Dag Prawitz's theory of grounds

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AIX-MARSEILLE UNIVERSITY

“SAPIENZA” UNIVERSITY OF ROME

ÉCOLE DOCTORALE 356

CENTRE GILLES GASTON GRANGER UMR 7304

Thèse présentée pour obtenir le grade universitaire de docteur

Discipline : Philosophie

Spécialité : Logique

Antonio PICCOLOMINI D'ARAGONA

Dag Prawitz's theory of grounds

Soutenue le 16/12/2019 devant le jury composé de :

Enrico MORICONI Pisa University Rapporteur Peter SCHROEDER-HEISTER Tübingen University Rapporteur Myriam QUATRINI Aix-Marseille University Examinateur Göran SUNDHOLM Leiden University Examinateur Luca TRANCHINI Tübingen University Examinateur Cesare COZZO “Sapienza” University of Rome Directeur de thèse Gabriella CROCCO Aix-Marseille University Directeur de thèse

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Résumé

Dans la récente théorie des grounds, Prawitz développe ses investigations sé-mantiques dans la direction d’une analyse, à la fois philosophique et formelle, de l’origine et de la nature du pouvoir que les inférences valides, ainsi que les démonstrations où ces inférences figurent, exercent sur des agents engagés dans l’activité déductive ; à savoir, le pouvoir d’obliger épistémiquement à accepter les conclusions, si l’on en a accepté les prémisses ou les hypothèses. Il s’agit de la plus ancienne des questions à laquelle s’intéresse la logique, depuis sa naissance avec Aristote - on pourrait dire, presque de la raison d’être de cette discipline.

La notion de base est celle de ground. Un ground, grosso modo, est ce dont on est en possession lorsqu’on est justifié à affirmer un certain énoncé. Les grounds peuvent être construits en accomplissant des opérations qui permet-tent le passage d’un état de justification à un autre ; il s’agit, donc, d’objets abstraits mais épistémiques, qui contiennent des opérations abstraites mais calculables. Un acte d’inférence consiste à l’application d’une opération des grounds pour les prémisses aux grounds pour la conclusion, acte qui sera valide si l’opération accomplie produit effectivement des grounds pour la conclusion quand appliquée aux grounds pour les prémisses. Finalement, une démonstration est une concaténation d’inférences valides.

Relativement à son objectif de fond, la théorie des grounds présente des avancements indubitables par rapport à la précédente approche de Prawitz, la proof-theoretic semantics. En particulier, la théorie des grounds offre une définition du concept d’inférence valide en vertu de laquelle il devient possi-ble de faire dépendre la contrainte épistémique des démonstrations de celle des inférences valides dont ces démonstrations se composent. Dans la proof-theoretic semantics, la notion d’inférence valide dépend de celle de démon-stration, alors que la caractérisation suggérée apparaît difficile - sinon impos-sible.

Mais la théorie des grounds et la proof-theoretic semantics partagent un problème ; dans l’une comme dans l’autre, inférences valides et démon-strations pourraient être telles qu’il est impossible, pour des agents qui les

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utilisent, de reconnaître le fait qu’elles justifient leur conclusion. Si une inférence valide ne peut pas être reconnue comme valide, et si une démon-strations ne peut pas être reconnue comme une démonstration, aucun agent qui les utilisent ne sera obligé à accepter la conclusion par le seul fait de les avoir accomplies. Il doit aussi reconnaître que ce qu’il a fait sert à fonder épistémiquement les résultats auxquels il vise.

Nous allons développer le cadre formel de la proposition de Prawitz en introduisant, d’un côté, un « univers » de grounds et opérations sur grounds et, de l’autre, des langages formels de grounding dont les termes dénotent grounds ou opérations sur grounds. Tout langage de grounding doit être indéfiniment ouvert à l’ajout des nouvelles ressources expressives. De plus, en raison du théorème d’incomplétude de Gödel, il ne peut pas exister un langage de grounding clôt capable de décrire tous les grounds ou toutes les opérations sur grounds. Par conséquent, nous allons également introduire une notion d’expansion de langage de grounding, ce qui nous permettra de créer une hiérarchie de langages et de fonctions de dénotation. Ainsi, il deviendra possible de décrire propriétés et résultats dont les langages de grounding jouissent, à la fois singulièrement et par rapport à leurs expansions.

À côté des langages de grounding, nous proposerons aussi des systèmes de grounding, à l’aide desquels démontrer des propriétés significatives des termes des langages de grounding (par exemple le fait qu’un terme dénote un ground ou une opération sur grounds, ou le fait que deux termes dénotent le même ground ou la même opération sur grounds), ou des composantes syntaxiques qui figurent dans ces termes (par exemple le fait qu’un certain symbol opéra-tionnel est défini de façon qu’il dénote une opération sur grounds avec un certain domaine et un certain co-domaine, ou qu’il est traduisible dans les symboles opérationnels d’un sous-langage du langage auquel il appartient).

Finalement, nous allons aborder deux questions concernant langages et systèmes. Tout d’abord, celle de la complétude de la logique intuitionniste par rapport à la théorie des grounds ; en d’autres mots, nous discuterons la conjecture de Prawitz dans le cadre formel que nous avons proposé. En second lieu, nous poursuivrons une analyse du problème de reconnaissabilité déjà évoqué, à la lumière des acquisitions formelles permises par langages et systèmes de grounding.

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Riassunto

Nella recente teoria dei grounds, Prawitz sviluppa le sue indagini semantiche nella direzione di un’analisi, al contempo filosofica e formale, dell’origine e della natura di quella speciale forza che le inferenze valide, e le dimostrazioni in cui tali inferenze sono coinvolte, esercitano su agenti impegnati nell’attività deduttiva: la forza di costringere epistemicamente ad accettare le conclusioni dell’inferenza o della dimostrazione, se se ne sono accettate le premesse o le ipotesi. Si tratta della più antica delle questioni di cui la logica si interessa fin dai tempi della sua nascita con Aristotele - diremmo quasi della raison d’être di tale disciplina.

La nozione di fondo è quella di ground. Un ground è, grosso modo, ciò di cui si è in possesso quando si è giustificati nell’asserire un certo enunciato. I grounds possono essere costruiti compiendo operazioni che consentano il passaggio da uno stato di giustificazione all’altro; si tratta, perciò, di oggetti astratti ma epistemici, in cui sono coinvolte operazioni astratte ma com-putabili. Un atto inferenziale consiste nell’applicazione di un’operazione dai grounds per le premesse ai grounds per la conclusione, atto che risulterà legittimo quando l’operazione compiuta è effettivamente tale da produrre grounds per la conclusione quando applicata a grounds per le premesse. Una dimostrazione, infine, è una concatenazione di inferenze valide.

