Priority Nominalism – IMAGUIRE (M)

1 de abril de 2019

IMAGUIRE, G.. Priority Nominalism. Springer Verlag, 2018. 171p. Resenha de: CICCARELLI, Vicenzo. Manuscrito, Campinas, v.42 n.2 Apr./June 2019.

In his most recent book ‘Priority Nominalism’ (2018, Springer Verlag), Guido Imaguire presents a new sort of nominalism considered both as a development and an enrichment of Quine’s view on universals notoriously labeled by Armstrong Ostrich Nominalism: like the Ostrich that buries his head under the ground, the Quinean nominalist gets rid of universals by not taking predicates with the adequate ontological seriousness.

The basic idea of the book is that questions of existence must be answered by applying what Imaguire calls the criterion of Grounded Ontological Commitment. According to this criterion, Quine’s criterion of ontological commitment must be applied not to all sentences composing the theories that we are open to accept as true, but only to sentences of the same theories – or suitable paraphrases – expressing fundamental facts, i.e. facts that are not grounded in any other fact. This criterion combined with the view that predication is fundamental, i.e. not grounded in any other fact about properties, classes, platonic forms, etc… – results in a strengthened version of Ostrich Nominalism: there are no universals for there are no sentences expressing fundamental facts in which expressions allegedly standing for universals occur in a position into which we may quantify. Thus the notion of metaphysical grounding represents the additional and crucial ingredient of Priority Nominalism that makes it a strengthened version of Quine’s Ostrich Nominalism. Moreover, Priority Nominalism besides gaining more argumentative strength than Ostrich Nominalism is also wider in scope: not just properties, but also entities like classes, numbers, meanings, and word types can be explained away using the same strategy.

The book has also another purpose: to show that the attitude of the Ostrich Nominalist is not dismissive regarding ontological questions about universals when he is endowed with the theoretical tool of the notion of grounding. The ostrich nominalist does not reject the problem of the grounds of predication: he simply argues that predication is not grounded in anything else, that is fundamental, and thus he does have an answer. Moreover, Imaguire argues that even Quine’s Ostrich Nominalism is not dismissive and consequently the label Ostrich is unfair: according to Imaguire, what is achieved with the introduction of the notion of metaphysical grounding is a way of making explicit some aspects of Quine’s nominalism that many times are justified by appealing to ontological intuitions.

In the first chapter, Imaguire presents the Problem of Universals. According to him, the expression ‘Problem of Universals’ has been traditionally used to denote different issues and formulations. Imaguire lists five formulations: (1) The existence question, (2) The grounding of predication, (3) The One-Over-Many Problem, (4) The Many-Over-One Problem, and (5) the Similar-But-Different Problem. (1) is summarized by the question “Do universals exist?”; (2) is the problem of explaining predication, i.e. of specifying in virtue of what a certain particular a is F. Formulations (3), (4), and (5) are deeply related and amount to answering questions introduced by the “how is it possible” clause: (3) “How is it possible that two different particulars a and b are both F?”; (4) “How is it possible that the same particular a is both F and G?”; (5) “How is it possible that two distinct particulars a and b are similar – insofar as they are both F – and different at the same time?”. As the author remarks, many traditional solutions to the Problem of Universals deal with one particular formulation, while his proposal has the ambition of providing answers to all formulations (1)-(5). The author defends that the logical core of the Problem of Universals is the type-token problem: the problem of accepting the existence of types just because there are tokens of the same type. The type-token problem represents a generalization of the Problem of Universals, for it extends the problem of existence to other suspicious abstract entities that are not properties. In another section, Imaguire argues that the type-token problem is often confused with the sparse-abundant problem. The sparse-abundant problem in spite of being connected with the problem of universals (existence of universal is defended on the ground of non-arbitrary special sorts of similarities, properties, classes) is not its logical core.

The chapter ends with a list of the main solutions to the Problem of Universals so far proposed:

  • Trascendental realism
  • Immanent realism
  • Class nominalism
  • Predicate/concept nominalism
  • Trope theory
  • Priority nominalism

In chapter 2 friends and foes of Ostrich Nominalism are considered. Concerning the foes, after examining the main objections in the literature, the author concludes that the Ostrich nominalist is subject to four main charges: 1) he does not take the Problem of Universals as a genuine metaphysical problem, 2) he does not take predicates with ontological seriousness, 3) he does not offer an analysis of predication, 3) he does not offer an explanation for commonality of properties. On the other hand, the Ostrich has friends, namely authors that partially agree with some crucial aspects of Ostrich Nominalism: e.g. the fact that predication is fundamental (Van Cleve, Summerford) and the fact that not just properties are suspicious entities but also other abstract entities (Melia, Azzouni).

In chapter 3 it is discussed what kind of explanation the Problem of Universals demands. Three main candidates for the role of explanans are considered:

  • Ontological commitment
  • Grounding
  • Truthmaking

The priority nominalist offers a complex explanation based on the combination of ontological commitment and grounding.

