O pagine editoriali.indd 1 10/04/19 09:10 pagine editoriali.indd 2 10/04/19 09:10 comitato scientifico / scientific committee Michael Arndt (Eberhard Karls Universität Tübingen), Luca Bellotti (Università di Pisa), Luca Gili (Université du Québec à Montréal), Mauro Mariani (Università di Pisa), Carlo Marletti (Università di Pisa), Pierluigi Minari (Università di Firenze), Enrico Moriconi (Università di Pisa), Giacomo Turbanti (Università di Pisa), Gabriele Usberti (Università di Siena) a–n–a–l–i–t–i–c–a 15 Analitica propone una serie di testi – classici, monografie, strumenti antologici e manuali – dedicati ai più importanti temi della ricerca filosofica, con particolare riferimento alla logica, all'epistemologia e alla filosofia del linguaggio. Destinati allo studio, alla documentazione e all'aggiornamento critico, i volumi di Analitica intendono toccare sia i temi istituzionali dei vari campi di indagine, sia le questioni emergenti collocate nei punti di intersezione fra le varie aree di ricerca. Analitica includes reprints of epoch-making monographs, original researches, edited collections and textbooks on major philosophical themes, with a special focus on logic, epistemology and the philosophy of language. The volumes of the series are designed for undergraduate students and for researchers, to whom they offer updated scholarship. They deal with both the fundamental topics of the various research areas and with the emerging questions to be found at their junctions. pagine editoriali.indd 3 10/04/19 09:10 pagine editoriali.indd 4 10/04/19 09:10 Third Pisa Colloquium in Logic, Language and Epistemology Essays in Honour of Mauro Mariani and Carlo Marletti Edited by Luca Bellotti, Luca Gili, Enrico Moriconi, Giacomo Turbanti Edizioni ETS pagine editoriali.indd 5 10/04/19 09:10 www.edizioniets.com © Copyright 2019 Edizioni ETS Palazzo Roncioni Lungarno Mediceo, 16, I-56127 Pisa info@edizioniets.com www.edizioniets.com Distribuzione Messaggerie Libri SPA Sede legale: via G. Verdi 8 20090 Assago (MI) Promozione PDE PROMOZIONE SRL via Zago 2/2 40128 Bologna ISBN 978-884675519-3 pagine editoriali.indd 6 10/04/19 09:10 Third Pisa Colloquium in Logic, Language and Epistemology Essays in Honour of Mauro Mariani and Carlo Marletti pagine editoriali.indd 7 10/04/19 09:10 pagine editoriali.indd 8 10/04/19 09:10 i i "Libro" - 2019/4/10 - 8:25 - page 9 - #1 i i i i i i CONTENTS Preface 13 Enrico Moriconi The philosophical work of Mauro Mariani 15 Luca Gili The philosophical work of Carlo Marletti 21 Giacomo Turbanti An informal exposition of von Neumann's consistency proof 25 Luca Bellotti Paradoxes and set existence 47 Andrea Cantini On false antecedent in dialetheic entailment 59 Massimiliano Carrara Future contingents, Supervaluationism, and relative truth 69 Roberto Ciuni and Carlo Proietti An informational approach to feasible deduction 89 Marcello D'Agostino From logistiké to logistique: the long travel of a word 117 Miriam Franchella and Anna Linda Callow On the size of infinite sets: some Wittgensteinian themes 139 Pasquale Frascolla i i "Libro" - 2019/4/10 - 8:25 - page 10 - #2 i i i i i i 10 CONTENTS Sellars and Carnap on emergence. Some preliminary remarks 153 Carlo Gabbani Is Aristotle's matter ordinary stuff? 167 Gabriele Galluzzo Knowledge and Ockhamist branching time 181 Pierdaniele Giaretta and Giuseppe Spolaore A dialectical analysis of Metaphysics Θ 3 199 Luca Gili In defense of theories and structures in semantics. Reflections on vector models of meaning 215 Alessandro Lenci Structure versus whole versus one 239 Anna Marmodoro Pragmatism and the limits of science 251 Michele Marsonet A note on the logic of distributed knowledge 263 Pierluigi Minari and Stefania Centrone On Popper's decomposition of logical notions 275 Enrico Moriconi Properties and parts in Williams's trope theory 303 Valentina Morotti Kripke's puzzle. A puzzle about belief? 