Relativamente al suo obiettivo di fondo, la teoria dei grounds presenta indubbi avanzamenti rispetto al precedente approccio di Prawitz, la proof-theoretic semantics. In particolare, la teoria dei grounds offre una definizione della nozione di inferenza valida in virtù della quale diventa possibile far dipendere la costrizione epistemica esercitata delle dimostrazioni da quella esercitata dalle inferenze valide di cui le dimostrazioni si compongono. Nella proof-theoretic semantics, al contrario, la nozione di inferenza valida dipende da quella di dimostrazione, sicché la caratterizzazione suggerita risulta diffi-cile - se non impossibile.

Ma teoria dei grounds e proof-theoretic semantics condividono un prob-lema; nell’una come nell’altra, inferenze valide e dimostrazioni potrebbero essere tali da risultare impossibile, ad agenti che ne facciano uso nella

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conc-reta pratica deduttiva, un riconoscimento del fatto che esse giustificano la loro conclusione. Se un’inferenza valida non può essere riconosciuta come valida, e se una dimostrazione non può essere riconosciuta come dimostrazione, nes-sun agente che ne faccia uso si sentirà costretto ad accettare la conclusione per il solo fatto di averle compiute. Egli deve anche riconoscere che quanto fatto serve a sostanziare epistemicamente i risultati cui egli mira.

Il quadro formale della proposta di Prawitz sarà da noi articolato intro-ducendo un "universo" di grounds ed operazioni su grounds e, poi, linguaggi formali di grounding i cui termini denotano grounds od operazioni su grounds. Ogni linguaggio di grounding deve essere indefinitamente aperto all’aggiunta di nuove risorse espressive né, a causa dell’incompletezza di Gödel, può es-istere un linguaggio di grounding chiuso capace di descrivere tutti i possibili grounds o tutte le possibili operazioni su grounds. Pertanto, introdurremo an-che una nozione di espansione di linguaggio di grounding, il an-che ci consentirà di creare una gerarchia di linguaggi e di funzioni di denotazione. Diventerà così possibile descrivere proprietà e risultati di cui i linguaggi di grounding godono, tanto singolarmente, quanto in relazione alle loro espansioni.

Accanto ai linguaggi di grounding, proporremo anche sistemi di ing, in cui dimostrare proprietà rilevanti dei termini dei linguaggi di ground-ing (ad esempio, il fatto che un termine denoti un ground o un’operazione su grounds, o che due termini denotino lo stesso ground o la stessa operazione su grounds), o di alcune componenti sintattiche in essi coinvolte (ad esempio, che un certo simbolo funzionale è definito in modo da denotare un’operazione su grounds con un certo dominio ed un certo co-dominio, o in modo da es-sere riscrivibile in termini di simboli operazionali di un sotto-linguaggio del linguaggio cui esso appartiene).

Ci occuperemo infine di due questioni relative a linguaggi e sistemi. In-nanzitutto, la questione della completezza della logica intuizionista rispetto alla teoria dei grounds; in altre parole, discuteremo una riformulazione della congettura di Prawitz nel quadro formale da noi proposto. In secondo luogo, perseguiremo una disamina del succitato problema di riconoscibilità alla luce delle acquisizioni formali consentite dai linguaggi e dai sistemi di grounding.

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Acknowledgments

The drafting of this thesis has involved quite difficult moments. However, I had the good fortune to meet, along a sometimes uneven path, persons who have wanted to honour me with their indispensable help, or their invaluable advice.

Two of these are my advisors, Gabriella Crocco and Cesare Cozzo. Their constant presence, their readiness to listen, and above all the criticisms and corrections that, with ever-profound humanity, they have given me in long and frequent interviews, are as important as a doctoral student may ask. Thanks to them, I have never felt alone on a long and often impervious research path. I will be forever in their debt in my future researches.

It is a little more difficult to find the right words to thank Dag Prawitz, because my gratitude towards him is enormous: as a student of logic, for the fundamental contributions he has given to this discipline, but also, in the small field of my personal research, for the kindness with which he has always answered my questions, for the attention that he has devoted to them, and above all for the wisdom and rigour with which, through his answers, I have been able then to nurture my thoughts. All this has been so important to explain alone the sense of the entire doctorate.

I cannot fail to give a special thanks to Peter Schroeder-Heister, Göran Sundholm, Luca Tranchini and Gabriele Usberti, for the patience with which they have listened to my many questions, and for the kindness and warmth with which they were able to answer. Without their works, without their research, without their help, I would know much less than what I know today and, above all, I would not be aware of the importance and essentiality of certain issues. They gave me, not only a considerable amount of their time, but above all the essential knowledge to find the key of the problem in the tangled moments of my research.

A special thanks goes to Myriam Quatrini for the valuable advice and the fruitful remarks that have guided, not only part of my survey on the theory of grounds, but also a parallel project, concerning a possible link between the theory of grounds and Girard’s Ludics. A project that I have had the pleasure

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and honor of conceiving and developing with Davide Catta, a friend as well as a colleague. I thank him too, obviously, for his enthusiasm, support, and ability to keep things light.

I am also deeply and sincerely grateful to Enrico Moriconi. I have had the pleasure to meet him for the first time during the FilMat 2018 Workshop in Mussomeli. I asked him some boring questions while we were climbing a hill up, to visit a beautiful medieval castle. He patiently answered, with the same kindness (and perhaps the same patience) as the one he is now honoring me with, in accepting to review my dissertation and being part of the examining board.

I owe a lot to many other people, so that I dedicate to each one of them the most heartfelt and dear of my thoughts; these people are Miloš Adžić, Claudio Bernardi, Constantin Brîncuş, Paola Cantù, Carlo Cellucci, Nissim Francez, Ansten Klev, Emiliano Ippoliti, Francesco Montesi, Peter Pagin and Paolo Pistone.

Antonio Tuzzi has remarkably helped me, to say the least, for the trans-lation of the thesis, originally written in Italian. He did it with his usual kindness, and with the affection he has always shown to me – totally re-turned. I think this is the right time to confess to him how important and irreplaceable it has been his presence in my curse of study, and even more so in my life.

I am also indebted to Simona de Leoni, who helped me to revise some articles I wrote during my PhD. Simona is one of the most passionate and enthusiastic persons I have ever met, and for this reason she is precious to me. She has a deep, sincere and witty interest in others, a quality that makes her capable to convey and give rise to positive thoughts. Many nice memories bind me to Simona since I was a child, and I am very happy she has been there again.

Let me also thank the members of the Center Gilles Gaston Granger (formerly CEPERC). Each of them in his/her own way has meant that I created a bond by now unbreakable with Aix-en-Provence and Marseille, and more generally with the French civilization and culture. I cannot fail to thank also the A*MIDEX foundation, that gave me a beautiful opportunity to carry out my research, providing me with all the means and comfort that a student may dream of.

