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Introduction: Kit Fine on Truthmakers,Relevance, and Non-Classical LogicFederico L.G. Faroldi and Frederik Van De PutteAbstract Kit Fine’s contribution to logic is vast and diverse; the chapters in thisbook deal with a signi?cant part of it. In this introductory chapter, we clarify andcontextualize the main themes of Fine’s work that are centre stage in this book, afterwhich we give a summary of each chapter.1 Introducing the IntroductionThis book appears in the series Outstanding Contributions to Logic and is devotedto Kit Fine (1946, Farnborough - England). It consists of 15 original research papersdealing with the formal and philosophical aspects of various themes at the center ofFine’s work. Some are critical, in-depth discussions of his published work, othersapply his ideas to new problems, and still others use Fine’s contribution to developnew perspectives on a classical topic. Each of the chapters is accompanied by anin-depth reply by Fine himself. Finally, the book also features a brief autobiographyand an exhaustive list of Kit’s publications.As editors, it has been a great pleasure to notice that the contributions are currentand they engage with Fine’s research in a creative, critical, and constructive man-ner. It has been a privilege to work with such talented philosophers and logicians.All contributions have moreover been thoroughly peer-reviewed and we thereforewish to acknowledge the crucial role played by the reviewers in the genesis ofthis book: Guillermo Badia, Willem Conradie, Louis deRosset, David Fernandez-Duque, Jeremy Goodman, Berta Grimau, Jesse Heyninck, John Horty, AndrewIrvine, Harvey Lederman, Stephan Leuenberger, David Makinson, Ondrej Majer, EdName of First AuthorName, Address of Institute, e-mail: name@email.addressName of Second AuthorName, Address of Institute e-mail: name@email.address1
2 Federico L.G. Faroldi and Frederik Van De PutteMares, Hiroakira Ono, Francesco Paoli, Pawel Pawlowski, Bryan Pickel, FrancescaPoggiolesi, Adam Prenosil, Tudor Protopopescu, Rasmus Rendsvig, Arthur Schip-per, Sebastian Sequoiah-Grayson, Shawn Standefer, Johannes Stern, Eric Swanson,Zach Weber, Yale Weiss, Malte Willer, Robert Williams.Some of the groundbreaking contributions of Kit Fine to logic are well-known,such as the discovery of an incomplete logic containing S4 (Fine, 1974) and thecanonicity of elementary modal logics (Fine, 1975b). Fine also did pioneering workconcerning the now standard method of ?ltration in modal logic (as highlighted inChapter 3 of this book) and developed a highly original semantics for relevancelogic (as reviewed in Chapter 7 of this book).However, Kit Fine has not only solved open problems in logic and related disci-plines, but also created (or discovered!) new problems and questions. While not thefocus of this book, Fine is a towering ?gure in contemporary metaphysics as well.More generally, he introduced or transformed ideas that have shaped the debate andinquiry, some of which are taken up by contributors to this book. Notwithstandingthe diversity of topics he worked on, there is great unity in his method, using for-mal logic where it helps to clarify certain distinctions and to develop his views inrigorous terms, but always driven by the underlying philosophical issues.In the remainder of this chapter we will brie?y introduce the central themestouched upon later in the book, and highlight how the following chapters contributeto one or another of them, without any claim to have exhausted the multifaceted,continuing conversation that unfolds in these pages. In particular, Section 2 is a gen-tle introduction to some of the main topics of the rest of the book. Section 3 presentsa short outline of each of the original chapters of the book. Section 4 concludes theintroduction with some lighter, but not less important, notes.2 Finean Themes of This BookAs noted above, Kit Fine’s in?uence on contemporary philosophy is enormous andhis contributions are fundamental. In what follows, we focus on those themes inhis work that are particularly relevant for the chapters that follow. An overview ofthe many topics we don’t even touch upon in this introduction, such as essence,semantic relationism, abstraction, metametaphysics and the foundations of mathe-matics, can be found in a recent encyclopedic entry devoted to Fine’s work (Raven,forthcoming).2.1 Truthmakers: A Hyperintensional RevolutionIn the past half century, possible worlds have played a key role not just as a toolin the semantics of modal logics, but also in the philosophy of language and meta-
Introduction: Kit Fine on Truthmakers, Relevance, and Non-Classical Logic 3physics. Fine has argued extensively that the job done by possible worlds is oftenbetter realized with other means, notably (albeit not exclusively) with truthmakers.The concept of a truthmaker is not entirely novel, neither did Fine invent it. Be-fore him, however, it was mostly used within a metaphysical project: what it is,out there in the world, that makes true what we take to be truth-apt (sentences,propositions, etc.). Kit Fine’s interest, when it comes to truthmakers, is mostly ina semantical project: how it is that sentences (propositions, etc.), by virtue of theirvery meaning, are made true by what’s out there, in the world.1We are used to thinking of a world as a complete, consistent state-of-affairs thatsettles the truth value of every proposition. Thus, ’Verdi composed Aida’ is true ata world just in case at that world Verdi composed Aida. But that world will alsocontain a lot of irrelevant material to Verdi and Aida, such as that Bach composedcello sonatas, or that Peano arithmetic is incomplete. In contrast, truthmakers canbe thought of as parts of (possible) worlds that verify or falsify propositions andare wholly or exactly relevant to it. They are neither necessarily complete nor con-sistent.2However, just as with possible worlds, states can be regarded as arbitrarypoints with little additional structure but for the fact that they are equipped with amereological (“parthood”) relation that is required to ful?ll a completeness condi-tion. This makes the framework of truthmakers very abstract and ?exible.Once such a state and parthood relation is ?xed, one still needs to determine howstates make statements true. As Kit Fine points out, the truthmaking relation comesin three forms: exact, inexact, and loose:Loose veri?cation is a purely modal notion. A state or situation will loosely verify a state-ment just in case it is necessary that if the state obtains then the statement will be true.Exact and inexact veri?cation, by contrast, require that there be a relevant connection be-tween state and statement. With inexact veri?cation, the state should be at least partiallyrelevant to the statement; and with exact veri?cation, it should be wholly relevant (Fine,2017d).The state of Sicily being an Italian island is an exact truthmaker of the sentence’Sicily is an Italian island’, but the state of Sicily being an Italian island and Sicilyhaving amazing architecture is only an inexact truthmaker of the sentence ’Sicily isan Italian island’.“Exact” truthmaking clauses were given for the ?rst time in van Fraassen, 1969.In the heyday of situation semantics, inexact veri?cation was common. To illustrateone instance where exact and inexact truthmaking differ on the syntactic level, takeAand A?(A?B). In classical logic, these two expressions are obviously equivalent.Moreover, they have the same inexact truthmakers. Yet, they don’t have the sameexact truthmakers: an exact truthmaker for A?Bmay be “too speci?c”, and henceonly be an inexact trutmaker for A. If one takes the sameness of exact truthmakersas a criterion of equivalence, then Aand A?(A?B)are not necessarily equivalent.1For this distinction and an introduction to the semantical project, see Fine, 2017d. For a uni?ed(i.e. metaphysical and semantical) use of truthmakers see Jago, 2018.2Some of these ideas have also been used in situation semantics, see Barwise and Perry, 1981 andBarwise and Etchemendy, 1990.