Given that the Problem of Universals is an ontological problem, ontological commitment seems at first glance the best candidate. Moreover, the One-Over-Many Problem, when framed as an argument, is based on the criterion of ontological commitment. The argument may be briefly resumed as follows: from the fact that a is F and b is F we deduce that a and b have something in common; thus there is something that a and b have in common. Hence, the fact that a is F and b is F is committed to an entity that a and b have in common.

The Priority Nominalist does not frame his explanation in terms of the truthmaking relation for several reasons. Truthmaking is a relation from reality to language while ontological commitment goes in the opposite direction. Yet the problem of universals is the problem of accounting for the truth of predication, thus it goes from language to reality. In this sense ontological commitment seems to be a more adequate explanans than truthmaking.

Another problem of the truthmaking relation is that – differently from ontological commitment – it is non-factive: the fact that a sentence S is made true by something does not ensure that there is a unique truth-maker and does not say anything at all on what sort of entities all possible truth-makers of S might be. Thus from the fact that ‘Fa‘ is true, we cannot conclude by invoking the truthmaking relation that there is a universal for which F stands.

Grounding also seems to be more adequate than truthmaking. The main reason the author offers is that to construct the ontological hierarchy that is required for the formulation of the problem by the Priority Nominalist, the transitivity of the relation between different ontological categories is required. Truthmaking is not transitive whereas grounding is.

The final part of the chapter is dedicated to introduce the notions that are crucial to the explanation offered by the Priority Nominalist. The notion of ontological categories as the most general categories of reality is introduced and ontological categories are divided into fundamental and derivative. A metaphysical theory is considered as a System of Ontological Categories. Four important claims are made regarding ontological categories:

  • Categories of the same level are pairwise disjoint.
  • To accept a range of entities as ontological category does not mean to accept it as a fundamental ontological category
  • The mere existence of instances of an ontological category does not imply that the category is fundamental
  • When at least one entity eof a category C is fundamental, then C is fundamental

With this conception of ontological category in mind the Priority Nominalist may state his position: properties, states of affairs, tropes are instantiated ontological categories, yet they are only derivative. The only fundamental ontological category is the category of concrete particulars.

In chapter 4 the author deals with the Problem of Predication, i.e. the problem of explaining in virtue of what a particular a is F. A solution to this problem is necessary to deal with the One-Over-Many Problem, for the One-Over-Many Problem is formulated in terms of multiple predications. The chapter has a negative purpose: to show that any view so far proposed according to which predication is not fundamental and thus any attempt to ground predication irremediably falls into an infinite regress.

The author starts by listing how the so-called Bradley’s regress occurs with the main approaches to the Problem of Predication: Transcendental Realism, Immanent Realism, Class Nominalism, Concept Nominalism, Predicate Nominalism, Resemblance Nominalism, and Trope Theory. Exception made for Trope Theory, the structure of the reasoning that gives rise to the regress is almost the same.

Successively, Imaguire passes to consider some possible strategies to block Bradley’s regress that have been proposed and points out their main flaws:

  • Identity of levels strategy: all levels of explanation are only apparently different. However, the author objects that each level amounts to the introduction of a different relation (e.g. relation of different order, different linguistic level) thus different levels of grounding correspond to different facts.
  • Quantificational Strategy:this is a strategy available only for the nominalist. The idea is that there is no need to quantify over the relation introduced to ground predication, thus there is no effective ontological commitment to such a relation. The Priority Nominalist replies that it is unclear why we should not take with ontological seriousness the relation introduced to ground predication while we should do the opposite with the property appearing at the initial level.
  • Formal Relations:the relation appearing at each level of explanation is a mere formal ontological relation, thus we are not ontologically committed to it. Yet – as the author remarks – which criterion do we have to distinguish between properties and relations that need to be grounded and a formal ontological relation?
  • Internal Relations:the relation between a particular and a universal (whatever it may taken to be) is an internal relation, and internal relations are given as ontological free lunch whenever the relata are given. However – the author objects – the notion of internal relation cannot be strictly defined without appealing to intrinsic features of the relata. As a consequence, the internal relation of e.g. instantiantion is grounded on the fact that a certain particular has certain properties. How could it possibly ground predication without begging the question?
  • Truthmakers:Armstrong proposed an argument to block the regress based on the notion of truthmaker. As Armstrong argues, all steps of explanation in Bradley’s regress have the same truthmaker, thus the explanation may arrest at the first step. The author objects that in Armstrong’s argument there is an illegitimate switch between the relation of truthmaking and that of grounding: why at the first level, i.e. the level of predication, we must ask for grounding and at the following levels we must ask for a truthmaker?

The conclusion Imaguire draws is that all attempts to ground predication so far proposed are unsatisfactory; albeit this fact does not represent a knock-down argument against the view that predication may be grounded, it supports the hypothesis that predication is fundamental. And precisely this is the position of the Priority Nominalist. Moreover, the author stresses that such a position does not amount to refuse to deal with the Problem of Predication, yet it corresponds just to a particular answer to this problem, i.e. that predication is not grounded in anything else.