I think I must point out my debt to Nicola Grana; ten years ago, he led my first steps in the world of logic in Naples, making me passionate with his engaging lectures, and accompanying me little by little to my bachelor degree in philosophy.

The memory that shows no sign of dying out, and that burns more and more every day alive, is finally my very deep thanks to Kosta Došen. In

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the uncertain future of the researcher I would like to be, his smile and his indispensable research are the extraordinary example of how humanity and rigor cannot live but next to each another.

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Introduction

According to a rather widespread interpretation, logic is to be understood as the science of correct reasoning. Far from being a definition, however, this expression is of a mere indicative nature. It raises more questions than the ones it answers. Even leaving out the problematic issue on what we can and should consider as science, it is far from clear what a reasoning is, and even more what a correct reasoning is.

From a general point of view, we can follow a long and well-established tradition according to which a reasoning is a concatenation of passages from certain premises to certain conclusions, called inferences. This position is for example present and aware in Descartes who, in a well-known passage of his Rules for the direction of the mind, equates reasoning to

a continuous and uninterrupted movement of thought in which each individual proposition is clearly intuited. This is similar to the way in which we know that the last link in a long chain is connected to the first: even if we cannot take in at one glance all the intermediate links on which the connection depends, we can have knowledge of the connection provided we survey the links one after the other, and keep in mind that each link from the first to the last is attached to its neighbour. (Descartes 1985, 15)

It is also obvious, however, that this idea can be further declined in many different alternative ways; a more specific determination will vary depending on the point of view adopted on the nature of premises and conclusions, and on the passage itself, as well as on the basis of which of these factors are considered really relevant to the logical investigation.

Also in relation to the narrower notion of correct reasoning, fortunately we have a basic intuition to which we can hold on, even if also such intuition is, in the final analysis, partial and liable to many different ramifications. It can be illustrated with the famous words that Aristotle binds to the key notion of the system – the first in its kind – developed in the Organon:

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the syllogism is a discourse in which, certain things being laid down, something follows of necessity from them. (Aristotle 1949, 287)

As a paradigm of correct reasoning, a syllogism therefore has a feature that Aristotle emphasizes: necessity. On the other hand, if the question "what did Aristotle mean by necessity?" concerns the history of logic or the history of philosophy, instead, it is very meaningful for what concerns us the more general question "what kind of necessity do we refer to when we talk about correct reasoning?". We could think that logic, in the cloak of the name of science, has a unanimously shared vision on the nature of necessity. But this is by no means the case.

It would obviously be impossible, in the restricted framework of this in-troduction, to review if only in general the multiple reflections which, over the centuries, have concerned the notion of necessity. Nor would such review be of any use with respect to the theme of our work: Dag Prawitz’s theory of grounds. On the other hand, necessity plays in Prawitz, and it will play for us, a decisive role to say the least. The theory of grounds, in fact, as-sumes the shape of an attempt at the same time philosophical and formal to respond, with sufficient and satisfactory articulation, to what is perhaps the most original among the questions of logic: how and why do some inferences, commonly called deductively correct, have the epistemic power to force us to accept their conclusion, assuming we have epistemically accepted their premises? If we adopt the aforementioned point of view, that an argument is a chain of inferences, to answer this crucial question also means explaining how and why deductively correct reasoning can exert that force that, since very distant times, has made it the main source of conclusive knowledge, and a source of irrefutable epistemic certainty for human beings.

In focusing on the power of epistemic binding of deductively correct infer-ences and reasoning, Prawitz emphasizes a very particular kind of necessity. In a paper that is in many ways a watershed between his previous research and the theory of grounds, the Swedish logician uses the fitting expression of necessity of thought, specifying that with it he means the circumstance in which

one is committed to holding α true, having accepted the truth of the sentences of Γ; one is compelled to hold α true, given that one holds all the sentences of Γ true; on pain of irrationality, one must accept the truth of α, having accepted the truth of the sentences of Γ. (Prawitz 2005, 677)

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This kind of necessity is diametrically different from another, equally well known and perhaps more practiced in logic, which is based on the notion of possible world and according to which necessity means truth in all possible worlds. The fact that something is true in all possible worlds is obviously alien to issues relating to knowledge: it could be a circumstance of which we are simply unaware or, even if we are aware of it, we might not see why it occurs. In the latter case, it is an occurrence that we must accept not by virtue of an epistemic binding but, so to speak, of a mere factual statement. However problematic, the notion of possible world is often used to sub-stantiate the idea that model-theory, the child of the pioneer works of Bolzano and Tarski, as well as for many years the standard formal semantics of con-temporary mathematical logic, actually captures that notion of necessity that characterizes, giving them a decisive modal structure, the concepts of validity and logical consequence. However, many criticisms have been raised against the thesis that model-theory contains modal ingredients of any kind and, given in such general terms, the question is still widely debated. What we can say with certainty, is that the modality captured by model-theory, although present, is undoubtedly not of the type Prawitz is interested in, that is, not of an epistemic type.

It is therefore not a case if, starting from some important results in proof theory, Prawitz has developed a formal semantics alternative to model-theory, today known as proof-theoretic semantics. Proof-theoretic semantics offers definitions of the concepts of validity and logical consequence that, in accor-dance with necessity of thought, replaces the notion of truth, as a semantic core, with those of proof or valid argument. The idea would seem obvious: the epistemic binding that one experiences in necessity of thought can occur only when we are in possession of a proof or a valid argument.

Proof-theoretic semantics is therefore based on mathematically rigorous characterizations of the concepts of proof and valid argument, from which to move forward to reach the more general semantic definitions. This happens, it seems to us, by following essentially three lines of research: the BHK se-mantics for first-order intuitionist logic, which originated with Heyting, some observations by Gentzen on the relationship between introduction and elim-ination rules for first-order logical constants in natural deduction systems, and Dummett’s investigations in theory of meaning, with a particular focus on a verificationist theory of meaning. These three sources are obviously mutually linked, and in turn, each of them, or all three jointly, are bound to reflections of another type, more or less explicitly recognized. By setting itself at the crossroads of so many suggestions, and in a sense harmonizing them all, proof-theoretic semantics is therefore of an interest more widely philo-sophical, which goes beyond the formal results, albeit fundamental, that it

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allows to reach.

