Divine Contradiction (Oxford Studies in Analytic Theology) 9780192845436, 0192845438

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OXFORD STUDIES IN ANALYTIC THEOLOGY Series Editors Michael C. Rea

Oliver D. Crisp

OXFORD STUDIES IN ANALYTIC THEOLOGY Analytic Theology utilizes the tools and methods of contemporary analytic philosophy for the purposes of constructive Christian theology, paying attention to the Christian tradition and development of doctrine. This innovative series of studies showcases high quality, cutting-edge research in this area, in monographs and symposia. p ublished titles in c luded: Essays in Analytic Theology Volume 1 & 2 Michael C. Rea The Contradictory Christ Jc Beall Analytic Theology and the Academic Study of Religion William Wood Divine Holiness and Divine Action Mark C. Murphy Analytic Christology and the Theological Interpretation of the New Testament Thomas H. McCall Fallenness and Flourishing Hud Hudson Analyzing Prayer Theological and Philosophical Essays Edited by Oliver D. Crisp, James M. Arcadi, and Jordan Wessling Explorations in Analytic Ecclesiology That They May be One Joshua Cockayne Gracious Forgiveness A Theological Retrieval Cristian F. Mihut Accountability to God Andrew B. Torrance

Divine Contradiction JC BEALL

Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Jc Beall 2023 The moral rights of the author have been asserted All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2022950229 ISBN 978–0–19–284543–6 DOI: 10.1093/oso/9780192845436.001.0001 Printed and bound in the UK by Clays Ltd, Elcograf S.p.A. Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

. . . not for those who seek the stage; nor for those whose aim is Win! just for those whose aim is truth; right until the very end. . . .

Preface Aim The aim of this book is to augment the theology developed in its predecessor (viz., The Contradictory Christ, 2021). The predecessor responded to the so-called logical problem of God incarnate; this book responds to the so-called logical (and, relatedly, 3-1-ness) problem of trinitarian reality. Just as in the predecessor, this book aims to accommodate the ‘standard christian theory’ of God – ‘orthodox’ christian theology – as constrained at least by Chalcedon 451 and the ‘axioms’ of the Athanasian Creed. Given said aims, a great deal of important work in other directions is not discussed. Important along these lines is Dale Tuggy’s arguments against the standard christian theory, and in particular against trinitarian theology (Tuggy, 2003, 2004, 2021). Tuggy’s work demands engagement but the engagement does not happen in this work. Instead, this work just assumes the standard axioms of trinitarian theology, and assumes them, per the first chapter, in the simple, flat-footed fashion from which much (if not all) work on ‘the logical problem’ takes off. Additionally, work in the direction of contradictory islamic theology, such as that of Chowdhury (2021), is not discussed, not because it’s not important from a christian-theological framework, but rather in the name of keeping the book focused specifically in christian analytic theology. Finally, in addition to mountains of work directly on ‘the logical problem’ that, with an aim of streamlining and concise presentation, I could not adequately address, there appears to be precedent, at least in some respects, to the contradictory theology that I advance. If Basil Lourié is correct (2019a; 2019b), Byzantine patristics advanced not only a contradictory christology but also a contradictory account of trinitarian reality. My sense from Lourié’s work, which is but the sense of one unschooled in medieval language, is that the Byzantine patristics were indeed advancing some sort of contradictory theology but one that is

viii preface along different lines from my work. I am not competent to judge Lourié’s theses about the Byzantine patristics, but I flag them here as additional work that demands further engagement in a different venue.

Previous work As emphasized throughout, this book is the second in a two-book series (of sorts). The jumping point of this book is its predecessor: namely, The Contradictory Christ (2021). For the most part, this work assumes much of its predecessor.

Hope I hope that theologians and theologically informed philosophers engage with this book. I believe that the book advances discussion of ‘the logical problem’ (and the 3-1-ness problem) in various ways. One way is simply a framework for discussing logic, identity relations, theology (qua theory of God) and counting conventions in the context of trinitarian theory. Another advance, I hope, is that the account is closer to the truth than previous accounts. Needless but still sad to say, a lot of academics these days just publish to be noticed or to get promotion or the like. In the world of contemporary academia, which has nearly completed its transformation into typical corporate models, this is not surprising. A troubling effect of such practices is that the output is often flashy without substance, driven by some sense of competition or ‘debate winning’ that I have never understood. I have no time or energy for such pursuits. Readers, I hope, will engage this book in the spirit in which it was written: the cold, hard, and humble pursuit of truth – no flash, no whistles, no ‘ding the debater wins’. The book undoubtedly fails on many fronts, but I hope that its main theses, ideas, and for-later questions help in what I hope is our joint quest not only to get at the truth but to understand it.

preface ix

Target audience As with The Contradictory Christ (2021) the target audience is broad: theologians, philosophers, students, church leaders, and the many nonacademic readers who are as aware of target ‘logical’ problems as the academics and reflective church leaders. With such a varied audience in mind, the writing shoots for userfriendliness and big pictures over the nitty-gritty details. This is not to say that details are not given. Details must be given in the context of systematic theology, particularly when it comes to obvious ‘logical problems’ confronting core axioms. But there are details and there are details. Those seeking more detail than what is provided are ones for whom pointers to other work are given.

Conventions governing style The conventions governing style are exactly per the preface of The Contradictory Christ (2021); they are repeated almost verbatim here. Religious names and adjectives: spelling. While its rationale remains unclear to me I nonetheless follow standard English convention in using uppercase ‘C’, ‘J’, and ‘M’ when talking about Christians, Jews, and Muslims, understood as those who, respectively, accept target christian doctrine (for my purposes, doctrine and creeds affirmed at least up through the Council of Chalcedon in 451), jewish doctrine, and islamic doctrine. In turn, and as reflected in the previous sentence, I use lowercase letters when the target terms are in adjectival position. One rationale for this is simply aesthetic. Scattering a multitude of uppercase letters all over a Page makes for Aesthetically displeasing Pages. Another rationale is that my chosen convention is in keeping with related conventions, including the common observation that one can be a Platonist without spending one’s life dreaming of platonic essences or engaging in platonic relations or even endorsing all platonic doctrines. The same considerations apply to theological theories, which are systematic accounts of divine reality. Sometimes, the name of a specific theological theory or

x preface creed involves uppercase letters, but I avoid this when possible. Finally, when it comes to the Athanasian Creed, I spell it just so; however, for at least the reasons just given, the spelling of terms such as ‘athanasian axioms’ or the like is just so. Quotation marks. I use single quotation marks in what, at least in philosophy, is often called the Analysis convention. On this convention, single quotation marks are used in three very different ways, where the given usage is always clear from context. First, I generally use single quotation marks to mention words, as when I mention the word ‘mention’. I also use single quotation marks for direct quotation, as when I directly quote Professor Dr. Greg Restall who, in conversation, once replied to something I said by saying nothing more nor less than ‘Fairdinkum.’ (The only exception to the direct-quotation convention is exceedingly rare: viz., a direct quotation within a direct quotation within a direct quotation. This is so rare that further comment on the convention is unnecessary.) Finally, I use single quotation marks for scare quotes or (equivalently) shudder quotes, as when I say that enjoyable ‘pains’ might not be pains at all. As always, communication rests largely on context, and when it comes to quotation conventions context is the cure. And if one wants a rationale for adhering to the given Analysis convention, I turn to two rationales. First, once again, there is an aesthetic one: single quotation marks do the job without cluttering pages. Second is Ockhamish: why use more marks when fewer will do? Em dashes. Here again, I follow the given Analysis style by spelling an em dash as an en dash flanked by the single-space character – as I do in this sentence. (The other, perhaps more common, way of spelling em dashes does so without spaces—as done in this parenthetical remark.) The Analysis spelling is cleaner; it uses less ink; and most important is that it’s aesthetically more pleasing, at least by my lights. Since both spellings of the em dash do the trick, I use the cleaner, less-ink-using and aesthetically more pleasing one. Displaying or labeling/listing sentences. I often either mention (occasionally use) sentences by either displaying them, as in This sentence is displayed. or labeling or listing them, as in

preface xi 1. Some sentence is true. A14. Some tractors are green. CD. Some trees are green like some tractors. Always (except, perhaps, in this initial example), context makes plain what’s going on – whether the sentences are merely mentioned for some reason, or whether they’re mentioned for purposes of labeling, but then either used in some way (e.g., indirectly used via ascribing truth to the sentence so labeled) or just mentioned by way of their new tags. The reason I’m highlighting this convention is not that it’s unfamiliar or requires explanation; I’m highlighting the convention to explain another convention used a lot in the book. The target convention concerns use of labels. In particular, I refer to (1) – the first sentence above labeled by the numeral ‘1’ followed by a period (the full-stop dot) – by ‘(1)’ but refer to A14 and CD just so – no parentheses required. The difference is that the numeral ‘1’, much like the full-stop dot (the period), has a very standard use in English while neither ‘A14’ nor ‘CD’ do. The latter tags are always interpreted per the given context of use; the numeral for the number one and, likewise, the full-stop dot are always interpreted per their standard and widespread meanings unless the context otherwise makes the matter abundantly clear. Of course, there may be an argument to the effect that flanking ‘A14’ and ‘CD’ with parentheses unifies the target convention with the parentheses flanking ‘1’ in ‘(1)’. That’s true, but here again is where aesthetics – and general ease on the eyes – swims to the surface: why clutter the page with further marks when none would do? The unifies-convention argument doesn’t tip the scales. Chapter relativity of labels. There is one more convention related to using labels for sentences: namely, that if sentences are enumerated using standard numerals, such as (1) above, the label is chapter-relative. If one were to flip through subsequent chapters and see ‘(1)’ in the pages, one would be mistaken to think that (1) above – in the preface (viz., ‘Some sentence is true’) – is being discussed. The convention requires that one stay within the chapter and look for the explicit introduction of the label ‘(1)’ in that chapter, and the sentence so labeled is the sentence being discussed. Chapters, subchapters, etc. Except in this preface, reference to chapters, subchapters, etc. are almost always given by ‘§’ followed by the target

xii preface number, so that ‘§2’ refers to Chapter 2, ‘§2.1’ refers to Section 1 of Chapter 2, and so on: the first numeral following ‘§’ is the chapter, the second (if a second) the section in said chapter, the third (if a third) the subsection, and so on. Authors and their work. Except when referring to my own work, I refer to an author (e.g., Michael Rea) and a piece of the author’s work when I truly say, for example, that Rea (2009) gives a succinct but highly illuminating and equally influential discussion of the so-called logical and threeness-oneness problems in trinitarian theology – fundamental problems confronting the core christian doctrine of trinitarian reality. Here, ‘Rea’ in ‘Rea (2009)’ denotes Rea while the parenthetical item denotes the given work, namely (and here is another convention), Rea 2009. When an author’s (sur-/last/family) name immediately precedes a numeral (usually, the name of a year) or a numeral-cum-letter (e.g., ‘2020a’ or the like) – separated only by one occurrence of the space character (viz., ‘ ’) – the resulting expression is being used to denote the given author’s work, not the author, so that, as above, ‘Rea 2009’ denotes only Rea’s given work, not Rea. In short, ‘Rea (2009)’ denotes both Michael Rea and also Rea 2009, but ‘Rea 2009’ denotes only Rea’s given work, namely, Rea 2009. Similarly, when parentheses enclose a comma flanked by an author’s name and the name of a year (with or without an alphabetical subscript), such as ‘(Rea, 2009)’, the entire expression – parentheses and all – refers to the work and only the work. A similar convention applies when multiple works are cited, except that semicolons are used to separate target items, as in ‘(Coakley, 2002; Cross, 2011; McCall, 2015; Pawl, 2016; Rea, 2003; Stump, 2003)’. In the special case of referring to my own work, I typically use only the year for citation purposes, as I do when I truly say either that the current book is a sequel to one concerning christology (2021) or, equivalently, that 2021 is the prequel (so to speak) of the current work. The only exception to this special yearonly convention is when context demands otherwise.

An unsolicited note on reading the book Mostly repeated from The Contradictory Christ (2021): The chapters are ordered, more or less, in a typical dependency series. There’s one

preface xiii exception. §2 of this book talks about logical consequence (‘logic’, for short), its role in true theories (and, hence, in the true theology), and what, by my lights, is the correct account of logical consequence. Despite my efforts to make §2 a user-friendly, from-scratch presentation, some readers who’ve never studied the mainstream account of logical consequence may find §2 to be a very-low-gear chapter. If you are such a reader, my advice is to read §2 for the big ideas, and slide past the details. The actual details are relegated to Appendix B (p. 141) and, more explicitly and in a much more leisurely and user-friendly fashion, in Chapter 2 of this book’s predecessor. The core theses of the book can be understood without having full competence of the given account of logical consequence.

Acknowledgements All of those acknowledged in this book’s predecessor (viz., The Contradictory Christ, 2021) are hereby sincerely acknowledged again. Without their help in the first stage of this project the current stage would be naught. Three people have been especially active throughout all stages of this book. • Mike DeVito – student, fellow seeker, and invaluable interlocutor. • Mike Rea – colleague and analytic-theological cutting stone. • Jonathan Rutledge – collaborator and source of valuable feedback. Four people were especially helpful in late stages of the book. • James Anderson – philosopher, theologian, and generous critic. • Tom McCall – teacher, source of sharp challenge and equal encouragement. • Greg Restall – friend, collaborator, and a logician’s logician. • Andrew Torrance – theological guide, colleague, and sage sounding board. Others who’ve been helpful with this particular book are Rebecca Brewster Stevenson, Franz Berto, Laura Frances Callahan, Safaruk Chowdhury, A. J. Cotnoir, Oliver Crisp, Richard Cross, Brian Cutter, Franca D’Agostini, Elena Ficara, Luis Estrada González, Joseph Jedwab, Andrew Jaeger, Jeremiah Joven Joaquin, David Lincicum, Basil Lourié, Dylan MacFarlane, Hitoshi Omori, M. d. R. Martínez-Ordaz, Daniel Nolan, Stephen Ogden, Tim Pawl, Martin Pleitz, Graham Priest, PhilipNeri Reese O.P., Dave Ripley, Gill Russell, Rachael Elizabeth Thompson, Sharon Southwell, Andrew Tedder, Dale Tuggy, Zach Weber, and Eric Yang.

xvi acknowledgements Logos Workshop: I’m grateful to the Logos Institute at St Andrews for hosting a workshop on the book, and especially grateful to Dennis Bray, Jon Kelly, Tom McCall, Greg Restall, Jace Snodgrass, and Andrew Torrance for help. Podcast engagement: I’m also grateful to the hosts of various podcasts for questions and engagement that fed the book: • • • • • • • •

Analytic Christian Capturing Christianity Friction Philosophy Furthering Christendom The Logos Podcast London Lycium Parker’s Pensees Trinities

Copyright/overlap: In an effort to make this book accessible on its own, there is some inevitable overlap with The Contradictory Christ (2021). Institutional acknowledgement: I’m grateful to the University of Notre Dame for research support; to the Notre Dame Center for Philosophy of Religion for valuable sessions on the book; to UNAM’s Institute for Philosophy for ongoing engagement; and to the University of St Andrews where, during the last stages of this book, I was a St Andrews Global Fellow. Indexing: I’m very grateful to Abby Concha for indexing the book, taking on the job in the thick of a busy semester. Without Concha’s help the index would be of no help to anyone. Jc Beall Notre Dame 2022

Contents 1. Aim, Scope, Limits, and Main Thesis 1.1 1.2 1.3 1.4 1.5 1.6 1.7

1

Guiding constraints Contradictions true of Christ Theology as logic-bound The Athanasian Creed Robust monotheism The target ‘logical’ problem The target thesis

2 3 5 9 11 13 17

2. Logical and Extra-Logical Entailment

20

2.1 2.2 2.3 2.4 2.5 2.6 2.7

True, false, and beyond Atomic attributions Logical compounds: sentential and quantification Entailment in general: consequence relations Logical entailment: universal Extra-logical entailment: theory-specific Big picture: logic and true theories

3. Trinitarian Identity 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Big picture: God, trinitarian identity, and contradiction Identity and true theories The leibnizian recipe The ingredients Trinitarian identity 3-1-ness entailments and trinitarian identity On the 3-1-ness problem: counting conventions

4. Seven Virtues 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Unified solution: christological and trinitarian contradiction Simplicity No need for analogical or metaphorical gesturing No new-fangled approach to identity relations No unmarked equivocation Metaphysical and epistemological neutrality The mystery of trinitarian reality

20 21 23 25 26 31 34

36 36 43 44 46 51 52 56

69 69 70 71 73 74 75 77

xviii contents

5. Seven Objections 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Contradiction is not a perfection! Divine simplicity! Counting divine reality! Trinitarian entailments should be unrestricted! Clashing christology and trinitarian theory! Heresy! Theology is at most analogical or mere model building!

6. Measuring Some Non-Contradictory Accounts 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Social-trinitarian accounts Pure relative-identity accounts Impure relative-identity accounts: constitution Epistemic-mystery accounts Gap-theoretic accounts Piecemeal theology: losing logic – one more time On the consistency-questing field

7. Towards Future Contradictory Theology 7.1 7.2 7.3 7.4 7.5 7.6 7.7

On omni-property problems Free will and determinism God’s transcendence God’s love God’s creation Denominationally distinct doctrines . . . apart from divine-incarnate and trinitarian reality?

82 82 84 86 90 91 94 95

99 99 101 112 121 125 128 130

131 131 135 136 136 136 137 137

Appendix A Athanasian Creed (tr. Philip Neri Reese, O.P.) Appendix B §2 Appendix: Formal Sketch of FDE

139 141

Bibliography Index

145 151

. . . the Father is God and the Son is God and the Holy Spirit is God; And yet they are not three gods but one god [viz., God]. Athanasian Creed

1 Aim, Scope, Limits, and Main Thesis The problem is that the doctrine [of trinitarian reality] seems to be logically inconsistent and thus necessarily false. (McCall, 2010, p. 11)

This book responds to the so-called logical problem of trinitarian reality (and also to the associated but distinct 3-1-ness problem). Before getting to the problem(s) I flag some salient limits of the project. Limits. The limits of this project are firm, broad, and notable, clearly refuting any suggestion that the current work approaches a full account of trinitarian reality. The literature on trinitarian reality is vast and importantly multi-disciplinary, spanning biblical studies, theology, history, and philosophy, to list a few conspicuous players. The full truth of trinitarian reality, however far our theories successfully approach it, contains answers to issues that go beyond the target ‘logical’ (and associated 3-1-ness) problem(s). A full account contains answers to difficult hermeneutical or interpretation questions that swirl around the target problem (Anatolios, 2011; Ayres, 2004; Barnes, 1998; Coakley, 2004, 1999; Wolfson, 1964). And even beyond detailed historical-cumhermeneutical issues are driving methodological issues that, for good or bad, have driven loud wedges between apparently different approaches to trinitarian reality (Ayres, 2004; Barnes, 1998; Coakley, 1999; Cross, 2002). Such issues are important and must be resolved before a full account of trinitarian reality is achievable. But such issues remain largely untouched in the current work. Other issues on which the current work only implicitly (versus explicitly) touches are ones that every account of trinitarian reality – no matter how partial – must address, certainly if the account applies to the target problem. One such issue is whether the theses that express trinitarian identity (e.g., in the Athanasian Creed, on which more below) involve an Divine Contradiction. Jc Beall, Oxford University Press. © Jc Beall 2023. DOI: 10.1093/oso/9780192845436.003.0001

2 divine contradiction ‘absolute’ or ‘relative’ or some sort of hybrid ‘absolute-relative’ relation (Brower and Rea, 2005; Coakley, 2013; Martinich, 1978, 1979; McCall and Rea, 2009; McCall, 2010; Rea, 2003, 2009). Another issue concerns a commitment to orthodox or standard tradition, avoiding the familiar trio of trinitarian heresies (viz., subordination, modalism, and polytheism), heresies that arise in response to the target ‘logical’ problem (Anatolios, 2011; Crisp and Sanders, 2014; Holmes, 2012; McCall and Rea, 2009). On these issues the current work implies answers, though not via a lengthy rehearsal of alternative answers – leaving that discussion to the enormously large pile of pages already available on trinitarian reality.1 Despite the given limits, the current work provides an account of the target problem. Moreover, the account herein advanced is one driven by christology, as many theologians have demanded. Indeed, the account provides a unified account of structurally similar ‘problems’ confronting the reality of Jesus Christ – who, via contradiction (more on which below), is not only God who is limitless but who, at the same time, walked, sweated, was ignorant, suffered, was able to sin, and died. The apparent contradictions of God incarnate are not the same as those confronting trinitarian reality, but a christology-informed account of the latter – where the incarnation informs apparent contradiction in trinitarian reality – is both natural and theologically sound. After all, it is through the incarnation that trinitarian truths were revealed.

1.1 Guiding constraints Four salient constraints guide the account advanced in this book: • contradictions true of Christ; • theology bound by logic; • the Athanasian Creed (qua widely professed import of the Nicene Creed); • robust monotheism. 1 Tuggy (2021) provides not only a useful bibliographic source but a very accessible and useful discussion of basic issues and positions on trinitarian reality, and Wood (2022) likewise gives useful bibliographic information and accessible discussion of highly relevant surrounding issues.

aim, scope, limits, and main thesis 3 Each of the given constraints is taken as read – a premise of the account. What the constraints come to is elaborated below.

1.2 Contradictions true of Christ Christian theology is driven by the revelations of Jesus Christ. The revelations are as reliable as God’s own word: Jesus Christ is God incarnate, at once divine, and so God, and at the same time human – not human-like, not a look-alike, but human. Being both divine and human is radically strange. That Christ is one person who is divine and human looks to be flatly contradictory, as many non-christian thinkers and christian thinkers have observed. After all, Christ’s being divine entails the essential limitlessness of being God, while Christ’s being human entails the essential limits of being human. At the very crux of christian doctrine appears to be a contradiction: it’s true that Christ is limitless (e.g., without ignorance) but it’s false that Christ is limitless (e.g., clearly ignorant). Put just so, the contradictions of Christ are jarring – a stumbling block to mainstream thinking. God incarnate, who sweats, eats fruit, fishes, stubs his toe and, the biggest stumbling block, dies for those who sinned against him, appears to buck consistent description. But the struggle to both consistently and fully describe Christ is a struggle that has defined christian thinkers for as long as christian thinkers have thought systematically. The so-called christological heresies (e.g., that Jesus Christ is actually two persons exactly one of whom is divine and the other human, or that Jesus Christ is one person but not really human, or that Jesus Christ is one person who is human but not really divine, or one person but neither human nor divine but instead some hybrid human-divine thing, or variations thereof), charitably interpreted, result from sincere, truth-seeking attempts to fully but consistently describe Jesus Christ. The heresies, so understood, now define nonstandard christology; they deviate from the theological axioms of christian theory. But the quest for a consistent account of Jesus Christ continues. The ‘fundamental problem’ (Cross, 2011) at which the given quest aims is just that: the struggle to maintain the christological axioms (viz., that Christ is one person who is truly divine and truly human) while avoiding

4 divine contradiction theological heresy – while avoiding, that is, deviating from the standard theory of Christ (and broader theological theory) that one aims to accommodate. Christian theologians throughout the ages, some more explicitly than others, have subscribed to an important methodological principle: M0.