In Chapter 5 a solution is offered to the One-Over-Many Problem and its variations (i.e. the Many-Over-One and the Similar-But-Different Problems). As previously mentioned, the One-Over-Many Problem may be formulated as an argument whose crucial point is the criterion of ontological commitment: the existence of universals is justified by the fact that we are allowed to quantify over what two particulars being both F have in common. Imaguire highlights that the principle of ontological commitment presents a fundamental difficulty which he calls the Paraphrase Symmetry Problem:

“suppose that sentence S commits us to the existence of entity E, but its paraphrase S* does not. Why should we conclude that the commitment of S to E is only apparent? What reason could we have to prefer S to S*? Since ‘is a paraphrase of’ is plausibly a symmetrical relation between sentences, one could also conclude that the non-commitment of S* to E is only apparent.” Imaguire (2018), p. 87

According to the Priority Nominalism, such a difficulty may be overcome by combining the criterion of ontological commitment with the notion of metaphysical grounding. In other words, given the two sentences S and S* of the quoted passage, the criterion of ontological commitment must be applied to the sentence (if any) that expresses a fundamental fact. The combination of ontological commitment with grounding results in the Criterion of Grounded Ontological Commitment which may be summarized by the motto:

“To be is to be a value of a bound variable of a fundamental truth.” Imaguire (2018), p. 89

The Criterion of Grounded Ontological Commitment is firstly used to analyze the One-Over-Many Problem. Imaguire considers a battery of sentences and the corresponding paraphrases that are relevant to the issue: for instance, he considers both the sentence ‘a is F‘ and the sentence ‘a has the property F”. The Priority Nominalist is willing to say that ‘a is F‘ expresses a fact that is more fundamental than that expressed by the sentence ‘a has the property F‘, and thus, given that in ‘a is F‘ there is no expression allegedly standing for a universal that occurs in a position over which we may perform a first-order quantification, there is no grounded ontological commitment to universals. To achieve this goal Imaguire uses Fine’s logical theory of the grounding relation and in particular Fine’s treatment of two lambda operators: the predicate abstraction and the property abstraction operator. Thus Imaguire formalizes the aforementioned battery of sentences and their paraphrases using Fine’s rules for the lambda operators and the immediate result is that ‘a is F‘ (formalized using the predicate abstraction operator) grounds ‘a has the property F‘ (formalized using the property abstraction operator). The same strategy is used to dissolve the apparent ontological commitment to universals in the Many-Over-One Problem and the Similar-But-Different Problem.

The remaining part of the chapter is dedicated to two important issues: the status of sentences apparently committed to abstract entities and the specification of a truthmaker for sentences of the form ‘a is F‘. Concerning sentences like ‘a has the property F ‘, ‘a and b have the property F in common’, etc… the priority nominalist has a position that notably differs from many nominalistic views: many nominalists would say that the sentence ‘a has the property F‘ is false or truthvalueless at best, for there are no such entities as properties; the priority nominalist holds that ‘a has the property F ‘ is “a common but misleading manner of saying that things are in a certain way” (Imaguire (2018), p. 100), i.e that a is in the F-way. The same applies to other abstract terms, as in the case of numbers: sentences containing number terms are not false, they are true even if they do not express fundamental facts; they are just ontologically misleading insofar as they suggests that there are numbers. By proposing a hierarchical structure of reality in which derivative facts are included with a thin notion of existence, the Priority Nominalist does not have to pay the price of avoiding property talk, or number talk, or meaning talk, etc…

Concerning the problem of specifying a truthmaker for ‘a is F’, the Priority Nominalism invokes the notion of thick particular: what makes true ‘a is F‘ is precisely the thick particular a with its being in a particular way expressed by the predicate F. In other words, the state of affairs corresponding to ‘a is F‘ has as only constituent the thick particular a.

Chapter 6 is about second-order quantification. Imaguire distinguishes two different senses of second-order quantification: the metaphysical sense and the logical sense. A metaphysical second-order quantification is a quantification over properties that may be instantiated by particulars: ‘There is a property that Napoleon and Julius Caesar have in common’ is an example of this sort of quantification. Notice that from a strictly logical point of view, this sort of quantification corresponds to a first-order quantification, for expressions allegedly standing for properties are nominalized: ‘There is an x such that x is a property and Napoleon and Julius Caesar have x in common’. A logical second-order quantification is a quantification into the position of first-order predicates, e.g. ‘ There is something that both Napoleon and Julius Caesar are’. It is crucial for the priority nominalist to explain away both logical and metaphysical second-order quantification, for if second-order existential quantified sentences were as fundamental as first-order ones, then there would be an unavoidable (grounded) ontological commitment to type entities.

The author considers three contexts of abstract reference to properties that allow for metaphysical second-order quantification: exemplification, intensional, and classificatory contexts. Exemplification and intensional contexts are easily dismissed, thus he concentrates on classificatory contexts and in particular to two examples: the sentences `Humility is a virtue’ and ‘Red is more similar to orange than blue’. The strategy Imaguire uses to paraphrase away abstract reference is based on the following guiding principle: every time we talk about a property F we are not really talking about F-ness but about F things. The one of the simplest examples of this kind is when we paraphrase the sentence ‘the concept F is a subconcept of the concept G‘ as ‘For every x, if x is F, then x is G‘. Clearly, cases like ‘Humility is a virtue’ are far more complex and require additional theoretical tools. Imaguire uses the method of grounded paraphrases, i.e. a method of paraphrasing second-order sentences based on both Fine’s logic of grounding (including the relation of partial grounding) and a sort of negative form of grounding that he calls despite operator.