However, the proof-theoretic approach is not entirely free of problems, of which the main one, related to the issues mentioned above, concerns precisely the possibility of accounting for the epistemic power of deductively correct inferences and reasoning. If we intend to explain how and why a correct reasoning can force epistemically to accept its conclusions, thus providing justification towards them, there seems to be no other choice but to make this force depend on that, of analogous nature, enjoyed by the inferential passages occurring in the reasoning itself. To make this explanation work, however, the notion of correct inference must conceptually take priority over that of correct reasoning, thus following the same explanatory order that makes reasoning a chain of inferences. In proof-theoretic semantics, though, the notion of correct inference is defined in terms of proofs or valid arguments, by stating that an inference is correct when it preserves provability, or the validity of the structure in which it occurs. It is in this sense not surprising that, with the theory of grounds, Prawitz renounces a characterization of this type, returning to the correct inferences their pivotal role.

This inversion of the natural relationship among the notions of correct inference, proof and valid argument, refers to another point on which the theory of the grounds offers, in our opinion, a doubtless progress. To be forced to validate the correctness of judgments or assertions that convey or express knowledge is something we experience when, for example, we follow the steps made by someone who is proving something, or when we personally perform a proof act. The epistemic binding is something we "feel", and of which we are aware; in this "feeling", in this being aware, however, we are not in a condition of passivity, but, on the contrary, in order to accomplish this experience we must do something, carry out appropriate acts. In the words of Cozzo (Cozzo 2015), necessity of thought has a phenomenal character, and assumes for us the form of an active experience.

Strictly speaking, therefore, correct reasoning leads to a state of epistemic justification, but it is not itself, as such, an epistemic justification; the proof act is what by virtue of which the binding manifests itself, acting upon us, but it is the result of this act, what the act leads to, that qualifies as the condition of epistemic success. Subtle, but crucial in the reconstruction of the epistemic force of correct inferences and reasoning, this distinction can be summed up as a dichotomy between proof-objects, on the one hand, and acts, on the other (see mainly Sundholm 1998). In this regard, proof-theoretic semantics seems ambiguous, since it deals with proofs and valid arguments at the same time as objects and as acts, while much more precise is the theory of grounds, in which Prawitz distinguishes between states of justification – reifying, he actually speaks of objects, called precisely grounds,

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of which we are in possession when we are justified in judging or asserting – and acts that enable us to enter a state of justification – namely, proofs that produce grounds.

In the theory of grounds, as just mentioned, "ground" is the expression used by Prawitz to indicate what we have when we are epistemically justified. Gounds are objects that reify states of epistemic success, and are obtained by performing inferential acts – single, or concatenated so as to form a proof. The most original trait of the theory of grounds is perhaps the reconstruc-tion that Prawitz offers of how an inferential act can produce grounds and, therefore, of what an inferential act actually is. In the commonly accepted meaning, which we have also referred to at the beginning of this introduction, an inference is simply identified by certain premises and certain conclusions, since it appears as a passage from the ones to the others. In the light of convincing arguments, however, Prawitz believes that this reconstruction is far too poor to allow an adequate explanation of the phenomenon of epis-temic compulsion. He adds to it the crucial idea that to make an inference means applying an operation that transforms constructively grounds for the premises into grounds for the conclusion, and that therefore an inference, in addition to premises and conclusions, must also involve an operation of this kind. A deductively correct inference is therefore an inference the oper-ation of which actually returns grounds for the conclusion, when applied to grounds for the premises.

Joining the aforementioned distinction between objects and proof-acts, and merging into the notion of deductively correct inference, this in-novative way to characterize inferential acts makes the theory of grounds a solid apparatus for a rigorous explanation, not circular and philosophi-cally meaningful, of the relationship between correct inferences and correct reasoning, as well as of the epistemic power of both. Both proof-theoretic semantics and the theory of grounds attribute a crucial role to the so-called canonical cases, primitive in contrast to the non-canonical ones, that on the contrary need justification. However, while in proof-theoretic semantics the objects themselves, as not distinguished from the acts, can be canonical or non-canonical, in the theory of grounds the distinction applies only to the acts, whereas the grounds are specified solely by virtue of primitive opera-tions, by simple induction on the complexity of the formulas for which they are grounds. From this perspective, the fact that the deductive correctness of the inferential acts is explained on the base of objects of this type, and not in relation to the acts in which these inferences occur, allows to overcome some problems of circularity, from which analogous attempts in the setup of proof-theoretic semantics suffer.

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gro-unds has also, in our opinion, an important role in relation to a problem that, this time, the theory of grounds shares in full with proof-theoretic se-mantics. If a proof or valid argument must give justification, it seems we cannot help but request that it be recognizable that they have this power. In the same way, if the possession of a ground coincides with being in a state of justification, we need to be able to recognize that the inference that gives possession of that ground is deductively correct. Otherwise, the fact that we have made correct deductions would correspond to the possession of ab-stract objects, so that we could be totally unaware of their epistemic weight; no justification, let alone any epistemic binding, seems to be possible under such circumstances. Similar issues are often raised in relation to the clauses for the implication and the universal quantification in BHK semantics. As regards the possibility itself of arguing that certain constructs, ultimately formal, are capable of respecting the wishes concerning epistemic justifica-tion and binding, the point is obviously crucial also and above all in Prawitz. Unfortunately, a precise delineation of how this recognizability can occur, assuming that it actually can, remains decidedly elusive. A good starting point could be to clarify what the recognition in question is, but already here positions are disparate and discordant, going from "strong" readings, in terms of decidability, to more "weak" readings, that involve pragmatic elements.

In the two frameworks, proof-theoretic semantics and theory of grounds, the problem of recognizability arises for the non-canonical cases; since they are not primitive in the explanation of meaning, namely not primitive in the determination of what counts as justification for judgments or assertions about propositions or sentences of different logical form, they must be justi-fied. And we need to be able to recognize that the justification given works, fulfills the task requested. In this perspective, the ground-theoretic idea of inferences as applications of operations on grounds allows a minimum, albeit limited, progress. In certain specific circumstances – when the inference is performed starting from premises for which we already have grounds – what an agent is in possession of at the end of the inference is not something that can be canonical or non-canonical, but an object defined only by primitive operations. All this because to perform the inference means for the agent ap-plying an operation, that is "to compute it" on the grounds of which he/she is already in possession, so as to get a ground for the conclusion. The same cannot be said of proofs or valid arguments in proof-theoretic semantics, where to make an inference only means to ensure that a conclusion follows certain premises, and where the structure resulting from the deductive activ-ity is therefore susceptible to the canonical/non-canonical distinction; when the structure is non-canonical, the fact that it is valid will be visible only

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after the subsequent application of justifications that show how to reduce it to a canonical structure. Nor, due to structural reasons, can the application of such justifications be understood as simultaneous to the completion of the inferential passages, as instead happens in the theory of grounds.