317 Ernesto Napoli i i "Libro" - 2019/4/10 - 8:25 - page 11 - #3 i i i i i i CONTENTS 11 Ruth Millikan on Gottlob Frege: dismantling an apparent clash 329 Carlo Penco The simplicity of the simple approach to personal identity 347 Andrea Sauchelli Category mistakes and the problem of their semantic status 359 Francesco Spada At the roots of rational expressivism 373 Giacomo Turbanti A notion of internalistic logical validity 389 Gabriele Usberti i i "Libro" - 2019/4/10 - 8:25 - page 12 - #4 i i i i i i i i "Libro" - 2019/4/10 - 8:25 - page 13 - #5 i i i i i i PREFACE ENRICO MORICONI In the fall of 1969, I moved to Pisa to start my undergraduate studies and there I met Mauro Mariani and Carlo Marletti. They were in their second year of university and we were all enrolled in the Scuola Normale Superiore. The atmosphere of the Scuola is special in that students live in residences and spend most of their time together, thereby learning from each other and forming lasting friendships. Carlo and Mauro made an immediate impression on me. Already then Carlo was insightful and brilliant and Mauro was a bibliophile, I daresay he was a bookworm. Despite their capabilities and broad knowledge, they were down to earth and eager to help those who approached them with a philosophical question. Mauro and Carlo were studying logic, epistemology and philosophy of language and they were finding their research paths in these fields. At the beginning of my second university year, when I was looking for a study topic in the same broad domain of all things logical, I naturally spent more time with them, benefitting from their insights and suggestions. Thanks to their inputs, I was prompted to widen my research interests and they provided me with answers to the many doubts I had while I was studying logic, philosophy of mathematics and, more generally, philosophy. At that time, they were focusing on W. V. O. Quine's philosophy. Later, Carlo developed an interest in nominalism and Mauro in modal logics. They eventually broadened their research topics to include Aristotle's logic, philosophy of language, linguistics, and Kripke's semantics for modal logics. Years passing, thanks to the special atmosphere of the Scuola Normale, our friendship became ever deeper and together with Lello Frascolla, Ernesto Napoli, and the late Paolo Casalegno we formed a close group that shared a common research agenda. In the 1980s, Carlo, Mauro and I landed jobs at the Department of Philosophy of the University of Pisa, where our mentors Francesco Barone and Vittorio Sainati were the already established scholars working on logic, philosophy of science, and Aristotle. More recently, we were joined by the much younger Luca Bellotti, who is co-editing this volume. Carlo and Mauro were excellent teachers and their classes included innovative approaches that went beyond the traditional syllabus. Yes, the students had to overcome some difficulties of communication, and not only those raised by the complexities of the philosophical topics treated: Mauro's teaching style was ciri i "Libro" - 2019/4/10 - 8:25 - page 14 - #6 i i i i i i 14 ENRICO MORICONI cuitous and Carlo's was concise, at times elliptical. But they were effective and many of their former students have since secured academic positions all over the world. Two of their former students, Luca Gili and Giacomo Turbanti, together with Luca Bellotti and me, are editing this volume in honor of Carlo and Mauro. It is our pleasure to present this collection of essays in this year 2019 as Carlo and Mauro are turning 70. We thank friends and former students who contributed papers on the favourite