of classical logic. That is, ST + is not only non-trivial, but it has exactly the same valid inferences as classical logic for the T-free fragment. How is this possible? Well, because ST + preserves all classically valid inferences but not some classical metainferences. The question then arises of exactly which are the metainferences of ST +. In a recent paper, Eduardo Barrio, Lucas Rosenblatt and Diego Tajer show that ST + metainferences are closely related to LP inferences. In this paper we review their result and put the connection in a broader context.

Inferences and metainferences in ST

The logic ST has been proposed to deal with paradoxes of vagueness and with the semantic paradoxe... more The logic ST has been proposed to deal with paradoxes of vagueness and with the semantic paradoxes. There is something very distinctive about ST: namely, it is classical logic for a classical language, but it provides ways of strengthening classical logic to deal with paradoxes. For example, the logic ST + (ST for a language with a transparent truth predicate and self-referential sentences) is a conservative extension of classical logic. That is, ST + is not only non-trivial, but it has exactly the same valid inferences as classical logic for the T-free fragment. How is this possible? Well, because ST + preserves all classically valid inferences but not some classical metainferences. The question then arises of exactly which are the metainferences of ST +. In a recent paper, Eduardo Barrio, Lucas Rosenblatt and Diego Tajer show that ST + metainferences are closely related to LP inferences. In this paper we review their result and put the connection in a broader context.

That there are finitely many sentences (in natural language

Perhaps it's peacefully accepted that there must be finitely many sentences in natural language. ... more

That there are finitely many sentences (in natural language)

I believe it can be easily shown that there are finitely many sentences in natural language. This... more

Superevaluacionismo y la solución de no-temporalidad al problema de la presciencia

If God knew I were going to write this paper, was I able to refrain from writing it this morning?... more If God knew I were going to write this paper, was I able to refrain from writing it this morning? One possible response to this question is that God's knowledge does not take place in time and therefore He does not properly fore-know. According to this response, God knows absolutely everything, it's just that He knows everything outside of time. The so-called timeless solution was one of the influential responses to the foreknowledge problem in classical Christian Theology. This solution, however, seemed to lose support in the recent debate. For example, Pike claims that "the doctrine of God's timelessness entered Christian Theology (only) because Platonic thought was stylish at the time" (Pike, 1970, 190) and Hasker (2001) catalogues this as one of the minor solutions to the problem. One possible source for this general attitude towards timelessness is the thought that the very idea of timelessness is incoherent. In this paper I argue that that the timeless solution to the foreknowledge problem is congenial with the supervaluationist theory of branching time and that this formal framework provides, in fact, a precise characterization of the timeless solution to the foreknowledge problem. The views presented in this paper are in line with those of Kretzmann and Stump (1981), Leftow (1991) and De Florio and Frigerio (2015).

Una oda a los árboles

Se trata de una introducción en español a algunos temas de la filosofía de la lógica y las lógica... more