However, the theory of grounds still suffers from a problem of recogniz-ability of the good position of certain equations. Since inferences are intended as applications of operations on grounds, non-canonical inferences are to be understood as applications of non-primitive operations on grounds. The lat-ter are in turn defined, not only by a domain and a codomain, but also by an equation that shows how the operation behaves. The equation provides a method of "computation", or "transformation", through which, when ex-ecuted, it is constructively possible to pass from grounds for the premises – arguments of the operations – to grounds for the conclusion – the value of the operation on those arguments. The equations are a "functional" ver-sion of the reduction or justification procedures of non-canonical inferences in proof-theoretic semantics. In this sense, we could say that the theory of grounds proposes a sort of "internalization" of such procedures to the process of construction of the argument structure. In appropriate circumstances, to prove means "computing" reductions of non-canonical structures.

Obviously, in his writings on the theory of grounds, Prawitz does not limit himself to state the ideas that, in a very general line, we have been listing so far. On the contrary, he indicates their formal articulation, which in turn seems to go in two distinct directions, albeit connected. The first consists of a more accurate characterization of grounds and operations on grounds as objects typed on formulaa of a background language. The typing establishes a link between the object and the judgment or assertion for which that object constitutes justification. The second aims at the development of a formal language, that includes terms denoting the aforementioned objects, and formulas that indicate their main properties. The resulting picture seems to come close to approaches of similar intuitionistic or, more generally, con-structivist inspiration, such as the Kreisel-Goodman theory of constructions (Kreisel 1962, 1965; Goodman 1968, 1970, 1973) or Martin-Löf intuition-istic type theory (Martin-Löf 1984). It is not surprising, then, that both the typing of objects, and the formal languages the terms of which denote such objects, come near the analogues in the typed λ-calculus; the general directives of the theory of grounds can therefore be taken into account also, and perhaps mainly, in the perspective of the formulas-as-types conception, a cornerstone of the Curry-Howard isomorphism (Howard 1982).

Despite these precious suggestions, however, the more formal side is, in the writings that Prawitz has so far dedicated to the subject, in an only embryonic state. On the other hand, the importance of the demand and

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of the basic objectives of the theory of grounds, the wide range of theories, traditions and hints to which it is connected, both in the philosophical field and with respect to contemporary mathematical logic, and last but not least, the progress it allows in many respects, are, in our opinion, more than enough reasons to reveal the need for an in-depth development of the "technical" area of the theory. Of course, this in-depth analysis does not have only a purpose of systematization, since it allows also to better understand some of the basic assumptions of the theory itself, to enlighten some of its not secondary philosophical aspects, to draw non-trivial consequences, and to indicate further possible theorizations.

In this context, our proposal will move along two main directives. First, the delineation of a class of formal languages of grounding, and the related definition of a denotation function that allows to associate terms with gro-unds and operations on grogro-unds. Secondly, the development of a class of formal systems of grounding, which allow to establish, deductively, relevant properties, expressed by means of appropriate formulas, of the terms and of some of the symbols of the languages of grounding. In either case, these primary intentions will be accompanied by a certain refinement of the anal-ysis, with the aim of perfecting general definitions, by introducing concepts and proving results that allow a more specific application of the definitions themselves. Far from claiming to be exhaustive, our contribution is to be understood as the first draft of a fully "mathematized" theory of grounds, so as to highlight fundamental characteristics which, in our opinion, should ap-ply in a complete formalization. Starting from this core, it will also become clearer how and where further advancements could differ, depending on the different needs, in specific choices or alternative characterizations.

The work is divided into three parts. The first illustrates the theoretical background, with its results and its open problems, on which the theory of grounds is based, or to which, more or less directly, it is linked. The second part, which corresponds entirely to the fourth chapter, aims at a reconstruc-tion of the theory of grounds as it has so far been presented by Prawitz, showing the progresses it allows, the problems shared with the background illustrated in the first part, and the points liable to further refinement. These points are finally taken into account in the third part, with a view to a first attempt at formalization, as systematic and pregnant as possible.

The first part is in turn divided into three chapters. The first raises the fundamental problem of the theory of grounds, namely the explanation of the power of epistemic compulsion of deductively correct inferences and rea-soning, providing a clarification of the fundamental concepts involved – i.e. inference, proof, premises/conclusion, and so on. The second chapter, start-ing from the consolidated bond that, in contemporary mathematical logic,

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usually exists between the concepts of (logically) valid inference and (logical) consequence, explains the main reasons for the inadequacy of model-theory in capturing a notion of necessity epistemically understood. After that, we will introduce the alternative Prawitz’s proof-theoretic semantics, following the different configurations it has taken over the years, and focus on some of its weak points from which it seems to suffer compared to a satisfactory expla-nation of the epistemic force of deduction. The third Chapter of the first part offers a general description of two theories conceptually and formally similar to the theory of grounds – Kreisel-Goodman theory of constructions (Kreisel 1962, 1965; Goodman 1968, 1970, 1973) and Martin-Löf’s intuitionistic type theory (Martin-Löf 1984) - and discusses some of Prawitz’s observations on these theories, concerning issues that then will be crucial in the theory of grounds.

The third part is again divided into three chapters, going from the fifth to the seventh. In the fifth Chapter we introduce, related to the notions of first-order logical language and atomic base on such a language, a class of languages of grounding that include only terms; the class is developed by ex-pansions of a core language which contains operational symbols correspond-ing to primitive operations on grounds, and the expansions are classified, according to different properties, in relation to a denotation function that associates grounds or operations on grounds to the terms of the language. In the following chapter, languages of grounding are enriched with predicates that allow us to construct formulas which in turn allow to express properties of the terms and of some of the symbols of the alphabet. The provability of these properties is reached then by means of a class of formal systems of grounding, each equipped with a set of rules identified starting from the same principles that had led to the characterization of the denotation functions. Finally, in the seventh and final Chapter two questions are discussed: first, the completeness of first-order intuitionist logic with respect to the apparatus developed in the two previous chapters, and this in the form of a conjecture - in a "weak" and "strong" version – which transposes the conjecture elabo-rated by Prawitz (Prawitz 1973) for his proof-theoretic semantics; the second discusses the aforementioned problem of recognizability, as it appears in the theory of grounds, in relation to the theme of the general form of the equa-tions that set the behaviour of non-primitive operaequa-tions on grounds, so as to define them.

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Part I

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Chapter 1 Inferences and proofs

1.1 Nature of inferences

Our mental activity is characterized by a series of processes and acts through which, by elaborating information, knowledge, thoughts and beliefs, we pass to other information, other knowledge, other thoughts and other beliefs. These operations, as well as their purposes, can concern several levels of awareness and voluntariness, varying degrees of complexity, a different use of time, different memory resources, thus implying a greater or lesser force in the results obtained. Frequently, an absolute unconsciousness is accom-panied by unintentionality and automatism, and immediacy, rapidity and relative simplicity produce/generate uncertain or fallible acquisitions. At the opposite extreme, the completion of operations can be totally conscious and voluntary, often complex, long and tiring, and lead to a state of which the epistemic content seems conclusive and not refutable. Obviously, be-tween these two poles there are many intermediate stages, in which various elements are combined in a partial and heterogeneous way.