True theology – the true theory of God – must be done in light of God’s enfleshed revelation, the incarnation of God, Jesus Christ.

Exactly how M0 (so called for ‘methodological rule zero’, the base rule) is to be implemented can be debated. But the gist is clear: let the true christology – the true and full (as possible) account of Christ – strongly inform the rest of the theology, including an account of trinitarian reality. I accept M0 as an important methodological rule. In The Contradictory Christ (2021) I argue that the quest for a logically consistent account of Christ should end. Instead, a contradictory account should be accepted. In short, the appearance of contradiction in Christ is best explained by its being veridical: Christ is a contradictory being, one of whom some contradictions are true (e.g., as above, that Christ knows everything but it’s false that Christ knows everything). Theologians at the Council of Chalcedon (451) faced the same struggle that systematic theologians face today: how to truly and fully describe Christ. To say just that Christ was human – an exemplary human but still human – is to leave too many truths out. To say just that Christ was divine – and so God – is to leave too many truths out. One must say it all: that Christ is without limits because Christ is divine; that Christ is limited because Christ is human; that Christ knows all that there is to know because Christ is divine; that Christ is ignorant of some things because Christ is human; and so on. Reflective thinkers, pursuing not just some of the truth but all of it, are knocked back and forth between contrary properties that the full truth of Christ appears to demand. The Council of Chalcedon (451), which is herein taken to define the standard christology (and broader theology), took the plunge despite the apparently contradictory result: namely, that Christ is human (with all of the limits thereby entailed) and Christ is divine (with all of the limitlessness thereby entailed). Chalcedon, as

aim, scope, limits, and main thesis 5 Coakley (2002) observes, left the contradiction as both strongly apparent but unexplained. And this is not surprising. Again, when confronted with the radical reality of God incarnate, together with the quest to truly and fully describe the event, one easily finds oneself in a familiar position: ‘I don’t see exactly how it all works,’ one reflects, ‘but the simple core – as contradictory as it appears – is just so, and saying anything less is to say too little, to leave fundamentally important truths out.’ My view, spelled out in The Contradictory Christ, is that the true christology – and christian theology more broadly – contains contradictions. Such contradictions don’t take away from the truth of Christ; they add to it. Before the incarnation, we knew at most half the story of God (e.g., God’s limitlessness, omniscience, impeccability, omnipotence, and more); the incarnation, who is Christ, revealed the other half (e.g., God’s limits, ignorance, peccability, weakness, and more). How such a view is logically coherent is spelled out in The Contradictory Christ but the key details, because relevant to the target problem, are presented in the next chapter (viz., §2). The first constraint on the current work is that christological truths – including the contradictions of Christ – play a guiding role in any adequate account of trinitarian reality. When it comes to the target ‘logical’ problem, the contradictory truths of Christ are not only relevant; they serve as one fundamental guiding constraint. How, if at all, the contradictions of Christ affect the target ‘logical’ problem is not obvious, but it would be surprising were such contradictions not to play a role in so-called ‘logical’ problems of trinitarian reality.

1.3 Theology as logic-bound Theology is a truth-seeking discipline; the aim is a true and complete (as possible) theory of the target phenomenon. Part of achieving a complete or full theory, in the relevant sense, involves a consequence or entailment relation that ‘closes’ the theory. These ideas are spelled out in §2; however, the basic idea is straightforward. A consequence relation for a theory churns out all consequences or implications of whatever’s in the theory. Logical consequence (equivalently, logical entailment) looks at the sparse logical vocabulary (e.g., logical conjunction ‘and’, logical

6 divine contradiction negation ‘it is false that’) and churns out the logical consequences of claims in the theory. Example: if the sentence (or claim, or proposition, or what have you) Jesus is human and Jesus is divine. is in the theory, and if the theory is ‘closed under logical consequence’ (i.e., ‘closed under logic’, for short), then so too are both conjuncts, namely, Jesus is human. and Jesus is divine. In other words, a theory is closed under logic (-al consequence) if and only if every logical consequence of anything that’s in the theory is also in the theory. Logical consequence is only one consequence or entailment relation, but it’s a special one tied to the universal logical vocabulary (discussed further in §2). All true and complete theories have consequence relations that build on top of logical consequence, covering the entailments or implications of the theory’s special extra-logical vocabulary. For example, the true theory of metaphysical necessity has an entailment relation that puts any sentence A into the theory when the sentence It is metaphysically necessary that A is in the theory. Similarly, the theory of knowledge (or at least of what’s known) has a relation that puts sentence A into the theory whenever the sentence It is known that A is in the theory. And so on. The central fact is that all true theories are closed under logic (i.e., logical consequence), and most are also closed

aim, scope, limits, and main thesis 7 under an extra-logical or theory-specific consequence relation too – one that puts all theory-specific consequences into the theory. Details of the foregoing ideas are given in §2. Enough is in place to convey the second guiding constraint on the advanced account of trinitarian reality, a constraint tied to the first (viz., the contradictions of Christ). Given that the truth about Christ involves contradictions (per §1.2), any true theological theory avoids all-out absurdity in one of three ways:2 • First Way: the theory is not closed at all – floats free of logical and theory-specific implications of the theory. • Second Way: the theory is not closed under logic but is closed under an extra-logical, theory-specific relation. • Third Way: the theory is closed under logic but logic is weaker than the mainstream account – weaker than so-called classical logic. The first way provides an escape from all-out absurdity in the face of Christ’s contradictions; however, it does so by turning its back on the demands of logical and theology-specific consequences of those contradictions. Indeed, the first way drains theological statements of any entailments – in effect, drains them of content. The second way is similar: it escapes all-out absurdity by refusing to close under logic’s demands, while nonetheless closing under (but only under) theology-specific implications. The second way appears to be more responsible than the first way, and perhaps, in some sense, it is. But ultimately the second way walks away from genuine truth-seeking theology by claiming, on one hand, that all-out absurdity logically follows from the contradictions in the theory but the theology is saved from such an end by simply refusing to listen (so to speak) to logical entailment. What drives the given first and second ways is a refusal to reflect on the mainstream theory of logical consequence. The mainstream theory 2 One may think of an all-out absurd theory as the so-called trivial theory, the theory that contains all sentences of the language of the theory – that is, the theory according to which all sentences (of the language of the theory) are true. For further discussion of the three given ways of responding to theological contradiction see Beall and DeVito 2022, which is on entailment, contradiction, and christian theology.

8 divine contradiction has it that any contradiction in your theory logically entails all sentences in the language of the theory. Hence, if the mainstream theory of logical consequence is true then closing the true but contradictory theology under logic reduces the theology to the all-out absurd one – the trivial theology according to which all claims in the language are true (including, e.g., that the Father is identical to the Son, the Son identical to the Spirit, the Spirit identical to Satan, that evil is the aim of God’s creation, and every other sentence in the language of theology). Whence, inasmuch as Christ is a contradictory being and logical consequence is per the mainstream account, the only responsible option is one of the first two ways. But why think that the mainstream account of logical consequence is the true account of logical consequence? As far as I can see – and I have genuinely looked far – there’s no strong argument for the view that the mainstream account of logical consequence is the true account. Moreover, when weighing the mainstream account versus a slightly weaker, so-called subclassical account of logical consequence, there’s a simple and strong argument for the latter over the former: namely, we lose no true theories but we gain important and viable candidates for true theories (2013b; 2013c; 2018). This argument is particularly germane in the face of the longstanding appearance of Christ’s contradictory being – and perhaps equally so in the face of the target trinity-related problem (more on which below). My way is the third way. Unlike the theologian Dahms (1978), I take theology, qua activity, to be a systematic truth-seeking enterprise that is bound by logical (and likewise extra-logical, theology-specific) entailment. The difference is that I see no reason to think that the mainstream account of logic is anything more than a common restriction of the true account – an important restriction, but a restriction of the true account all the same, and a restriction that simply doesn’t apply to the true theology. The second constraint that guides the current account is just that: namely, that theology is logic-bound, but that logic is weaker than the standard story claims. Sufficient details on the correct account of logic (-al consequence) are given in §2.

aim, scope, limits, and main thesis 9

1.4 The Athanasian Creed While its official voice is in the Niceno-Constantinopolitan Creed (381), the ‘axioms’ (so to speak) of trinitarian reality are widely professed via the so-called Athanasian Creed,3 which begins with the familiar claims that 1. we venerate one God in trinity, and trinity in unity; 2. neither confounding the persons nor dividing the substance. 3. For there is one person of the Father, another of the Son, and another of the Holy Spirit. and ending with the very familiar athanasian identity axioms (so to speak) and an important counting convention, namely, that 4. . . . the Father is God, the Son is God, and the Holy Spirit is God; 5. and yet they are not three gods, but one god (viz., God). The simple, flat-footed – and charitable – reading of the creed begins with a familiar assumption of univocity about predicates and names, as follows. • Predicates and names are univocal throughout. ∘ There is a univocal use of predication (the ‘is’ of predication): the same predication – or exemplification, or instantiation – relation is involved throughout. ∘ There is a univocal use of identity (the ‘is’ of identity): the same identity relation is involved throughout. • ‘God’ is a singular term in some some of the axioms. 3 The creed is so called because it is taken to voice the key views of Athanasias of Alexandria, who, while probably not the author, was a key champion of Nicene theological theory, both with respect to christology and trinitarian theology. Augustinian views have also been identified in the Athanasian Creed. On historical roots see the work of Ware (1997), Pelikan (1971), Kelly (1964, 1977) and references therein.

10 divine contradiction • ‘God’, qua singular term, is in the central identity claims (viz., that Father is God, Son is God, Spirit is God). • The central non-identity claims are logically negated identity claims.⁴ Of course, while it strongly resonates with christian tradition, it’s precisely this simple and flat-footed (and, again, charitable) reading of the creed that delivers the appearance of contradiction, one that arises from standard paraphrase of the key athanasian axioms: A1. A2.

Non-identity: the divine persons are pairwise distinct (i.e., nonidentical). [From axiom (2) above.] Identity: each divine person is (identical to) God. [From axiom (4) above.]

Each divine person, per claims of the Athanasian Creed,⁵ has all essential divine properties (e.g., eternality, omnipotence/almighty-ness, and so on). But, per explicit axioms (1) and (5), the trinitarian theory of God is robustly monotheistic; it contains the explicit affirmation of one god: A3.

Positive Uniqueness: the number of gods is 1 (viz., God).

All of this leads directly to the target problem – the so-called ‘logical’ (and, additionally, associated 3-1-ness) problem – explicitly taken up in §1.6 below. The third notable constraint is not only a commitment to the athanasian axioms so understood; the constraint is also a commitment to the simple, flat-footed reading of the claims – unless, as many have

⁴ On this assumption the target non-identity claims entailed by (2), (3), and (5), when explicitly written out, are: • It’s false that Father is Son. • It’s false that Son is Spirit. • It’s false that Father is Spirit. And so the given assumption is that the ‘is’ in both the ‘positive’ identity claims and the ‘negative’ identity (i.e., the non-identity) claims, so understood, is univocal throughout. ⁵ Example: But the Godhead [divinity, ‘godness’] of the Father, Son and Holy Spirit is one . . . The Father is eternal, the Son is eternal, and the Holy Spirit is eternal. . . . Likewise, the Father is almighty, the Son is almighty, the Holy Spirit is almighty. . . .

aim, scope, limits, and main thesis 11 argued,⁶ there’s very strong reason to give up the simple and flat-footed account.

1.5 Robust monotheism Another aspect of the target ‘logical’ problem is an unwavering commitment to robust monotheism. The god of whom christian theology is written just is the god of Abraham, Isaac, and Jacob, who just is the god revealed in Christ, who just is God. While Jews, like Muslims, reject the divinity of Jesus, christian theology affirms it. But christian theology also affirms more: the divinity of Spirit, and the divinity of Father, each of whom is distinct from the other and distinct from Jesus Christ (Son). And that’s where questions have long persisted, including the target problem. The target ‘logical’ (and, relatedly, 3-1-ness) problem is partly a question of robust monotheism:⁷ How is christian theology as robustly monotheistic as jewish or islamic theology? On one hand, christian theology, per the athanasian axioms, contains the positive-uniqueness claim: the number of gods is 1 (viz., God). On that claim Jews and Muslims and Christians agree. The disagreement, which is fundamental, is the necessary biconditional peculiar to christian theology:⁸ TG. Necessarily, the number of gods is 1 if and only if the number of triune gods is 1. Before the incarnate revelation of God, the only available truth was the ‘lefthand side’ of TG: target monotheists accepted as much. According to christian theology, God incarnate (viz., Christ) revealed the triune nature of God, ushering in the ‘righthand side’ of TG and the biconditional itself.⁹ The necessity of TG partly explains its foundational role in christian theology (specifically, christian trinitarian theory). ⁶ Some of the standard consistency-driven accounts are discussed in §6. ⁷ See too the work of Brower and Rea (2005) and Rea (2009) for the constraint of a robust monotheism. ⁸ One can see this as a bi-entailment principle similar to lines illustrated in §2 and §3. (If to some readers the term ‘bi-entailment’ is an unfamiliar one, familiarity might come in §3.) ⁹ Axiom A3 in §1.4 is thereby equivalent to the claim that number of triune gods is 1. Much more in §3.

12 divine contradiction But doesn’t TG step away from a robust monotheism? Even apart from the contradictions of Christ, the apparent rub against robust monotheism comes with the other axioms: christian theology also affirms the existence of three divine persons who are distinct from each other. And now every reflective person immediately asks: How is that monotheistic – let alone robustly so? My answer invokes identity.1⁰ Christians have – or, at least, should have – long thought that being divine is not something in terms of which God is defined; rather, being divine is no more nor less than being identical to God. And that’s pretty much it. You call anyone ‘divine’ and, on the robust monotheism that infuses core christian theology, you thereby claim that the given one is identical to God. This, on my view, is part of the very heart of robust monotheism: that ‘being divine’ is defined by being identical to God. And such equivalence is another constraint on the current project: that robust monotheism, so understood, be satisfied. To be clear, robust monotheism, as constraining what I take to be an adequate account of the ‘logical’ problem, must satisfy – and can simply be thought of as – the following intersubstitutability rule (called ‘M’ for robust monotheism): M.

x is divine and x is identical to God are intersubstitutable in all contexts.11

Hence, were one to say that Jesus is divine one would thereby be implicitly (though, depending on the context, very nearly explicitly) saying that Jesus is identical to God – a claim that standard christian theology firmly contains while its robustly monotheistic cousins (viz., jewish and islamic theology) firmly lack.

1⁰ The answer also involves a few familiar and fundamental trinitarian entailment patterns, but these are left to the account in §3. 11 Technical note (only for those who might already be wondering): strictly, the rule applies only to what philosophers call ‘non-intensional’ (sometimes ‘non-opaque’) contexts. Such contexts are defined relative to a given identity relation. A typical illustration of such a context involves ‘belief ’ contexts. Agnes can believe that Superman flies but not believe that Clark Kent flies even though Superman is identical to Clark Kent. Accordingly, substitution of ‘Superman’ and ‘Clark Kent’ fails in ‘belief ’ (and related intensional or opaque) contexts. It is possible, of course, that the substitution goes through in many otherwise intensional/opaque contexts; it’s just that the substitution won’t be sanctioned by whatever entailment relation defines valid substitution. End technical note.

aim, scope, limits, and main thesis 13 That being divine just is being identical to God is intimately tied to the athanasian axioms whereby something is divine just if it is thereby identical to God, whence the divinity of Father, Son, and Spirit entail that, as the target claims (‘axioms’) go, • Father is God; • Son is God; • Spirit is God. This account of divinity and its definition in terms of being identical to God is neither complicated nor unnatural; it seems to be apparent in the simple, flat-footed reading of the Athanasian Creed. Are there different senses of ‘divine’? Yes, and those are the senses that deliver polytheism or lighter versions of monotheism in contrast to the robust monotheism of christian theology. Indeed, uses of ‘divine’ span the spectrum from hyper-royalty to para-/extra-human to, ultimately, the highest usage – namely, being identical to God. It is only the highest usage, so understood, that figures in trinitarian theory, at least as I advance it. On the robust monotheism of christian theology, divinity is defined as being identical to God. And that’s the core of robust monotheism on the account I advance.

1.6 The target ‘logical’ problem Robust monotheism, per §1.5, demands that any divine being simply be God – be identical to God. And this is so even where, as in christian theology, there are two or more divine persons who are not identical with each other. The so-called logical problem is the apparent contradiction that arises from there being pairwise-non-identical subjects each of whom is divine, and so each of whom is (identical to) God. Were the theological theory not robustly monotheistic, the apparent contradiction wouldn’t so conspicuously jump out. But it is robustly monotheistic. But what exactly is the problem? Where exactly is the contradiction? Though there are variations on it, the usual derivation of trinitarian contradiction makes three assumptions, as follows.

14 divine contradiction • The ‘is’ of identity is univocal in the trinitarian axioms. • The ‘is’ of predication is univocal in the trinitarian axioms. • The ‘is’ of identity is the is-of-identity relation defined via classical logic.12 The first two of these assumptions are part of the simple, flat-footed reading of the athanasian account (see §1.4), and indeed are part of charitable reading (since otherwise fallacies of unmarked equivocation are involved in the athanasian account). The third assumption jumps out as prima facie curious. Why should the classical-logic identity relation be the ‘trinitarian-identity’ relation – that is, the theory’s identity relation that underwrites the key axioms? The classical-logic ‘is’ of identity is transitive.13 Hence, from • Father is God. • Son is God. we get • Father is Son. from transitivity of the given ‘is’ of identity.1⁴ But the athanasian axioms also contain logically negated identity claims (henceforth, ‘non-identity claims’), including, among others, • It is false that Father is Son. [From A1 in §1.4.]

12 The familiar ‘is’ of identity is modeled by the first-order classical-logic account of identity – basically, a leibnizian ‘indiscernability’ account that in turn relies on classical logic to govern the logical vocabulary and, in particular, the material conditional. (Details of these points are discussed in §2 and especially §3.) 13 On transitivity: a binary relation R is transitive on a domain of objects just if, for any such objects x, y, and z, x’s standing in R to y together with y’s standing in R to z entails that x stands in R to z. Hence, if R is the classical-logic ‘is’ of identity (say, =) then that x = y and that y = z jointly entail that x = z, since the given relation is transitive. 1⁴ On standard definitions of transitivity, the given derivation of ‘Father is Son’ also requires symmetry of the familiar ‘is’ of identity – a feature that the familiar relation satisfies. A relation R is symmetric on a domain of objects if and only if (henceforth, iff) for any such objects x and y, x’s standing in R to y entails the converse, namely, that y stands in R to x.

aim, scope, limits, and main thesis 15 And so the following contradiction is a typical example of the target ‘logical’ problem: • It is true that Father is Son and it is false that Father is Son. But this apparent contradiction – and any of the ones similarly derived via a familiar transitive ‘is’ of identity – is not terribly apparent. If anything, the derivation looks to be forced. By my lights, the apparent contradiction above is at most only very weakly apparent. The strength of the apparent contradiction rests on the strength of the apparent transitivity. But why think that the target identity relation – the trinitarian-identity relation that underwrites the central axioms of the trinitarian account – is transitive? I see no good reason to think as much – none at all, not even a tiny sliver. The fact that one very familiar is-of-identity relation is transitive doesn’t imply that the trinitarian-identity relation is transitive. And nothing in the core trinitarian axioms, per the creeds, demands transitivity of trinitarian identity either. So why demand it? Again, I see no good reason. Not only is there no good reason to demand that the target relation be transitive; the appearance, at least so far as I see, is exactly the opposite. In particular, when you have a true theory according to which It’s true that a is b. It’s true that b is c. but, also explicitly, It is false that a is c. the immediate and most natural – and simplest – appearance is that the given true theory’s target identity relation is decidedly not the familiar is-of-identity relation; it’s some other relation, some non-transitive relation or the like. By my lights, the apparent contradictions of trinitarian reality are not the ones derived, as above, via the would-be transitivity of trinitarian identity. Absent some strong reason to think that trinitarian identity must be transitive, the so-called logical problem isn’t a would-be

16 divine contradiction derivation of the identity of Son and Father, or Spirit and Father, or Spirit and Son – or, for that matter, any other claim that turns on the unmotivated imposition of transitivity on the key identity relation.1⁵ What, then, is the target ‘logical’ problem? Answer: The problem is an identity problem. The problem, as above, is not apparent contradictions allegedly arising from the (nonexistent) transitivity of trinitarian identity; the problem is the identity of trinitarian identity itself. The target ‘logical’ problem is just to answer three basic questions: What is the relation? How is it defined? And – importantly – what features of trinitarian reality explain the non-transitive nature of the relation? Answering these questions is the central task of the target problem. And that’s the principal aim of the book: to resolve the ‘logical’ problem by specifying the details of the non-transitive trinitarianidentity relation, and by explaining what it is about trinitarian reality that motivates or demands its non-transitive identity relation.1⁶ * * Parenthetical note on 3-1-ness problem. Closely related to the ‘logical’ problem is the so-called 3-1-ness problem. Some analytic theologians and philosophers, such as Rea (2009), conflate the two problems because they are in fact very intimately connected; however, it is fruitful to explicitly treat them as distinct problems. The ‘logical’ problem, as above, is an identity problem: identify trinitarian identity. The 3-1-ness problem is an important follow-up problem: a counting-convention problem. In short: how do you count relevant phenomena given the target identity 1⁵ I note that Rea (2009) formulates a metaphysics-heavy derivation of apparent contradiction in terms of alleged entailments between the metaphysical notion of ‘consubstantial’ and standard notions of identity and exemplification. The Brower-Rea positive account of the target problem is discussed in §6. 1⁶ I am not the first to recognize the conspicuous non-transitivity of trinitarian identity. Thom (2011), for example, also observes the conspicuous non-transitivity, and, in a different direction, Uckelman (2010) reports on an anonymous text that presents a non-transitive (and, likewise, non-contradictory) account. My account differs in substantial ways, not only being driven by christology (and the contradictions of Christ) but an overall account of logical consequence, qua universal entailment relation in all true theories (see §2), together with a general account of (leibnizian) identity relations (per §3) and, especially, a natural and simple metaphysics-free explanation of the target non-transitivity (namely, divine contradiction). One whose work is closer to mine, though not driven by christology or a broader account of logic or identity, is that of Pleitz (2015), whose work is driven by substantive metaphysics. Along such lines, some philosophers, such as Priest (2014), are monists about identity who think that the true metaphysics demands the non-transitivity of identity. I remain entirely neutral on metaphysics in the current work, and, as §3 makes plain, I also think that there are many (not one) identity relations some (but only some) of which are non-transitive.

aim, scope, limits, and main thesis 17 relation? This is not peculiar to trinitarian theory. Given an identity relation in any theory (be it set theory, theology, physics, what have you), a counting convention associated with the relation must be given. The convention specifies how to count a given predicate (e.g., ‘divine persons’, ‘quarks’, what have you). Normally, the target identity relation is used to define relevant duplicates. All of this is discussed in §3 but, of necessity, only after the account of trinitarian identity is in place. End parenthetical. * *

1.7 The target thesis The simple thesis advanced in this book is that trinitarian identity is nontransitive in virtue of contradictions true of divine reality – in virtue of divine contradiction, contradictions arising not just from Christ’s contradictory being but also from differences among the persons each of whom is identical to God. That’s the main thesis. And saying just that much is to say very little. The rest of the book says more. One pressing question, discussed in §3, is whether the trinitarianidentity relation bucks the standard leibnizian recipe for identity. The leibnizian recipe defines an identity predicate ‘x ∼ y’ via a biconditional scheme of the form 𝜑(x) ⇔ 𝜑(y) such that,1⁷ where 𝜑 can be any relevant unary predicate in the language of the theory, the given identity claim is true of the pair x and y just when all ‘instances of the scheme’ (i.e., replacing 𝜑 with every predicate in the target range of the scheme) are true. This recipe is more familiar and friendlier than it may look. In effect, the standard leibnizian

1⁷ A biconditional, say the material one ≡ (or some other ↔ or just ‘if and only if ’ in English or the like), is a forward-and-back conditional built from a conditional and logical conjunction. Letting ⇒ be a conditional in the language and ∧ logical conjunction, the corresponding biconditional is the conjunction (A ⇒ B) ∧ (B ⇒ A) normally then abbreviated or expressed by, in this case, a double arrow as in A ⇔ B.