4 Federico L.G. Faroldi and Frederik Van De PutteAn important question is which of these three notions (exact, inexact, loose)is fundamental, in the sense of being irreducible to the other(s). Fine argues thatthe notion of exact truthmaking is fundamental (Fine, 2017d, p. 565), others argueinstead that it can be de?ned via the inexact notion (cf. e.g. Deigan, 2019). Boththe inexact and the exact notion of truthmaking are used in various chapters of thisvolume.3With truthmaker semantics, one can also get a better formal grip on the phe-nomenon of hyperintensionality. Very generally, a context is hyperintensional whenclassically logically (or necessarily) equivalent contents cannot be substituted salvaveritate in its scope. (Sameness of) exact truthmaking provides a semantic basis foraccepting hyperintensionality. It therefore cannot come as a surprise that this per-spective on truthmaking has started to be widely used to capture many phenomenathat have been argued to require a hyperintensional analysis. Truthmaker seman-tics, both in inexact and exact forms, has been applied by Kit Fine to counterfactualconditionals (Fine, 2012a), the notion of “ground” (Fine, 2012c), intuitionistic logic(Fine, 2014b), and analytic equivalence (Fine, 2016a). Fine studies how to applytruthmaker semantics to permission (Fine, 2014a) and to imperatives and from thereto deontic modals, in an ongoing series of papers (Fine, 2015a,b). We introducesome of these applications in the next subsections (namely counterfactual condi-tionals, grounding, and subject matter). After that, we move to other Finean topics,viz. arbitrary objects and vagueness.2.2 Counterfactual ConditionalsA counterfactual conditional is a conditional of the form: ”If Xwere (had been) soand so, then Ywould be (would have been) such and such.” Alongside the indica-tive conditionals, it has been known for a long time that counterfactual conditionalscannot be captured by material or (classical) strict implication. Most notably, theyviolate strengthening of the antecedent: from “If it had rained, the grass in my gar-den would have been wet”, one cannot infer “If it had rained and I had put up a gianttent covering my entire garden, the grass in my garden would have been wet”.The very in?uential Stalnaker-Lewis (cf. e.g. Lewis, 1973) treatment maintains,in one of its forms, that such a counterfactual is true if and only if the consequent istrue at all the closest possible worlds in which the antecedent is true. Given a clas-sical account of possible worlds, this however implies that counterfactuals satisfyreplacement of tautological equivalents, both in the antecedent and the consequent.Fine argued against this principle for counterfactuals (Fine, 1975a). More recently,he developed a truthmaker-theoretic account of counterfactuals Fine, 2012a.In his contribution to the present book, Andrew Bacon (Chapter 17) criticizesFine’s account, studying two paradoxes and arguing for a possible worlds analysisof counterfactuals, though distinct from the Stalnaker-Lewis account.3Cf. also possibility semantics (van Benthem et al., forthcoming) and Leitgeb’s semantics for hisHYPE system (Leitgeb, 2018).
Introduction: Kit Fine on Truthmakers, Relevance, and Non-Classical Logic 52.3 Subject Matter and AboutnessThe subject matter of a proposition is, roughly, what the proposition is about.Truthmakers ?gure prominently in more recent theories of subject matter (seee.g. Yablo, 2014 and Fine, 2017a and the ensuing discussion Fine, 2020; Yablo,2018). For Fine, the subject matter of a proposition may be identi?ed with its max-imal truthmaker, i.e. the fusion of all its veri?ers. Mereological relations on subjectmatters will then be had quite easily via the mereological relations already presenton states.In Chapter 19 of the present book, Alessandro Giordani compares the Finean andthe classical account of subject matter, and proposes a synthesis of both. Aboutnessis also central in Chapter 23, in which it is argued that truthmakers can help us geta grip on what it means that a given statement is “the whole truth regarding a givensubject matter”.2.4 GroundingGrounding is a much studied concept that is thought to capture considerations offundamentality in metaphysics and in other domains. For instance, we can say thata conjunction is grounded in its conjuncts, or that the singleton of Socrates exists invirtue of Socrates existing. How to de?ne grounding more accurately is in fact partof the debate on grounding itself.The contributions of Fine to the debate on grounding, as a non-causal, explana-tory, primitive, hyperintensional notion, are numerous and well-documented (for anentry point see e.g. Fine, 2012b). Grounding is mostly thought of in metaphysicalterms, but Fine has suggested also natural and normative grounding notions.How to express grounding? Two options are the relational and the operationalapproach. The former expresses grounding with a predicate, thus suggesting thatgrounding is a relation between that which is grounded, and that which grounds.The latter expresses grounding as a variably polyadic operator between sentences.An advantage of the operational approach would be that it does not commit one toa speci?c position on the metaphysical status of grounding.In the context of formally modeling the grounding relata, several additional dis-tinctions can be introduced. While a conjunction is fully grounded in its conjuncts, itis only partially grounded in each conjunct. One can study the pure logic of ground-ing and the impure logic of ground. The former is concerned only with the structuralprinciples of the grounding operator, the latter also takes into account the internalarticulation of what grounds and what is grounded. One can moreover distinguisha worldly notion of grounding from a conceptual, or representational, notion ofgrounding.In his contribution to the present book, Fabrice Correia offers a novel semanticframework for the worldly notion that is intended to improve upon his own (Cor-reia, 2010) and Fine’s (Fine, 2012c) previous accounts (Chapter 25). Some formula-
6 Federico L.G. Faroldi and Frederik Van De Puttetions of ?ne-grained grounding principles lead to inconsistencies and paradoxes. InChapter 21 of this book, Peter Fritz shows how one can solve some of these issuesin broadly Finean terms.2.5 Arbitrary ObjectsArbitrary objects have been criticized as either useless or inconsistent since at leastLocke and Berkeley. Yet, ordinary reasoning uses arbitrary objects for quanti?ca-tional inferences, which are then suitably regimented e.g. in natural deduction sys-tems. For instance, we conclude that all individuals have a certain property by show-ing that an arbitrary individual has that property. This rule of universal generaliza-tion allows us to infer ?xf(x)from f(a), given certain restrictions, and it is part andparcel of mathematical reasoning too.Fine (Fine, 1985) gave rigorous foundations to our reasoning (e.g. in mathemat-ics, natural deduction, anaphora) with arbitrary objects. He developed a theory thatis far better developed and coherent that was on the market, and he applied it tomany ?avors of generality. More recently, Fine applied these foundations to accountfor the much discussed Cantorian abstractionist constructions of cardinal numbersand order types (Fine, 1998), to a general account of types or forms (Fine, 2017b),to identity criteria (Fine, 2016b), and to providing uni?ed foundations for essenceand ground (Fine, 2015c).In their contribution to the present book, Leon Horsten and Ryo Ito criticallydiscuss Fine’s conception of arbitrary objects by comparing it to Russelll’s (Chapter29). Remarks on arbitrary objects also appear in the contributions of Jago (Chapter9 and Bimb´o and Dunn (Chapter 7).2.6 VaguenessA signi?cant portion of natural language is vague. The predicate ’is bald’, for in-stance, is commonly thought to admit of borderline instantiations: its meaning doesnot settle whether a particular man, Al, with thinning hair is bald or not. In thissimple version, this is a problem for classical logic, because it does not make senseto say that ”Al is bald or is not bald” if ’is bald’ is vague. This raises the problemof formulating a suitable logic and semantics that can handle vague sentences in anappropriate way.Fine, 1975c formulated an in?uential supervaluationist account of vagueness, ac-cording to which, roughly, a vague sentence is true if and only if it is true for all waysof making it completely precise. More recently, however, Fine stressed a distinctionbetween local vagueness – it is indeterminate whether a predicate applies in a singlegiven case – and global vagueness – it is indeterminate whether a predicate appliesacross a range of cases (Fine, 2008). In particular, he showed that it is impossible to
Introduction: Kit Fine on Truthmakers, Relevance, and Non-Classical Logic 7understand global indeterminacy in terms of local indeterminacy (while leaving theother direction open).(Fine, 2017c) suggests a novel account that focuses on the global character ofvagueness, and maintains that it can be captured by logical means alone (i.e. withoutemploying a distinctive vagueness-theoretic notion). Andreas Ditter challenges thisnew possibility result in his contribution to the present book (Chapter 31).3 Outlines of the ChaptersAs we explained above, Kit Fine’s contribution to logic does not merely consist ina range of particular results in various sub?elds of logic; rather, what makes himstand out is the fact that he managed to relate each of these, and to make fruitfuluse of ideas and solutions for one topic, in dealing with another. The same canbe said for the chapters in this book: each of them relates to various others, be itin methodology, formal apparatus, or conceptual targets. This will already partlybecome salient in the summary that follows.We start with two chapters that relate directly to Kit Fine’s older contributions inmodal and relevant logic. Chapter 3 zooms in on the well-known ?ltration method,on which Fine did pioneering work in the 1970s (cf. supra). Johan van Benthemand Nick Bezhanishvili investigate this method from a wide range of perspectives,including model-theoretic, proof theoretic, and dynamic-epistemic ones. The resultis both a solid overview of existing work, and an inspiring venture into unknownterritory, related to ?ltration.In Chapter 5, Vladimir Lifschitz outlines the history of the central idea of “stablemodels” in the context of the Prolog programming language and, later, answer setprogramming. As explained in this chapter, Fine occupies a special place in thishistory, with his notion of “felicitous models” as one alternative formulation of thestability idea. This chapter thus provides a nice illustration of how fundamental workin philosophical logic may turn out to be of importance for more applied work incomputer science.Katalin Bimb´o and J. Michael Dunn (Chapter 7) recall Fine’s two-sorted seman-tics for relevance logic and related work. Their chapter leads to a detailed compari-son of Fine’s semantics to more well-known Routley-Meyer semantics for relevancelogics. Interestingly, it is also shown how this work relates to the notion of arbitraryobjects (cf. supra and Chapter 29).This brings us to a number of chapters that all use truthmaker semantics in oneway or another. Chapter 9, by Mark Jago, consists of two related parts. The ?rst partdiscusses the logical relation of disjunctive parthood and links it to the semanticnotion of re?nement and the philosophical determinable-determinate relation. In thesecond part, the author develops a formal logic on the basis of these ideas, whichincludes a relevant conditional. This chapter moreover touches on the subject ofvagueness (cf. Chapter 31) and arbitrary objects (cf. Chapter 29), and Jago’s formalsemantics is clearly related to Chapter 13 in its treatment of disjunction.