Linguistic practices, on the other hand, are largely, if not essentially, aimed at the exchange of information and knowledge. The linguistic her-itage, which a more or less extensive community uses for communication purposes, is, it seems, closely linked to the mental activity of each speaker; the two levels, only apparently distinct in public and private, intersect each other in such a narrow way as to be, from certain points of view, hardly, or at least not significantly, separable. When we are willing and able to do it, we express what we think, sometimes remarking that what has been said depends on something else, from previous elements in our possession, and maybe that we have arrived there through a certain path. Our interlocutor can share, take with him/her what we have made known, in turn

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commu-nicate to us; but also, he can criticize us, ask for further explanations, and indicate errors we had not noticed. This urges us, or should urge us, as it were, on retracing our steps, reviewing our line of thought, modifying it, by expanding our intellectual baggage or rejecting its elements no longer sus-tainable. Dialectics, then, can continue, in an overall activity of which it is probably impossible (and in any case not required here) to give a precise picture. In this context, a role of primary importance is played by those that, using a widespread term of technical literature, we call inferences.

The notion of inference seems to rest on at least three indisputable points. First, an inference involves a passage from certain data, generally called premises, to another datum (or more), the conclusion (or the conclusions). Secondly, inferences must be connected to reasoning. In fact, they can be described as minimum units which can be reached by breaking a reasoning down into progressively simpler parts; in turn, a reasoning can be understood as a chain of inferences. Finally, inferences are important to logic. According to a fairly widespread reading, in fact, the latter is to be understood as the science of correct reasoning. It must therefore deal with inferences and reasoning in general, providing tools by which to establish which inferences are valid and which pieces of reasoning are correct, as well as a further analysis of the notions of validity and correctness themselves, that is to say, when and why an inference can be said valid, and a reasoning correct. The choice of the theoretical armament, in this sense, will have to be adequate with regard to informal desiderata.

From this point of view, their minimal character clearly does not exclude that the inferential units can be further analyzed. However, this circumstance could collide with the intention of finding a notion of inference that does not change depending on the historical and scientific context. Similarly, if the idea that a reasoning is a chain is not accompanied by the requirement that its steps have a minimal complexity, an enormously complicated reasoning can be reduced to an inference having as premises its hypothesis, and as conclusion its conclusion. This, for its part, could not go together with the epistemic desiderata of an analysis that looks at inferences as acts carried out consciously and voluntarily by agents with limited time and resources. In fact, some transitions could be too difficult, so that agents of the type described may be never willing to carry them out.

In both cases, the description of the nature of inferences will influence the description of the nature of reasoning and, if we are too generous in our way of looking at inferences, the link could be affected. If we accept that some inferences may be unconscious, involuntary and even automatic steps, we should also be willing to accept that steps of this type do not occur in reasoning, or be equally generous in reasoning, so as to authorize

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unconscious, involuntary and automatic components in it. More generally, the interdependence between inferences and reasoning makes it impossible to adopt the above description as a definition of the concepts involved: one cannot, on pain of vicious circles, postulate that an inference is the minimum unit of a reasoning, and at the same time that a reasoning is a chain of inferences. The aforementioned intertwinement is therefore a result to aspire to, and not a given from which to start. We will have the opportunity to discuss more extensively these problems later.

Cesare Cozzo has argued that different conceptions about the nature of inferences are not necessarily incompatible, since they can rather serve dif-ferent purposes. In particular,

the question "what conception of inference ought we to adopt?" thus leads to the question "what is the problem?". (Cozzo 2014, 165)

Given the three previous points, the answer to the question on the nature of inferences, therefore, cannot and should not be univocal. How to articulate it, then? Cozzo himself indicates, in a commendable way, seven relevant factors.

The first one concerns the nature of premises and conclusion. If some au-thors maintain that the description given above, according to which premises and conclusion are mere data, is satisfying, others consider it too permissive. Premises and conclusion are truth-bearers, that is, entities that are liable to be true or false. And in turn, this can be understood in at least three man-ners; truth-bearers can have an abstract nature, in which case one usually speaks of propositions, or a linguistic nature, being therefore sentences, or, finally, they can assume the form of mental states, or beliefs. However, the range of possible answers does not end here. Following yet another approach, we could in fact require that

premises and conclusions are not objects or states in which we happen to find ourselves, but responsible acts or actions, which we do. (Cozzo 2014, 162)

Such acts or actions can, once again, stand out on a mental level, as judg-ments of various kinds, or on a linguistic level, as in the case of assertions (but also, perhaps, of questions or commands). But, more precisely, what are propositions, sentences, beliefs, judgments and assertions? Here too, the answers are heterogeneous, and so we have numerous, further subramifica-tions. The second factor is given by the inferential agent. As in the previous case, we can reason in a more or less exclusive way:

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if you think that only a person can make an inference, you have a narrow conception of the subject of inference. If you believe that not only a person, but also a machine, or a non-personal biological entity can infer, then you have a broad conception of the subject. (Cozzo 2014, 162 - 163)

The third ingredient involves the relation between the agent and the set of premises and conclusion, and depends, or it is expected to depend, on the way in which the first and the second factor have been settled. Thus, we could say that

the data X are stored in S; S is in the representational state X; S is in the neural state X; S performs the act X, etc. (Cozzo 2014, 163)

Probably, the most important factor, no doubt central to the logical in-quiry, is what Cozzo labels as the fourth, and concerns the relation between premises and conclusion. Here, we move from an extremely inclusive re-sponse, according to which an inference is simply a pair where the first ele-ment is the set of premises and the second the conclusion, to answers instead insisting on the fact that, in an inference, premises and conclusion cannot be completely untied, for some sort of connection must occur among them. What this connection is, however, is anything but unquestionable. Is it an abstract relation, or maybe a causal relation dependent on a psychological, possibly unaware, involuntary and automatic transition? Perhaps, none of these two things, but in a more epistemic sense,

a conscious and deliberate act on the part of the subject. (Cozzo 2014, 163 - 164)

The fifth element is the stability of the premises-conclusion relation. This relation in fact can be completely aleatory, or substantial but refutable ac-cording to future circumstances. However, some inferences, commonly called deductive, seem to be such that

the connection between premises and conclusion is stable and can never be subverted by a new piece of information. (Cozzo 2014, 164)

At the sixth point, we find the more or less public character of inferences. If inferences are conceived as subpersonal psychological transitions, it will

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be very difficult to think of them as something publicly communicable. Ac-cording to many, however, this view must be rejected: inferences, and the pieces of reasoning in which they are involved, must be able to materialize in practices accessible to all members of the community. Finally, the seventh factor concerns what Cozzo defines

the context in which premises and conclusion are placed. (Cozzo 2014, 165)

When we describe an inference, shall we limit ourselves to describing only its premises, conclusion, agent, and their relations? Should we not, perhaps, take into account also the broader context in which the inference is accom-plished? Maybe other inferences, or pieces of reasoning, to which it is con-nected? Or the whole information or knowledge in possession of the agent? Or even the whole set of co-agents able to perform and accept inferences?