The case of logical second-order quantification is treated differently. Imaguire argues that logical second-order quantification is not intelligible, for every attempted construction of a grammatical sentence in ordinary language translating the formal second-order quantification ends up altering the syntactic category of the quantified expressions. Moreover, even if an intelligible reading of second-order quantification were available, it could not result in an ontological commitment, for there would still be a difference between second and first-order quantification based on the different semantic behaviors between naming expressions and predicative expressions. Such a difference is explained in terms of Dudman’s distinction between the representational and the semantic role conceptions of reference. Predicative expressions have reference in the sense that they fulfill a certain semantic role in the determination of the truth-value of an entire sentence and not in the sense that they stand for existing entities. On the contrary, proper names have reference in the sense that they stand for an existing possibly extra-linguistic entity: according to Imaguire’s terminology, proper names have referents, predicates have references. The conclusion of the chapter is that provided that both metaphysical and logical second-order quantification may be respectively explained away and dismissed, intelligible second-order sentences express facts that are not fundamental for they are grounded in facts expressed by the correspondent first-order grounded paraphrases. As a consequence, according to the criterion of grounded ontological commitment, second-order sentences do not have any additional ontological import compared to their first-order paraphrases.

In the last chapter the author resumes his position by presenting the big picture of reality resulting from Priority Nominalism. Not just properties are dismissed as non-fundamental entities, but also word types, meaning, sets, numbers, etc… Even concrete particulars like mereological aggregates do not exist insofar as they are not fundamental: Priority Nominalism entails a form of mereological atomism. The chapter is also concerned with the contextualization of Priority Nominalism in the theoretical debate on the status of universals. Imaguire considers traditional realist and nominalist approaches to the Problem of Universals and remarks that most of them do not make explicit the distinctive features of Priority Nominalism, for the proposal of the book besides offering a new solution to the Problem of Universals, represents a relatively new way of understanding and formulating the problem itself. He argues that the approach the priority nominalist favors is what he calls the hierarchical approach. According to this approach, realists and nominalists disagree about the hierarchical structure of reality. The realist holds that properties are treated like objects, while the nominalist holds that they belong to a different level and only the level zero (i.e. the level at which we are allowed to quantify over in fundamental truths) is the level of existing entities. Thus the realist is accused of confusing different levels of the hierarchical structure, by reifying what does not belong to the level zero.

Finally the author lists the main advantages of Priority Nominalism:

  • Agreement with pre-theoretical intuitions: the apple is red is a fundamental fact, is not grounded in the existence of properties or classes, or mysterious relations of resemblance.
  • Few metaphysical assumptions: does not need modal realism, does not need universals, does not need sets.
  • Qualitatively and quantitatively economical: there is just one category of existence, i.e. particulars.
  • Provides an objectivist solution to the problem of universals: there are no abstract entities, yet there are objective truths about them.

‘Priority Nominalism‘ is an excellent example of how a book may be concerned with vexed problems and vast topics in a relatively small number of pages. The author goes constantly straight to the point and avoids as much as possible any sort of digression; moreover, the purpose of book and the general view of the author on the matter are frequently resumed and re-proposed, so that the reader should not get lost in the dialectic of the arguments. At every crucial point of the book, a long list of examples facilitates the understanding; the style of writing is simple, direct, and pleasant.

The book is an original and interesting work in two senses: it contains a new solution to the Problem of Universals and represents a new formulation and understanding of the problem itself. For instance, in chapter 1 the reader may find a remarkably clear exposition of the Problem of Universals that is re-formulated in many versions according to fine and profound philosophical distinctions; even the reader who is not so familiar with the relevant literature may read the introductory parts of the book without big efforts.

The basic idea of the proposed view – the combination of grounding with ontological commitment – is simple and at first glance seems to be fruitful. However, the fact that complex arguments and notions are often resumed in the space of few pages may give the false impression that notions such as that of metaphysical grounding are less controversial than they actually are. Perhaps such an impression is given by the fact that while the author is extremely clear regarding the main difficulties and objections to Quine’s criterion of ontogical commitment, he does not seem to do the same with the notion of metaphysical grounding which is definitely a controversial notion; in recent times almost any aspect of grounding has been questioned and put under debate: its modal status (Trogdon 2013), the formal properties of the grounding relation (Rodriguez-Pereyra 2015), its theoretical role and indispensability (Wilson 2014). Given that it is not clear from the book what is the author’s position regarding the main problems of the notion of grounding, possible difficulties for Priority Nominalism may arise if grounding is understood in one way or the other. For the sake of clarity, I will try to sketch a problem for the priority nominalist that may arise from the controversial nature of the notion of grounding. The priority nominalist may be in a predicament engendered by the very strategy he uses to dismiss universals, i.e. the principle of grounded ontological commitment. Consider the sentences ‘a is F‘ and ‘a has the property F‘; let A be the fact expressed by the former and B the fact the latter expresses. Consider now the sentence ‘A grounds B’: the priority nominalist is happy to say that ‘A grounds B’ is true. Let C be the fact expressed by ‘A grounds B’. For the priority nominalist to say that C is a fundamental fact is not an option: for given that B is a fact included in C and given that C is fundamental, then B must also be fundamental (this is a version of what has been known as the collapse problem (Sider 2011)). Yet if B is fundamental, then by the criterion of grounded ontological commitment there are properties. Thus the priority nominalist owes us an argument to the effect that C is not a fundamental fact and is grounded in another fact D not including B. Perhaps C is grounded on A (as suggested by adapting deRosset’s strategy to avoid the collapse problem to the present case (deRosset 2013)); nevertheless, the author does not say anything in the book about facts expressed by sentences of the form ‘X grounds Y’.