18 divine contradiction recipe – discussed further in §3 – is just this: take the predicates of your precise theory; take a biconditional from your theory; and then define identity to be indiscernibility with respect to all relevant predicates in the theory, where such indiscernibility is expressed via the truth of all instances of the target biconditional scheme. Basically, the idea is that x and y are identical just if, for any relevant predicate 𝜑 in the language of the theory, 𝜑 is true of x if and only if 𝜑 is true of y, where the biconditional ‘if and only if ’ is the target one in the theory.1⁸ The question is whether trinitarian identity deviates from the standard ‘leibnizian recipe for identity’ (much more on which in §3). While I see no obvious theological or philosophical reason that the true account of trinitarian identity must follow the standard leibnizian recipe, I equally see no special reason to buck the recipe – unless trinitarian reality clearly demands it. Accordingly, my central thesis (viz., that trinitarian identity is non-transitive in virtue of divine contradiction) follows the leibnizian recipe for identity, understood as above. Many readers immediately respond: ‘But the standard leibnizian recipe for identity results in a transitive identity relation!’ Too true. But the symptomatic transitivity is not the fault of the standard recipe; it’s the fault of the ingredients that are standardly used. If your ingredients come only from the mainstream supply (viz., the classical-logic account of logical consequence) you’re thereby stuck with transitive identity relations. While such ingredients are just the ticket for many true theories, they are not demanded in all true theories. On the correct view of logical consequence, as discussed in §2, the logical ingredients available for the leibnizian recipe are more flexible than the restricted ingredients described by the mainstream account. And this is a good thing. Reality has very strange phenomena, some of which, like the incarnation, are strikingly unique and call for the full range of available ingredients, not just the usual restricted list of ingredients. The identity relations involved in the true and full description of those phenomena may need – in my view, in fact need – to draw on the unrestricted power of logical 1⁸ For experts (though all of this is explained in §3): for some (many) true theories, the biconditional in question may be only in the target theory’s ‘metatheory’. In §3 the target biconditional is logical vocabulary (viz., logic’s material biconditional), and so the relation is defined in terms of the truth of that biconditional schema (even if the definition, of course, is strictly in a metatheory).

aim, scope, limits, and main thesis 19 vocabulary. On the thesis I advance, the standard leibnizian recipe, using logic’s material conditional, still applies to trinitarian identity. *** So go the aim, constraints, and basic thesis. The rest of the book fills in the details, with §2 rehearsing the logical machinery and §3 – the central chapter – advancing details of both trinitarian identity and the associated counting convention.

2 Logical and Extra-Logical Entailment This chapter reviews, in very broad terms, what I take to be the correct account of logical consequence (‘logic’ for short) and its relation to true theories. Formal details of the account are sketched in an appendix to this chapter.1 A longer, more leisurely presentation is available in this book’s predecessor (2021, §2) and in works cited throughout.

2.1 True, false, and beyond There are two fundamental semantic properties for sentences: namely, truth and falsity. On the standard theory of logic (more on which below), these fundamental properties deliver exactly two semantic statuses for sentences, statuses that are jointly exhaustive, in that every sentence enjoys at least one of the given statuses, and mutually exclusive, in that no sentence enjoys both statuses.2 The standard theory thereby overlooks two combinatorial statuses that accompany the given two: neither true nor false (i.e., having neither fundamental status), which is the ‘gappy’ status (e.g., falling in a gap between truth and falsity), and both true and false (i.e., exemplifying both fundamental statuses), which is the ‘glutty’ status (e.g., having a glut of truth and falsity). Why are the additional two statuses often overlooked? The most charitable explanation is that because much of reality is truly and fully described without the glutty or gappy options, such options are, in effect, forgotten – if noticed at all. Our true mathematical theories and true biological theories, for example, like the true theory of tractors and the 1 See Appendix B, p. 141. 2 By ‘sentence’ is meant meaningful, declarative sentence – a sentence that declares that suchn-so is such-n-such. For present purposes, one can think of sentences as ‘statements’ about reality, ‘claims’ about reality, or even (with a bit of translation and not too much theoretical baggage) ‘propositions’ (about reality). Divine Contradiction. Jc Beall, Oxford University Press. © Jc Beall 2023. DOI: 10.1093/oso/9780192845436.003.0002

logical and extra-logical entailment 21 true theory of moonshine, rule out the option of gluts and gaps as options that are theoretically useless or, more strongly, theoretical impossibilities (i.e., impossible as far as the theory allows). That much – perhaps most – of reality is truly and fully described without gaps or gluts primes a standard inductive stretch to the thought that all of reality is truly and fully described without gaps or gluts. In the absence of recalcitrant data, perhaps the given stretch would be fruitful, reliable, and universal. But, per §1, the stretch is a bad one: the incarnation, despite its uniqueness, belies its would-be universality. While many true theories fully describe their respective target phenomena (e.g., algebraic groups, salamanders, velocity, shoelaces, what have you) without gaps or gluts – and my view is that the many here is in fact most – some involve gluts and some gaps (and some both gluts and gaps). Per §1, a premise of this book is that true theology (i.e., the true theological theory) is contradictory: it contains gluts, notably gluts that express the contradictory truth about Christ. In what follows, the four semantic statuses arising from the two fundamental semantic properties are taken as given.3 This chapter rehearses the place of the four semantic statuses in a broader account of logic, language, and true theories.⁴

2.2 Atomic attributions All four statuses for sentences (per §2.1) mirror corresponding statuses for exemplification. Truth and falsity, as two fundamental properties of sentences, reflect two fundamental ways of exemplification or instantiation or, simply, predication: a predicate is truly (falsely) satisfied by x just when the predicate is true of (respectively, false of ) x. Most thinkers overlook these two ways because, for all we can see, so much of reality collapses things into exactly one way: a predicate is either true-of an object or false-of an object but never – ever, anywhere, anyhow – both true-of and false-of an object. Alas, that much of reality is truly and fully described in such an either-or-but-not-both fashion has led many 3 For theology-independent arguments for the four properties, see 2013a; 2017; 2018. ⁴ Much of this chapter overlaps with Chapter 2 of The Contradictory Christ (2021).

22 divine contradiction thinkers – including theologians – to erroneously generalize too far to the thesis that all of reality is just so. But reality is not just so, as the radical revelation of God – walking and sweating and crucified and dying among us on earth – shows in the contradictions manifest therein (2021). In a formal ‘model language’ the two-way exemplification relation (viz., truly exemplifying, falsely exemplifying) is modeled via the interpretation (or meaning, or denotation) of predicates. Where P is some unary predicate (i.e., it takes a single singular term or name or the like to make a sentence), the interpretation of P is a pair ⟨P+ , P− ⟩, where P+ contains everything of which P is true (i.e., contains all objects that truly exemplify P) while P− contains everything of which P is false (i.e., contains all objects that falsely exemplify P).⁵ Terminology: P+ is called the extension of P, and P− is called the antiextension of P. Predicate-name (or, generally, predicate-singular-term) sentences that do nothing more than truly or falsely predicate something of an object – that is, truly or falsely attribute a property to an object – are called atomic sentences. They’re the basic building blocks of a language’s so-called molecular or compound sentences. Once atomic sentences and exemplification of predicates are in place, relations of truth and falsity for sentences fall out in the usual fashion. In particular, an atomic sentence is true iff the object denoted by the name (generally, singular term) is in the extension of the given predicate, and an atomic sentence is false iff the object denoted by the name (generally, singular term) is in the antiextension of the given predicate. In other (very familiar) words: an atomic sentence is true iff it truly attributes something of the given object; an atomic sentence is false iff it falsely attributes something of the given object. In logical studies, and especially when defining ‘logical consequence’ or ‘logical entailment’ (more on which in §2.4–§2.5), the central relations are not merely truth and falsity but rather truth-in-a-possibility

⁵ Where P is a binary, ternary, etc. predicate (i.e., takes two singular terms to make sentence, or three, etc.) the objects in P+ and P− are themselves pairs of objects, triples of objects, etc. (In general, an n-ary predicate is interpreted via a pair of n-tuples.) I suppress details of the full formal picture to an appendix. Note: Chapter 2 of 2021 contains a leisurely discussion of the entire language and account of logic rehearsed in this chapter.

logical and extra-logical entailment 23 (or truth-in-a-model) and falsity-in-a-possibility (or falsity-in-a-model).⁶ The reason is that entailment or consequence relations are defined as absence of counterexample, and counterexamples don’t need to be actual (in the sense of the actual world); they need merely be in the space of possibilities recognized by the relation. (More in §2.4 below.) Accordingly, the relations of true-in-a-possibility and false-in-a-possibility (for sentences) are critical, but their definitions naturally follow the special case of truth and falsity (for sentences) above. In particular: where m is some relevant ‘possibility’ (or, formally, ‘model’), an atomic sentence is true-in-m iff the object denoted by the name (generally, singular term) is in the predicate’s extension at possibility m, and an atomic sentence is false-in-m iff the object denoted by the name (generally, singular term) is in the predicate’s antiextension at m. Hence, technical issues aside, in the special possibility that is the actual one, truth (falsity) for sentences and truth-in-a-model (falsity-in-a-model) coincide.

2.3 Logical compounds: sentential and quantification Atomic sentences are the building blocks. So-called logical vocabulary connects such atoms to make molecular or compound sentences.

2.3.1 The basic logical vocabulary The logical vocabulary divides into two salient categories: sentential (or propositional) and quantification. The basic picture – focusing on (and only on) the logical vocabulary – is this:⁷ • Logical vocabulary contains two primitive unary connectives: ∘ Truth (or nullation) connective: It is true that. . . , formalized as †. ∘ Falsity (or negation) connective: It is false that. . . , formalized as ¬.

⁶ See, again, Chapter 2 in 2021 for a leisurely and more detailed discussion. ⁷ See Appendix B on p. 141 for more details.

24 divine contradiction • Logical vocabulary contains two primitive binary connectives:⁸ ∘ Conjunction connective: It is true that . . . and it is true that . . . , formalized as ∧.⁹ ∘ Disjunction connective: Either it is true that . . . or it is true that . . . (or both), formalized as ∨.1⁰ • Logical vocabulary contains various derived or defined connectives, most notably: ∘ Material Conditional: If . . . then. . . defined as Either it is false that. . . or it is true that. . . (or both), formalized as →.11 So go the sentential or propositional bits of logical vocabulary. But there are also two logical quantifiers. • Logical vocabulary contains two primitive quantifiers: ∘ Universal: Every object is such that . . . , formalized as ∀. ∘ Existential: Some object or other is such that . . . , formalized as ∃.

2.3.2 Truth and falsity conditions for logical compounds Exemplification conditions pave the ground for truth and falsity of atomic sentences. (See above.) The relations of truth-in-a-possibility and falsity-in-a-possibility are extended via the so-called truth and falsity conditions. The basic (though, for simplicity, herein slightly imprecise) picture for the logical connectives is this: • A nullation †A is true-in-m iff A is true-in-m. • A nullation †A is false-in-m iff A is false-in-m. • A negation ¬A is true-in-m iff A is false-in-m. • A negation ¬A is false-in-m iff A is true-in-m. ⁸ Lest one doubt: taking both of these as primitive is in the service of simplified discussion. One or the other can be defined, as usual, in terms of the other and negation (and the implicit but entirely redundant primitive truth connective). Ditto for quantifiers below. ⁹ Strictly speaking, the truth connective isn’t explicitly in the logical conjunction connective, but in English it helps to explicitly use it so as to highlight logical conjunction (versus other sorts of conjunction that care about things beyond just truth and falsity, e.g., like temporal orderings). 1⁰ Again, the truth connective is used here to distinguish from other non-logical disjunctions in English. 11 In notation above: where A and B are any sentences of the language, A → B is defined as ¬A ∨ †B or, logically equivalent, ¬A ∨ B.

logical and extra-logical entailment 25 • A conjunction A ∧ B is true-in-m iff A is true-in-m and B is true-in-m. • A conjunction A ∧ B is false-in-m iff either A is false-in-m or B is false-in-m. • A disjunction A ∨ B is true-in-m iff either A is true-in-m or B is true-in-m. • A disjunction A ∨ B is false-in-m iff A is false-in-m and B is false-in-m. The quantifiers (viz., ∀ and ∃) behave, in effect, like conjunction and disjunction. While these are very important, details are left to an appendix.12 The general idea is that a universal sentence, which says that every object is so-n-such, is true-in-m (respectively, false-in-m) just if the given ‘so-n-such’ condition is truly (respectively, falsely) exemplified by all (respectively, some) of the objects over which the claim is made.13 Dually, an existential sentence, which says that some object or other is so-n-such, is true-in-m (respectively, false-in-m) just if the given ‘so-n-such’ condition is truly (respectively, falsely) exemplified by at least one (respectively, all) of the objects over which the claim is made.

2.4 Entailment in general: consequence relations Some claims follow from others. For example, 1. The tractor is colored. follows from 2. The tractor is red. In other words, (2) entails (1): there’s no relevant possibility in which (2) is true but (1) untrue. In the true theory of tractors, the pair ⟨(2), (1)⟩ is a valid one; (1) is true according to the theory if (2) is true according to 12 See Appendix B on page 141. 13 Talk of objects over which the claim is made is precisely cashed out in terms of the so-called domain of a theory in which the claim is made. (See the domain D of an FDE model spelled out in Appendix B on p. 141.) For present purposes, one can think of this as being all objects (as far as the given theory or claim can see).

26 divine contradiction the theory; (1) is a consequence of (2) according to the given entailment or validity (or consequence) relation. There are many different entailment relations on any given natural language like English, each keyed to specific fragments of vocabulary and a specific space of possibilities. In a theory of tractors, predicates like ‘is a tractor’ and ‘is red’ are central predicates, and the entailment relation for the language of the theory validates set-sentence patterns involving such terms, patterns such as (2)-to-(1).

2.5 Logical entailment: universal Logical entailment or logical validity or logical consequence – henceforth, for short, logic – is an entailment relation keyed to (and only to) the logical vocabulary (per §2.3.1). The specialness of logical entailment rests, in part, with the specialness of the sparse set of logical vocabulary to which it is keyed. The logical vocabulary is topic-neutral in the sense that no matter what topic a true theory concerns, the language of the true (and complete as possible) theory contains the logical vocabulary. While the languages of true theories rarely overlap with respect to their central predicates or other topic-specific vocabulary (e.g., the language of arithmetic doesn’t contain key biological vocabulary), all such languages have a very minimal common core: logical vocabulary. Whether it’s the true theory of tractors, quantum reality, divine reality, walnut trees, mathematical categories, fictional objects, or what have you, the language of the true theory contains the sparse, topic-neutral logical vocabulary.

2.5.1 Counterexamples and logical consequence The definition of logical entailment – or, equivalently, logical consequence, logical validity, or, in the current context, logic – is familiar. At the core of the definition is absence of counterexample. Definition 2.5.1 (Counterexample) Let X and A be a set of sentences and, respectively, a sentence from a given language. A counterexample to

logical and extra-logical entailment 27 the pair ⟨X, A⟩ is a relevant possibility in which every sentence in X is true but A is untrue. Here, ‘relevant’ in ‘relevant counterexample’ is tied to the given entailment relation in question. Many true theories preclude lots of logical possibilities – possibilities recognized by logical entailment as candidate counterexamples – and so have a smaller space of relevant possibilities than logic itself. (More on this below.) But in general, a counterexample to ⟨X, A⟩ is a possibility, recognized by the target entailment relation, in which everything in X is true but A untrue. A counterexample is the sign of invalidity: its existence belies the claim that X entails A, since the counterexample is a possibility in which everything in X is true but A untrue. Logical entailment – like any entailment relation – is defined as absence of counterexample. Definition 2.5.2 (Logical Entailment) Set of sentences X logically entails sentence A iff there’s no counterexample among logical possibilities to ⟨X, A⟩. The difference between logical entailment and other entailment relations partly involves its space of possibilities – its space of relevant candidate counterexamples, the space of logical possibilities.

2.5.2 Salient logical possibilities Logic (-al entailment) recognizes the broadest space of possibilities. In particular, as far as logic is concerned, any predicate in any language enjoys the four (logical) possibilities for exemplification – the four possibilities around truth and falsity. • Just true: a predicate is only truly exemplified by an object. • Just false: a predicate is only falsely exemplified by an object. • Gappy: a predicate is neither truly nor falsely exemplified by an object. • Glutty: a predicate is both truly and falsely exemplified by an object.

28 divine contradiction As far as logic is concerned, no matter the predicate, there’s a possibility m in which an attribution of a predicate to something is ‘just true-in-m’, ‘just false-in-m’, ‘gappy in m’ (i.e., neither true-in-m nor false-in-m), and ‘glutty in m’ (i.e., both true-in-m and false-in-m). That logic recognizes all four given possibilities for any given predicate does not imply that all entailment relations recognize all four possibilities for any (let alone all) predicates. Every possibility recognized by any entailment relation (in any true theory) is a logical possibility, but many logical possibilities are ruled out as theoretically impossible – i.e., impossible as far as the theory sees – by many true theories. (More on this in §2.6 and §2.7.) The account of logical consequence that results from the standard truth and falsity conditions for logical vocabulary together with the full space of the four given logical possibilities (and the given absence-ofcounterexample definition of consequence) is FDE, further details of which are in the appendix (p. 141).1⁴

2.5.3 Some notable logical validities and invalidities A few patterns are notable with respect to logical validity and logical invalidity. Each in turn.

2.5.3.1 Notable logical validities At the heart of logic, on the account advanced in this book, is socalled DeMorgan behavior. Letting ⊢ be logical entailment and ⊣⊢ be two-way logical entailment (i.e., A ⊣⊢ B iff A logically entails B and vice versa), and letting A and B be any sentences, common examples of DeMorgan behavior may be illustrated as follows. For example, one common DeMorgan pattern is that the conjunction of two sentences is logically equivalent to the negated disjunction of the negations of each conjunct (i.e., each of the sentences conjoined by logical conjunction): A ∧ B ⊣⊢ ¬(¬A ∨ ¬B) 1⁴ For many more details of FDE, both philosophical and technical, see the work of Anderson and Belnap (1975); Anderson et al. (1992); Bimbó and Dunn (2001); Dunn (1966, 1976); Omori and Wansing (2017, 2019); Shramko et al. (2017) as well as 2013a; 2017; 2020; 2021.

logical and extra-logical entailment 29 and, likewise, A ∨ B ⊣⊢ ¬(¬A ∧ ¬B) And so on.1⁵ All such DeMorgan patterns are valid according to logic. Indeed, the validity of such DeMorgan patterns is largely the extent to which logic validates anything. One might accordingly think that the validation of other patterns comes from some source other than logical entailment. And that’s correct, as §2.6 and §2.7 discuss.

2.5.3.2 Notable logical invalidities There are some very important patterns that are not logically valid; they’re logically invalid. Two jump out:1⁶ • A ∧ ¬A ∴ B. This pattern or ‘form’ is sometimes called ‘ex contradictione quodlibet’ or, for short, ‘exclusion’ – the idea being that the cost of an arbitrary contradiction in a theory is the exclusion of all theories except for the unique trivial theory (whereby every sentence in the language of the theory is true).1⁷ • B ∴ ¬A ∨ A. This pattern or ‘form’ is sometimes called ‘excluded middle’ or, for short, ‘exhaustion’ – the idea being that an arbitrary sentence, no matter what it is, demands that the (prime) theory contain every sentence or its negation.1⁸

1⁵ For logic experts: logical negation behaves, at bottom, as a ‘DeMorgan mapper’ (so to speak): it takes a sentence to its dual, where the truth operator † is the dual of the falsity operator ¬ (i.e., negation), and disjunction and conjunction are duals (and likewise the universal and existential quantifier are duals). Hence, the dual of ¬A ∨ ¬B is †A ∧ †B, which is why ¬(¬A ∨ ¬B) is logically equivalent to A ∧ B. Indeed, as in 2017 and similarly Paoli 2015, even familiar double-negation equivalence – that is, the logical equivalence of ¬¬A and A itself – is just more DeMorgan: the dual of ¬A is †A, which is equivalent to A, and hence ¬, doing its ‘DeMorgan mapping’ job, underwrites the equivalence of ¬¬A and A. 1⁶ Note that, as above, while entailment is a set-sentence relation, an easy-on-the-eyes practice of dropping set braces when confusion won’t occur is useful and herein adopted. 1⁷ This pattern – from arbitrary contradiction to arbitrary sentence – is also sometimes called ‘explosion’, the idea being, again, that an arbitrary contradiction in a theory ‘explodes’ the theory into the unique trivial one. 1⁸ The term ‘prime’ here is technical, but it basically demands that logical disjunction have the truth and falsity conditions that it in fact has, per §2.3.2. More precisely: a theory T is prime iff a logical disjunction A ∨ B is in T (i.e., true according to T) iff either A is in T or B is in T. Hence, the validity of Exhaustion (EM) requires that a prime theory contain either A or ¬A for each sentence A in the language.