8 Federico L.G. Faroldi and Frederik Van De PutteChapter 11 starts from Fine’s exact truthmaker semantics for intuitionistic logicand asks whether it can be generalized to other non-classical logics. In this chapter,Ondrej Majer, V´it Pun?coch´a?r and Igor Sedl´ar provide an exact truthmaker seman-tics for the nonassociative Lambek calculus and some of its extensions, such as theimplicational fragment of the relevant logic R. They draw interesting similarities be-tween their generalization of Fine’s semantics and Urquharts semilattice semanticsfor R.In a similar vein, Peter Verd´ee combines the notion of exact truthmaking withRestall’s closure frames semantics for substructural logics without distribution anda modal operator, to obtain a truthmaker semantics for four non-transitive relevantlogics (Chapter 13). Following Verd´ee’s earlier work (cf. Peter Verd´ee and Samonek,2019; Verd´ee and Bal, 2015), the four logics are de?ned syntactically as the rele-vant cores of classical logic, Priest’s logic of paradox, strong Kleene logic, and ?rstdegree entailment respectively. The proof of soundness and completeness moreoverproceeds via a third characterization of the logics using sequent calculi, and in-volves an innovative view on relevant proofs in terms of the network structure of thesequents in those proofs.In their chapter, Peter Hawke and Ayb¨uke ¨Ozg¨un ask whether truthmaker se-mantics can be used to solve certain well-known problems of traditional, classicallogic-based epistemic logic. Starting from a rich formal language that distinguishesa priori knowability, suf?ciency for knowledge, and a priori (knowledge-level) im-plication, the authors of this chapter focus on a range of ?rst-order logical princi-ples and discuss counterexamples to those principles. They note that each of theprinciples are valid on the traditional, Hintikka-style semantics for epistemic logic.Finally, they study the behavior of those principles in six different truthmaker se-mantics for the same formal language, that are distinguished by their treatment ofsuf?ciency for knowledge.Chapter 17 deals with the semantics of counterfactual conditionals. Here, An-drew Bacon starts by explaining two puzzles that involve such conditionals: Yablo’sbutton and Bernadete’s paradox. After carefully formalizing the arguments that areinvolved in both, he goes over the various principles that could be given up in orderto avoid the paradoxes. While Fine’s diagnosis is that replacement of equivalentsshould be weakened, thus pointing towards a truthmaker semantics of counterfactu-als (cf. supra), Bacon argues that this move still leads to serious problems as long ascertain very weak logical principles are in place. Instead, he suggests rejecting whathe calls the “Disjunction” principle for counterfactuals, and to endorse a semanticsin terms of a speci?c possible worlds semantics that works with selection functions.Chapter 19 compares Fine’s theory of subject matter to Lewis’s, and it proposes acharacterization based on the information structure model. Giordani aims to developa theory of subject matter that is able to capture both hyperintensionality (a keyfeature of Fine’s account) and context-variation (a key feature of Lewis’s), amongother features.In his contribution to this volume, Peter Fritz sets out to discuss various inconsis-tency results arising from a ?ne-grained conception of propositions, results similarto those arising from naive set comprehension (Chapter 21). Fritz draws on Fine’s
Introduction: Kit Fine on Truthmakers, Relevance, and Non-Classical Logic 9potentialist theory when it comes to set-theoretic inconsistency to explore what asimilarly potentialist theory of propositions might look like. The principles he pro-poses for such a theory are shown to be consistent with a model construction. Aninteresting expressive limitation emerges when trying to apply this theory to ground-ing. The author also discusses using truthmaker semantics to give a theory of propo-sitions, noting along the way some dif?culties in understanding the notion of ’state’used in truthmaker semantics.Chapter 23 puts a fundamental, but often overlooked concept on the table: that ofthe whole truth on a given subject-matter. As Stephan Kr¨amer argues in this chapter,this notion is intricately linked to the modality “and that’s it”, whereby one expressesthat a given statement covers everything there is to say on the subject at hand. It isshown that both presuppose a form of relevance, and that possible worlds semanticscannot adequately capture them. Instead, the author proposes a truthmaker accountof “the whole truth” and the corresponding modality.Fabrice Correia’s chapter deals with the logic of worldly grounding. In particular,it develops a synthesis of the author’s own earlier semantic approach (Correia, 2010)and Fine’s (Fine, 2012b, Fine, 2012c), in order to overcome various defects of both.After a detailed comparison and critique of the existing approaches, the new one isworked out and applied to the semantic characterization of various proof systemsfor ground-theoretic notions.In Chapter 27, Daniel Rothschild and Stephen Yablo focus on the notion of apermissive update, i.e., the information change caused by granting a permissionin a context where before it may have been forbidden. They argue quite convinc-ingly that the various existing approaches to this notion – all based on classicallogic or possible worlds semantics – fall short in one way or another, and developa truthmaker-based approach instead. This approach includes a subtle update mech-anism on sets of truthmakers, which may in turn inspire new work on the logic ofbelief change.In Chapter 29, arbitrary objects take centre stage. Here, Leon Horsten and RyoIto argue that Fine’s theory of arbitrary objects has an early precursor in Russelll’snotion of variables in his Principles of Mathematics, be it that they differ in theway the dependence relation between variables is treated. This observation goessomewhat against earlier remarks by Fine himself “A Defence of Arbitrary Objects”,and is used as the starting point for an in-depth historical overview of Russell’s workon variables, which may help to revive the philosophical debate on arbitrary objects.Chapter 31 raises a problem for Fine’s recent theory of vagueness and the re-lated possibility result concerning the af?rmation or denial of all propositions in aSorites-like sequence (Fine, 2017b). In his contribution, Andreas Ditter argues thatFine’s possibility result as well as his novel solution to the sorites paradox fail incertain legitimate extensions of the formal language employed. Ditter shows thatone can prove a new impossibility result in extensions of the language containinga negation operator that obeys reductio ad absurdum, and argues that at least somesuch extensions are unobjectionable. In particular, it is shown that an operator thatbehaves exactly like intuitionistic negation, which obeys reductio, can be de?ned ina natural propositionally quanti?ed extension of Fine’s logic.