In spite of possible controversies regarding Imaguire’s proposal, ‘Priority Nominalist‘ is a highly recommended reading for everyone who is interested in contemporary metaphysics. It is at the same time an excellent introduction to the Problem of Universals, a capillary survey of the recent literature on the issue, and an extremely interesting attempt of vindicating some ontological intuitions that are implicit in Quine’s nominalism.

References

DEROSSET, L. Grounding Explanations, Philosophers’ Imprint 13, 2013. [ Links ]

IMAGUIRE, G. Priority Nominalism, Springer Verlag, 2018. [ Links ]

RODRIGUEZ-PEREYRA, G. Grounding is not a Strict Order, Journal of the American Philosophical Association 1 (3):517-534, 2015. [ Links ]

SIDER, T. Writing the Book of the World, Oxford University Press, 2011. [ Links ]

WILSON, J. No Work for a Theory of Grounding, Inquiry: An Interdisciplinary Journal of Philosophy 57 (5-6):535-579, 2014. [ Links ]

Vincenzo Ciccarelli – University of Campinas, Department of Philosophy, Campinas, SP, Brazil. E-mail: [email protected]

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Paraconsistent Logic: Consistency, Contradiction and Negation – CARNIELLI; CONIGLIO (M)

1 de abril de 2018

CARNIELLI, W.; CONIGLIO, M.. Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science Series. New York: Springer, 2016. Resenha de: ANTUNES, Henrique; CICCARELLI, Vicenzo. Manuscrito, Campinas, v.41 n.2 Apr./June 2018.

The principle of explosion (also known as ex contradictione sequitur quodlibet) states that a pair of contradictory formulas entails any formula whatsoever of the relevant language and, accordingly, any theory regimented on the basis of a logic for which this principle holds (such as classical and intuitionistic logic) will turn out to be trivial if it contains a pair of theorems of the form A and ¬A (where ¬ is a negation operator). A logic is paraconsistent if it rejects the principle of explosion, allowing thus for the possibility of contradictory and yet non-trivial theories.

Among the several paraconsistent logics that have been proposed in the literature, there is a particular family of (propositional and quantified) systems known as Logics of Formal Inconsistency (LFIs), developed and thoroughly studied within the Brazilian tradition on paraconsistency. A distinguishing feature of the LFIs is that although they reject the general validity of the principle of explosion, as all other paraconsistent logics do, they admit a a restrcited version of it known as principle of gentle explosion. This principle asserts that a contradiction that concerns a consistent formula logically entails any other formula of the language. The expression ‘consistent’ here is a generic term susceptible to several alternative interpretations (not necessarily coinciding with non-contradiction), depending on the particular LFI under consideration. Another (related) feature that distinguishes the LFIs from other paraconsistent logics is that they internalize this unspecified notion of consistency inside the object language by means of a unary sentential operator ○ (called ‘consistency operator’ or simply ‘circle’). When prefixed to a formula A, ○ expresses that A is consistent or well behaved, however these expressions are to be interpreted in each particular case.

Paraconsistent Logic: Consistency, Contradiction and Negation, by Walter Carnielli and Marcelo Coniglio, is entirely devoted to the Logics of Formal Inconsistency. The book covers the main achievements in the field in the past 50 years or so, presenting them in a systematic and (to a great extend) self-contained way. Although the book is mostly concerned with particular logical systems, the relations among them, and their corresponding metatheoretical properties, it also sets the basis of a new philosophical interpretation of paraconsistent logics.

The book contains nine chapters, which altogether cover several topics about the LFIs. In Chapter 1 the authors explain the rationales behind paraconsistent logics in general and the LFIs in particular, and discuss the philosophical problems related to paraconsistency under the light of some general issues in the philosophy of logic (such as the nature of logic and the nature of contradictions). It is argued that since there are some real life situations in which contradictions do actually turn up, paraconsistent logics are justified, no matter how those contradictions are interpreted – whether they are seen as concerning reality or knowledge. The chapter also discusses the relation between paracomplete and paraconsistent logics and analyzes some key notions related to paraconsistency, such as consistency, contradiction (and the principle of non-contradiction) and negation.