research topics of the two honorandi. This volume contains essays originally written for this celebration, and eleven of them are by former students of Carlo and Mauro. I thank all the people who enthusiastically contributed to the project. I thank Valentina Morotti for her precious help in drafting Carlo's and Mauro's bibliographies and Laura Tesconi for editing and type-setting the volume. This Festschrift is a token of friendship and gratitude from us all. Cari Carlo e Mauro, buon compleanno! i i "Libro" - 2019/4/10 - 8:25 - page 59 - #51 i i i i i i ON FALSE ANTECEDENT IN DIALETHEIC ENTAILMENT MASSIMILIANO CARRARA∗ massimiliano.carrara@unipd.it Department FISPPA, University of Padua Abstract: Aim of the paper is to analyze Priest's dialetheic solution to Curry's paradox. It has been shown that a solution refuting ABS, accepting MPP and consequently refuting CP meets some difficulties. Here I just concentrate on one difficulty: one obtains the validity of MPP just using FA in the metalanguage, an invalid rule for a dialetheist. Keywords: Dialetheism, entailment, false antecedent. 1 Introduction Consider a paradigmatic case of self-reference paradox, the strengthened liar, having the form: (a): (a) is untrue. A solution to the strengthened liar is notoriously hard to find. If we admit, by the law of excluded middle, that (a) is determinately true or untrue it is immediately inferred that it is true if and only if it is untrue. Moreover, the strengthened liar, differently from the simple one, is a paradox also for those who argue for the socalled gap solutions to the paradox: contrary to the law of excluded middle there are some sentences that are neither true nor false.1 If you assume (a) as neither true nor false one can conclude that, it is, in particular untrue, what the paradoxical sentence says, being so true. There are different solutions to the paradox. Just to mention two of them one can consider Tarski's solution based on the difference between language and metalanguage and Kripke's notion of semantic foundation.2 Aim of this paper is to discuss the dialetheic solution to the paradox, a solution proposed by Priest, for example in (Priest, 1979, 2002a,b, 2006a,b). The dialetheic solution simply consists in accepting the conclusion that (a) is both true and untrue. It is a dialetheia, i.e., a sentence having the form (A∧¬A).3 ∗ I would like to thank Enrico Martino for useful discussions on this paper. Parts of it are in (Carrara and Martino, 2014b). 1 For a discussion on gap theories solutions to the strengthened liar see (Field, 2008). 2 Tarski's solution is in (Tarski, 1956), Kripke's solution is in (Kripke, 1975). 3 G. Priest uses the terms 'dialetheiae' and 'true contradictions' to indicate 'gluts', which in turn is a term coined by K. Fine in (Fine, 1975). i i "Libro" - 2019/4/10 - 8:25 - page 60 - #52 i i i i i i 60 MASSIMILIANO CARRARA In classical logic, the presence of dialetheiae entails trivialism and – from a deductive point of view – explosion. Explosion is produced using the rule ex contradictione quodlibet (ECQ). The classical justification for ECQ rests on the alleged evidence that no contradiction can be true, which is rejected by dialetheists. Observe that in standard natural deduction ECQ can be derived using reductio ad absurdum (RAA) and other apparently non-problematic rules. And RAA is indeed rejected by a dialetheist. However, the banish of RAA is insufficient to avoid trivialism: Curry's paradox, from which trivialism follows, can be generated without the help of RAA. In the Logic of Paradox (LP)4 Priest observes that, in a semantically closed theory, using modus ponens (MPP) and absorption (ABS), i.e.: φ → (φ → ψ) ABS φ → ψ a version of Curry's paradox is derivable.5 In LP, (A→B) is defined as (¬A∨B) (the material conditional), which suffices to establish that MPP can't in general be valid: if A is a dialetheia, (¬A∨B) is true even if B is not. MPP is labeled in LP as a quasi-valid rule, a rule that is valid provided that all truth-values involved are classical (i.e., solely true or solely false). However, Priest realizes that the material conditional, just because it invalidates MPP, is not a genuine conditional. He emphasizes that "any conditional worth its salt should satisfy the modus ponens principle" (Priest, 2006b, p. 83). So, in subsequent works, for example in (Priest, 2006b, 2008), he introduces a new conditional satisfying MPP, the entailment connective, and tries to escape Curry's paradox by rejecting ABS. Since in natural deduction ABS is derived from CP and MPP to reject ABS and accept MPP implies to reject CP. The above strategy should be compatible with the following two general claims Priest also makes: 1. the presence of dialetheias does not entail trivialism; 2. the meaning of logical constants should be dialethically acceptable both in the object language and in the metalanguage. 4 For a general introduction to LP see (Asenjo, 1966; Asenjo and Tamburino, 1975; Routley, 1979; Beall, 2009). For an introduction to dialetheism see (Berto, 2007, Ch. 8). 5 Formulations of Curry's paradox that do not rely on ABS typically make an appeal to the structural version of the rule, Structural Contraction, viz. that if Γ,A,A ` B, then Γ,A ` B. On this, see, e.g., (Beall and Murzi, 2013). i i "Libro" - 2019/4/10 - 8:25 - page 61 - #53 i i i i i i FALSE ANTECEDENT AND DIALETHEIC ENTAILMENT 61 In the rest of this paper, I critically assess the foregoing approach to Curry's Paradox. I just focalize on a single difficulty: Priest's strategy for recovering a genuine conditional, the entailment connective or conditional (A⇒ B) – one that allows him to recover MPP – is such at the cost of a use of the conditional rule of false antecedent FA in the metalanguage. This use is highly problematic, however, because of Priest's requirement that the inference rules used in the metalanguage should be dialetheically acceptable. 2 Curry's paradox and its arithmetical formalization Curry's paradox6 is derived in natural language from sentences like the following: (b): If sentence (b) is true, then Santa Claus exists. Suppose the antecedent of the conditional in (b) is true, i.e., sentence (b) is true. Then, by MPP, Santa Claus exists. So, we have proved the consequent of (b) under the assumption of its antecedent. By CP, we have then proved (b), i.e., sentence (b) is true. We can now apply MPP once more, and conclude that Santa Claus exists. Of course, we could substitute any arbitrary sentence for 'Santa Claus exists'. As a result, every sentence can be proved and trivialism follows. I reconstruct Curry's argument in the language of first order arithmetic with a truth predicate. Let L be the language of first order arithmetic and N its standard model. Now extend L to L * by introducing a new predicate T . Assume a codification of the syntax of L * by natural numbers and extend N to a model N * of L * by interpreting T as the truth predicate of L *. So, for all n ∈N , T (n) is true if and only if n is the code of a true sentence A of L *, in symbols n = dAe. To be sure, classically such an interpretation is impossible, since the theory obtained by adding to Peano arithmetic the truth predicate for the extended language L * (with Tarski's shema) is inconsistent. This is not so for a dialetheist, however, who accepts inconsistent models. At this point one can show that, if one uses the classical rules of the conditional in natural deduction and Tarski's scheme T (dAe)↔ A, the model N * turns out to be trivial. In fact, let A be any