30 divine contradiction Neither of these patterns is logically valid. A counterexample to exclusion is a logical possibility in which A is a glut and B untrue (i.e., either just false in the possibility or gappy there). A counterexample to exhaustion is any logical possibility in which some B is true while A is a gap (i.e., neither true in the possibility nor false in the possibility). In true theology, wherein Christ (The Son) is a contradictory being, the logical invalidity of exclusion is likewise a theological invalidity: an arbitrary glut in theology does not ‘explode’ the theology into the unique trivial theology (whereby every sentence in the language of theology is true). The extent to which the true theology is gappy is largely an open issue – one not settled in this book. Logic itself recognizes such gappy possibilities for every predicate in the language of theology, but whether true theology also recognizes such logical possibilities as theological possibilities remains open. (The issue is discussed at various points in subsequent chapters.) In addition to the logical invalidity of exhaustion and exclusion, other notable logical invalidities concern logic’s lone (and derivative) conditional. The following two patterns jump out: • A, A → B ∴ B. This pattern or ‘form’ is sometimes called ‘modus ponendo ponens’ or, for short, ‘detachment’ – the idea being that an arbitrary material conditional and its antecedent jointly entail the consequent. • B ∴ A → A. This pattern or ‘form’ is something called ‘materialconditional identity’ or, for short (provided the context is clear), ‘identity’ – the idea being that reflexivity of a material conditional is demanded by every sentence in the language, no matter the topic of the sentence. These are logically invalid; they are not sanctioned by logic alone. A counterexample to detachment is any logical possibility in which A is a glut (i.e., both true in the possibility and false in the possibility) and B is untrue (i.e., either just false in the possibility or a gap in the possibility). A counterexample to identity is a logical possibility in which B is true and A is a gap.

logical and extra-logical entailment 31

2.6 Extra-logical entailment: theory-specific One would rightly think that patterns such as detachment and identity, as in §2.5.3.2, if not also both exclusion and exhaustion, are so strongly involved in many of our true theories that their logical invalidity immediately undermines or topples truth-seeking theorizing in general. Not so. The critical distinction concerns logical invalidity (similarly, validity) and extra-logical or theory-specific invalidity (similarly, validity).1⁹

2.6.1 Logical validity and extra-logical validity Restricting attention to true theories and their respective entailment relations, every logically valid pattern is T-valid too, where T-validity is simply the entailment relation involved in a specific theory T. After all, if a pattern survives the broadest space of would-be counterexamples (viz., logical space) then any other entailment relation, at least governing a true theory, won’t find a challenger to the pattern in a proper subspace of the broadest space.

2.6.2 Logical invalidity and extra-logical validity Logically invalid patterns needn’t be T-invalid for all true theories T. (NB: this is both very important and, alas, very often overlooked.) Any entailment relation that governs a true theory must, except in the limit case,2⁰ go beyond what logical validity itself sanctions; otherwise, the only implications or consequences of any true theory would be the mere logical consequences, and these miss a great deal of important consequences. Example: in the true theory of color, the predicate ‘is colored’ is entailed by the predicate ‘is red’ in that, according to the theory’s entailment relation ⊢T ,

1⁹ For further discussion of the given distinctions see Beall and DeVito 2022. 2⁰ The limit case is the ‘true theory of everything’, which can be thought of as a sort of union of all individual true theories. (I say ‘sort of union’ because there may be issues of so-called cardinality that arise. I set this aside here.)

32 divine contradiction Object b is red. entails Object b is colored. This pattern is logically invalid for a simple and dull reason: namely, the only vocabulary to which logic attributes any substantive weight is the logical vocabulary, and there is no logical vocabulary in the given pattern; hence, the pattern – as far as logical vocabulary goes – is little more than A ∴ B, a pattern for which there are many familiar counterexamples.21 Hence, the target theory T must ‘extend’ the universal (logical) entailment relation by adding specific patterns that are keyed to T’s key vocabulary (e.g., ‘colored’, ‘red’, what have you). Once the language is set, the theory, except in said limit case, is tied to a proper subspace of logical space – some proper family of logical possibilities that are deemed by the theory to be theoretically possible (i.e., relevantly possible as far as the theory’s target phenomenon goes). And this proper family of logical possibilities – the theory’s ‘T possibilities’ (for lack of a better term) – are the ones, and the only ones, over which theory T’s extra-logical (i.e., goes-beyond-logical) consequence relation is defined. There is a special case of extra-logical entailment: the case involving logical vocabulary itself. This is a case relevant to the driving concern over toppling or undermining truth-seeking theories in the face of the logical invalidity of detachment, identity, exclusion, and exhaustion. While details are left for elsewhere (1966; 2015; 2013c; 2013b; 2013a; 2014), the basic facts involve shrieking and shrugging predicates, which are simply ways of paring down the logical possibilities to, in effect, classical possibilities. Definition 2.6.1 (Shrieking) Let P be any predicate in the language of theory T, and let ⊢T be T’s (extra-logical) entailment relation, and let ⊥ be an ‘explosive sentence’ in T in the sense that ⊥ ⊢T A for all A in the language

21 There is the invisible or implicit truth operator in the pattern, but it is logically redundant and accordingly impotent with respect to validating any patterns that aren’t validated without it.

logical and extra-logical entailment 33 of the theory. To shriek predicate P in theory T is to impose the condition that Px ∧ ¬Px T-entails ⊥ for all objects x in the domain of the theory. To shriek a theory is to shriek all predicates in the language of the theory. Definition 2.6.2 (Shrugging) Let P be any predicate in the language of theory T, and let ⊢T be T’s (extra-logical) entailment relation, and let ⊤ be an ‘impotent sentence’ in T in the sense that A ⊢T ⊤ for all A in the language of the theory. To shrug predicate P in theory T is to impose the condition that Px ∨ ¬Px is T-entailed by ⊤ for all objects x in the domain of the theory. To shrug a theory is to shrug all predicates in the language of the theory. Again, leaving precise details to other work, shrieking and shrugging predicates in a theory is a common practice (even if not recognized as such). Shrieking and shrugging a particular predicate in the language of a given theory treats the predicate as an in-effect ‘classical’ predicate; its only non-trivial exemplification options are just-true and just-false (just as the so-called classical picture demands). And there’s an important further fact: Fact 2.6.3 (Shrieking and Shrugging Theory T) The result of (fully) shrieking and shrugging a theory is a classically closed theory, that is, a theory whereby the consequences of logical vocabulary according to the classical account of logic are exactly the consequences of logical vocabulary in the theory, even though logic itself is weaker than classical logic (2013a; 2013c). While there is much more that can be said, the central response to the driving worry about the notable weakness of logic (e.g., logical invalidity of detachment) is just this: the validity of such patterns remains in all true theories in which they are indeed valid; it’s just that the validity of such patterns comes not from logic but rather from extra-logical entailments that arise from either extra-logical vocabulary or shrieking/shrugging logical vocabulary in the confines of the theory. Truth-seeking theories carry on as before. The difference is in the source of the valid patterns, if indeed valid patterns at all.22 22 Related discussion of these issues is Chapter 4 of 2021.

34 divine contradiction

2.7 Big picture: logic and true theories The big picture is a traditional one wherein the role of logical validity, distinguished from the extra-logical validity relations in our true theories, is the universal one: it’s part of all validity (consequence, entailment) relations that govern all of our true theories. The relation of logical consequence enjoys such a privileged, universal (topic-neutral, etc.) status in virtue of enjoying the sparsest set of (topic-neutral, universal) vocabulary and the biggest space of possibilities over which it’s defined. There are many – in fact, perhaps most – true theories wherein logically invalid patterns of logical vocabulary are valid (according to the extra-logical validity relation). For example, true mathematics, on my view, is classically closed; every true mathematical theory validates whatever classical logic validates for the logical vocabulary. How so? Per §2.6.2, such theories chop off large swaths of logical possibilities, treating them as T-impossible according to the given theory T; in particular, they chop off all of the glutty regions and gappy regions of logical space (the space of logic’s many possibilities). But doing that is what’s called for if, as current mathematics seems to reflect, the reality itself is without gaps or gluts. And all of this is perfectly natural with logic’s recognition of a wider space. Logic itself doesn’t dictate the truth of any theory; it adds no truths to any true theory. (Indeed, even the true theory of logic is one according to which there are no logical truths – no sentences that are logically valid. Witness for counterexample: the unique vacuous logical possibility, dual to the unique trivial one, wherein all sentences are gappy.) Logic sets the boundaries of possibilities. Some of the possibilities are so wildly remote that few if any true theories recognize them as T-possible (i.e., possible according to the theory T). But so too are much closer logical possibilities that are treated as T-impossible by many true theories (e.g., think of so-called science-fiction possibilities that are simply ruled out as T-impossible according to a given true science). That logic recognizes many bizarre possibilities is nothing more nor less than that: namely, logic’s recognition of them as candidate counterexamples to patterns of logical vocabulary. The good thing about such bizarre possibilities is that reality itself can be bizarre in ways that the standard story of logic neglects for no good reason. (The standard story of logic, as discussed in 2021, was written very recently, and written

logical and extra-logical entailment 35 not as the true account of logic qua universal consequence in all true theories; it was written as the true account of extra-logical consequence governing logical vocabulary in true mathematics! Wonderful as it is, there is more to reality than mathematical reality.) One strikingly bizarre event in reality is the incarnation of God: one person, who is divine (with all of the limitlessness thereby entailed) and who is human (with all of the limits thereby entailed). Trying to squeeze the incarnation into a consistent theory has long been the wrong direction, a direction fueled by the under-motivated insistence that logical consequence is per the socalled classical-logic account. Shaking off such unmotivated constraints does not in any way demand the acceptance of contradictory (or gappy) theories, but in the face of theology’s persistent appearance of central contradictions, such shaking off opens up a better first step: acceptance of the contradictory being who is God. *** Much more on the foregoing account of logic (-al consequence) and its role is available in cited sources. What is in this chapter is just enough to get to the central topic: trinitarian identity and the ‘logical’ or 3-1-ness problem.

3 Trinitarian Identity [W]e venerate one god in trinity and trinity in unity. . . . [I]n all things the unity in trinity and trinity in unity is to be venerated. A.C., trans. Philip-Neri Reese, O.P.

Per §1 the ‘logical’ problem is to answer three basic questions: What is the relation involved in the athanasian identity and non-identity axioms? How is it defined? And importantly: what features of trinitarian reality explain the non-transitive nature of the relation? Answering the given questions is the central task of the target problem. Closely related is the 3-1-ness problem, where the task is to specify the counting convention associated with the central trinitarian relation. All of these questions are answered in this chapter, with remaining chapters elaborating and defending the account. When it comes to both the target ‘logical’ problem, and likewise the associated 3-1-ness problem, details matter. But the details matter because the big picture matters. Lest the proverbial forest be lost for its trees, I first sketch the big picture (§3.1) – and the centrality of divine contradiction therein – followed by the critical details. As with any such broad sketch, the following big picture is telling in a big-picture sort of way; however, its exact implications are to be understood via the details that follow it.

3.1 Big picture: God, trinitarian identity, and contradiction The target ‘logical’ problem, per §1, is ultimately an identity problem: what is the identity relation involved in the athanasian axioms? My answer, spelled out in §3.2–§3.6.2 and given succinctly in §3.5, is Divine Contradiction. Jc Beall, Oxford University Press. © Jc Beall 2023. DOI: 10.1093/oso/9780192845436.003.0003

trinitarian identity 37 trinitarian identity, the relation defined by the familiar ‘leibnizian recipe’ for identity relations (Def 3.3.1) using the proper logical ingredients (§3.4.3) – notably, logic’s material conditional, which is non-transitive. (All of these things are explained from scratch in the following sections.) Trinity in unity, guiding the athanasian axioms, is the identity of each divine person to God; the unity in trinity is the identity of God to each divine person. Trinitarian identity is the identity relation that unifies divine reality; it’s the relation whereby Father is identical to God and Son is identical to God but it’s untrue that Father is identical to Son. (Mutatis mutandis for Spirit.) These truths involve contradiction. In an important sense, trinitarian contradictions naturally arise ‘across’ trinitarian reality. And those essentially trinitarian contradictions explain, in a direct and illuminating fashion, the non-transitivity of trinitarian identity. The triune god (viz., God), to whom Son is identical, to whom Father is identical, and to whom Spirit is identical, is truly described only via contradiction. The truth of Father’s divinity, of Son’s divinity, and of Spirit’s divinity just is the truth of each being identical to God. That’s oneness – as the slogan goes. But whereof, then, the other slogan – threeness? How is it that, for example, Son (Christ) is identical to God, and Father is identical to God, but it’s just false that Son is identical to Father (and, mutatis mutandis, Father and Spirit)? Start with the given non-identity: namely, it’s just false that Father is identical to Son (Christ). What explains this? Well, given that trinitarian identity is defined in the leibnizian-recipe fashion using logic’s biconditional (much more on which in §3.3.1 and §3.5), there must be some predicate 𝜑 in the theory that’s just true of Father while being just false of Son (or vice versa), in which case the defining biconditional (for fromscratch details see §3.3.1) 𝜑(Son) ≡ 𝜑(Father) is itself just false, in which case the target trinitarian identity (viz., ≃) claim, Father is Christ, namely, Father ≃ Son is just false, in which case the axiomatic non-identity claim

38 divine contradiction It’s false that Father ≃ Son is just true. Is there such a predicate 𝜑? Yes; there are many but ‘begets Son’ (just true of Father but just false of Son) is one; ‘begotten’, flipping things (just true of Son and just false of Father), is another. Any such predicate is sufficient to explain the given non-identity truth.1 What of the relevant identity? Since it’s just false that Father and Son (Christ) are identical but it’s true that each is identical to God, there must be no predicate 𝜑 in the theory that’s just true of Son but just false of God, and likewise no predicate 𝜑 in the theory that’s just true of Father but just false of God. (Likewise, mutatis mutandis, for Spirit.) And that’s exactly what trinitarian reality reflects. Whenever one speaks truly of Christ one speaks truly of God; whenever one speaks truly of Father one speaks truly of God; whenever one speaks truly of Spirit one speaks truly of God. And such speaking of God is not merely speaking in an indirect way that reminds one of God or the like; it’s downright identity: anything true of Christ (Son), who is God, is true of God; and anything true of Father, who is God, is true of God; and anything true of Spirit, who is God, is true of God. That’s trinitarian reality. That’s divine reality. Along the same lines, trinitarian reality also reflects that the full truth of God – not that we’ll know it in full – is the pool of truths of Father, Son (Christ), and Spirit together.2 What else could it be given that Christ is identical to God, and Father is identical to God, and Spirit is identical to God? Whatever is true of Christ, Father, or Spirit is true of God – full stop. That’s just what divinity, in the robustly monotheistic sense, demands. On the other hand, to speak of Christ is not thereby to speak of Father or Spirit; for each of Father and Spirit is non-identical to Christ (and to each other). Contradiction underwrites the relation. The claim that Christ atoned for sin is true of Christ, and so true of God; but it is just false of Father and Spirit, neither of whom underwent crucifixion. 1 The trio of fundamental trinitarian entailments, given in §3.6.2 (p. 54), are also in play in all such facts. 2 To those readers who therein detect an air of so-called social-trinitarian thinking I point to two things: first, principle M whereby, per §1, being divine just is being identical to God; second, I point to the details of trinitarian identity below (§3.5) and also the thrust of robust christian monotheism as reflected in the Athanasian-Creed-constrained counting convention (§3.7.3.1). On the other hand, inasmuch as there’s something important about social-trinitarian leanings that can be cut apart from their historical flight from divine contradiction, perhaps the foregoing point about the full truth of God is exactly the valuable gem. Perhaps. I leave further debate to tell.

trinitarian identity 39 The incarnation, through which otherwise unseen truths of God were revealed, is instructive. True claims about Christ are very often contradictory (i.e., they entail a contradiction given other truths). That Christ is ignorant is contradictory given that Christ is omniscient. That Christ can sin is contradictory given that Christ cannot sin. And so on for many – perhaps most – central truths of Christ: they are contradictory. What, though, of claims concerning Father? What of Spirit? Are claims of Father or Spirit just as often contradictory as claims of Christ? No. That Father begot Son is just true; there is no contradiction therein. That Father is omniscient, impeccable, omnipotent, merciful, holy, loving, and sovereign involves no contradiction; all such claims – and vastly many others – are just true. And likewise for Spirit. But claims across trinitarian reality – the fundamental, relational core of divine reality – are contradictory, and naturally so. It’s just false that Father hung on the cross (and likewise for Spirit), but Father is God, and God is identical to Christ who hung on the cross. Christ’s hanging on the cross is God’s hanging on the cross. Father’s not hanging on the cross is God’s not hanging on the cross. The identity claims that speak of Father’s identity to God, Christ’s identity to God, and Spirit’s identity to God are naturally contradictory; their truth essentially involves contradiction; for their falsity is ensured by the differences between Father, Christ, and Spirit. To say that Father begot Son is to speak truly – with no contradiction. To say that Father is God is to speak truly – a fundamental, axiomatic truth – but the identity is also false; for Spirit is God and it’s just false that Spirit begot Son. That Spirit is God is true – fundamentally, axiomatically true – but also false; for Christ is God and Christ is begotten, but it’s just false that Spirit is begotten. The contradictions are natural but only salient when talking ‘across’ trinitarian reality. God, who is Father, and who is Son, and who is Spirit, is the one to whom any divine being, in the robustly monotheistic sense of ‘divine’, is identical. But God is contradictory: what is true of Father is true of God; what is true of Christ is true of God; what is true of Spirit is true of God – including true negations (e.g., the falsity of Father’s hanging on the cross, the falsity of Christ’s begetting Father, and more). Divine contradiction unites the three divine persons each of whom is God and all of whom are pairwise non-identical to each other. Trinitarian identity is an intricate relation at the core of divine reality – at the very core of God. The relation reflects the oneness of God given

40 divine contradiction that each of Christ, Father, and Spirit is truly identical to the one and only triune being (viz., God). But the oneness of God is only part of the full truth. The relation also reflects the threeness of God given that each of Christ, Father, and Spirit are pairwise non-identical to each other. Trinitarian identity exemplifies unity in trinity and trinity in unity, and it does so via contradiction. Divine contradiction is central to the very being of God who is triune in a straightforward way: God is identical to each of Christ, Father, and Spirit – three pairwise non-identical persons each of whom is divine (i.e., each of whom is God). And that’s just what it is to be triune: to be identical to each of Christ, Father, and Spirit. One would be incorrect to think that the given divine contradictions are peripheral to trinitarian reality. Far from it. The contradictions of God are at core of trinitarian reality. Truths of Christ are truths of God; truths of Father are truths of God; truths of Spirit are truths of God. But – lest this remain unclear – some truths of Christ (similarly, of Father, of Spirit) are just false of Father and of Spirit, differences that underwrite the axiomatic pairwise non-identities at the root of trinitarian reality: it’s false that Christ is Father; and it’s false that Father is Spirit; and it’s false that Spirit is Christ. These non-identities rest on differences that are manifest in God: the features of Christ that are just false of Father are thereby features true and false of God. As theologians have long said, the relation (viz., trinitarian identity) that unites the non-identical divine persons and the one and only triune god (viz., God) is a highly intricate and intimate relation; but what has been unsaid, for too long, is that the relation unites a

trinitarian identity 41 contradictory reality. To ignore the contradictions is to ignore trinitarian truths; to reject the contradictions is to reject such truths. Lest confusion from prevalent consistent theories creep in, one very important fact must be unforgettably clear: namely, the falsity of certain claims in the true theology does not in any way remove or take back the truth of such claims. That Christ is God is true is only part of the truth; the falsity of that claim, namely, it’s false that Christ is God is more of the full truth. The unfolding of surprising – and often contradictory – truths of God is the common pattern ushered in by God’s incarnation, namely, Christ. That God is omniscient, impassible, impeccable, and more are truths whose truth, however lightly or fully, was evident without the special revelation of God enfleshed; they were evident before God walked, sweated, and cried on earth. What Christ revealed is the astonishing full story: that God is ignorant, passible, as peccable as you and me, and more. But while the falsity of God’s being omniscient, impeccable, and more is of great consequence in christian theology, it doesn’t in any way take away from the truth of God’s omniscience, impeccability, or more. Again, the ‘positive’ truths of God were available, however dimly, before God’s incarnation; the ‘negative’ truths of God were ushered in by the fuller and incarnate story, one that remains an extraordinary stumbling block in juxtaposition with the full truth of familiar fragments of reality. The incarnate revelation bears directly on the trinity. Per the target theory, the incarnation reveals the truth of God’s triune nature by revealing that God is identical to Father, to Son (Christ), and to Spirit. And therein the core trinitarian contradiction arises. God’s being identical to both Father and Christ while Father and Christ are non-identical is explained via contradiction: it’s true that Father is God but it’s also false; and it’s true that Christ is God but it’s also false. What is entirely untrue is that Father is identical to Christ; the non-identity is just false – full

42 divine contradiction stop. But none of this undermines or takes away from the axiomatic truths that Father is identical to God and that God is identical to Christ. Were there no contradictions true of God there’d be no explanation of the explosive non-identity of Father and Christ – an explosive non-identity in the sense that the true theology recognizes no possibility in which Father is Christ. Likewise, mutatis mutandis, for Spirit and Father, and Spirit and Christ. * * Parenthetical note (especially for those familiar with Chapter 4 of my 2021). The example of Christ’s contradictory divinity carries an important methodological lesson for the interplay between christology and trinitarian theory. In my 2021 I bracketed out trinitarian reality (except for a brief reflection in the last chapter), and in doing so inadvertently bracketed out the essential relational aspect of divinity in the robustly monotheistic sense herein discussed. The essential relational aspect of divinity is reflected in principles M and TG in §1, the equivalence of being divine and being identical to God (and, likewise, of being ‘a god’ and being triune). But, as in the current work, once the target identity relation is carefully spelled out, it’s plain that the divinity of Christ – that is, the truth of ‘Christ is identical to God’ – is contradictory; it’s absolutely true, as advanced in said 2021 work, but, pace that work, its logical negation is also true. This is a stark reminder that while christology informs trinitarian theorizing, the role of Christ in trinitarian reality likewise informs back. End note. * * The contradictory axiomatic claims of the trinity do not ‘compete’ for the truth about God; they are the truth about God – the full and radical truth about God. They are not ‘in tension’ with each other; they are in contradiction. There is absolutely no tension in God – absolutely no competition in God – whatsoever. None. Ever. Nothing of the sort. But there is contradiction – divine contradiction – at the core of God’s triune being. For too long christian theology has sought to avoid the apparent contradictions at the root of divine reality – at the root of the one and only triune god, namely, God, who is triune in being identical to Father, Christ, and Spirit while Father, Christ, and Spirit are three pairwise nonidentical persons. These are not peripheral claims in the theory; they are core axioms – the oneness of God and the threeness of Father, Christ, and Spirit. And they are mutually explanatory: the pairwise non-identity

trinitarian identity 43 of Father, Christ, and Spirit, together with the identity of each to God, explains the contradictions true of divine reality. Likewise, the falsity of the target identities of Father, Son, and Spirit to God explains the pairwise non-identities; for were such identities to God not false the would-be identities of Father and Christ, Christ and Spirit, and Father and Spirit would quickly follow from a would-be transitive trinitarianidentity relation that would otherwise be in place. So goes the big picture. In the big picture, trinitarian identity unifies divine reality; it’s the identity relation whereby, above all, there is unity in trinity and trinity in unity. The pressing question concerns precise details. At least one pressing question is palpable: what, exactly, is this identity relation?