10 REFERENCESFinally, in Chapter 33, Kit Fine and Errol Martin provide a formal account ofnon-circular reasoning, i.e. of reasoning in which the conclusion of an argumentis not somehow presupposed in its premises. Martin (along with R. Meyer) hadpreviously shown that the implicational system P-W does not contain any theoremsof the form A?A. Fine and Martin then extend this result to systems that alsocontain conjunction.The book ends with an exhaustive bibliography of Kit Fine’s published works(Chapter 20).4 The ManThe preceding, and this book in its entirety, not to mention his collaborative spiritthroughout this endeavor, should already go some way towards indicating the impor-tance of Fine’s work for philosophical logic as it is today. But Fine’s contributionsto Logic (and Philosophy) are not just in the form of papers or books. For thosewho have had the pleasure to attend them, his presentations are remarkable for theirclarity and inspiring nature. As a witness of this, one would sometimes see logi-cians ?ght over who gets to take the ?ipcharts of his keynote lecture home. Finally,with his mentorship, sense of humor, love for music (and good food!), charitable,constructive questions, he contributed to many young logicians’ and philosophers’careers, scienti?c or not.ReferencesBarwise, J. and J. Etchemendy (1990), “Information, infons, and inference”, in Sit-uation Theory and Its Applications, ed. by R. Cooper, K. Mukai, and J. Perry,CSLI Lecture Notes, pp. 33-78. (Cited on p. 3.)Barwise, J. and J. Perry (1981), “Situations and Attitudes”, The Journal of Philoso-phy, 78, pp. 668-91. (Cited on p. 3.)Correia, Fabrice (2010), “Grounding and Truth-Functions”, Logique & Analyse, 53,211, pp. 251-79. (Cited on pp. 5, 9.)Deigan, Michael (2019), “A plea for inexact truthmaking”, Linguistics and Philos-ophy,DO I:10.1007/s10988-019-09279-2. (Cited on p. 4.)Fine, Kit (n.d.), “A Defence of Arbitrary Objects”, Proceedings of the AristotelianSociety, 57, pp. 55-77. (Cited on p. 9.)– (1974), “An Incomplete Logic Containing S4”, Theoria, 40, 1, pp. 23-29, DOI:10.1111/j.1755-2567.1974.tb00076.x. (Cited on p. 2.)– (1975a), “Critical Notice of Lewis, Counterfactuals”, Mind, 84, 335, pp. 451-458. (Cited on p. 4.)
REFERENCES 11– (1975b), “Some connections between elementary and modal logic”, in Proceed-ings of the Third Scandinavian Logic Symposium, ed. by Stig Kanger, North-Holland, Amsterdam, pp. 15-31. (Cited on p. 2.)– (1975c), “Vagueness, Truth and Logic”, Synthese, 30, 3-4, pp. 265-300, DOI:10.1007/BF00485047. (Cited on p. 6.)– (1985), Reasoning with Arbitrary Objects, Aristotelian Society. (Cited on p. 6.)– (1998), “Cantorian Abstraction: A Reconstruction and Defense”, Journal of Phi-losophy, 95, 12, pp. 599-634, DOI:jphil1998951230. (Cited on p. 6.)– (2008), “The Impossibility of Vagueness”, Philosophical Perspectives, 22, 1,pp. 111-136, DOI:10 . 1111 / j . 1520 - 8583 . 2008 . 00143 . x. (Citedon p. 6.)– (2012a), “Counterfactuals without Possible Worlds”, Journal of Philosophy, 109,3, pp. 221-246. (Cited on p. 4.)– (2012b), “Guide to Ground”, in Metaphysical Grounding, ed. by Fabrice Correiaand Benjamin Schnieder, Cambridge University Press, pp. 37-80. (Cited on pp. 5,9.)– (2012c), “The Pure Logic of Ground”, The Review of Symbolic Logic, 25, 1,pp. 1-25. (Cited on pp. 4, 5, 9.)– (2014a), “Permission and Possible Worlds”, Dialectica, 68, 3, pp. 317-336.(Cited on p. 4.)– (2014b), “Truth-Maker Semantics for Intuitionistic Logic”, Journal of Philo-sophical Logic, 43, 2-3, pp. 549-577. (Cited on p. 4.)– (2015a), “Compliance and Command, I”, ms. (Cited on p. 4.)– (2015b), “Compliance and Command, II”, ms. (Cited on p. 4.)– (2015c), “Uni?ed Foundations for Essence and Ground”, Journal of the Ameri-can Philosophical Association, 1, 2, pp. 296-311, DO I:10.1017/apa.2014.26. (Cited on p. 6.)– (2016a), “Angellic Content”, Journal of Philosophical Logic, 45, 2, pp. 199-226.(Cited on p. 4.)– (2016b), “Identity Criteria and Ground”, Philosophical Studies, 173, 1, pp. 1-19,DO I:10.1007/s11098-014-0440-7. (Cited on p. 6.)– (2017a), “A Theory of Truthmaker Content II: Subject-Matter, Common Con-tent, Remainder and Ground”, Journal of Philosophical Logic, 46, 6, pp. 675-702. (Cited on p. 5.)– (2017b), “Form”, Journal of Philosophy, 114, 10, pp. 509-535, DOI:10.5840/jphil20171141036. (Cited on p. 6.)– (2017c), “The Possibility of Vagueness”, Synthese, 194, 10, pp. 3699-3725, D OI :10.1007/s11229-014-0625-9. (Cited on p. 7.)– (2017d), “Truthmaker Semantics”, in A Companion to the Philosophy of Lan-guage, ed. by Bob Hale, Crispin Wright, and Alexander Miller, 2nd ed., ms,Blackwell, London, pp. 556-77. (Cited on pp. 3, 4.)– (2020), “Yablo on Subject-Matter”, Philosophical Studies, 177, pp. 129-171.(Cited on p. 5.)Jago, Mark (2018), What Truth Is, Oxford University Press, Oxford. (Cited on p. 3.)Leitgeb, Hannes (2018), “HYPE”, Journal of Philosophical Logic. (Cited on p. 4.)
12 REFERENCESLewis, David Kellogg (1973), Counterfactuals, Harvard University Press, Cam-bridge (MA). (Cited on p. 4.)Peter Verd´ee, Inge De Bal and Aleksandra Samonek (2019), “A non-transitive rel-evant implication corresponding to classical logic consequence”, AustralasianJournal of Logic, 16, 2, pp. 10-40. (Cited on p. 8.)Raven, Mike (forthcoming), “Kit Fine”, Entry for the Internet Encyclopedia of Phi-losophy. (Cited on p. 2.)Van Benthem, Johan, Wesley H. Holliday, and Nick Bezhanishvili (forthcoming),“A Bimodal Perspective on Possibility Semantics”, Journal of Logic and Com-putation. (Cited on p. 4.)van Fraassen, Bas C. (1969), “Facts and Tautological Entailment”, Journal of Phi-losophy, 66, 15, pp. 477-487. (Cited on p. 3.)Verd´ee, Peter and Inge De Bal (2015), “A New Approach to Classical Relevance”,Studia Logica, 103, 5, pp. 919-954. (Cited on p. 8.)Yablo, Stephen (2014), Aboutness, Princeton University Press, Princeton. (Cited onp. 5.)– (2018), “Reply to Fine on Aboutness”, Philosophical Studies, 175, 6, pp. 1495-1512. (Cited on p. 5.)