In Chapter 2 the concept of LFI is precisely defined, as well as other basic technical notions employed throughout the book. A minimal propositional LFI, called mbC, is introduced by means of an axiomatic system. mbC results from positive classical propositional logic by the inclusion of two additional axioms: the principles of excluded middle and gentle explosion – A ∨ A and ○A → (A → (¬A → B), respectively. mbC is then provided with a valuation semantics with respect to which it is proved to be sound and complete. The relations between mbC and classical propositional logic are carefully analyzed. The analysis reveals that mbC can be viewed both as a sublogic and as an extension of classical logic, when these terms are suitably qualified.

Chapter 3 presents several extensions of mbC and analyzes the relations between the notions of consistency/inconsistency and contradictoriness/non-contradictoriness – formally expressed by the formulas ○A/¬○A and A ∧ ¬A/¬(A ∧ ¬A), respectively. As it turns out, although consistency and non-contradictoriness (and inconsistency and contradictoriness) are partially independent in mbC, they may or may not coincide in some of its extensions. In addition, the notion of a C-system is introduced. Despite the complexity of the relevant definition, a C-system simply amounts to an LFI within which the consistency operator is definable in terms of the other connectives of the language. Da Costa’s hierarchy of paraconsistent logics – a family of paradigm examples of C-systems – is briefly presented and explained. The chapter also deals with the important notions of propagation and retro-propagation of the consistency operator.

The first part of Chapter 4 is devoted to the problem of the algebraizability of some LFIs, and the second part discusses some many-valued LFI-systems. In Section 4.1 some preliminary concepts concerning logical matrices are introduced. Section 4.2 contains a Dugundji-style proof of the uncharacterizability by finite matrices of the LFIs presented so far. Section 4.3 contains a proof of the algebraizability of some extension of mbC in the broader sense of Block and Pigozzi. The remaining sections deal separately with different many-valued LFIs, most of which were proposed several decades before the emergence of the concept of Logic of Formal Inconsistency.

Chapter 5 represents a partial detour from the main exposition, for the systems presented therein are not extensions of positive classical propositional logic. The first case considered by the authors is that of intuitionistic logic: more specifically, it is shown how a consistency operator ○ can be defined within Nelson’s logic N4 in terms of a strong negation ~ operator (i.e., ○A ≡ ~(A ∧ ¬A)). Another interesting case covered by the chapter is that of modal logic, where the consistency operator is shown to be interpretable as having a sort of “modal flavor”. In particular, the definition ○A ≡ A → □A can be introduced in normal non-degenerate modal logics. Some systems of fuzzy logic are also analyzed in the chapter. In all of the aforementioned logics, the strategy pursued by the authors consists in defining a consistency operator within the system in question and then showing that it satisfies the general definition of an LFI.

Chapter 6 is devoted to the problem of defining non-deterministic semantics for non-algebraizable systems (even in the broader sense of Block and Pigozzi). It presents three main formal semantics – based, respectively, on F -structures, non-deterministic logical matrices, and possible translations. Of particular interest, especially from a more philosophical point of view, is the so-called possible translation semantics, whose main idea is to translate a given logic into logics whose semantics are well known and deterministic. The relevant notion of translation is that of a mapping preserving logical consequences and the rationale for this approach is the interpretation of a logic as a combination of “possible world views”.

Chapter 7 concerns first-order LFIs. The chapter is mainly devoted to two systems: QmbC, the first-order extension of mbC, and QLFI1. Due to the non-deterministic nature of mbC, a non-standard semantics is defined for its first-order extension: the authors introduce the notion of a Tarskian paraconsistent structure, defined as an ordered pair composed of a Tarskian structure (in the classical sense) together with a non-deterministic valuation. Concerning QLFI1, the approach is twofold: on one hand, it is shown how the language may be interpreted in a suitable Tarskian paraconsistent structure; on the other hand, a different semantics is proposed, given that the propositional fragment of QLFI1 can be characterized by a three-valued matrix. The semantics is represented by a partial structure, defined in a similar way to a classical Tarskian structure, except for the fact that all predicate symbols are interpreted as partial relations. Both QmbC and QLFI1 are proved to be sound and complete with respect to the corresponding semantics. Compactness and Lowenhëim-Skolem theorems are proved for QmbC.

Chapter 8 concerns one of the most straightforward applications of paraconsistent logics: set theory. Nevertheless, the authors’ approach to the subject is substantially different from what has been traditionally done in the field of paraconsistent set theory – namely, to formulate a non-trivial naïve set theory countenancing the unrestricted comprehension principle for sets. The systems presented in the chapter include all of Zermelo-Fraenkel set theory’s axioms (except for the axiom of foundation, which is replaced by a weaker version of it) with an LFI as the underlying logic. Another distinguishing feature of those systems is that they include a consistency predicate for sets whose behavior is governed by a set of additional axioms. Hence, whereas in a propositional LFI the property of consistency applies only to formulas, in the corresponding paraconsistent set theories it applies to both formulas and sets. The main results of the chapter are the derivability adjustment theorem (establishing that any derivation in ZF can be recovered within its paraconsistent counterpart) and a proof of the non-triviality of the strongest system presented in the chapter.