3.2 Identity and true theories Per §2 logical vocabulary contains no predicates; it contains only the sentential connectives and two first-order quantifiers. Hence, logical vocabulary contains no necessity predicate, no possibility predicate, no truth predicate, no falsity predicate, not even a validity predicate, and, of most relevance to the current chapter, no identity predicate. Unlike notions of necessity, possibility, truth, falsity, and validity, most true theories contain some relation of identity or other. But exactly what the identity relation should look like is a matter for reality to demand. While perhaps many parts of reality are truly and fully described by theories that involve fairly familiar identity relations, it’s far from clear that all parts of reality are just so. Indeed, given the elementary (athanasian) axioms of trinitarian reality, it’s fairly clear that some true theories involve fairly unfamiliar identity relations. When it comes to producing true and complete theories, logic provides the elementary, topic-neutral vocabulary; the rest of the theory’s language is a matter of discovery. And this applies to a theory’s identity relation(s) as much as it applies to other predicates in the theory’s language. The task in general and simplified form: figure out the target part of reality; specify the language to truly and fully describe it (including identity relations for the target reality); and specify the consequence (entailment) relation that completes the theory with respect to all truths expressible in the language. Such is the way of systematic, truth-seeking theory building.

44 divine contradiction But are identity relations really so theory-specific? Yes and no. Yes: one true theory’s identity relation needn’t – though certainly could – be another’s. And two or more true theories might both involve a common identity relation, at least restricted to the respective domains of the theories, while also containing identity relations that the other doesn’t contain. In this way, identity relations are very much theory-specific. But no: there’s also a clear sense in which identity relations – at least in many true theories – enjoy a common recipe, often called the ‘leibnizian’ recipe (from which the well-known ‘Leibniz Law’ term arises).3

3.3 The leibnizian recipe Though very familiar the leibnizian recipe demands rehearsal. Let ⇒ be a conditional in the language of a given theory T. Let A and B be any sentences in the language of T. Let ∧ be logical conjunction. Then the biconditional corresponding to ⇒, namely, ⇔, is defined as usual via conjunction: A⇔B is defined to be (A ⇒ B) ∧ (B ⇒ A) Now, these are all resources within the language of the given theory T. Also in the language of theory T are predicates; these are tied to the theory; they’re used to express the truths about objects in the theory’s domain – whether, for example, a predicate is true of an object or false of an object. The leibnizian recipe comes into play only after the foregoing pieces are on the table.⁴ The leibnizian recipe is a general recipe for generating 3 The recipe is called ‘leibnizian’ with a small ‘l’ rather than a large ‘L’ because it’s tracking the general recipe, not only familiar specific instances that Leibniz himself might’ve endorsed (or not). ⁴ Strictly speaking, the given conditional need not be in the theory; it could instead be strictly in the so-called metatheory, which is the theory of the identity relation being defined. Since, ultimately, I define the relation in terms of logic’s material biconditional, I’m presenting the recipe in terms of a given (‘object’) theory’s conditional – though in fact much of the definition

trinitarian identity 45 identity relations in a theory. The key idea is to define the target identity relation, which is a binary relation ∼,⁵ in terms of ‘indiscernibility’ with respect to some set of properties. How does the leibnizian recipe go? First, let ‘𝜑’ be a so-called schematic variable over the entire set 𝒫 of predicates in theory T, and let ‘t’ and ‘t′ ’ be (meta-) variables ranging over singular terms (for simplicity, names) in the language of T.⁶ With this in place, we can talk directly about a bunch of sentences of the same form. Example (of no general importance except as a random illustration): we might talk about a bunch of axioms in theory T by saying something like • Where 𝜑 is in 𝒬 (for some subset 𝒬 of the set 𝒫 of predicates), every sentence of the form 𝜑(t) is an axiom. or, using T’s conditional ⇒, we might say • Where 𝜑 is in 𝒬, for subset 𝒬 of 𝒫, no sentence of the form 𝜑(t) ⇒ 𝜑(t′ ) is false. though, again, these are merely random illustrations; they are not in any way put forward as claiming anything about theology or any other theory. The leibnizian recipe, as above, defines identity relations via ‘indiscernibility’. Specifically, a target identity relation ∼ is defined via a so-called sentential schema 𝜑(⋅) ⇔ 𝜑(⋅), namely (where T is the target theory in question): Definition 3.3.1 (The leibnizian recipe.) Where 𝜑 ranges over the set 𝒫 of T’s (non-identity) predicates:⁷ is truly expressed only in the theory’s metalanguage. NB: Readers for whom the distinction between ‘object language’ and ‘metalanguage’ is unfamiliar are not missing any critical ideas. The point here is largely for experts who breathe the distinction daily. For historical discussion of the distinction see the work of Tarski (1956). ⁵ A relation R is binary if and only if it relates exactly two (not necessarily distinct) things. ⁶ A slightly technical note: ‘𝜑’, in this context, is a so-called metavariable; it is a variable in a metatheory of T, namely, the metatheory in which T’s target identity relation is defined. Generally, a theory’s identity relation is not defined within the given theory even if the relation is defined in terms of other expressions in the language of the given theory (e.g., logical vocabulary). ⁷ Identity is defined over a set of predicates that doesn’t contain the identity predicate being thereby defined. The relation is not only thereby well-defined; it’s also so-called well-founded.

46 divine contradiction • t ∼ t′ is true according to T iff every instance of 𝜑(t) ⇔ 𝜑(t′ ) is true according to T; • t ∼ t′ is false according to T iff some instance of 𝜑(t) ⇔ 𝜑(t′ ) is false according to T. That’s the basic – and familiar – leibnizian recipe. But ingredients are telling.

3.4 The ingredients What the familiar leibnizian recipe produces depends on the ingredients involved.

3.4.1 The usual ingredients One of the key ingredients in the leibnizian recipe is the biconditional involved in the defining schema 𝜑(t) ⇔ 𝜑(t′ ). The most familiar ingredients involve biconditionals that are reflexive, symmetric, and transitive: • Reflexive: A ⇔ A is true in the theory, for all sentences A in the theory. • Symmetric: If A ⇔ B is true in the theory then B ⇔ A is true in the theory. • Transitive: If both A ⇔ B and B ⇔ C are true in the theory then A ⇔ C is true in the theory. There are many conditionals whose corresponding biconditionals, formed as above (p. 44), have all such features, even though the underlying conditionals differ from one another in different ways (e.g., one might have ‘modal strength’, or one might contrapose while the other doesn’t, or so on). Regardless of differences, the usual ingredients used in the leibnizian recipe for defining identity predicates involve biconditionals that are reflexive, symmetric, and transitive. Symmetry of the biconditional basically falls out of the definition of biconditionals and features of logical conjunction: for example, A ⇔ B,

trinitarian identity 47 by definition, just is (A ⇒ B) ∧ (B ⇒ A), which, given features of logical conjunction ∧, delivers symmetry of the biconditional (since logical conjunction ‘commutes’): (B ⇒ A) ∧ (A ⇒ B). Reflexivity and transitivity of a biconditional are functions of the underlying conditional (and features of logical conjunction): the biconditional ⇔ is reflexive if and only if the given conditional ⇒ is reflexive. Likewise, the biconditional ⇔ is transitive if and only if the given conditional ⇒ is transitive.⁸ Important to note is that if the leibnizian-recipe schema involves a reflexive, symmetric and transitive biconditional then the resulting identity relation ∼ itself is reflexive, symmetric and transitive; it’s a socalled equivalence relation (over its defined range): • Reflexive: x ∼ x is true in the theory for all x over which ∼ is defined. • Symmetric: If x ∼ y is true in the theory then y ∼ x is true in the theory. • Transitive: If both x ∼ y and y ∼ z are true in the theory then x ∼ z is true in the theory. And this is why the most familiar accounts of identity are equivalence relations: they result from using reflexive, symmetric, and transitive biconditionals in the leibnizian-recipe definition of the given identity predicates.

3.4.1.1 Note on restricting the schema’s range The leibnizian recipe defines an identity relation in terms of some subset 𝒬 of the given theory’s set 𝒫 of predicates. If a reflexive, symmetric, and transitive biconditional is used in the leibnizian-recipe schema then not only is the identity relation, defined over 𝒬, an equivalence relation, but so too for any restricted relation so defined only over a proper subset 𝒮 of 𝒬. (This is notable mostly in comparison with leibnizian-recipe relations that involve biconditionals that are either not reflexive or not transitive. More below.)

⁸ Reflexivity, symmetry, and transitivity of ⇒ are defined exactly as for ⇔ except that ⇒ replaces ⇔ in the definition. (While biconditionals are generally symmetric, conditionals themselves rarely are.)

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3.4.2 Beyond the usual ingredients Varying the biconditional varies the identity relation baked by the leibnizian recipe. In particular, if, per §3.4.1, the leibnizian recipe uses a non-reflexive or non-transitive conditional whose non-reflexivity or non-transitivity shows up over the subset 𝒬 of T’s predicates in terms of which the leibnizian-recipe schema is defined, then the resulting identity relation will also fail to be reflexive or fail to be transitive – at least over the given set 𝒬 of predicates.

3.4.2.1 Note on restricting the schema’s range Let the target schema be defined over subset 𝒬 of theory T’s predicates, and let the conditional underwriting the schema be non-reflexive or nontransitive. Unlike with the usual ingredients (per §3.4.1.1), restricting the range of the leibnizian-recipe schema’s target predicates (i.e., restricting the range of the metavariables in the schema) can make a difference to the relation’s behavior; it may well be an equivalence relation over certain proper subsets of 𝒬 even though not an equivalence relation over all of 𝒬. Indeed, it might, for example, be reflexive and symmetric but nontransitive over one subset of 𝒬 while, say, not reflexive but still symmetric and non-transitive over another, and yet a full-on equivalence relation over another subset.

3.4.3 Towards the target ingredients of trinitarian identity The usual leibnizian recipe invokes logic’s conditional, namely, the derivative material conditional A → B defined per §2 via logical negation and logical disjunction: namely, A → B is defined to be ¬A∨B. In turn, on the usual leibnizian recipe the target schema involves logic’s biconditional, namely, the material biconditional A≡B defined, as above, via the material conditional and logical conjunction: (A → B) ∧ (B → A).

trinitarian identity 49 I follow the standard suit: the leibnizian-recipe schema involves logic’s biconditional, the material biconditional. Where the dress is different is not in using some extra-logical conditional to underwrite the leibnizian-recipe identity schema; the difference is in the account of logic itself, and in particular the features of logic’s biconditional. On my view, per §2, logical consequence is per FDE, not per the standard (so-called classical) account. And logic, so understood, does not itself validate either reflexivity or transitivity of its biconditional: • Not logically reflexive: logic itself does not validate the truth of A ≡ A for all A in the language of all true theories. • Not logically transitive: logic itself does not validate the truth of A ≡ C given only the truth of A ≡ B and B ≡ C. Such reflexivity and transitivity of the biconditional may well be – in fact, on my view, are – very widely validated by many true theories; however, it’s not logic itself that forces the issue; it’s the extra-logical validity (consequence, entailment) relation involved in such theories. Per §3.4.1, since ≡ is not logically reflexive or logically transitive (i.e., such features are not validated by logic alone), the leibnizian-recipe schema can deliver an identity relation that is neither reflexive nor transitive, depending on the set of (the theory’s) predicates over which the schema is defined. If (and only if) the non-reflexivity or non-transitivity of ≡ shows up over the set 𝒬 of predicates over which 𝜑(t) ≡ 𝜑(t′ ) is defined, the resulting identity relation is respectively non-reflexive or non-transitive (or both). Critical to recall is that while neither reflexivity nor transitivity of identity, so understood, is logically forced, each is available for a theory to deliver as axiomatic or fundamental (theory-specific) entailments. This is a virtue of the account: on its own logic itself doesn’t contribute truths to any theory; logic, on its own, remains entirely topic-neutral. And this neutrality nicely spills over into identity relations defined in terms of logic’s ingredients. In the end, if reflexivity or transitivity (or both) is demanded by the target reality then the given true theory adds reflexivity or transitivity (or both) as an extra-logical, theory-specific axiom or entailment.

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3.4.3.1 Extension without gluts Logic itself, qua FDE consequence, recognizes the possibilities of both gaps and gluts. (See §2.) But portions of reality might be entirely free of gluts while still exhibiting gaps. In such a case, the true and complete theory of the given reality excludes the logical possibility of gluts, treating it as a theoretical impossibility – simply off the chart of relevant models or out of the space of relevant possibilities. Extending logical consequence to be glut-free results in a relation often called Strong Kleene or K3.⁹ On this relation, reflexivity of the material biconditional is not valid; however, transitivity and symmetry are validated. Accordingly, running the leibnizian-recipe schema in such a context delivers a transitive and symmetric identity relation – and can but, as noted, needn’t result in a reflexive one too, depending on the set 𝒬 of predicates over which the leibnizian-recipe schema is defined. 3.4.3.2 Extension without gaps Again, logic itself recognizes the possibilities of both gaps and gluts. But just as portions of reality might be entirely free of gluts while exhibiting gaps, so too portions of reality might exhibit gluts while being entirely free of gaps. In such a case, the true and complete theory of the given reality excludes the logical possibility of gaps, treating it as a theoretical impossibility. Extending logical consequence to be gap-free results in a relation often called ‘LP’.1⁰ On this relation, reflexivity and symmetry of the material biconditional is valid; however, transitivity is not valid. Accordingly, running the leibnizian-recipe schema in such a context delivers a reflexive and symmetric identity relation – and can but, as noted, needn’t result in a transitive one too, depending on the set 𝒬 of predicates over which the leibnizian-recipe schema is defined.

⁹ See Kleene 1952 or Beall and Logan 2017. This is an ‘extension’ of FDE consequence because it (properly) augments – properly extends – the set of valid patterns that logic itself validates; it recognizes more validities than the logical ones because it removes – from the theory’s space of possibilities – some of the logically recognized possibilities that otherwise invalidate various patterns (e.g., in the case at hand, glutty possibilities). 1⁰ Further details are available elsewhere (Asenjo, 1966; Asenjo and Tamburino, 1975; Priest, 1979), including elementary, user-friendly discussion (Beall and Logan, 2017).

trinitarian identity 51

3.4.3.3 Extension without gluts and without gaps Just as there are portions of reality whose true theories involve gaps but no gluts, and some that involve gluts but no gaps, some – indeed, by far the most familiar – true theories have neither gaps nor gluts. Extending logical consequence to be both gap-free and glut-free results in a relation often called ‘classical logic’.11 On this relation, reflexivity, symmetry, and transitivity of the material biconditional are valid. Accordingly, running the leibnizian-recipe schema in such a context delivers a reflexive, symmetric, and transitive identity relation – no matter the (non-empty) set 𝒬 of predicates over which the leibnizianrecipe schema is defined. For present purposes, this particular (so-called classical) extension is important to remember: namely, that even though logic itself doesn’t validate reflexivity or transitivity of its biconditional, when we’re in a theory with no gaps and no gluts the leibnizian-recipe schema results in an identity relation that just is the familiar first-order identity relation underwritten by classical logic.

3.5 Trinitarian identity My thesis is that the relation of trinitarian identity – henceforth symbolized ‘≃’ to distinguish it from an arbitrary identity relation ∼ – is per §3.4.3: the relation that follows the leibnizian recipe using logic’s ingredients and, in particular, logic’s biconditional, where, as in §2, logic is per FDE. (The exact trinitarian-identity relation is constrained by fundamental ‘trinitarian entailments’ discussed in §3.6 but the relation is per the leibnizian recipe of §3.4.3.) Since trinitarian identity, so understood, is non-transitive provided that gluts aren’t ruled out of theology’s space of possibilities (and they aren’t – because cannot be – ruled out without losing important theological truths), trinitarian identity is non-transitive in the trinitarian theory, as expected. So goes the main thesis and the explanation of why, in the theory, the central identity relation is non-transitive. Two pressing questions 11 Though there are at least three gazillion introductions to so-called classical logic, for the sake of comparison and ease of use see Beall and Logan 2017.

52 divine contradiction remain. The first: what, if any, salient entailment patterns figure in trinitarian reality and its central identity relation? The second: what counting convention is associated with trinitarian identity? Both questions in turn (viz., §3.6.1 and §3.7, respectively).

3.6 3-1-ness entailments and trinitarian identity Two of three fundamental trinitarian entailment are 3-1-ness patterns. (The third, discussed in §3.6.2, concerns ‘freedom from gaps’ in trinitarian reality.) For both clarity and ease of exposition the terms ‘𝔣’, ‘𝔰’, ‘𝔥’, and ‘𝔤’ are used, respectively, for Father, Son, Spirit, and God. As throughout, ⊢𝜃 is theological consequence – the consequence or entailment relation that ‘completes’ (technically, closes) the theory with respect to all claims that follow from any claims in the theory – and ⊣⊢𝜃 is two-way theological consequence (or, equivalently, theological bi-entailment), so that A ⊣⊢𝜃 B is the logical conjunction of A ⊢𝜃 B and B ⊢𝜃 A.

3.6.1 The 3-1-ness patterns The two 3-1-ness patterns are as follows, with 𝜑 being any predicate (in the language of the theory) within the range of the definition of trinitarian identity.12 The entailment patterns are called ‘T+ ’ and ‘T− ’, with the former concerning ‘nullations’ (i.e., dual-of-negation claims governed by the implicit null or truth connective), and the latter concerning ‘negations’ (i.e., claims explicitly governed by the explicit negation connective). +

Definition 3.6.1 (Trinitarian entailment: T .) Let 𝜑 be any (nonidentity) predicate in the range of (the definition of) trinitarian identity. Let ⊣⊢𝜃 be theological bi-entailment (i.e., the theological entailment relation in both directions). T+ . 𝜑(𝔣) ∨ 𝜑(𝔰) ∨ 𝜑(𝔥) ⊣⊢𝜃 𝜑(𝔤)

12 Per §3.3.1 (p. 45) trinitarian identity, being well-defined and, technically, well-founded, is defined only over non-identity predicates.

trinitarian identity 53 T+ reflects the 3-1-ness fact that, with respect to target non-identity features 𝜑, anything true of Father, Son, or Spirit is true of God, and vice versa. Anything less than T+ diminishes the fact that each of Father, Son, and Spirit is God, and, exhibiting the uniqueness of trinitarian reality, God is each of Father, Son, and Spirit – despite their pairwise nonidentity. Likewise for the other 3-1-ness pattern: −

Definition 3.6.2 (Trinitarian entailment: T .) Let 𝜑 be any (nonidentity) predicate in the range of (the definition of) trinitarian identity. Let ⊣⊢𝜃 be theological bi-entailment (i.e., the theological entailment relation in both directions). T− . ¬𝜑(𝔣) ∨ ¬𝜑(𝔰) ∨ ¬𝜑(𝔥) ⊣⊢𝜃 ¬𝜑(𝔤) T− reflects the 3-1-ness fact that, with respect to target non-identity features 𝜑, anything false of Father, Son, or Spirit is false of God, and vice versa. Anything less than T− diminishes the fact that each of Father, Son, and Spirit is God, and that God is each of them – again, despite their pairwise non-identity. T+ and T− are two fundamental entailments that jointly reflect some of the central structure and fundamental patterns of the trinitarian identity relation in divine reality. The patterns are not only fundamental; they are familiar, even if rarely recorded in detail.13

3.6.2 Trinity without gaps The third fundamental entailment pattern reflects the absence of gaps in trinitarian reality. Logic, per §2, recognizes the possibility of gaps 13 One might think that the entailment patterns are – or somehow ought to be – entirely unrestricted, governing all predicates in the language of theology, including trinitarian-identity predicates, rather than governing just the predicates over which trinitarian identity is defined. This is a natural thought in the abstract, unmoored from axioms of trinitarian reality; however, the axioms themselves – particularly axiomatic non-identities – clearly reflect the restricted range of T+ and T− . In short: if such entailment patterns were in fact ranging over all predicates – including trinitarian-identity predicates – then, for but one example, Father would be identical to Son. But the explicit proscription against ‘confusing the persons’ in the Creed proscribe against the identity of Father and Son, and likewise for Spirit. (This point is rehearsed again in §5.4.)