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Introduction: Kit Fine on Truthmakers,Relevance, and Non-Classical LogicFederico L.G. Faroldi and Frederik Van De PutteAbstract Kit Fine’s contribution to logic is vast and diverse; the chapters in thisbook deal with a signi?cant part of it. In this introductory chapter, we clarify andcontextualize the main themes of Fine’s work that are centre stage in this book, afterwhich we give a summary of each chapter.1 Introducing the IntroductionThis book appears in the series Outstanding Contributions to Logic and is devotedto Kit Fine (1946, Farnborough - England). It consists of 15 original research papersdealing with the formal and philosophical aspects of various themes at the center ofFine’s work. Some are critical, in-depth discussions of his published work, othersapply his ideas to new problems, and still others use Fine’s contribution to developnew perspectives on a classical topic. Each of the chapters is accompanied by anin-depth reply by Fine himself. Finally, the book also features a brief autobiographyand an exhaustive list of Kit’s publications.As editors, it has been a great pleasure to notice that the contributions are currentand they engage with Fine’s research in a creative, critical, and constructive man-ner. It has been a privilege to work with such talented philosophers and logicians.All contributions have moreover been thoroughly peer-reviewed and we thereforewish to acknowledge the crucial role played by the reviewers in the genesis ofthis book: Guillermo Badia, Willem Conradie, Louis deRosset, David Fernandez-Duque, Jeremy Goodman, Berta Grimau, Jesse Heyninck, John Horty, AndrewIrvine, Harvey Lederman, Stephan Leuenberger, David Makinson, Ondrej Majer, EdName of First AuthorName, Address of Institute, e-mail: name@email.addressName of Second AuthorName, Address of Institute e-mail: name@email.address1
2 Federico L.G. Faroldi and Frederik Van De PutteMares, Hiroakira Ono, Francesco Paoli, Pawel Pawlowski, Bryan Pickel, FrancescaPoggiolesi, Adam Prenosil, Tudor Protopopescu, Rasmus Rendsvig, Arthur Schip-per, Sebastian Sequoiah-Grayson, Shawn Standefer, Johannes Stern, Eric Swanson,Zach Weber, Yale Weiss, Malte Willer, Robert Williams.Some of the groundbreaking contributions of Kit Fine to logic are well-known,such as the discovery of an incomplete logic containing S4 (Fine, 1974) and thecanonicity of elementary modal logics (Fine, 1975b). Fine also did pioneering workconcerning the now standard method of ?ltration in modal logic (as highlighted inChapter 3 of this book) and developed a highly original semantics for relevancelogic (as reviewed in Chapter 7 of this book).However, Kit Fine has not only solved open problems in logic and related disci-plines, but also created (or discovered!) new problems and questions. While not thefocus of this book, Fine is a towering ?gure in contemporary metaphysics as well.More generally, he introduced or transformed ideas that have shaped the debate andinquiry, some of which are taken up by contributors to this book. Notwithstandingthe diversity of topics he worked on, there is great unity in his method, using for-mal logic where it helps to clarify certain distinctions and to develop his views inrigorous terms, but always driven by the underlying philosophical issues.In the remainder of this chapter we will brie?y introduce the central themestouched upon later in the book, and highlight how the following chapters contributeto one or another of them, without any claim to have exhausted the multifaceted,continuing conversation that unfolds in these pages. In particular, Section 2 is a gen-tle introduction to some of the main topics of the rest of the book. Section 3 presentsa short outline of each of the original chapters of the book. Section 4 concludes theintroduction with some lighter, but not less important, notes.2 Finean Themes of This BookAs noted above, Kit Fine’s in?uence on contemporary philosophy is enormous andhis contributions are fundamental. In what follows, we focus on those themes inhis work that are particularly relevant for the chapters that follow. An overview ofthe many topics we don’t even touch upon in this introduction, such as essence,semantic relationism, abstraction, metametaphysics and the foundations of mathe-matics, can be found in a recent encyclopedic entry devoted to Fine’s work (Raven,forthcoming).2.1 Truthmakers: A Hyperintensional RevolutionIn the past half century, possible worlds have played a key role not just as a toolin the semantics of modal logics, but also in the philosophy of language and meta-
Introduction: Kit Fine on Truthmakers, Relevance, and Non-Classical Logic 3physics. Fine has argued extensively that the job done by possible worlds is oftenbetter realized with other means, notably (albeit not exclusively) with truthmakers.The concept of a truthmaker is not entirely novel, neither did Fine invent it. Be-fore him, however, it was mostly used within a metaphysical project: what it is,out there in the world, that makes true what we take to be truth-apt (sentences,propositions, etc.). Kit Fine’s interest, when it comes to truthmakers, is mostly ina semantical project: how it is that sentences (propositions, etc.), by virtue of theirvery meaning, are made true by what’s out there, in the world.1We are used to thinking of a world as a complete, consistent state-of-affairs thatsettles the truth value of every proposition. Thus, ’Verdi composed Aida’ is true ata world just in case at that world Verdi composed Aida. But that world will alsocontain a lot of irrelevant material to Verdi and Aida, such as that Bach composedcello sonatas, or that Peano arithmetic is incomplete. In contrast, truthmakers canbe thought of as parts of (possible) worlds that verify or falsify propositions andare wholly or exactly relevant to it. They are neither necessarily complete nor con-sistent.2However, just as with possible worlds, states can be regarded as arbitrarypoints with little additional structure but for the fact that they are equipped with amereological (“parthood”) relation that is required to ful?ll a completeness condi-tion. This makes the framework of truthmakers very abstract and ?exible.Once such a state and parthood relation is ?xed, one still needs to determine howstates make statements true. As Kit Fine points out, the truthmaking relation comesin three forms: exact, inexact, and loose:Loose veri?cation is a purely modal notion. A state or situation will loosely verify a state-ment just in case it is necessary that if the state obtains then the statement will be true.Exact and inexact veri?cation, by contrast, require that there be a relevant connection be-tween state and statement. With inexact veri?cation, the state should be at least partiallyrelevant to the statement; and with exact veri?cation, it should be wholly relevant (Fine,2017d).The state of Sicily being an Italian island is an exact truthmaker of the sentence’Sicily is an Italian island’, but the state of Sicily being an Italian island and Sicilyhaving amazing architecture is only an inexact truthmaker of the sentence ’Sicily isan Italian island’.“Exact” truthmaking clauses were given for the ?rst time in van Fraassen, 1969.In the heyday of situation semantics, inexact veri?cation was common. To illustrateone instance where exact and inexact truthmaking differ on the syntactic level, takeAand A?(A?B). In classical logic, these two expressions are obviously equivalent.Moreover, they have the same inexact truthmakers. Yet, they don’t have the sameexact truthmakers: an exact truthmaker for A?Bmay be “too speci?c”, and henceonly be an inexact trutmaker for A. If one takes the sameness of exact truthmakersas a criterion of equivalence, then Aand A?(A?B)are not necessarily equivalent.1For this distinction and an introduction to the semantical project, see Fine, 2017d. For a uni?ed(i.e. metaphysical and semantical) use of truthmakers see Jago, 2018.2Some of these ideas have also been used in situation semantics, see Barwise and Perry, 1981 andBarwise and Etchemendy, 1990.