Chapter 9 discusses the significance of contradictions for science, describing some historical paradigm examples where contradictions seem to have played an important role in the development of scientific theories. It also proposes an interpretation of paraconsistent logics according to which they are better viewed as possessing an epistemological, rather than an ontological, character; in a nutshell, this means that they are not supposed to deal primarily with reality and truth (as in the case of classical logic), but with the epistemic notion of evidence. This interpretation is meant to be a more palatable alternative to dialetheism (the thesis that there are true contradictions), since it neither affirms the existence of true contradiction nor rejects classical logic as incoherent – adhering thus to logical pluralism.

One of the main virtues of Paraconsistent Logic: Consistency, Contradiction and Negation is that it keenly highlights the pervasiveness and generality of the notion of logic of formal inconsistency. Firstly, because it shows through the definition of an LFI how several systems of paraconsistent logic proposed in the literature – which at first sight might have appeared to be quite unrelated with one another – can be framed under a single unifying concept. Secondly, because it emphasizes that the definition of an LFI is applicable to systems based on logics of various different kinds, such as classical, intuitionistic, fuzzy, and modal logic. The resulting multiplicity of systems allows for various alternative semantic approaches, which are carefully described in several chapters of the book (e.g., valuation semantics, deterministic and non-deterministic matrices, F-structures, swap structures, possible translations semantics).

The book is mainly devoted to the taxonomy of LFI-systems, leaving little room for a more detailed discussion of the intrinsic properties of each particular system. This is understandable, though, since it is not meant to be a textbook. However, it is possible to use the book as an introductory text on formal paraconsistency by skipping some of the more technical chapters (e.g., a reader merely interested in those LFIs based on positive classical propositional logic may well skip chapters 5, 6 and possibly 8).

Concerning the more philosophical chapters of the book (chapters 1 and 9), the reader might think that the issues discussed therein would have deserved a more extended and rigorous analysis, especially when compared to the painstakingness of the other chapters. In particular, she might find the epistemic interpretation of paraconsistent logics wanting, despite its initial plausibility, this view in not sufficiently argued for. Moreover, specific relations between the epistemic interpretation and the particular features of the LFIs are missing. Nevertheless, this apparent shallowness is presumably due to the fact the purpose of those chapters is not to thoroughly develop a philosophical theory about paraconsistency, but merely to indicate some conceptual possibilities. After all, Paraconsistent Logic is mainly a technical piece of work.

So much for the general considerations. There are two specific points that we think would deserve a more detailed discussion. The first one concerns the cumbersome notation employed in the characterization of the semantics of first-order LFIs (Chapter 7): the strategy adopted by the authors in that chapter consists in extending the (non-deterministic) propositional valuations to the first-order case, combining these with a (classical) Tarskian structure – characterized, as usual, by a non-empty domain together with an interpretation function. The resulting first-order valuations apply thus only to sentences and the notion of truth, as in the propositional case, is not defined in terms of assignments, sequences, or any other technical device usually employed in order to interpreted the variables. The absence of any of these devices leads the authors to locally indicate all the relevant substitutions of individual constants for the free variables of a given formula. In the case of QmbC, for example, the semantic value of a quantified formula ∀xA (under a structure ? and a valuation v) is defined by means of the following clause:

v(∀xA) = 1 iff v(A[x / ā]) = 1, for every a in the domain of ?

where A[x / ā] denotes the result of substituting the constant ā for all free occurrences of x in A, and where the language is supposed to have at least one individual constant ā for each elements a of the domain of ? (that is, the language is supposed to be diagrammatic). At first sight, the use of the notation [x / ā] (and its generalization [x 1,…, x n / ā 1,…, ā n] to multiple simultaneous substitutions) does not seem to compromise readability at all – in fact, they are usually employed in the definition of substitutional semantics for first-order logic. However, matters become much more complicated when it comes to the additional clauses introduced in the definition of v(A) in order to guarantee that the substitution lemma holds for Tarskian paraconsistent structures. One of these clauses, which concerns the negation operator, is formulated as follows:

(sNeg) For every contexts (x→ ; z) and (x→ ; y), for every sequence (a→ ; b→ ) in the domain of ? interpreting (x→ ; y→ ), for every A ∈ L(?) x→ ; z and every t ∈ T(?) x→ ; y→ such that t is free for z in A, if A[z/t] ∈ L(?) x→ ; y→ and c = (t[x→ ; y→ / a→ ; b→ ]) ?“ then:

If v((A[z/t])[x→ ; y→ / a→ ; b→ ]) = v(A[x→ ; z / a→ ; c]) then

v((¬A[z/t])[x→ ; y→ / a→ ; b→ ]) = v(¬A[x→ ; z / a→ ; c])

Without attempting to individually explain every piece of notation above, (sNeg) merely expresses that if the substitution lemma holds for a formula A, then it holds for its negation as well (the introduction of this clause, absent in the definition of classical first-order structures, is necessary given the non-deterministic behavior of the negation operator in mbC). Now, it is quite clear that the reader would probably take several minutes to read and understand (sNeg). Moreover, this situation is not restricted to (sNeg), but it also happens with the similar clause concerning the consistency operator and the formulation and proof of various semantic theorems enunciated in Chapter 7. The notational cumbersomeness of the chapter is further worsened by the introduction of the notion of extended valuation, which assigns a truth value to an arbitrary formula A (not necessarily a sentence) by indicating a sequence of individual constants with respect to which A is to be evaluated. More precisely, if the free variables in A are among x 1,…, x n (abbreviated by x → ) then the truth value of A under the extended valuation v x→ a→ is simply v(A[x 1,…, x n / ā 1,…, ā n]). This notion represents a simile of the notion of satisfaction and is necessary in order to provide an interpretation for the open formulas.