54 divine contradiction (i.e., claims that are neither true nor false), and does so just as much for theology as physics, mathematics, biology, ethics, and more. Most (if not all) of the true theories of such phenomena treat gaps as ‘merely logical possibilities’ but, importantly, not theoretical possibilities – that is, not possibilities recognized by the (entailment relation of the) given theory. What of theology – or at least the sub-theological theory of the incarnation and trinity? This is where the third fundamental entailment pattern enters concerning ‘distribution of gaps’ (DG). Some notation: where A is any sentence (or simply schematic for a sentence), let ¡A abbreviate A ∨ ¬A so that, for example, ‘¡𝜑(𝔣)’ abbreviates 𝜑(𝔣) ∨ ¬𝜑(𝔣).1⁴ The abbreviation notation affords an easier-on-the-eyes convenience for expressing the distribution-of-gaps entailment pattern, namely, Definition 3.6.3 (Distribution of gaps: DG.) Let 𝜑 be any (non-identity) predicate in the range of (the definition of) trinitarian identity. Let ⊣⊢𝜃 be theological bi-entailment (i.e., the theological entailment relation in both directions).1⁵ DG. ¡𝜑(𝔣) ∨ ¡𝜑(𝔰) ∨ ¡𝜑(𝔥) ⊣⊢𝜃 ¡𝜑(𝔣) ∧ ¡𝜑(𝔰) ∧ ¡𝜑(𝔥) DG reflects this trinitarian fact: a predicate that’s either true or false of Father, Son, or Spirit is either true or false of Father, Son, and Spirit. (Note well: this does not imply that a predicate true of Father is thereby true of Son and Spirit; it implies only that a predicate true of, for example, 1⁴ A common abbreviation for a contradiction A ∧ ¬A is ‘!A’, and since ¬A ∨ A is the dual (a sort of upside-down image) of A ∧ ¬A, the given upside-down-shriek notation (i.e., upsidedown ‘!’, viz., ‘¡’) is motivated. 1⁵ The right-to-left direction follows by logical entailment alone but it’s explicitly given here for uniformity with other key trinitarian (bi-) entailment patterns.

trinitarian identity 55 Father is thereby either true of or false of Son, and likewise true of or false of Spirit.) In effect, DG recognizes no predicate (in the range of trinitarian identity) that, on one hand, is either true or false of a divine person but, on the other, somehow fails to be either true or false of another. Were there special reason to think that trinitarian reality involves gaps of some but not all divine persons, DG would thereby be undermotivated. But, as far as I can see, there is no special reason to recognize gaps in trinitarian reality, at least if the predicate is non-gappy with respect to some divine person. Moreover, DG has solid methodological backing. One strong methodological rule of thumb pushes for gapfree theories, just as another equally strong methodological rule of thumb pushes for glut-free theories. (Another equally important methodological rule of thumb pushes for simple and natural theories.) Unlike gluts – which, in theology, have special motivation at least from Christ’s contradictory being – gaps have no special motivation in trinitarian theory. Perhaps future work will motivate a rejection of DG. For now, DG stands as the third of three fundamental entailment patterns, augmenting the two 3-1-ness patterns. * * Long parenthetical note. A slightly technical question concerning gaps remains: does DG alone – or even in concert with T+ and T− – preclude gaps from trinitarian identity? The answer: almost but no. The absence of gaps is ensured by the joint work of all three fundamental entailment patterns (viz., T+ , T− , DG) together with the key axiomatic identities and non-identities concerning divinity (i.e., any entity identical to God). For example, the three fundamental entailment patterns are compatible with the logical possibility – call it uniform-𝜑-gappiness – in which some relevant predicate 𝜑 (in the range of trinitarian identity) is gappy with respect to all divine persons (viz., Father, Son, Spirit, God); there’s nothing in the three fundamental entailment patterns that precludes, for some 𝜑 in the range of trinitarian identity, uniform-𝜑gappiness in divine reality. But such a logical possibility is theologically impossible – ruled out of the space of theology’s possibilities – by the axiomatic identities and non-identities. Such a logical possibility would be one wherein for some 𝜑 we have |𝜑(𝔣)| = |𝜑(𝔰)| = |𝜑(𝔥)| = |𝜑(𝔤)| = ∅

56 divine contradiction in which case, as one can verify from §2, |𝜑(t) ≡ 𝜑(𝔤)| = ∅ for any relevant terms t (e.g., ‘Father’, ‘God’, ‘Son’, ‘Spirit’), and hence, by definition of ≃, the axiomatic identities (e.g., 𝔣 ≃ 𝔤) are one and all untrue (since at best gappy). A similar result holds for key axiomatic non-identities (e.g., 𝔣 ≄ 𝔰). Accordingly, the joint work of all trinitarian entailments together with the fundamental identities and non-identities suffice to rule out gaps in divine reality (at least for features over which trinitarian identity is defined). End note. * *

3.7 On the 3-1-ness problem: counting conventions Divine contradiction is partially reflected in the interplay between the fundamental trinitarian entailments and trinitarian identity. In the end, per §3.1, God is contradictory precisely because God is identical to three divine beings who are not likewise identical to each other (i.e., three divine persons no two of whom truly stand in the given identity relation together). Enter the 3-1-ness problem. Any theory involving an identity relation must specify the counting convention associated with the relation. A complete counting convention associated with a given identity relation is a rule whereby, for any relevant predicate 𝜑 over which the identity relation is defined, the number of 𝜑s is determined. Specifying such a rule for trinitarian identity is at the heart of the 3-1-ness problem. My view is that the exact details of the complete counting convention governing trinitarian identity – and involved in the true trinitarian theology in general – remain open. The Athanasian Creed, for example, appears to reflect different conventions, depending (at least) on the subject of predication (and also, e.g., whether the incarnation or trinity is salient under discussion). My aim is not to advance a complete counting convention for trinitarian theory, important and pressing as the task is for the theory. My aim, in this work, is only to explain part of the counting convention governing a few critical subjects involved in the 3-1-ness (or,

trinitarian identity 57 simply, ‘counting’) problem, including, the number of (triune) gods and the number of divine persons. What should be flagged from the start, and independent of theology, is that expressing the count of a predicate (i.e., ‘the number of 𝜑’) – just like expressing identity in a particular theory – enjoys a wealth of resources, depending on the language of the theory. Since logical vocabulary is universal, appearing in the language of all true and complete theories regardless of subject matter, logic’s quantifiers are always inexorably on hand. Hence, regardless of the target identity relation ∼ in the theory, claims of the form ∃x∃y [ 𝜑(x) ∧ 𝜑(y) ∧ x ≁ y ] are available for expressing one sort of ‘count’ claim: in standard paraphrase, there are at least two beings (or things or objects) of which 𝜑 is true – in shorthand, ‘there are at least two 𝜑s’.1⁶ But whether this claim, using logic’s quantifiers, adequately expresses the important ‘number’ claim concerning the target objects is a matter for the theory to specify. If the claim doesn’t adequately express the target number or count claim, other resources are required. The other resources may be extralogical quantifiers of some sort, or perhaps logical quantifiers combined with additional identity relations, or – importantly – some primitive expression ‘the number of ’. Indeed, combinations of these (and other) options are not only available to theology; they are likely in use – even if only imprecisely or merely implicitly so. My aim, to repeat, is not to chart a complete counting convention for trinitarian identity as I advance it in this book. My principle aim, to repeat again, is to focus on two salient ‘number issues’ in the 3-1ness (or counting) problem – divine persons and triune gods. But worth remembering is that, as above, there are many different resources that the true theological theory may ultimately involve, and the complete counting convention in theology may well involve a number of different counting conventions. 1⁶ Any readers who might be thrown off by the loose use of ‘𝜑’ in this context are welcome to use so-called Quine quotes if it helps. As throughout, I keep things as simple and easy-onthe-eyes as possible, trusting context to illuminate. (Any readers to whom, for example, ‘Quine quotes’ are unfamiliar should not think on this footnote again. No loss.)

58 divine contradiction

3.7.1 Counting, double counting, and relevant duplicates Let 𝜑 be a predicate in the relevant language on which an identity relation ∼ is defined. What is the number of 𝜑s? Outside of a particular theory, the answer is wide open. But one issue is generally tantamount: avoiding double counts.

3.7.1.1 On double counting and ∼-duplicates Consider the extension of ‘is a prime number less than 3’, namely, {2, 1 + 1, 3 − 1, √4} What is the number of the predicate ‘is a prime number less than 3’? Answer: 1. Why not 2? (Or 3? Or 4?) Answer: counting to two, in this case, is double counting. (And counting to three is triple counting, etc.) Why? Answer: after counting the number 2 (so that the count is at 1), subsequently counting, say, the number √4 is just to count the number 2 again, since, according to the true theory of arithmetic, the number 2 and the number √4 are identical. Why, in the example of ‘is a prime number less than 3’, is 1+1 a duplicate of 2? Answer: because in the relevant portion of mathematical reality, the identity relation in terms of which relevant duplicates are defined is the classical-identity relation (or some restricted version thereof).1⁷ But that’s not the only identity relation one might use for defining relevant duplicates of 2. Example: concerning the world of numbers modulo 5, for example, 7 is a duplicate of 2. Standard counting conventions, regardless of the reality they govern, reflect a common practice: don’t double count by recounting relevant duplicates! When you count by some identity relation ∼ you’re thereby using ∼ to define the class of relevant duplicates. The definition of ∼-duplicates, for an identity relation ∼, is definable in terms of a restricted sort of duplicate, namely, a 𝜑-duplicate, for predicate 𝜑 and its extension 𝜑+ . 1⁷ Predicates like ‘is known by So-n-so to be less than 3’ might be true of 2 but not 1+1. (Consider the so-called hooded-man case: Agnes knows her father but Agnes does not know the hooded man before her, even though the latter is identical to the former. Etc.) The identity relation involved in 1+1’s being a relevant duplicate of 2 ignores such so-called intensional predicates.

trinitarian identity 59 Definition 3.7.1 (𝜑-duplicate.) Let 𝜑 be some predicate in the target language. Let ∼ be the target identity relation for the target counting convention. Then i is a 𝜑-duplicate iff x ∼ i for all x ∈ 𝜑+ . In other words: i is a 𝜑-duplicate, for some predicate 𝜑, just if i is ∼-identical (i.e., stands in the identity relation ∼) to all satisfiers of 𝜑 – that is, to everything of which 𝜑 is true (i.e., everything in 𝜑’s extension). Definition 3.7.2 (∼-duplicate.) Let 𝜑 be some predicate in the target language. Let ∼ be the target identity relation for the target counting convention. Then u is a ∼-duplicate iff i ∼ u for all 𝜑-duplicates i. In other words: something is a ∼-duplicate, for identity relation ∼, just if it’s a 𝜑-duplicate for all relevant predicates 𝜑. A notable difference between 𝜑-duplicates, for some 𝜑, and ∼-duplicates is that the former entities might not be 𝜑-duplicates for all 𝜑. A very simple example: even though 2 is a relevant prime less than 3-duplicate – since it’s identical, according to target relation, to all elements of the extension – it’s not a prime less than 5-duplicate since the extension of ‘prime less than 5’ is {2, 1 + 1, 3 − 1, √4, 3, 2 + 1, 4 − 1, √9} and 2 is not a duplicate of 3 or, for that matter, any of 3’s duplicates. A ∼-duplicate is different: even if you are a 𝜑-duplicate for some predicate 𝜑 in the language, you’re not a ∼-duplicate unless you’re a 𝜑-duplicate for all relevant predicates 𝜑 in the language. With definition in hand, the general principle involved in familiar counting conventions is the same regardless of the identity relation ∼ governing the target reality: do not double count by recounting ∼-duplicates! This applies to material objects, fictional characters, numbers, sets, and, of course, trinitarian reality.

3.7.1.2 Glutty ∼-duplicates One uncommon sort of ∼-duplicate is a glutty sort: u is both truly and falsely identical to all 𝜑-duplicates for all relevant predicates 𝜑. Given the rarity of contradictory entities, such glutty ∼-duplicates are uncommon,

60 divine contradiction and certainly absent from many true theories; however, the counting conventions are basically the same. The general do-not-double-count principle in §3.7.1.1 applies to ∼duplicates regardless of whether the entity in question is contradictory. The reason is that counting counts the truth of an identity over any would-be falsity. In particular, if i is truly and falsely identical to all satisfiers of 𝜑 then it’s a 𝜑-duplicate: being truly identical to all relevant satisfiers of 𝜑 is sufficient for being a 𝜑-duplicate even if, however rare you might be, you are also falsely so identical. Counting counts truth over falsity. That a ∼-duplicate u – glutty or otherwise – never raises the count of 𝜑 above 1 is just the general counting principle: count u unless you’ve already counted one of its relevant duplicates!

3.7.2 The trinitarian counting convention The trinitarian counting convention, regardless of its complete shape, at least partly involves a primitive, theory-specific count expression that is not definable only (if at all) in terms of logical quantifiers and trinitarian identity. In trinitarian theory – at least with respect to salient 3-1ness-problem targets (e.g., triune beings, divine persons, and more) – a primitive ‘the number of 𝜑’ device expresses correct counts. Despite not being reducible to only logical vocabulary and trinitarian identity, the target counting convention is familiar. In short: TC. Trinitarian Counting convention: Count the extension of the target predicate but don’t double count by recounting ≃-duplicates! In effect, the convention takes the extension 𝜑+ of a relevant predicate 𝜑 and counts it not by, say, classical identity but rather by the central identity at the core of trinitarian reality: namely, trinitarian identity (viz., ≃), in terms of which ≃-duplicates are defined. As with any identity relation, when you count by ≃ you’re thereby using ≃ to determine relevant duplicates. Accordingly, given the untruth of each of

trinitarian identity 61 • 𝔣≃𝔰 • 𝔰≃𝔥 • 𝔥≃𝔣 it’s plain that neither 𝔣, 𝔰, nor 𝔥 is a ≃-duplicate in the true theology. On the other hand, 𝔤 is identical to each of 𝔣, 𝔰, and 𝔥; hence, 𝔤 is a 𝜑-duplicate of each person for each relevant predicate 𝜑. Hence, 𝔤 is a ≃-duplicate in the theory – in the true theology. And so, in just that light, counting 𝔤 when one has already counted one or more of 𝔣, 𝔰, and 𝔥 is double counting, at least as far as the trinitarian counting convention goes. Important to observe is that in trinitarian reality 𝔤 is the unique ≃-duplicate: Fact 3.7.3 (Unique trinitarian duplicate.) There is no ≃-duplicate except God. Proof Suppose otherwise. The only relevant candidates are 𝔣, 𝔰, and 𝔥. But while each of them is identical to 𝔤 they are nonetheless pairwise nonidentical: it’s untrue that x ≃ 𝔣 for any x except 𝔣 and 𝔤. Same, mutatis mutandis, for 𝔰 and 𝔥. That there is exactly one ≃-duplicate plays a direct role in the shape of the trinitarian counting convention. In short, to double count by recounting ≃-duplicates is to recount 𝔤 after already counting one of 𝔣, 𝔰, and 𝔥. The convention TC (see p. 60) can be expressed cleanly and succinctly – though, because more explicitly, perhaps not more clearly – in basic mathematical English. Let ‘#(𝜑)’ abbreviate the theory’s target expression the number of 𝜑.1⁸ Moreover, let ‘C(𝜑+ )’ abbreviate the cardinality of extension {x ∶ 𝜑(x)}, where this is defined in standard (classically closed) set theory using classical identity (perhaps restricted to nonintensional predicates etc.).1⁹ Finally, let 𝒰≃ contain all ≃-duplicates 1⁸ ‘#’, so understood, expresses a function: it takes an ‘input’ (in this case, a predicate) and delivers an output (in this case, a natural number). 1⁹ Let the relevant set theory be so-called ZFC. Then C(𝜑+ ) is basically what you’d expect in standard mathematics (at least for finite sets): it’s pretty much the number of classical-identity+ distinct objects in the extension of 𝜑. So, e.g., C(‘is a prime number strictly less than 5’ ) is the number of classical-identity-distinct objects in

62 divine contradiction according to the theory. Then the target (fragment of the) counting convention associated with trinitarian identity is just this: 0 if 𝜑+ = ∅ ⎧ ⎪ #(𝜑) = C(𝜑+ ) if 𝒰≃ = 𝜑+ ⎨ ⎪C(𝜑+ ) − 1 otherwise. ⎩ This convention is simply an equivalent way of expressing the familiar practice in TC: namely, count the extension of 𝜑 but don’t double count – and don’t under count either! Hence (first clause), if nothing at all satisfies 𝜑 then #(𝜑) = 0; nothing to count gives a count of zero. Furthermore (second clause), if just the ≃-duplicate u satisfies 𝜑 then the count is 1; anything less is an under count. Finally (third clause), in light of Fact 3.7.3 (see p. 61), if any non-≃-duplicates satisfy 𝜑 then recounting the ≃-duplicate is an over count.2⁰ In short, the contradictory account points to three facts – backed by the precise leibnizian recipe account of identity and the essential entailment patterns in T+ and T− – in accommodating (the church-stamped) counts of divine reality: • Fact One: the number of gods (equivalently, triune gods) is 1. (See §3.7.3.1.) • Fact Two: the number of divine persons is 3. (See §3.7.3.2.) • Fact Three: the precise trinitarian counting convention is simply the standard convention, reflected in many counting practices, by

{1 + 1, 2, 3 − 1, 2 + 1, 3, 4 − 1} and so, since two and three are the only classical-identity-distinct objects in the given set, +

C(‘is a prime number strictly less than 5’ ) = 2. Details of how cardinality is actually defined – via 1-1 functions – in standard set theory is more involved than necessary for present purposes. Every competent introduction to standard set theory has such details for readers who wish to dive deeper. (Note on notation: one standard notation for cardinality in set theory is the vertical-bar notation, viz., ‘| ⋅ |’, used in this book for semantic value and, so, not for set-theoretic cardinality.) 2⁰ Note that the trinitarian counting convention is not itself tied to a relative-identity relation, and is also at best a restricted instance of a more general convention beyond theology, details of which are not herein worked out.

trinitarian identity 63 which relevant duplicates are not (double-) counted, where relevant duplicates, in trinitarian theory, are defined – unsurprisingly – by trinitarian identity. What should be plain is that the trinitarian counting convention is indeed a common one, and not peculiar to divine reality except for the identity relation used to define relevant duplicates.

3.7.3 Salient examples In the context of trinitarian reality, where each of Father, Son, and Spirit is identical to God while Father, Son, and Spirit are pairwise non-identical, the trinitarian-identity counting convention is both natural and accurate. The proof is in key examples.

3.7.3.1 The number of triune beings The robust monotheism of christian theology is reflected not only in the fact that being divine and being identical to God are equivalent (per principle M in §1); it’s also reflected in the distinctively christian theological principle TG (viz., that the number of gods is 1 if and only if the number of triune gods is 1, per p. 11). What’s clear in the central athanasian axioms is the strict condition under which such monotheism (and any would-be polytheism) is to be understood: namely, one god in trinity and trinity in unity. In all things (per creed) this is the condition under which robust monotheism is expressed; it’s the go-to-whoa of robust christian monotheism. What the athanasian axioms and overall creed emphasize is that the relevant count of one, when it comes to counting ‘gods’, is the count of triune gods. (See TG on p. 11.) There is one and only one triune god (viz., God), even though each of the pairwise non-identical divine persons is identical to God. The rub of both the ‘logical’ and 3-1-ness problems comes together just there: the unity in trinity and trinity in unity; the one and only triune god to whom three pairwise non-identical beings are identical. Witness: God is triune; Father is identical to God; but it’s just false that Father is triune. (Likewise, mutatis mutandis, for Son and Spirit.) The axiomatic identities – though not axiomatic

64 divine contradiction non-identities – involve contradiction, but neither trinitarian reality nor its counting convention involves even a dose of incoherence.21 The central predicate ‘is triune’, like ‘is divine’ itself, is defined via trinitarian identity; indeed, it is defined via the (conjunctive) compound predicate being identical to Father and being identical to Son and being identical to Spirit: x≃𝔣∧x≃𝔰∧x≃𝔥 Critically, the key axiomatic non-identities of Father, Son, and Spirit, namely, • 𝔣≄𝔰 • 𝔰≄𝔥 • 𝔥≄𝔣 are one and all just true; they are not gluts. Hence, neither Father, nor Son, nor Spirit is a triune being in the foundational sense.22 Of course, given the axiomatic identities (e.g., 𝔣 ≃ 𝔤 and the others), each axiomatic non-identity speaks of God to whom each of Father, Son, and Spirit is identical (by the axiomatic identities). And this, again, is the central seam of both the ‘logical’ and 3-1-ness problems: God’s being identical to each of Father, Son, and Spirit while Father, Son, and Spirit are pairwise nonidentical. But how? Answer: God is a contradictory being: it is true that Father is God but also false; it is true that Son is God but also false; it is true that Spirit is God but also false. This, in its stripped-down and simplest terms, just is the triunity of God: namely, the truth of

21 One might erroneously think that since God is identical to Father (and to Son, and to Spirit) that substitution of identicals delivers that Father is triune (and likewise Son, Spirit). This is erroneous because such substitution principles, while valid in some true theories for some identity relations, are invalid for trinitarian identity. (Were such substitution validated for trinitarian identity, one could immediately substitute ‘Father’ for ‘God’ in ‘Son is God’ to get ‘Son is Father’ from ‘Father is God’, which, per §1, is the effect of a would-be transitive identity relation, which is clearly not trinitarian identity.) Invalidity of substitution for ≃ falls out of the definition of trinitarian identity. In other words, given the definition (which follows the standard leibnizian recipe, it’s provable that substitution of trinitarian identities is invalid. 22 Are there other senses of ‘triune’ around? Yes, but they’re at best derivative. Example: let x be *-triune iff x is identical to a triune being. And so on.

trinitarian identity 65 𝔤 ≃ 𝔣 ∧ 𝔤 ≃ 𝔰 ∧ 𝔤 ≃ 𝔥. Given the key axiomatic (and just-true) non-identities of Father, Son, and Spirit, the given truth of God’s triune being is true via contradiction, but via contradiction the truth is indeed true. The count question: what is the number of triune beings? Answer: #(‘triune being’) = #(‘x ≃ 𝔣 ∧ x ≃ 𝔰 ∧ x ≃ 𝔥’) = 1 And that being is God. * * Parenthetical remark on details. For any who wish to crunch through the trinitarian convention, let 𝕋 be the given triune predicate. 0 if 𝕋+ = ∅ ⎧ ⎪ #(𝕋) = C(𝕋+ ) if 𝒰≃ = 𝕋+ ⎨ ⎪C(𝕋+ ) − 1 otherwise. ⎩ The extension of ‘triune being’ (viz., 𝕋+ ) is {𝔤}. (Proof: God alone satisfies the conjunctive ‘triune’ predicate, as may be checked.) Hence, then, 𝒰≃ = 𝕋+ , and so per the second line: #(‘triune being’) = 1. End parenthetical remark. * *

3.7.3.2 The number of divine persons Just as the principal sense of ‘divine’, per principle M in §1, directly invokes the central relation of trinitarian identity (viz., x ≃ 𝔤), so too with the target sense of ‘divine person’. In particular, neither Father, Son, nor Spirit is defined in terms of some compound of ‘person’ and ‘divine’, but rather ‘divine person’ is defined in terms of identity to Father or to Son or to Spirit: being a divine person, in the target theological sense, just is being identical to Father or being identical to Son or being identical to Spirit. DP. x is a divine person is intersubstitutable with being identical to Father or being identical to Son or being identical to Spirit.

66 divine contradiction Accordingly, the predicate ‘x ≃ 𝔣 ∨ x ≃ 𝔰 ∨ x ≃ 𝔥’ is the target with respect to the number of divine persons. And the answer is three: #(‘x ≃ 𝔣 ∨ x ≃ 𝔰 ∨ x ≃ 𝔥’) = 3. Each of Father, Son, and Spirit truly satisfies the given predicate. Of course, per the athanasian axioms (e.g., 𝔣 ≃ 𝔤 and more), the predicate is true of God; however, per §3.7.1.1–§3.7.1.2, the number of divine beings is three – full stop. * * Parenthetical remark on details. For any who wish to crunch through the trinitarian convention (where, strictly for space considerations, ℙ is the target predicate ‘x ≃ 𝔣 ∨ x ≃ 𝔰 ∨ x ≃ 𝔥’): 0 if ℙ+ = ∅ ⎧ ⎪ #(ℙ) = C(ℙ+ ) if 𝒰≃ = ℙ+ ⎨ ⎪C(ℙ+ ) − 1 otherwise. ⎩ The extension of ‘divine person’ (viz., ℙ+ ) is {𝔣, 𝔰, 𝔥, 𝔤}, and so the first condition doesn’t apply. Since 𝒰≃ = {𝔤} ≠ ℙ+ , the second condition doesn’t apply. The third condition delivers the result: namely, #(‘divine person’) = 3. End parenthetical remark. * *

3.7.3.3 A different example: ‘loves’ Per the foregoing sections, the driving predicates in the 3-1-ness problem are defined via trinitarian identity. But what of (non-identity) predicates that themselves are just true of Father, Son, Spirit, and, therefore – via fundamental trinitarian entailments – true of God? Consider an example: loves.23 Let us say that ‘loves’ is just true (not also false) of each of Father, Son, Spirit, and, therefore, God. What, then, is the relevant number of ‘loves’ (restricted to divine reality)? This example nicely illustrates a peculiar but fascinating facet of trinitarian reality: God is loving (i.e., God loves), and while God is one triune

23 If this particular example is controversial then another can be used. The issue at hand concerns the count of a predicate that’s just true of Father, Son, Spirit, and, therefore, God.

trinitarian identity 67 being (see §3.7.3.1), God’s love is manifest only in three divine persons (see §3.7.3.2). In short, the count of ‘loves’, as above, is 3; it’s not 1, not 4, but 3. Perhaps not unlike many of God’s features, the love of God – the one and only triune being – is realized only via the love of three: Father, Son, and Spirit. So go some of the peculiar but fascinating facets of divine contradictions. * * Parenthetical remark on details. For any who wish to crunch through the trinitarian convention (where 𝕃 is the target): 0 if 𝕃+ = ∅ ⎧ ⎪ + #(𝕃) = C(𝕃 ) if 𝒰≃ = 𝕃+ ⎨ ⎪C(𝕃+ ) − 1 otherwise. ⎩ The extension of ‘loves’ (viz., 𝕃+ ) is {𝔣, 𝔰, 𝔥, 𝔤}, and so the first condition doesn’t apply. Since 𝒰≃ = {𝔤} ≠ 𝕃+ , the second condition doesn’t apply. The third condition delivers the result: namely, #(‘loves’) = 3. End parenthetical remark. * *

3.7.3.4 Another example: ‘crucified’ Consider ‘crucified’ (or, for present purposes, equivalently ‘crucified divine persons’). For convenience, let ℂ be the relevant predicate: 0 if ℂ+ = ∅ ⎧ ⎪ #(ℂ) = C(ℂ+ ) if 𝒰≃ = ℂ+ ⎨ ⎪C(ℂ+ ) − 1 otherwise. ⎩ The extension of ‘crucified’ (viz., ℂ+ ) is {𝔰, 𝔤}, and so the first condition doesn’t apply. Since 𝒰≃ = {𝔤} ≠ ℂ+ , the second condition doesn’t apply. The third condition delivers the result: namely, #(‘crucified’) = 1.

3.7.3.5 A final example: ‘unbegotten’ Lest one think that the trinitarian counting convention always gives either 1 or 3 as an answer, consider the well-known predicate ‘unbegotten’. For convenience, let 𝔹 be the relevant predicate (with the over-line notation doing duty for ‘un-’ treated as logical negation):

68 divine contradiction ⎧0 ⎪ + #(𝔹) = C(𝔹 ) ⎨ + ⎪ ⎩C(𝔹 ) − 1

+

if 𝔹 = ∅ +

if 𝒰≃ = 𝔹

otherwise. +

The extension of ‘unbegotten’ (viz., 𝔹 ) is {𝔣, 𝔥, 𝔤}, and so the first condition doesn’t apply. Since 𝒰≃ = {𝔤} ≠ {𝔣, 𝔥, 𝔤}, the second condition doesn’t apply. The third condition delivers the result: namely, #(‘unbegotten’) = 2. *** The central questions that make up both the ‘logical’ (identity) problem and the associated 3-1-ness (counting) problem have been answered. Supporting, defending, and elaborating those answers is the job of the next three chapters.

4 Seven Virtues §3 gives the basic account of trinitarian identity. Why accept the given account? What follows are seven virtues of the given contradictory account of the trinity. Not surprisingly, the given virtues overlap central virtues of the contradictory christology that informs the trinitarian account (2021, §3).

4.1 Unified solution: christological and trinitarian contradiction While the terminology is unfortunate, the so-called logical problem of the incarnation and the so-called logical problem of trinitarian reality share a prima facie similarity: the relevant axioms appear to directly and very simply entail a contradiction. Maneuvering around the strongly apparent contradictions has fueled important work in systematic, philosophical, and especially so-called analytic theology. At least on the surface, a unified response to both problems is better than a response on which the contradictions are treated as unrelated – or at best as related only in an unnatural or ad hoc fashion. The contradictory account, advanced herein on the back of The Contradictory Christ (2021), not only offers an intimately related account of both problems but does so in a traditionally important fashion: namely, driven by the reality – and the very contradiction – of Christ, who augmented pre-incarnation facts of God (e.g., that God is one god, omniscient, omnipotent, awesome, immutable, impeccable, and much more) by revealing, among many other poignant and scandalous facts, that God is essentially triune. The incarnation of God is the surprising fuller story of divine reality, one achieved via contradiction, and one that reveals the Divine Contradiction. Jc Beall, Oxford University Press. © Jc Beall 2023. DOI: 10.1093/oso/9780192845436.003.0004

70 divine contradiction core contradictions in divine reality: that each of Father, Son, and Spirit is God – the one and only god, namely, God – even though it’s false that Father is Son, false that Son is Spirit, and false that Father is Spirit. The apparent contradiction in trinitarian axioms, to which many christian and non-christian thinkers have long pointed, is as surprising as the contradictory revelation who revealed it. The natural unification of divine reality in a christology-first theology accepts the striking truth of Christ and that of trinitarian reality as intimately related: Christ is the contradiction through whose Word and action trinitarian facts are revealed. Prima facie, the apparent contradiction at the core of trinitarian reality is itself the truth: the oneness of God is God’s identity to Father, Son, and Spirit (as reflected in trinitarian identity and the trio of trinitarian entailments); the threeness of God is the falsity of otherwise expected identities between Father, Son, and Spirit. These facts exhibit contradiction at the core. But this is not a messy slop of contradiction or a ‘competition’ in God; the contradictions arise in plain and simple and, given their source, sacred ways: namely, via the identities and non-identities at the root of divine reality. A salient virtue of the contradictory account of trinitarian reality is its thoroughly natural unity. While the contradictions of Christ arise from God’s incarnation, and the contradictions in trinitarian reality arise from relations ‘across’ trinitarian identities, the two are so intimately bound that a non-contradictory account of just one of the two phenomena would screech of unnatural disunity. Except, perhaps, for so-called pure relative-identity accounts and so-called epistemic-mystery accounts (more on which in §6), no other non-contradictory account promises as natural a unified account of the incarnation and trinitarian reality as the advanced contradictory account. This is not insignificant, at least by my lights.

4.2 Simplicity For reasons having nothing special to do with theology (2018), the right account of logical consequence is in the vicinity of FDE per §2. Logic recognizes the possibilities of gluts and gaps, in addition to the very

seven virtues 71 familiar possibilities of being just true or just false. That a lot of observed reality is truly and fully described by a consistent and classical-logicclosed theory does not imply that all of reality is just so. Indeed, as christian theology has long affirmed in christological and trinitarian doctrines, there are central, foundational parts of reality that appear to buck clearly consistent description. For reasons that remain unclear, theologians have insisted that logic is per the classical-logic story, and tried in ever more elaborate ways to consistentize divine reality within the confines of the classical-logic account. But because that account should be rejected, at least as an account of logic itself (versus, as it was historically intended, an account of the entailment behavior of logical vocabulary in standard mathematics), the elaborate ways of ‘consistentizing’ divine reality are far from necessary. And when it comes to the persistence of apparent contradiction around the incarnation and trinitarian reality – and the core place of such phenomena in divine reality generally – a simpler direction is well-motivated: namely, accept the contradictions. By my lights, the contradictory account – of God incarnate and God in trinity – has no competitors with respect to simplicity. Even would-be close competitors that can approach a naturally unified account – such as ‘pure’ relative-identity (which, per §6, proliferates identity relations and positing of equivocations) or epistemic-mystery accounts (which, per §6, accept that our best theology is apparently contradictory but posits an inaccessible equivocation somewhere therein) – do so in ways that demand substantial complications (either in the proliferation of equivocations or in metaphysics or in epistemology). While simplicity of an account is not in any way sufficient for truth, it’s nonetheless an important feature that carries genuine weight. By my lights, the simplicity of the contradictory account, in light of the simplicity of the apparent contradiction at which it is aimed, should – and does – weigh heavily.

4.3 No need for analogical or metaphorical gesturing Trinitarian identity, per §3, is well-defined from the resources of logic and predicates involved in the language of true theology. There’s no obvious need for analogies or metaphors involving three-leaf clovers,

72 divine contradiction persons in community, triune-like dancers, statues and hunks of clay, or so on. Some theologians sometimes come across as thinking that the critical work in understanding (what we can understand about) trinitarian reality lies in mining metaphors or analogies that might – somehow – make sense of the otherwise ‘illogical’ trinitarian truths (e.g., athanasian identity and non-identity axioms). There is no question that mining for metaphors and analogies can be useful in applied or practical (or pastoral) theology; however, such mining is both unnecessary and unnecessarily distracting from truth-seeking theology that purports to truly describe trinitarian reality. (And such analogies can also be dangerous in promoting metaphysical pictures that often implicitly taint towards heretical accounts.) The true theology claims that • Father is God. • Son is God. • Spirit is God. while claiming that • It’s false that Father is Son. • It’s false that Son is Spirit. • It’s false that Father is Spirit. and while also claiming that • • • •

The number of gods is 1. The number of divine persons is 3. The number of crucified divine persons is 1. The number of unbegotten divine persons is 2.

and so on. To be sure, these given facts jointly appear to result in incoherence if one thinks that the given identity relation is transitive, or that the given counts are expressed (or otherwise reducible to) standard quantifier-cum-identity claims. But why – at all – think as much when even the surface of the facts suggests otherwise? The search for illuminating metaphors or analogies is misplaced if, as I maintain, the project is to

seven virtues 73 truly and precisely describe trinitarian identity and the target counting convention. Such tools (e.g., analogies) are very useful in some contexts but not for the project of true and precise description. Per §3, the relation of trinitarian identity is characterizable in a very, very familiar way (more on which in §4.4). And while the counting convention involved in trinitarian reality is peculiar to trinitarian reality (by using trinitarian identity for defining relevant duplicates), it too need not be propped up by analogies that are not part of the literal description. A virtue of the advanced contradictory account of trinitarian reality is just that: namely, trinitarian identity does not require vague gestures at analogy or metaphor for an understanding of (or even motivation for) its basic description; its precise description uses little more than standard logical vocabulary and leibnizian-identity tools.

4.4 No new-fangled approach to identity relations Not only can trinitarian identity be clearly and precisely defined without reliance on analogy or metaphor; it can be done so without any need whatsoever to reinvent the wheel. Prima facie apparent identity claims in the athanasian axioms are not only genuine identity claims; they are identity claims per the usual leibnizian recipe. The leibnizian recipe is the standard one for identity relations. While different conditionals, stronger than logic’s material conditional, can and sometimes are used for defining an identity relation along leibnizianrecipe lines, the most standard approach uses logic’s material conditional (with or without modal strengthening along so-called strict-conditional lines).1 And this is precisely the course of trinitarian identity: it’s defined using the standard leibnizian recipe – including logic’s material (bi-) conditional in the defining schema – per usual. The differences in the result (e.g., failure of transitivity) is not a difference in recipe but, rather,

1 Let 2 be the It is necessarily true that connective, for some notion of necessity or other. Then, where (as throughout) → is logic’s material conditional, a so-called strict-implication (which is unfortunately but standardly so called) is defined as 2(A → B), where A and B are any sentences in the relevant language.

74 divine contradiction a difference in ingredients. Given the correct account of logical consequence (per §2) the usual restrictions (e.g., transitivity) are an option; however, they’re not in any fashion a logical necessity. A virtue of the advanced contradictory account of trinitarian reality is just that: namely, there’s no new-fangled approach to identity in general. Instead, trinitarian reality is a rare fragment of reality that exhibits the non-transitive features of logic’s conditional. Trying to squeeze trinitarian reality into a standard transitive-identity account is a wrong turn. Logic already provides the possibilities that allow for non-transitivity; trinitarian reality is special in essentially exhibiting such possibilities in actuality.

4.5 No unmarked equivocation Not only is identity per the leibnizian recipe; the account need not rest on charges of unmarked equivocation. Per §1.4, a constraint on the current account of trinitarian reality is the prima facie simple, flat-footed reading of athanasian axioms: • Predicates and names are univocal throughout. ∘ There is a univocal use of predication (the ‘is’ of predication): the same predication – or exemplification, or instantiation – relation is involved throughout. ∘ There is a univocal use of identity (the ‘is’ of identity): the same identity relation is involved throughout. • ‘God’ is a singular term in some of the axioms. • ‘God’, qua singular term, is in the central identity claims (viz., that Father is God, Son is God, Spirit is God). • The central non-identity claims are logically negated identity claims. But none of the non-contradictory accounts that accept the standard axioms avoid unmarked equivocation (unless they reject some other item in the flat-footed reading). Any such account is an option if divine contradictions are precluded from the get-go. But why preclude divine contradictions, especially when the only way to do so is to essentially posit unmarked equivocation?

seven virtues 75 Sometimes, simple, flat-footed readings are wrong. No doubt. But when, as with the incarnation and trinitarian reality, attempts to consistentize the phenomena very often carry a whiff of heresy (i.e., going off of the standard theory in question), one naturally reflects on whether the simple, flat-footed account is the simple, flat-footed truth – not hiding some spectacularly hidden equivocation or new-fangled metaphysics or epistemology, but just the truth, plain and simple. The answer, in the end, is affirmative: it is the plain and simple truth. And in this special case, the plain and simple truth exhibits a strikingly rare feature: contradiction. But such is divine reality, remarkable and rare as it appears in an otherwise consistently described created world. A strong virtue of the contradictory account is that it accommodates the simple, flat-footed reading of athanasian axioms. This, by my lights, is a very important virtue.

4.6 Metaphysical and epistemological neutrality No theory is metaphysically neutral or without some epistemological implications. Obviously. Still, the so-called ‘logical’ problem of trinitarian reality (likewise, divine-incarnate reality) is not prima facie metaphysical or epistemological. While the contradictory account of divine reality carries metaphysical and epistemological implications, its central response to the ‘logical’ problem is not in any way a metaphysics-first or epistemology-first response. This is notable.

4.6.1 Metaphysical neutrality Neither Christ nor his church nor christian scripture explicitly stamps a metaphysical theory as the true metaphysics. Of course, there is a true metaphysics; it’s just not one that either Christ, his church, or christian scripture has explicitly stamped as true. While they, like the creedal statements from which they emerge, afford heavy-duty metaphysical and/or epistemological readings, the athanasian identity and nonidentity axioms are plainly expressed without a heavy poke of officially revealed or otherwise church-stamped metaphysics – unsurprisingly

76 divine contradiction so, given that an officially revealed or otherwise church-stamped metaphysics doesn’t yet exist. So what? Just this: pending an officially revealed or otherwise churchstamped metaphysics, a metaphysically neutral account of the apparent contradictions in trinitarian or divine-incarnate reality is prima facie better than a metaphysically heavy or metaphysics-driven account. Take any consistency-seeking, metaphysics-first response to either the incarnation or trinitarian reality (ideally, both, with the incarnate revelation informing the latter account). Suppose that it’s simple to state. Question: why, then, did the church leaders drop the ball and not just state it – given that it’s the key to consistency? Suppose, on the other hand, that the consistency of the target phenomena comes via a complicated metaphysical story. Question: is the complication genuinely worth rejecting the simplicity of the apparent contradictions when there’s no apparent problems that come from accepting the contradiction? Of course, there’s nothing necessarily untrue about a complicated metaphysics. Some things are complicated. Obviously. But still, when the standard axioms are themselves simple (and free of any obvious equivocation), why hammer the axioms in the name of consistency with a heavy, complicated metaphysics? A strong virtue of the contradictory account is that it is metaphysically neutral in ways that few, if any, consistent accounts can be.

4.6.2 Epistemological neutrality What is true of metaphysically neutral accounts is equally true of epistemologically neutral accounts. Indeed, this virtue is simply that of §4.6.1, mutatis mutandis with respect to ‘epistemology’. But one more small word can be added. Christ, who just is God, monumentally augmented knowledge of God. Before the incarnation, the truth of God’s omniscience, omnipotence, impeccability, and more was already known. Christ revealed the corresponding falsity – that God is also peccable, limited in knowledge, limited in power, and more. While the augmented knowledge remains not only novel but striking and utterly unique in its scope, the epistemology that explains knowledge-acquisition is not something out of the ordinary. Whatever explains knowledge of

seven virtues 77 divine reality and knowledge of human reality and, in short, reality in general is what explains the knowledge of divine contradiction. Theology doesn’t demand some peculiar epistemology to resolve the apparent contradictions of divine reality. A virtue of the contradictory account is that it is epistemologically neutral in ways that few, if any, consistent accounts can be.

4.7 The mystery of trinitarian reality The Catholic Church, and much of the christian church in general, not only acknowledges an apparent mystery in trinitarian reality; the mystery is an official fact about trinitarian reality. As the General Catechetical Directory (GCD) of the Catholic Church puts it: The mystery of the Most Holy Trinity is the central mystery of Christian faith and life. It is the mystery of God in himself. It is therefore the source of all the other mysteries of faith, the light that enlightens them. It is the most fundamental and essential teaching in the ‘hierarchy of the truths of faith’. GCD 43.

Mystery, in this context, entails at least that the full truth is something that can be known only via divine intervention. In short: while the truth-seeking aim of systematic theology is not just the truth but the whole truth of divine reality, part of the truth, at least per Catholic theology, is the given fact: namely, that trinitarian reality is mysterious. The unavoidable upshot: the full truth and its understanding is genuinely beyond our ken – at least short of further revelation. Exactly what the given mystery demands of the true theology is unclear. But one thing is clear: discussion of ‘the central mystery’ of trinitarian reality is intimately tied to trinitarian identity: how God can be identical to Father, Son, and Spirit even though Father, Son, and Spirit are pairwise non-identical. Similarly (3-1-ness): how the number of gods (ergo, by TG in §1, the number of triune gods) is 1 while the number of divine persons – each of whom by definition is identical to the 1 triune god – is 3. In short, much of the ‘mystery of the most holy’ revolves around the ‘logical’ and associated 3-1-ness problems. Preserving the

78 divine contradiction mystery while giving a precise solution to such problems is at least prima facie challenging. Suppose, as the dominant quest to consistentize trinitarian reality assumes, that there’s a precise account of the ‘logical’ and 3-1-ness problems that purports to explain, in very familiar terms, the consistency of trinitarian reality.2 Wherein lies the mystery? If the account isn’t clearly consistent then the account has failed its aim of providing a clearly consistent account. If, on the other hand, the account is clearly consistent, the place – indeed, existence – of ‘the central mystery’ is unclear. Perhaps Catholic theology gets things wrong on the matter of mystery. I leave that matter for debate elsewhere. For present purposes, the contradictory account of trinitarian reality aims to be at least compatible with the core of Catholic theology, including ‘the central mystery’. Along these lines, a contradictory account enjoys a virtue that most consistencyclinging accounts lack. The contradictory account appears to accommodate the central mystery more naturally than relevant alternatives.3 On the contradictory account, the location of contradiction in the theory is not mysterious; the contradictory claims are known, and they’re known to be contradictory. Where both the appearance and experience of mystery are conspicuous is in the attempt to behold the full contradictory truth at once. The mystery is a lived experience: the central mystery of christian life and faith (GCD 43). According to the contradictory account, the central relation of trinitarian reality – namely, trinitarian identity – involves contradiction at a fundamental level. To give the entire truth (or even approach some of the full truth) of God one must not only give the truth of each divine person’s identity to God but the falsity too.⁴ But to accept that the truth of God is contradictory is one thing; to accept any particular such contradiction 2 See §6 for discussion of some leading accounts in the quest for consistency. 3 A possible exception among alternatives is the explicitly epistemic-mystery account (Anderson, 2007), more on which in §6 and in this book’s predecessor (2021). But that account posits unknowable claims that consistentize otherwise apparently inconsistent divine reality. This is very different from the contradictory account. ⁴ Recall, on pain of fallacious inference from only consistent theories, that when a true theory is contradictory, some sentence A is true according to the theory and also its logical negation ¬A is true according to the theory. To say that A is a truth according to the theory is to say that A is in the theory; to say that A is a falsity according to the theory is to say that ¬A is in the theory.

seven virtues 79 ‘all at once’ plunges one into a sense of genuine mystery, a sense that many christian thinkers have experienced time and again, and again. Per §3, that Christ is God is true – axiomatically so. That Christ is God is also false; for, per §3, Father is God and it’s just false that Christ is Father (likewise, mutatis mutandis, Spirit). These are truths whose logical negations are also truths. A theology that contains only one or the other of the contradictory truths is thereby at best partial, robbing the full story of God not only of truths but of ‘the central mystery’. The familiar experience of mystery comes directly and forcefully in one’s attempt to hold both contradictory truths together at once: • It’s true that Christ is God. • It’s false that Christ is God. (Likewise, mutatis mutandis, for Father and Spirit.) Holding the one makes it difficult – if not doxastically impossible without divine intervention – to simultaneously hold the other.⁵ When one tries to simultaneously behold – in an explicit and salient way ‘before one’s mind’ (so to speak) – both conjuncts of central trinitarian contradictions one experiences a throwing, a jostling, a sudden bump against a hard limit to what one can so behold. In response to such common experience one says as many continue to say: . . . it’s a flat-out mystery! And that response to said experience doesn’t change on the contradictory account. In fact, the story may be enhanced. Christian theologians have often thought, in response to apparent trinitarian (or incarnation) contradiction, that the target axioms are true but can’t be true on their standard and flat-footed meanings. The quest then shoots off towards some complicated way of consistentizing the axioms. Ultimately, the central mystery accentuates the familiar faithseeking-understanding position: ‘I believe that these claims are true; I just don’t see how [given apparent contradiction].’ In response, the quest to avoid contradiction drives the theology.

⁵ Let me make one thing crystal clear: any would-be doxastic impossibility along this front is not in any way either a commitment or, let alone, an essential component of the contradictory account being advanced in this book. The difficulty of beholding an explicit contradictory truth ‘all at once’ is very familiar, but for all I know the difficulty is not the result of an impossibility.

80 divine contradiction While rejecting the quest for consistency (on such points) the contradictory account remains squarely in the exact same faith-seekingunderstanding position: ‘I believe that these claims are true (and false); I just can’t hold the full contradictory truth in mind at once.’ The mystery is not so much in what’s true (and/or false). The mystery, as above, is a very practical one: namely, trying to behold the full truth at once. As soon as one explicitly – in a salient and right-now-before-me sort of way – focuses on one of the contradictory conjuncts, the other one seems to slide away. Ultimately, beholding the full truth at one time is exceedingly difficult (if not doxastically impossible) without divine intervention. The ‘central mystery’ is a mystery ‘of life and faith’ (per GCD 43 above); it’s an experienced mystery that arises when one tries to behold the contradictory truth of God ‘all at once’ in its explicit contradictory form.⁶ The contradictory account of trinitarian reality responds to the ‘logical’ problem by accepting that ‘God in himself ’ (GCD 43) is contradictory. The mystery is not one over whether this or that claim is true. The central contradictory claims are known; and that they are contradictory is known. Wherein lies the mystery? Answer: ‘the mystery of the most holy’ is an experienced one in the face of known contradiction: a familiar and powerful mystery arising from the core contradictions of God and our hard limitations with respect to fully beholding them all at once. Notable is that many theologians resort to talk of paradox instead of apparent contradiction. If, as I assume of such usage, ‘paradox’ is used in its original sense of beyond belief then one must ask: what is beyond belief? Presumably, it’s the apparent contradiction that’s beyond belief (if anything is). But why? The answer above fits nicely: the full truth itself is ‘beyond belief ’ in – and only in – the sense that it’s beyond full-and-allat-once belief (as above). One might as well get to the point and call it what it is: a contradictory truth. Lest one think otherwise I should repeat: there is no tension in God; there is contradiction. But there is very familiar tension in christian practice: namely, trying to see beyond the veil all at once – to see, all at once and in full, even a single given contradictory truth of God. Knowing ⁶ The mystery of Christ – one person who is both divine and human – is the same: Christ’s being God (and all the limitlessness that being God entails) is one thing; but when juxtaposed with the commonness, ignorance, wretchedness, mortality, and other such limitations of Christ, the experience of hitting a hard limit on beholding such contradictory truths is familiar.

seven virtues 81 that it’s true is not the issue; beholding its truth – all at once and in full – is the ‘mystery of christian faith and life’, the hard mystery experienced in the struggle to so behold. The mystery, so understood, is not reason to reject the contradictory truth in favor of some perfectly consistent (however complicated) account of divine reality. The mystery resides in the experience of divine contradiction: hitting the hard limits of beholding the contradictory truth in full all at once, even when the full truth is known, even when the full truth hangs naked before you. *** This chapter answers the question: what, if any, virtues does the contradictory account enjoy? The next chapter answers the next question: what, if any, problems does the contradictory account carry?