4 Federico L.G. Faroldi and Frederik Van De PutteAn important question is which of these three notions (exact, inexact, loose)is fundamental, in the sense of being irreducible to the other(s). Fine argues thatthe notion of exact truthmaking is fundamental (Fine, 2017d, p. 565), others argueinstead that it can be de?ned via the inexact notion (cf. e.g. Deigan, 2019). Boththe inexact and the exact notion of truthmaking are used in various chapters of thisvolume.3With truthmaker semantics, one can also get a better formal grip on the phe-nomenon of hyperintensionality. Very generally, a context is hyperintensional whenclassically logically (or necessarily) equivalent contents cannot be substituted salvaveritate in its scope. (Sameness of) exact truthmaking provides a semantic basis foraccepting hyperintensionality. It therefore cannot come as a surprise that this per-spective on truthmaking has started to be widely used to capture many phenomenathat have been argued to require a hyperintensional analysis. Truthmaker seman-tics, both in inexact and exact forms, has been applied by Kit Fine to counterfactualconditionals (Fine, 2012a), the notion of “ground” (Fine, 2012c), intuitionistic logic(Fine, 2014b), and analytic equivalence (Fine, 2016a). Fine studies how to applytruthmaker semantics to permission (Fine, 2014a) and to imperatives and from thereto deontic modals, in an ongoing series of papers (Fine, 2015a,b). We introducesome of these applications in the next subsections (namely counterfactual condi-tionals, grounding, and subject matter). After that, we move to other Finean topics,viz. arbitrary objects and vagueness.2.2 Counterfactual ConditionalsA counterfactual conditional is a conditional of the form: ”If Xwere (had been) soand so, then Ywould be (would have been) such and such.” Alongside the indica-tive conditionals, it has been known for a long time that counterfactual conditionalscannot be captured by material or (classical) strict implication. Most notably, theyviolate strengthening of the antecedent: from “If it had rained, the grass in my gar-den would have been wet”, one cannot infer “If it had rained and I had put up a gianttent covering my entire garden, the grass in my garden would have been wet”.The very in?uential Stalnaker-Lewis (cf. e.g. Lewis, 1973) treatment maintains,in one of its forms, that such a counterfactual is true if and only if the consequent istrue at all the closest possible worlds in which the antecedent is true. Given a clas-sical account of possible worlds, this however implies that counterfactuals satisfyreplacement of tautological equivalents, both in the antecedent and the consequent.Fine argued against this principle for counterfactuals (Fine, 1975a). More recently,he developed a truthmaker-theoretic account of counterfactuals Fine, 2012a.In his contribution to the present book, Andrew Bacon (Chapter 17) criticizesFine’s account, studying two paradoxes and arguing for a possible worlds analysisof counterfactuals, though distinct from the Stalnaker-Lewis account.3Cf. also possibility semantics (van Benthem et al., forthcoming) and Leitgeb’s semantics for hisHYPE system (Leitgeb, 2018).
Introduction: Kit Fine on Truthmakers, Relevance, and Non-Classical Logic 52.3 Subject Matter and AboutnessThe subject matter of a proposition is, roughly, what the proposition is about.Truthmakers ?gure prominently in more recent theories of subject matter (seee.g. Yablo, 2014 and Fine, 2017a and the ensuing discussion Fine, 2020; Yablo,2018). For Fine, the subject matter of a proposition may be identi?ed with its max-imal truthmaker, i.e. the fusion of all its veri?ers. Mereological relations on subjectmatters will then be had quite easily via the mereological relations already presenton states.In Chapter 19 of the present book, Alessandro Giordani compares the Finean andthe classical account of subject matter, and proposes a synthesis of both. Aboutnessis also central in Chapter 23, in which it is argued that truthmakers can help us geta grip on what it means that a given statement is “the whole truth regarding a givensubject matter”.2.4 GroundingGrounding is a much studied concept that is thought to capture considerations offundamentality in metaphysics and in other domains. For instance, we can say thata conjunction is grounded in its conjuncts, or that the singleton of Socrates exists invirtue of Socrates existing. How to de?ne grounding more accurately is in fact partof the debate on grounding itself.The contributions of Fine to the debate on grounding, as a non-causal, explana-tory, primitive, hyperintensional notion, are numerous and well-documented (for anentry point see e.g. Fine, 2012b). Grounding is mostly thought of in metaphysicalterms, but Fine has suggested also natural and normative grounding notions.How to express grounding? Two options are the relational and the operationalapproach. The former expresses grounding with a predicate, thus suggesting thatgrounding is a relation between that which is grounded, and that which grounds.The latter expresses grounding as a variably polyadic operator between sentences.An advantage of the operational approach would be that it does not commit one toa speci?c position on the metaphysical status of grounding.In the context of formally modeling the grounding relata, several additional dis-tinctions can be introduced. While a conjunction is fully grounded in its conjuncts, itis only partially grounded in each conjunct. One can study the pure logic of ground-ing and the impure logic of ground. The former is concerned only with the structuralprinciples of the grounding operator, the latter also takes into account the internalarticulation of what grounds and what is grounded. One can moreover distinguisha worldly notion of grounding from a conceptual, or representational, notion ofgrounding.In his contribution to the present book, Fabrice Correia offers a novel semanticframework for the worldly notion that is intended to improve upon his own (Cor-reia, 2010) and Fine’s (Fine, 2012c) previous accounts (Chapter 25). Some formula-
6 Federico L.G. Faroldi and Frederik Van De Puttetions of ?ne-grained grounding principles lead to inconsistencies and paradoxes. InChapter 21 of this book, Peter Fritz shows how one can solve some of these issuesin broadly Finean terms.2.5 Arbitrary ObjectsArbitrary objects have been criticized as either useless or inconsistent since at leastLocke and Berkeley. Yet, ordinary reasoning uses arbitrary objects for quanti?ca-tional inferences, which are then suitably regimented e.g. in natural deduction sys-tems. For instance, we conclude that all individuals have a certain property by show-ing that an arbitrary individual has that property. This rule of universal generaliza-tion allows us to infer ?xf(x)from f(a), given certain restrictions, and it is part andparcel of mathematical reasoning too.Fine (Fine, 1985) gave rigorous foundations to our reasoning (e.g. in mathemat-ics, natural deduction, anaphora) with arbitrary objects. He developed a theory thatis far better developed and coherent that was on the market, and he applied it tomany ?avors of generality. More recently, Fine applied these foundations to accountfor the much discussed Cantorian abstractionist constructions of cardinal numbersand order types (Fine, 1998), to a general account of types or forms (Fine, 2017b),to identity criteria (Fine, 2016b), and to providing uni?ed foundations for essenceand ground (Fine, 2015c).In their contribution to the present book, Leon Horsten and Ryo Ito criticallydiscuss Fine’s conception of arbitrary objects by comparing it to Russelll’s (Chapter29). Remarks on arbitrary objects also appear in the contributions of Jago (Chapter9 and Bimb´o and Dunn (Chapter 7).2.6 VaguenessA signi?cant portion of natural language is vague. The predicate ’is bald’, for in-stance, is commonly thought to admit of borderline instantiations: its meaning doesnot settle whether a particular man, Al, with thinning hair is bald or not. In thissimple version, this is a problem for classical logic, because it does not make senseto say that ”Al is bald or is not bald” if ’is bald’ is vague. This raises the problemof formulating a suitable logic and semantics that can handle vague sentences in anappropriate way.Fine, 1975c formulated an in?uential supervaluationist account of vagueness, ac-cording to which, roughly, a vague sentence is true if and only if it is true for all waysof making it completely precise. More recently, however, Fine stressed a distinctionbetween local vagueness – it is indeterminate whether a predicate applies in a singlegiven case – and global vagueness – it is indeterminate whether a predicate appliesacross a range of cases (Fine, 2008). In particular, he showed that it is impossible to
Introduction: Kit Fine on Truthmakers, Relevance, and Non-Classical Logic 7understand global indeterminacy in terms of local indeterminacy (while leaving theother direction open).(Fine, 2017c) suggests a novel account that focuses on the global character ofvagueness, and maintains that it can be captured by logical means alone (i.e. withoutemploying a distinctive vagueness-theoretic notion). Andreas Ditter challenges thisnew possibility result in his contribution to the present book (Chapter 31).3 Outlines of the ChaptersAs we explained above, Kit Fine’s contribution to logic does not merely consist ina range of particular results in various sub?elds of logic; rather, what makes himstand out is the fact that he managed to relate each of these, and to make fruitfuluse of ideas and solutions for one topic, in dealing with another. The same canbe said for the chapters in this book: each of them relates to various others, be itin methodology, formal apparatus, or conceptual targets. This will already partlybecome salient in the summary that follows.We start with two chapters that relate directly to Kit Fine’s older contributions inmodal and relevant logic. Chapter 3 zooms in on the well-known ?ltration method,on which Fine did pioneering work in the 1970s (cf. supra). Johan van Benthemand Nick Bezhanishvili investigate this method from a wide range of perspectives,including model-theoretic, proof theoretic, and dynamic-epistemic ones. The resultis both a solid overview of existing work, and an inspiring venture into unknownterritory, related to ?ltration.In Chapter 5, Vladimir Lifschitz outlines the history of the central idea of “stablemodels” in the context of the Prolog programming language and, later, answer setprogramming. As explained in this chapter, Fine occupies a special place in thishistory, with his notion of “felicitous models” as one alternative formulation of thestability idea. This chapter thus provides a nice illustration of how fundamental workin philosophical logic may turn out to be of importance for more applied work incomputer science.Katalin Bimb´o and J. Michael Dunn (Chapter 7) recall Fine’s two-sorted seman-tics for relevance logic and related work. Their chapter leads to a detailed compari-son of Fine’s semantics to more well-known Routley-Meyer semantics for relevancelogics. Interestingly, it is also shown how this work relates to the notion of arbitraryobjects (cf. supra and Chapter 29).This brings us to a number of chapters that all use truthmaker semantics in oneway or another. Chapter 9, by Mark Jago, consists of two related parts. The ?rst partdiscusses the logical relation of disjunctive parthood and links it to the semanticnotion of re?nement and the philosophical determinable-determinate relation. In thesecond part, the author develops a formal logic on the basis of these ideas, whichincludes a relevant conditional. This chapter moreover touches on the subject ofvagueness (cf. Chapter 31) and arbitrary objects (cf. Chapter 29), and Jago’s formalsemantics is clearly related to Chapter 13 in its treatment of disjunction.