sentence of L *. By diagonalization, there is a natural number k such that 6 Curry's original paper in which the paradox was introduced is (Curry, 1942). i i "Libro" - 2019/4/10 - 8:25 - page 62 - #54 i i i i i i 62 MASSIMILIANO CARRARA k = dT (k)→ Ae. We can now prove A using natural deduction as follows: 1 (1) T (k)↔ (T (k)→ A) Tarski's schema 2 (2) T (k) Assumption 1, 2 (3) T (k)→ A 1, 2 MPP 1, 2 (4) A 2, 3 MPP 1 (5) T (k)→ A 2, 4 CP 1 (6) T (k) 1, 5 MPP 1 (7) A 5, 6 MPP The LP logic doesn't validate MPP: as we have already observed in §1, if A is a dialetheia, (¬A∨B) is true even if B is solely false. In this way Curry's paradox is blocked. 3 Entailment: logic and semantics As said, Priest is aware that the material conditional is inadequate to capture the intended meaning of the genuine conditional. If, following Priest, it is thought that MPP is constitutive of the meaning of 'if', one natural reaction to Curry's Paradox, then, is to define a conditional which validates MPP but not CP. For this reason in (Priest, 2006b, Ch. 6) Priest introduces a more sophisticated conditional (⇒) which he takes to be an entailment connective. Priest suggests to read (A⇒ B) as "B follows logically from A". The main feature of the entailment connective, ⇒, is that it is a modal connective invalidating ABS, preserving MPP while avoiding Curry's paradox. . The modal force of⇒, however, is quite different from the force of other modal conditionals, such as the strict conditional, or even the counterfactual conditional. Both, in fact, validate ABS. An interpretation I for a language L of propositional logic with ⇒ is a quadruple 〈W,R,G,v〉, where W is, as usual, an arbitrary set of objects ("possible worlds"), R is a dyadic relation between members of W ("the accessibility i i "Libro" - 2019/4/10 - 8:25 - page 63 - #55 i i i i i i FALSE ANTECEDENT AND DIALETHEIC ENTAILMENT 63 relation"), G is a designated member of W ("the actual world") and v is an evaluation function that assigns to each propositional atom and world w a non-empty subset of {0,1}, where 1 is the value "true", 0 is the value "false". Similarly for a first order language. The semantic clauses for a formula like φ ⇒ ψ are the following: • φ ⇒ ψ is true in w if, and only if, for every world w′ such that R(w,w′), if 1 ∈ vw′(φ), then 1 ∈ vw′(ψ) and if 0 ∈ vw′(ψ), then 0 ∈ vw′(φ). • φ ⇒ ψ is false in w if, and only if, for some world w′ such that R(w,w′), 1 ∈ vw′(φ) and 0 ∈ vw′(ψ). In short: φ ⇒ ψ is true in a world w if and only if, for every world w′ accessible from w, if φ is true in w′, so is ψ and if ψ is false in w′, so is φ . φ ⇒ ψ is false at a world w if and only if there is at least one accessible world w′ where φ is true and ψ is false.7 He defines semantic consequence and logical truth as follows. (SC) Γ |= α ifdf. for all I, if, for every β ∈ Γ, 1 ∈ vG(β ), then 1∈ vG(α), and if 0∈ vG(α) then 0∈ vG(β ) for some β ∈Γ. (LT) |= α if and only if, for every I, 1 ∈ vG(α). Counterexamples to ABS are obtained by means of interpretations with the following two features: • G is omniscient: for every w ∈W , R(G,w). • R is non-reflexive: there is at least one w ∈W such that ¬R(w,w). Consider now the following interpretation where the two above mentioned properties are at work: • W = {G,w} • R(G,w),¬R(w,w),R(G,G),R(w,G) • vG(φ) = {0};vG(ψ) = {0};vw(φ) = {1}; vw(ψ) = {0} In the above interpretation, we have that vG(φ ⇒ (φ ⇒ ψ)) = {1}, at least in the classical metalanguage. However, vG(φ⇒ψ) = {0}, since in w, which accessible from G, φ is true and ψ is false. 