1.2 Valid inferences

The seven factors outlined by Cozzo offer a general, neutral grid in which to frame different conceptions on the nature of inferences. However, as far as we are concerned, if the adoption of a determinate conception depends on the problem we intend to solve, what we have just affirmed must be reported to the object of this investigation, namely, Dag Prawitz’s theory of grounds, and to the fundamental question the latter aims to answer: in what sense and why do some inferences - often called valid - seem endowed with a power of epistemic compulsion?

1.2.1 Epistemic compulsion

Epistemic compulsion is something we often experience in everyday life. The bill for a dinner at the restaurant amounts to 45 euros, we have given the waiter a banknote of 50 and expect to receive 5 euros change, and so the waiter will have to do. An inference has been made, the premises of which concern the following circumstances: the bill is of 45 euros, we have given the waiter a banknote of 50, and 5 is what we get by subtracting 45 from 50 (being subtraction the relevant operation). Of course, we could feel magnanimous, or have made a mistake, and ask for a lower change. In the same way, the waiter might have been mixed up, or could refuse to give us his due, perhaps claiming to follow a strange arithmetic in which 45 and 50 indicate the same quantity. Except the details and the secondary aspects, however, the inference compels us to accept its conclusion: we feel authorized to claim

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5 euros, and the waiter should feel obliged to give us exactly that sum. Again, if we are reasonably convinced that A implies B, and we have ascertained A, we have to conclude B. Without this having consequences of any kind, we are obviously free to refuse the conclusion. There is though a clear sense in which anyone would maintain that such a behaviour would be wrong, or even irrational. Being free to do what one wants seems to fail if the correctness of reasoning or the sustainability of its conclusions are at stake: we are forced because we feel forced.

We are, in other words, in the presence of a very particular phenomenon, which originates from and depends on

a special force, which is neither the threat of violence, nor the charm of a seductive persuader: it is simply the sober force of reasoning. (Cozzo 2014, 165)

Reasons and modalities of epistemic compulsion have often been at the cen-ter of the reflection of philosophers and logicians. Precisely to this issue Prawitz has mainly and explicitly dedicated some of his latest works. In them, the Swedish logician proposes and develops the aforementioned the-ory of grounds, an answer, at the same time philosophical and formal, that passes through an innovative characterization of the nature of inferences and of their validity, of proofs, and of the conceptual content of such, intercon-nected, notions. From this perspective, the theory of grounds is, so to speak, intrinsically worthy of interest. However, its importance derives also from the solutions and advances that it is able to offer with respect to similar approaches, including the one that Prawitz himself developed in the past. After all, and as we shall see, the epistemic relevance of valid inferences and proofs has always been one of the pivotal points of Prawitz’s semantic in-vestigations; and the theory of grounds constitutes a significant change of perspective with respect to these previous conceptions.

Focusing on the problem of epistemic compulsion seems to impose some forced choices on the type of inferences to be considered, a quite binding se-lection with respect to the general framework provided by the seven factors identified by Cozzo. First, as regards the stability of the premises-conclusion relation, we are only interested in deductive inferences. Of course, we could also feel epistemically forced towards conclusions drawn on the base of in-ferences, the strength of which might decrease, or even vanish, in view of future occurrences. However, in this case, the inference accomplished does not provide a truly conclusive support, and inferences of this type may be of interest for logic only in relation to a notion of valid inference, as cases that do not respect the conditions of the corresponding definition.

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As for the nature of premises and conclusions, it seems reasonable to argue that their description in terms of data is decidedly unsatisfactory, or at the very least too general. There are certainly data for which it makes sense to speak of epistemic compulsion. It is a datum for those who accept the rules of usual arithmetic that every multiple of 3 is divisible by 3, and that 6 = 2 · 3, and it is therefore a datum to which one is epistemically compelled that 6 is divisible by 3. However, there are also data for which this discourse does not seem to be valid. Let us assume we are monitoring the different configurations that the iris of an observer O takes according to different colors projected on a screen; when the red color, the datum-premise, appears on the screen, the iris of O will assume a certain configuration, the datum-conclusion. From a certain point of view, O is forced, so to speak, to draw a certain conclusion, but it would be strained to speak of an epistemic compulsion. The constraint, in fact, is not generated by the sober force of reasoning, nor is O free, if he/she wants, to oppose it. On the contrary, the compulsion acts in an unconscious, involuntary and automatic way (the awareness that the red color has been projected arrives at a later moment of reflection). Premises and conclusion should therefore be, at the very least, truth-bearers, that is to say, propositions, sentences or beliefs. Of course, it is compatible with our purposes even the stronger circumstance that premises and conclusion are judgments or assertions.

In the light of the above, it does not even seem promising to claim that the agent of an epistemically compelling inference can be a generic biological entity; propositions, sentences, beliefs, judgments and assertions are objects or acts concerning an abstract sphere, conceptual or linguistic, which only with an extreme forcing we could attribute, for example, to a jellyfish. A human being is certainly more suitable, but what about the famous Chrysip-pus’ dog? Following its master, which is far and not visible, it arrives at a crossroads, sniffs one of the possible branches and, not smelling its master, confident and without sniffing takes the other path. Similarly, can the agent of an epistemically compelling inference be a machine? As Prawitz rightly remarks, the central point is, here, that epistemic compulsion involves a re-flective activity, lacking in animals and machines. In fact, it seems to go with a reflection through which the performed activity can be appropriately generalized, and understood as deductively reliable:

when we deliberate over an issue or are epistemically vigilant in general, we are conscious about our assumptions and are careful about the inference steps that we take, anxious to get good rea-sons for the conclusions we draw. [...] taking for granted the truth of a disjunction ‘A or B’, and getting evidence for the truth of

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not-A, we start to behave as if we held B true without noticing that we have made an inference [...] the Babylonian mathematicians were quite advanced, for instance knowing Pythagora’s theorem in some way, but, as far as we know, they never tried to prove theorems deductively. (Prawitz 2015, 67)

Essentially the same reasons that have guided the previous choices should at this point induce to demand that, between the agent of an epistemically compelling inference and the set of premises and conclusion of such an in-ference, there is something more than having stored certain information, or finding oneself in a neural state. The agent must be in a sense conscious of the content of premises and conclusions, if we consider them as propositions or sentences, or, in the case of a reading in terms of beliefs, find him/herself towards them in an intentional state, or, finally, when premises and conclu-sions are conceived as judgments or assertions, perform the acts to which they correspond. This, one would say, also favors a reading of the premises-conclusion relation in terms of a conscious and deliberate act on the part of the agent. The conclusion is (or binds to) the prefixed goal, which the agent shows to aim to achieve on the basis of some sort of support provided by the premises. In any case, such an act could be accomplished at a later stage, after the achieved awareness of a link between premises and conclusion. Therefore, it is not possible to exclude a priori that the premises-conclusion relation can be of an abstract type.