8 Federico L.G. Faroldi and Frederik Van De PutteChapter 11 starts from Fine’s exact truthmaker semantics for intuitionistic logicand asks whether it can be generalized to other non-classical logics. In this chapter,Ondrej Majer, V´it Pun?coch´a?r and Igor Sedl´ar provide an exact truthmaker seman-tics for the nonassociative Lambek calculus and some of its extensions, such as theimplicational fragment of the relevant logic R. They draw interesting similarities be-tween their generalization of Fine’s semantics and Urquharts semilattice semanticsfor R.In a similar vein, Peter Verd´ee combines the notion of exact truthmaking withRestall’s closure frames semantics for substructural logics without distribution anda modal operator, to obtain a truthmaker semantics for four non-transitive relevantlogics (Chapter 13). Following Verd´ee’s earlier work (cf. Peter Verd´ee and Samonek,2019; Verd´ee and Bal, 2015), the four logics are de?ned syntactically as the rele-vant cores of classical logic, Priest’s logic of paradox, strong Kleene logic, and ?rstdegree entailment respectively. The proof of soundness and completeness moreoverproceeds via a third characterization of the logics using sequent calculi, and in-volves an innovative view on relevant proofs in terms of the network structure of thesequents in those proofs.In their chapter, Peter Hawke and Ayb¨uke ¨Ozg¨un ask whether truthmaker se-mantics can be used to solve certain well-known problems of traditional, classicallogic-based epistemic logic. Starting from a rich formal language that distinguishesa priori knowability, suf?ciency for knowledge, and a priori (knowledge-level) im-plication, the authors of this chapter focus on a range of ?rst-order logical princi-ples and discuss counterexamples to those principles. They note that each of theprinciples are valid on the traditional, Hintikka-style semantics for epistemic logic.Finally, they study the behavior of those principles in six different truthmaker se-mantics for the same formal language, that are distinguished by their treatment ofsuf?ciency for knowledge.Chapter 17 deals with the semantics of counterfactual conditionals. Here, An-drew Bacon starts by explaining two puzzles that involve such conditionals: Yablo’sbutton and Bernadete’s paradox. After carefully formalizing the arguments that areinvolved in both, he goes over the various principles that could be given up in orderto avoid the paradoxes. While Fine’s diagnosis is that replacement of equivalentsshould be weakened, thus pointing towards a truthmaker semantics of counterfactu-als (cf. supra), Bacon argues that this move still leads to serious problems as long ascertain very weak logical principles are in place. Instead, he suggests rejecting whathe calls the “Disjunction” principle for counterfactuals, and to endorse a semanticsin terms of a speci?c possible worlds semantics that works with selection functions.Chapter 19 compares Fine’s theory of subject matter to Lewis’s, and it proposes acharacterization based on the information structure model. Giordani aims to developa theory of subject matter that is able to capture both hyperintensionality (a keyfeature of Fine’s account) and context-variation (a key feature of Lewis’s), amongother features.In his contribution to this volume, Peter Fritz sets out to discuss various inconsis-tency results arising from a ?ne-grained conception of propositions, results similarto those arising from naive set comprehension (Chapter 21). Fritz draws on Fine’s
Introduction: Kit Fine on Truthmakers, Relevance, and Non-Classical Logic 9potentialist theory when it comes to set-theoretic inconsistency to explore what asimilarly potentialist theory of propositions might look like. The principles he pro-poses for such a theory are shown to be consistent with a model construction. Aninteresting expressive limitation emerges when trying to apply this theory to ground-ing. The author also discusses using truthmaker semantics to give a theory of propo-sitions, noting along the way some dif?culties in understanding the notion of ’state’used in truthmaker semantics.Chapter 23 puts a fundamental, but often overlooked concept on the table: that ofthe whole truth on a given subject-matter. As Stephan Kr¨amer argues in this chapter,this notion is intricately linked to the modality “and that’s it”, whereby one expressesthat a given statement covers everything there is to say on the subject at hand. It isshown that both presuppose a form of relevance, and that possible worlds semanticscannot adequately capture them. Instead, the author proposes a truthmaker accountof “the whole truth” and the corresponding modality.Fabrice Correia’s chapter deals with the logic of worldly grounding. In particular,it develops a synthesis of the author’s own earlier semantic approach (Correia, 2010)and Fine’s (Fine, 2012b, Fine, 2012c), in order to overcome various defects of both.After a detailed comparison and critique of the existing approaches, the new one isworked out and applied to the semantic characterization of various proof systemsfor ground-theoretic notions.In Chapter 27, Daniel Rothschild and Stephen Yablo focus on the notion of apermissive update, i.e., the information change caused by granting a permissionin a context where before it may have been forbidden. They argue quite convinc-ingly that the various existing approaches to this notion – all based on classicallogic or possible worlds semantics – fall short in one way or another, and developa truthmaker-based approach instead. This approach includes a subtle update mech-anism on sets of truthmakers, which may in turn inspire new work on the logic ofbelief change.In Chapter 29, arbitrary objects take centre stage. Here, Leon Horsten and RyoIto argue that Fine’s theory of arbitrary objects has an early precursor in Russelll’snotion of variables in his Principles of Mathematics, be it that they differ in theway the dependence relation between variables is treated. This observation goessomewhat against earlier remarks by Fine himself “A Defence of Arbitrary Objects”,and is used as the starting point for an in-depth historical overview of Russell’s workon variables, which may help to revive the philosophical debate on arbitrary objects.Chapter 31 raises a problem for Fine’s recent theory of vagueness and the re-lated possibility result concerning the af?rmation or denial of all propositions in aSorites-like sequence (Fine, 2017b). In his contribution, Andreas Ditter argues thatFine’s possibility result as well as his novel solution to the sorites paradox fail incertain legitimate extensions of the formal language employed. Ditter shows thatone can prove a new impossibility result in extensions of the language containinga negation operator that obeys reductio ad absurdum, and argues that at least somesuch extensions are unobjectionable. In particular, it is shown that an operator thatbehaves exactly like intuitionistic negation, which obeys reductio, can be de?ned ina natural propositionally quanti?ed extension of Fine’s logic.