7 On this see also (Carrara et al., 2011, 2012; Carrara and Martino, 2014a). i i "Libro" - 2019/4/10 - 8:25 - page 64 - #56 i i i i i i 64 MASSIMILIANO CARRARA Graphically, the counterexample can be characterized in the following way: G w v(φ) = 0 v(φ) = 1 v(ψ) = 0 v(ψ) = 0 v(φ ⇒ ψ) = 0 v(φ ⇒ ψ) = 1 v(φ ⇔ (φ ⇒ ψ)) = 1 We can, then, solve Curry's paradox by holding that, if in a semantically closed language φ is false only, then the Curry sentence (Curry) φ ⇔ (φ ⇒ ψ) is true, but both φ and φ ⇒ ψ are false only and ψ does not follow by MPP. Observe that the presence of non reflexive worlds is essential for invalidating ABS. Suppose that all worlds are reflexive and prove ABS. Let 1 ∈ vG(φ ⇒ (φ ⇒ ψ)) and let w be any world. Suppose that 1∈ vw(φ). Then, 1∈ vw(φ⇒ψ) and, by reflexivity, 1∈ vw(ψ); besides, if 0∈ vw(ψ) then 0∈ vw(φ). Thus, 1∈ vG(φ⇒ψ). Moreover, note that no dialetheia is involved in the above solution of the paradox. It means that the foregoing solution to the paradox is not specifically a dialetheist solution. Finally, it is worth emphasising that the non-reflexivity of R is essential for falsifying ABS. 4 On false antecendent It is the aim of this section to argue that the validity of MPP for⇒ – as well as the above counterexample to ABS – are problematic in a dialetheic metalanguage. To see this, remember that the meaning of the logical constants should be dialethically acceptable both in the object language and in the metalanguage: claim 2 mentioned above. To prove that ⇒ satisfies MPP one must show that, given an arbitrary model M , the following holds: (*) if 1 ∈ vG(φ ∧ (φ ⇒ ψ)) then 1 ∈ vG(ψ). Now suppose that M has a unique world G and consider the evaluation: vG(φ) = vG(ψ) = {0}. How can we recognize that (*) holds? Since all we know from the evaluation v is that antecedent of (*) is only false, the only way to recognize the validity of (*) is to invoke the false antecedent rule: (FA) Any conditional with a false antecedent is true. i i "Libro" - 2019/4/10 - 8:25 - page 65 - #57 i i i i i i FALSE ANTECEDENT AND DIALETHEIC ENTAILMENT 65 Question: what kind of conditional is used in the metalanguage when proving that⇒ satisfies MPP? Here is the problem. As said before, according to Priest, even the metalinguistic logical constants are to be dialetheically understood. Since, as Priest maintains, any genuine conditional must validate MPP, it does invalidate, on pain of trivialism, FA. Consider the following train of thoughts. Dialetheism rejects FA by observing that, if φ is a dialetheia and ψ is only false, then the conditional φ → ψ is only false since it does not preserve truth from φ to ψ . So, one could think, at first sight, that – where dialetheias are not involved – FA dialetheically holds. Unfortunately, that is not the only case. Indeed, take the above countermodel to ABS: G w v(φ) = 0 v(φ) = 1 v(ψ) = 0 v(ψ) = 0 v(φ ⇒ ψ) = 0 v(φ ⇒ ψ) = 1 v(φ ⇔ (φ ⇒ ψ)) = 1 φ ⇒ψ is only false even if φ is only false. Thus, following Priest's semantics, FA is rejected independently of the presence of dialetheias. For this reason the metalinguistic conditional cannot be a genuine one. A typical non-genuine dialetheic conditional satisfying FA is the material conditional. If it is so, it would seem plausible to adopt the latter in the metalanguage. However, as said, the material conditional invalidates MPP in LP. Moreover, I am going to show that, if the material conditional is used in the metalanguage, the entailment connective φ ⇒ ψ no longer validates MPP. A preliminar observation. Consider that, though Priest does not identify falsity with untruth, however he holds that certain sentences are both true and untrue, the strengthened liar, for example, is one of these. Now, consider a model M with a unique world G, where φ is both true and untrue and ψ is only false. Since φ is untrue at G, the metalinguistic material conditionals: (**) If φ is true at G, then so is ψ If ψ is false at G, then so is φ are true. It follows that (φ ⇒ψ) is true. So, φ and (φ ⇒ψ) are true but ψ is only false; hence MPP does not hold. It follows, then, that⇒ fails to satisfy MPP. How to reply? A dialetheist may perhaps object to our use of FA in establishing the first conditional in (**) as follows: i i "Libro" - 2019/4/10 - 8:25 - page 66 - #58 i i i i i i 66 MASSIMILIANO CARRARA • "φ is true" is expressed by 1 ∈ v(φ), and hence 1 ∈ v(T (dφe)), while "φ is untrue" is expressed by 1 ∈ v(¬T (dφe)) i.e., 0 ∈ v(T (dφe)). And since, according to Priest, untruth implies falsity, 0∈ v(φ). Summing up, "φ is tue and untrue" is expressed by v(φ ) = v(T (dφe)) = {0,1}. That is, both φ and T (dφe) are dialetheias. So the appropriate truth-conditions of (φ ⇒ ψ) are: (***) If 1 ∈ v(φ) then 1 ∈ v(ψ); if 0 ∈ v(ψ) then 0 ∈ v(φ). With (***) in place, we can no longer resort to FA to establish the truth of the first conditional in (***). If φ is a dialetheia the v(φ ) = {0,1} and the conditionals are so interpreted: (***) If 1 ∈ {0,1} then 1 ∈ {0}; if 0 ∈ {0} then 0 ∈ {0,1}. To apply FA to the first conditional we have to negate its antecedent, but the negation of the antecedent is 1 /∈ v(φ) i.e., 1 /∈ {0,1} which is false. Notice, however that, the argument shows that the semantics at issue is inadequate to express the metalinguistic notion of untruth and hence to a dialetheic solution of the strengthened liar. In fact, if 1 ∈ v(φ) means that φ is true, the untruth of φ is properly expressed by 1 /∈ v(φ), while the truth of ¬(T (dφe)) is expressed by 1 ∈ v(¬(T (dφe)), i.e., 0 ∈ v(T (dφe)); and from the latter 1 /∈ v(φ) does not follow. In a reply to a criticism developed by Littman and Simmons (2004), Priest observes that the treatment of functions in a dialetheic framework is a "sensitive matter" (Priest, 2006b, p. 288). He suggests – a suggestion later used many times – to employ relations instead of functions. In particular, in the case of semantic values, instead of an evaluation function, one can take an evaluation relation R from the set of sentences to {0, 1}, such that, for any sentence φ ,R(φ ,0) or R(φ ,1). Moreover, he insists claiming that even the metalanguage may be inconsistent, so that R may both correlate and not correlate a sentence with a certain truth value. If we follow this suggestion, the evaluation of an untrue sentence φ must satisfy the condition ¬R(φ ,1); and if T must express metalinguistic truth R((T (dφe)),0) is to be equivalent to ¬R(φ ,1). However, Priest's suggestion does not help him circumvent the problem To see this, consider again our model M , this time using R instead of v. Then, the appropriate evaluation of a true and untrue sentence φ is i i "Libro" - 2019/4/10 - 8:25 - page 67 - #59 i i i i i i FALSE ANTECEDENT AND DIALETHEIC ENTAILMENT 67 R(φ ,1) and ¬R(φ ,1). Hence, the first metalinguistic material conditional if R(φ ,1) then R(ψ,1) is true by FA and the proposed conclusion follows. 5 Conclusions Priest's strategy of having a "true" conditional implies a counterintuitive modal semantics, which allows it to recover MPP at the cost of using the FA rule in metalanguage, a rule that it is not admitted by a dialectist. References Asenjo, F. G. (1966). A calculus of antinomies. Notre Dame Journal of Formal Logic, 16:103–105. Asenjo, F. G. and Tamburino, J. (1975). Logic of antinomies. Notre Dame Journal of Formal Logic, 16:17–44. Beall, J. C. (2009). Spandrels of Truth. Oxford University Press, Oxford. Beall, J. C. and Murzi, J. (2013). Two flavors of Curry's paradox. Journal of Philosophy, 110(3):143–165. Berto, F. (2007). How to Sell a Contradiction: the Logic and Metaphysics of Inconsistency. College Publications, London. Carrara, M., Gaio, S., and Martino, E. (2011). Can Priest's dialetheism avoid trivialism? 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