The notation of Chapter 7 could, however, be greatly simplified in the following way: instead of importing the notion of valuation from the corresponding propositional LFI, the authors could well have defined a new notion of valuation which assigns one of the truth values 0 or 1 to each pair (s, A), where s is an assignment of objects of the domain to first-order variables and A is an arbitrary formula (open or closed). All definitions and theorems of the chapter could then be easily adapted according to this strategy, yielding much simpler formulations. In particular, clause (sNeg) above would become:

(sNeg’) Let A be a formula with at least one free variable z and let t be a term free for z in A. Let s be an assignment in a structure ? and let s’ be the assignment which is just like s except that is assigns the interpretation of t under s to the variable z. Then:

If v(s’, A) = v(s, A[z / t]) then v(s’, ¬A) = v(s, ¬A[z / t])

In addition to the evident simplicity of this new formulation, it is worth mentioning that since the notion of valuation above applies to any formula whatsoever of the language (open or closed), it is unnecessary to introduce extended valuations, resulting in a significant conceptual simplification.

Our second criticism concerns the paraconsistent set theories of Chapter 8. In general, the main motivation for a paraconsistent set theory is to recover the intuitive notion of set codified in the unrestricted principle of comprehension – i.e., the idea that every property P determines a set of all and only those objects having P. Of course, this can only be achieved by renouncing to classical logic, since that principle classically entails the existence of contradictory sets (e.g., Russell’s set, universal set, etc.). On the other hand, classical set theories (such as ZF) maintain classical logic at the cost of imposing what seems to be ad hoc restrictions to the comprehension principle and countenancing additional principles whose justification seems also ad hoc. Hence, paraconsistent and classical set theories are symmetrically opposed to one another: what the former tries to achieve (i.e., preserve the intuitive notion of set) is given up by the latter, and what the latter preserves (i.e., classical logic) the former revises.

Nevertheless, the approach to paraconsistent set theory adopted by the authors diverges significantly from these two trends. Firstly, because the attempt to recover the intuitive notion of set codified in the principle of comprehension is explicitly given up once they opt for ZF-like axiomatizations of their theories – ruling out well-known inconsistent collections from the outset. Secondly, given that those theories are variations of ZF based on one or another LFI, the revision of the underlying logical theory is achieved by extending classical logic, rather than renouncing to it. In fact, each of the set theories of Chapter 8 is equivalent to ZF under the assumption that all sets enjoy the property of consistency.

This particular take on paraconsistency may leave the reader wondering what is the point of having a paraconsistent set theory that does not explicitly countenance contradictory collections (‘Why not just stick with ZF?’, she might ask.). The book does not provide an explicit answer to this question, though. However, it would not be difficult to imagine a scenario in which the systems of Chapter 8 would be vindicated: suppose that ZF is someday shown to be inconsistent. Under this circumstance, any of those systems could be used to preserve the strength of ZF while avoiding its triviality. Even though a paraconsistent set theory of this kind may turn out to be fruitful, its fruitfulness turns on an unlikely possibility, though – namely, that ZF could be inconsistent. In view of such a possible application, we suggest that the approach to paraconsistent set theory adopted by the authors is aimed at presenting alternative versions of ZF that are more “cautious” in the sense that they would be able to withstand contradictions, should they ever arise within ZF. For this reason, we believe that those theories should not be viewed as competitors to classical set theories, but rather as interesting and possibly useful variations of it, whose mathematical properties are nonetheless worth investigating.

Paraconsistent Logic: Consistency, Contradiction and Negation is a comprehensive text on the LFIs and fulfills an important gap in the literature on paraconsistency. A huge amount of significant results is presented for the first time in a single text, providing the reader with an extensive survey of the research in the area. Moreover, the content of the book is not limited to the achievements of the so-called Brazilian school of logic, but also encompasses contributions coming from other areas and research groups. As a result, it is highly recommended for everyone interested in both the formal and the philosophical aspects of paraconsistency, including mathematicians, linguistics, computer scientists, and philosophers of language, mathematics and science.

References

CARNIELLI, W., CONIGLIO, M. Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science Series. New York: Springer, 2016. [ Links ]

Henrique Antunes – State University of Campinas, Department of Philosophy, Campinas, SP, Brazil, antunes. E-mail: [email protected]

Vincenzo Ciccarelli – State University of Campinas, Department of Philosophy, Campinas, SP, Brazil. E-mail: [email protected]

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