10 REFERENCESFinally, in Chapter 33, Kit Fine and Errol Martin provide a formal account ofnon-circular reasoning, i.e. of reasoning in which the conclusion of an argumentis not somehow presupposed in its premises. Martin (along with R. Meyer) hadpreviously shown that the implicational system P-W does not contain any theoremsof the form A?A. Fine and Martin then extend this result to systems that alsocontain conjunction.The book ends with an exhaustive bibliography of Kit Fine’s published works(Chapter 20).4 The ManThe preceding, and this book in its entirety, not to mention his collaborative spiritthroughout this endeavor, should already go some way towards indicating the impor-tance of Fine’s work for philosophical logic as it is today. But Fine’s contributionsto Logic (and Philosophy) are not just in the form of papers or books. For thosewho have had the pleasure to attend them, his presentations are remarkable for theirclarity and inspiring nature. As a witness of this, one would sometimes see logi-cians ?ght over who gets to take the ?ipcharts of his keynote lecture home. Finally,with his mentorship, sense of humor, love for music (and good food!), charitable,constructive questions, he contributed to many young logicians’ and philosophers’careers, scienti?c or not.ReferencesBarwise, J. and J. Etchemendy (1990), “Information, infons, and inference”, in Sit-uation Theory and Its Applications, ed. by R. Cooper, K. Mukai, and J. Perry,CSLI Lecture Notes, pp. 33-78. (Cited on p. 3.)Barwise, J. and J. Perry (1981), “Situations and Attitudes”, The Journal of Philoso-phy, 78, pp. 668-91. (Cited on p. 3.)Correia, Fabrice (2010), “Grounding and Truth-Functions”, Logique & Analyse, 53,211, pp. 251-79. (Cited on pp. 5, 9.)Deigan, Michael (2019), “A plea for inexact truthmaking”, Linguistics and Philos-ophy,DO I:10.1007/s10988-019-09279-2. (Cited on p. 4.)Fine, Kit (n.d.), “A Defence of Arbitrary Objects”, Proceedings of the AristotelianSociety, 57, pp. 55-77. (Cited on p. 9.)– (1974), “An Incomplete Logic Containing S4”, Theoria, 40, 1, pp. 23-29, DOI:10.1111/j.1755-2567.1974.tb00076.x. (Cited on p. 2.)– (1975a), “Critical Notice of Lewis, Counterfactuals”, Mind, 84, 335, pp. 451-458. (Cited on p. 4.)
REFERENCES 11– (1975b), “Some connections between elementary and modal logic”, in Proceed-ings of the Third Scandinavian Logic Symposium, ed. by Stig Kanger, North-Holland, Amsterdam, pp. 15-31. (Cited on p. 2.)– (1975c), “Vagueness, Truth and Logic”, Synthese, 30, 3-4, pp. 265-300, DOI:10.1007/BF00485047. (Cited on p. 6.)– (1985), Reasoning with Arbitrary Objects, Aristotelian Society. (Cited on p. 6.)– (1998), “Cantorian Abstraction: A Reconstruction and Defense”, Journal of Phi-losophy, 95, 12, pp. 599-634, DOI:jphil1998951230. (Cited on p. 6.)– (2008), “The Impossibility of Vagueness”, Philosophical Perspectives, 22, 1,pp. 111-136, DOI:10 . 1111 / j . 1520 - 8583 . 2008 . 00143 . x. (Citedon p. 6.)– (2012a), “Counterfactuals without Possible Worlds”, Journal of Philosophy, 109,3, pp. 221-246. (Cited on p. 4.)– (2012b), “Guide to Ground”, in Metaphysical Grounding, ed. by Fabrice Correiaand Benjamin Schnieder, Cambridge University Press, pp. 37-80. (Cited on pp. 5,9.)– (2012c), “The Pure Logic of Ground”, The Review of Symbolic Logic, 25, 1,pp. 1-25. (Cited on pp. 4, 5, 9.)– (2014a), “Permission and Possible Worlds”, Dialectica, 68, 3, pp. 317-336.(Cited on p. 4.)– (2014b), “Truth-Maker Semantics for Intuitionistic Logic”, Journal of Philo-sophical Logic, 43, 2-3, pp. 549-577. (Cited on p. 4.)– (2015a), “Compliance and Command, I”, ms. (Cited on p. 4.)– (2015b), “Compliance and Command, II”, ms. (Cited on p. 4.)– (2015c), “Uni?ed Foundations for Essence and Ground”, Journal of the Ameri-can Philosophical Association, 1, 2, pp. 296-311, DO I:10.1017/apa.2014.26. (Cited on p. 6.)– (2016a), “Angellic Content”, Journal of Philosophical Logic, 45, 2, pp. 199-226.(Cited on p. 4.)– (2016b), “Identity Criteria and Ground”, Philosophical Studies, 173, 1, pp. 1-19,DO I:10.1007/s11098-014-0440-7. (Cited on p. 6.)– (2017a), “A Theory of Truthmaker Content II: Subject-Matter, Common Con-tent, Remainder and Ground”, Journal of Philosophical Logic, 46, 6, pp. 675-702. (Cited on p. 5.)– (2017b), “Form”, Journal of Philosophy, 114, 10, pp. 509-535, DOI:10.5840/jphil20171141036. (Cited on p. 6.)– (2017c), “The Possibility of Vagueness”, Synthese, 194, 10, pp. 3699-3725, D OI :10.1007/s11229-014-0625-9. (Cited on p. 7.)– (2017d), “Truthmaker Semantics”, in A Companion to the Philosophy of Lan-guage, ed. by Bob Hale, Crispin Wright, and Alexander Miller, 2nd ed., ms,Blackwell, London, pp. 556-77. (Cited on pp. 3, 4.)– (2020), “Yablo on Subject-Matter”, Philosophical Studies, 177, pp. 129-171.(Cited on p. 5.)Jago, Mark (2018), What Truth Is, Oxford University Press, Oxford. (Cited on p. 3.)Leitgeb, Hannes (2018), “HYPE”, Journal of Philosophical Logic. (Cited on p. 4.)
12 REFERENCESLewis, David Kellogg (1973), Counterfactuals, Harvard University Press, Cam-bridge (MA). (Cited on p. 4.)Peter Verd´ee, Inge De Bal and Aleksandra Samonek (2019), “A non-transitive rel-evant implication corresponding to classical logic consequence”, AustralasianJournal of Logic, 16, 2, pp. 10-40. (Cited on p. 8.)Raven, Mike (forthcoming), “Kit Fine”, Entry for the Internet Encyclopedia of Phi-losophy. (Cited on p. 2.)Van Benthem, Johan, Wesley H. Holliday, and Nick Bezhanishvili (forthcoming),“A Bimodal Perspective on Possibility Semantics”, Journal of Logic and Com-putation. (Cited on p. 4.)van Fraassen, Bas C. (1969), “Facts and Tautological Entailment”, Journal of Phi-losophy, 66, 15, pp. 477-487. (Cited on p. 3.)Verd´ee, Peter and Inge De Bal (2015), “A New Approach to Classical Relevance”,Studia Logica, 103, 5, pp. 919-954. (Cited on p. 8.)Yablo, Stephen (2014), Aboutness, Princeton University Press, Princeton. (Cited onp. 5.)– (2018), “Reply to Fine on Aboutness”, Philosophical Studies, 175, 6, pp. 1495-1512. (Cited on p. 5.)
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