Finally, the question relating to epistemic compulsion is undoubtedly compatible with a public view of inferences, and of the reasoning in which they occur. In fact, the reflective character of the inferences in question seems to imply that we can give of them, and of the reasoning in which they are used, a testimony, and this without denying that the phenomenon can in-volve - also, or mainly - mental dynamics. Certainly, there is a long tradition, dating back to at least René Descartes (Descartes 1985), according to which some acts, essentially intuitive and private, induce an epistemic compulsion. A discussion on the plausibility of this thesis, or on the way it can be artic-ulated, would obviously take us too far; therefore, we will restrict ourselves here to emphasize that the compelling character of such acts must in any case refer to certain propositions or sentences, and ultimately depend on the meaning attributed to them. But then, the analytical character of such sup-posed intuitive knowledge is something that can manifest itself in practices that testify the acceptance and sharing of meanings. This observation leads us to the final case of the context in which inferences are accomplished, or located. In this case, however, the possible alternatives are different, and all mutually compatible.

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1.2.2 Justification

That some inferences are commonly held to have a power of epistemic com-pulsion is certainly a fact. On the other hand, the existence of valid inferences can be a matter of doubt. The deductive ability to support conclusively propositions, sentences, beliefs, judgments or assertions, might be nothing but an illusion, and some forms of scepticism seem to be oriented exactly towards this thesis. Prawitz himself observes, however, that

to justify deduction in order to dispel sceptical doubts about the phenomenon is not likely to succeed, since such an attempt can hardly avoid using deductive inference, the very thing that is to be justified. This should not prevent us from trying to explain why and how deductive inference is able to attain its aims. Deduction should thus be explained rather than justified. (Prawitz 2015, 65 - 66)

If we successfully fulfill the task of explaining how and why some inferences force epistemically, we are also in possession of a weapon against the sceptic. On the other hand, in order to express more than a mere opinion, the sceptic should show that no plausible conception on the nature of inferences, and of their validity, is such as to attribute to valid inferences something that can count as a power of epistemic compulsion. Even assuming that this does not go through the use of inferences that the sceptic treats as epistemically compelling, it should still result in something very similar to what Prawitz, although with opposite objectives, suggests to do. Therefore, the basic ques-tion of the theory of grounds is, it seems to us, important even for the sceptic. Be that as it may, Prawitz does not give too much credit to possible sceptical positions, arguing rather that

we take for granted that some inferences have such a power, and there is no reason to doubt that they have. But what gives them this power? This should be explained. (Prawitz 2015, 73)

Far more important, especially for the development of our reasoning, is instead another consideration. The questions related to what we are here calling epistemic compulsion, in fact, have often been formulated as questions related to the ability that valid inferences have to justify who performs them with respect to their conclusion. It is often said that valid inferences preserve, or transmit, justification, in the sense that, if one is justified with regard to the premises, the same will be true for the conclusion. What “to be justified with regard to ..." means obviously depends on the adopted conception of

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inference, but it is clear that the justification we are talking about is of the strongest possible type: a conclusive and non-refutable justification, a definitive reason for the truth of propositions, sentences or beliefs, or for the correctness of judgments or assertions. The idea seems to be implicit, or at the very least relevant, even when valid inferences are conceived in the most abstract sense of necessary truth-preservation: it cannot occur that the premises are true, and the conclusion is false. The manner in which, from the necessary preservation of truth one passes to the preservation or transmission of justification becomes then, in general, a central problem in the frameworks, philosophical as well as formal, which adopt such an approach.

As Cozzo further points out (Cozzo 2014), epistemic compulsion and preservation or transmission of justification are in any case two sides of the same coin. The general argument in favor of this position is based on a

plausible assumption concerning the relation between the notion of justification and the notion of argumentative context with ra-tional disputants, the idea that justification is something publicly acknowledgeable by rational subjects: if a person is justified in asserting a sentence, or an inference has the power to transmit justification, then both facts must be acknowledged by the dis-putants involved, if they are rational. (Cozzo 2014, 165)

From this assumption it derives the equivalence between the circumstance that someone is forced or justified in accepting something, and the circum-stance that each of his/her interlocutors is equally forced on pain of irra-tionality, or justified to do the same. Therefore, given two interlocutors A and B, let us suppose that A is justified in accepting the premises in a set Γ of an inference I from Γ to the conclusion α that preserves or transmits justification, and let us suppose furthermore that A performs I. According to the public nature of justification, B recognizes that A is justified in accepting the premises in Γ and, according to the public nature of the preservation or transmission of justification, B also recognizes that A is justified in accepting α. Therefore, on pain of irrationality, B is forced to accept α. Vice versa, let us suppose that I is epistemically compelling and, again, that A is justified in accepting the premises in Γ, and that he/she performs I. According to the public nature of justification, B recognizes that A is justified in accepting the premises in Γ, and therefore he/she is him/herself obliged to accept them; but then, B will be forced, on pain of irrationality, to accept α. It follows that, for the above equivalence, A will be justified in accepting α.

Although the capacity to force epistemically and the capability to preserve or convey justification are equivalent concepts, we will have the opportunity

Dag Prawitz's theory of grounds

AIX-MARSEILLE UNIVERSITY

“SAPIENZA” UNIVERSITY OF ROME

ÉCOLE DOCTORALE 356

CENTRE GILLES GASTON GRANGER UMR 7304

Thèse présentée pour obtenir le grade universitaire de docteur

Discipline : Philosophie

Spécialité : Logique

Antonio PICCOLOMINI D'ARAGONA

Dag Prawitz's theory of grounds

Soutenue le 16/12/2019 devant le jury composé de :

Résumé

Riassunto

Acknowledgments

Contents

I

Theoretical Background

21

II

Prawitz’s theory of grounds

122

III

A formal theory of grounding

213

Introduction

Part I

Chapter 1

Inferences and proofs

1.1

Nature of inferences

1.2

Valid inferences

1.2.1

Epistemic compulsion

1.2.2

Justification