5 Seven Objections There are many objections to the contradictory christology that informs the target contradictory account of the trinity. Most would-be objections to the former are equally objections to the latter; they are very general objections to a glut-theoretic theology. All of those objections – together with replies – are in Chapter 4 of The Contradictory Christ (2021). There are seven big-picture objections discussed in this chapter, a few overlapping with Chapter 4 of 2021, but most peculiar to trinitarian reality.1

5.1 Contradiction is not a perfection! Objection: God is perfect. To suggest that God is contradictory is not only to directly posit a tension in God; it’s to diminish God’s utter perfection. Reply. To repeat from both 2021 and §4: there is absolutely no tension in God; there is contradiction. The central question is whether contradiction is thereby imperfection. The reply is twofold. First, there’s an obvious ambiguity in the claim that contradiction is not a perfection. On one hand, an arbitrary contradiction is not thereby a mark of perfection. True. On the other hand, any contradiction true of God is a mark of perfection – no question. God is who God is. That some contradictions are true of God is thereby yet another surprising fact about God’s perfect being. Indeed, if any account of divine reality has shown that divine perfection is more than what pre-incarnation reflection delivers, it’s the christian account. Just as the robust monotheism of

1 Worth repeating from §1: the current trinitarian account treats the contradictory christological account as read. Readers of the current book who correctly think that the number of important objections exceeds seven are directed, again, to Chapter 4 of 2021. Divine Contradiction. Jc Beall, Oxford University Press. © Jc Beall 2023. DOI: 10.1093/oso/9780192845436.003.0005

seven objections 83 christian theology takes identity to God to be the long and short of being divine, so too with perfections: God is not defined to meet some notion of perfection discernible apart from revelation; the perfections are defined in terms of God. What is true of God is thereby a perfection, including the fundamental, trinitarian-identity-involving contradictions of God.2 Second, while any divine contradiction is a perfection, the startling truth of God also involves imperfection – the falsity of God’s very perfection. That God is perfect is without question a fundamental claim in the true theology. Were a theology to lack the claim that God is perfect the theology would thereby at best be critically incomplete. What, though, of the falsity of God’s perfection? Before the fuller revelation of God – whereby, among other things, God’s triune being was revealed – the falsity of God’s perfection would’ve been rejected as flat-out blasphemy or perhaps even nonsense. But then came the surprise of surprises: God enfleshed, a walking, talking, limited, and clearly imperfect being who was stabbed, crucified, and died. Imperfect? Yes. Were Christ not imperfect (i.e., were ‘perfection’ not false of him), Christ would thereby lack what only imperfect experiences can teach of wretchedness, of limited knowledge, of the terrible frustrations of imperfectly trying to understand perfection, and more.3 To achieve as much Christ – God incarnate – didn’t somehow lose God’s perfection but ‘took on’ the falsity of such perfection too: the limitations of human ignorance; human weakness; human frailty. How can Christ, who is God, be perfect and yet truly exemplify imperfection? The answer points to the full contradictory truth of Christ (2021), from whence, again, the full trinitarian truths of God were revealed. Contradiction is therefore at the very crux of God’s fuller revelation. To suggest that contradictions true of God somehow diminish God’s perfection is to forget a fact about God that should never be forgotten: namely, that being God just is being perfect. That God is also imperfect, as God enfleshed made plain for any willing to see, is no diminishment of God’s perfection; it’s a surprise that’s no bigger or smaller than immortal God’s being killed by the very imperfect beings for whom immortal God died. 2 For closely related discussion, see §5.2 concerning divine perfections and divine properties. 3 That Christ was ‘perfect human’ doesn’t diminish his imperfections. Attempts to say otherwise (Morris, 1986), as far as I can see, are driven mainly by a firm – but, in my view, unfounded – rejection of Christ’s uniquely contradictory being.

84 divine contradiction For too long, an entirely unmotivated clinging to the so-called classical account of logical consequence (or a classical-logic-fueled metaphysics or the like) has kept christian thinkers from recognizing that the apparent contradictions of God are in fact true. That God is contradictory is not itself an imperfection; it is at the core of God’s being, which is perfect, and perfectly contradictory.⁴

5.2 Divine simplicity! Objection: The contradictory account of divine reality purports to accommodate ‘orthodox’ or standard christian theology, including Catholic theology. That God truly exemplifies divine simplicity is a traditional part of such standard theology. But any contradiction involves features that are in contradiction. Core facets of the contradictory account involve, for example, the claims that the number of divine persons is 3, and that the divine persons are pairwise non-identical while individually identical to God. Divine contradiction arises, on the contradictory account, ‘across’ trinitarian identity via the identities that are true and false. But none of these differences arise on any conception of divine simplicity. Reply. The notion of divine simplicity is both traditional and standard, but it’s not just one notion; it’s many.⁵ Which notion is ‘the’ correct notion is not obvious. That God enjoys some important sense of ‘divine simplicity’ is clear. One sense is mereological: there are no parts to God, apart from any that are essential to being human. That alone – which I accept – is sufficient to conclude that the truth of God’s contradictions is not ‘made true’ (whatever that relation may be) by divine parts. Characterized negatively just so, few should reject God’s divine simplicity (i.e., the lack of divine parts).

⁴ Michael DeVito, in correspondence, suggested that there may be an argument from the image of God to the perfection of contradiction. Depending on details, such a suggestion might motivate a revised response to the current objection. Details are first required. ⁵ For a recent glimpse into the wide array of notions of ‘divine simplicity’ see, for example, (Brower, 2008, 2009; Mullins, 2016) and especially references therein. In addition, McCall (2014) gives examples of various senses of ‘simplicity’ in christian tradition, and Wood (2021) provides relevant discussion of ‘ontotheological error’.

seven objections 85 The more fruitful question concerns positive characterization (versus the negative no-divine-parts characterization). Is there a natural notion of divine simplicity that can be positively defined while being clearly true of God?⁶ The contradictory account, as currently advanced, remains neutral on this question; however, there is a promising direction for future exploration. The target account of divine simplicity involves the fundamental relation of trinitarian identity. Consider, first, features such as being a divine person, being triune and so on. Per §3, all such features are defined in terms of the unique relation of trinitarian identity. Accordingly, there’s a straightforward sense in which all identity-defined features of God are ‘simple’ in that, in the given sense, they one and all entail identity to God. And this is a positive, fruitful, and natural way of thinking of divine simplicity: the relevant properties one and all entail identity to God. But what of properties – such as love, crucified, sweated, experienced the pain of hatred, and more – that are not defined via trinitarian identity? Is there any way in which these might likewise be ‘divinely simplified’ to identity to God? No. But there are very close cousins of such properties that take the ‘divine’ in ‘divine simplicity’ seriously; such cousins are ‘divinely simplifiable’ (so to speak). Witness: let 𝜑 be any of the target features that are not defined in terms of trinitarian identity. To ‘divinely simplify’ 𝜑 one conjoins it with divinity itself (i.e., conjoin with ‘x ≃ 𝔤’): 𝜑(x) ∧ x ≃ 𝔤 where ∧ is logical conjunction (per §2). Since logical conjunction logically simplifies – and since logically so, thereby theologically so too – the ‘divinely simplified’ predicate entails identity to God, just as each identity-defined feature does (e.g., divine person, triune). Accordingly, while loves does not entail identity to God, divine love, namely, Loves(x) ∧ x ≃ 𝔤 ⁶ Some thinkers might say that any talk of a feature of God, or a predicate being true (false) of God, is thereby a rejection of divine simplicity. I reject such thinking, if any there be, on the grounds that theology aims to convey the truth of God, and the only way to do so requires truths, which require predicates being true of the subject of whom they are true, etc.

86 divine contradiction does so entail identity to God. Notable is that divine perfections are themselves instances of divine properties – that is, ‘divinely simplified’ properties – so understood: they are properties the exemplification of which entails identity to God. And that’s precisely what one would expect of divine perfections: they are truly exemplified only by God. Contrary to the objection, there is not only a straightforward and familiar negative sense of ‘divine simplicity’ that is true of God; there is a promising positive sense too.⁷

5.3 Counting divine reality! Objection. The counting convention, per §3.7.2, may get some salient counts right; however, it clearly goes wrong in some cases that are explicit in the Athanasian Creed. Take ‘omnipotent’, as explicitly discussed in the given creed (tr. Philip-Neri Reese, O.P.): Likewise, the father [is] omnipotent, the son [is] omnipotent, and the holy spirit [is] omnipotent. And yet [there are] not three omnipotents, but one omnipotent.

The convention of §3.7.2 has it that the number of ‘omnipotent’ is 3 since the extension of ‘omnipotent’ is {𝔣, 𝔰, 𝔥, 𝔤} and hence, not double counting ≃-duplicates, #(‘omnipotent’) = 3. As above, the creed clearly says that #(‘omnipotent’) = 1. This not only refutes the accuracy of the given counting convention in the given case; it casts doubt on the convention as it applies more generally. Reply. There is a lot to say but I’ll say just a little of the lot. When applying any counting convention, including the convention of §3, one must be clear about the predicate whose count is being calculated (so to speak). The question of ‘omnipotent’, like the question of any predicate in the context of trinitarian theory, is governed ‘in all things’ by ‘unity in trinity ⁷ To repeat: the contradictory account, as I advance it, is not committed to the positive sense of ‘divine simplicity’ sketched above, but it is an account worthy of further exploration.

seven objections 87 and trinity in unity’ – that is, above all, by ‘one god in trinity’ (i.e., the one triune god) and ‘trinity in unity’ (i.e., three divine persons). The relevant half of the creed begins and ends with veneration and illustration of the one and only triune god, who is each of three divine persons, and whose count is exactly one. In short, as the creed makes plain, there are indeed three divine persons each of whom is omnipotent; however, the principal trinitarian count concerning ‘omnipotent’ is the principal count of the triune god. For purposes of the principal count, the relevant predicate, in the context of the creed, is not merely ‘omnipotent’; it’s ‘omnipotent and triune’. In short, following notation from §3 wherein 𝕋 is triune, namely, x≃𝔣∧x≃𝔰∧x≃𝔥 and 𝕆 is omnipotent or even, ‘divinely simplified’, divine omnipotent (i.e., per §5.2, 𝕆 ∧ x ≃ 𝔤), the principal count concerning ‘omnipotent’ is the count of 𝕋x ∧ 𝕆x which, as the creed makes plain, is 1, just as the §3.7.2 convention delivers: 0 ⎧ ⎪ #(𝕋 ∧ 𝕆) = C(𝕋 ∧ 𝕆+ ) ⎨ ⎪C(𝕋 ∧ 𝕆+ ) − 1 ⎩

if 𝕋 ∧ 𝕆+ = ∅ if 𝒰≃ = 𝕋 ∧ 𝕆+ otherwise.

The extension of ‘triune omnipotent’ (viz., 𝕋 ∧ 𝕆+ ) is {𝔤}. (Proof: God alone satisfies the conjunctive ‘triune’ predicate, as may be checked, and via 3-1 entailment pattern T+ God satisfies ‘omnipotent’.) Hence, 𝒰≃ = {𝔤} = 𝕋 ∧ 𝕆+ , and so per the second line: #(‘triune omnipotent’) = 1. A related (counting) objection.⁸ According to the target counting convention, the number of gods, if expressed as #(x ≃ 𝔤), turns out to be 3 rather than 1, which is not the expected result, at least on the requisite

⁸ I’m grateful to James Anderson for pressing the current count objection.

88 divine contradiction ‘flat-footed’ reading of the question how many gods. If the response is as above with ‘omnipotent’, then the relevant question supposedly concerns the count of triune gods, where the count is 1 rather than 3; however, this stands out as special pleading and at odds with a univocal, ‘flat-footed’ reading of the Athanasian Creed. In short, the account treats the creed’s ‘one god’ claim as the number of triune gods, in which case, to avoid equivocation in the creed, the account must attach the ‘triune’ qualifier to the initial axiom(s) that ‘Father is God’ (likewise, Son, Spirit) – which is a prima facie awkward and unnatural reading of the target creed, contrary to virtues the contradictory account purports to reflect. A reply to the related (counting) objection. The reply, in central effect, is the same as the reply to the counting objection canvassed above, but a few more lines should be said, beginning with a reminder of the central import of the target creed. Reminder. The only sense of ‘a god’ or ‘gods’ that’s true of God and figures in (true) trinitiarian theory is triune god; hence, while ‘divine’ comes to identity to God (i.e., ‘x ≃ 𝔤’), ‘is a god’ is definitely not expressed ‘x ≃ 𝔤’.⁹ This is the very beginning and very end of the Athanasian Creed; it’s the key motif; it’s the critical lesson; it’s the principal point – highlighting the strikingly distinctive difference between close cousins of christian monotheism and christian monotheism itself. With respect to the trinitarian counting convention, the point is reflected in principle TG (see p. 11): the number of gods is 1 iff the number of triune gods is 1.1⁰ There may be some sense of ‘is a god’ that refutes versions of TG but it’s not a sense that is true of a divine person and is relevant to true trinitarian theology. The distinctive equivalence of ‘a god’ and ‘triune god’ in christian – versus jewish or islamic – theology, which is exactly ‘trinity in unity and unity in trinity’, is something that, per the target creed, is to be remembered ‘in all things’. The main reply. As above, ‘gods’ is not the trinitarian predicate ‘x ≃ 𝔤’, whose number, according to the trinitarian counting convention, is indeed 3 (since the number of divine persons, each of whom is divine ⁹ If it were, the first commandment (‘no other gods before me’) would be an exceedingly strange commandment to give: You shall have nobody identical to me before me. 1⁰ If one prefers, TG may be expressed as a bi-entailment: that the number of gods is 1 entails (and is entailed by) that the number of triune gods is 1.

seven objections 89 and hence identical to God, is 3); instead, ‘a god’ (or, for short, ‘god’) is triune god, that is, x≃𝔣∧x≃𝔰∧x≃𝔥 whose number, according to the trinitarian counting convention, is 1. And this is why the creed is emphatic that while each of Father, Son, and Spirit is identical to God (i.e., satisfies ‘x ≃ 𝔤’), not one of them is ‘a god’ or the like. The number of persons identical to God is 3; the number of gods – ergo, triune gods – is 1. Where is the equivocation? It’s not there. The alleged equivocation is supposed to emerge in the following creedal claims (herein paraphrased for ease): g1. Father (likewise Son, Spirit) is (identical to) God. g2. But the number of gods is not three; the number of gods is 1. The objection is that if ‘x ≃ 𝔤’ is the predicate in g1 then – on pain of equivocation – it’s likewise the predicate in g2. But, as rehearsed above, that’s incorrect; ‘x ≃ 𝔤’ is the predicate in g1 (or, at least, ‘≃’ is flanked by ‘Father’ and ‘God’) but it is not the trinitarian predicate expressed by ‘god’ or ‘gods’ or ‘a god’ – and hence isn’t the predicate in g2. What, then, is the ‘But’ in g2 flagging if not to flag that the g1-predicate count is not what one might expect? A key point of the target creed is to remind (per g2) that ‘gods’ and ‘triune god’ are equivalent in the true theology, and also to remind (per g1) that each of the non-identical divine persons is identical to God – the one and only god, that is, identical to the unique triune god. This is not equivocation; it’s precisely the central import of the creed. In short, g2 is indeed equivalent, in true trinitarian theory, to g2*. But the number of triune gods is not 3; the number of triune gods is 1. The objection raises the question as to why g2 is worth saying if there’s an entirely different predicate involved in g1. But the answer is apparent: when one reads that Father is identical to God (who is the unique god iff

90 divine contradiction the unique triune god), and Son is identical to God (who is the unique god iff the unique triune god), and Spirit is identical to God (who is the unique god iff the unique triune god), one naturally infers – as charges from monotheistic cousins reflect – that Father is Son is Spirit, and that each is ‘a triune god’. What is striking about trinitarian reality is that while Father is identical to God (who is triune), it’s untrue that Father is triune – and likewise, mutatis mutandis, for Son and Spirit. Trinitarian identity is as peculiar as the divine persons who stand in it. There are indeed three divine persons each of whom satisfies ‘x ≃ 𝔤’ (i.e., each of whom is divine; each of whom is identical to God); there are not three gods, as there are not three triune gods. The joint role of g1 and g2, standing as they do in the creed, is to hammer home such central truths, and to highlight the central and distinctive differences with close monotheistic cousin theologies. Contrary to the given counting objection(s), the counting convention, when applied to the principal predicate involved in the target count, gets things right. While the convention remains partial and incomplete, and perhaps in many ways in need of improvement, the allegedly problematic counts (e.g., ‘omnipotent’, ‘gods’), when taken as the creed demands – ‘above all things’ – to focus on the triune god, do not obviously demand the convention’s revision.

5.4 Trinitarian entailments should be unrestricted! Objection. It’s one thing for trinitarian identity to be defined only in terms of non-identity predicates, since otherwise circularity or unfruitful unfoundedness arises. But the familiar, fundamental trinitarian entailment patterns are different; they should be unrestricted! Specifically, the 3-1-ness patterns T+ and T− , from Definitions 3.6.1 and 3.6.2, respectively, T+ . 𝜑(𝔣) ∨ 𝜑(𝔰) ∨ 𝜑(𝔥) ⊣⊢𝜃 𝜑(𝔤) and T− . ¬𝜑(𝔣) ∨ ¬𝜑(𝔰) ∨ ¬𝜑(𝔥) ⊣⊢𝜃 ¬𝜑(𝔤)

seven objections 91 should range over all predicates in the language of true theology, not just non-identity predicates. Any such restriction is unmotivated. Reply. As per footnote 13 on page 53, the objection arises from a natural instinct abstracted away from constraints of trinitarian axioms. Entailment patterns are often – though not at all always – unrestricted over the language of the theory they govern. But once constraints are imposed on the theory the range of entailment patterns is often thereby constrained in turn. And that’s precisely what the truth about trinitarian reality does. In short, unless all claims in the language of the true theology – including the logical negations of all claims – are true (and, hence, thereby false too), the familiar trinitarian entailment patterns are restricted to non-identity predicates. Since the true theory of divine reality is not the trivial theology (i.e., the theology according to which every sentence in the language of the theory is true) the familiar 3-1-ness patterns are restricted just so. The itch to lift all restrictions ignores the central unity in trinity and trinity in unity constraints: three pairwise distinct persons each of whom is identical to God, where the pairwise distinctness is explosive distinctness: the identity of, for example, Father and Son explodes the theology into triviality, and likewise, mutatis mutandis, for Spirit. The familiar 3-1-ness patterns illustrate the identities of God to Father, Son, and Spirit; however, any ‘trinity in unity’ is obliterated were the range of such patterns to involve trinitarian identity itself. (Lest this be unclear: every identity claim would thereby be true of all divine persons. This route towards ‘unity in trinity’, while quickly reaching ‘unity’, is far too blunt to preserve the essential differences that true theology preserves.) The trivial theology is a logical possibility, but that’s saying very little. Pending some reason to accept that the trivial theology is a theological possibility – that is, recognized as a possibility in the true theology – any march towards lifting 3-1-ness restrictions is a march unmoored from core trinitarian truths and untruths.

5.5 Clashing christology and trinitarian theory! The Contradictory Christ (2021) highlights two different paths that a contradictory theory of Christ might take. On the first, it’s just true that

92 divine contradiction Christ is divine and just true that Christ is human. On the second, divinity is both true and false of Christ. The Contradictory Christ advances the first route. But given principle M in §1, ‘Christ is divine’ and ‘Christ is God’ are intersubstitutable. Per §3, the identity claim ‘Christ is God’ is true and false (i.e., glutty); hence, ‘Christ is divine’ should be glutty too; but in The Contradictory Christ ‘Christ is divine’ isn’t glutty. Accordingly, either the overall contradictory theology – including both christology and trinitarian theory – takes divinity to be true and false of Christ or the overall contradictory theology should be rejected as incoherent. A theory’s membership relation (i.e., the relation of a claim’s being in a theory) is itself perfectly consistent (not even possibly glutty). Hence, if it’s true that Christ is divine is just true, as The Contradictory Christ affirms, then only it is in the theory; its negation it’s false that Christ is divine is not also in the theory. But if ‘divinity’ is true and false of Christ, as principle M in §1 seems to demand (given that ‘𝔰 ≃ 𝔤’ is glutty), then both of the given claims are in the overall combined theory of divine reality. Since, as above, theory membership isn’t glutty, either the overall theology is incoherent or The Contradictory Christ is wrong on the given point. Reply. On whether divinity is true and false of Christ, The Contradictory Christ was driven by caution against recognizing unmotivated gluts. Rightly so. And since theological entailment doesn’t contrapose (a feature common and, indeed, expected of extralogical consequence relations in glutty theories), there was no clear pressing reason to affirm the gluttiness of Christ’s divinity in The Contradictory Christ.11 11 Other philosophers, at least in correspondence, disagreed. For example, Franca D’Agostini, Elena Ficara, Thomas Hofweber, Timothy Pawl, Mike Rea, Sam Newlands and others suggested in conversation that a more natural account of the contradictory Christ involves his divinity and humanity too, not just, as The Contradictory Christ advanced, the properties entailed by being divine and being human. (That theological entailment doesn’t contrapose is discussed at length in Chapter 4 of 2021.)

seven objections 93 Caution can sometimes keep one from seeing the proverbial forest for the trees. A turn towards trinitarian reality, Christ’s role in trinitarian reality, together with the paramount principle of robust monotheism, jointly portray truths that were otherwise hidden. In particular, given that being divine just is being identical to God, the issue of whether divinity is true and false of Christ turns on the status of the central (axiomatic) identity, namely, Christ is God. Since this is intersubstitutable with ‘Christ is divine’ the status of the latter turns on the status of the former. The former – namely, that Christ is God – is glutty. (See §3.) This is very good reason to accept that ‘Christ is divine’ is equally glutty, that is, that ‘is divine’ is both true and false of Christ. The objection is not only correct in its claims; it’s critically important in correcting an error in The Contradictory Christ. As briefly discussed in §3, there is something interesting, and possibly instructive, about the error in question. Before Christ, there was no obvious reason to think that the full truth of God involved contradiction. Christ changed that, revealing a host of ‘negative’ truths of God – for example, ignorance, passibility, peccability, and much more.12 Trinitarian reality, also revealed via Christ, unfolds even more: namely, that even Christ’s divinity is not just true of him but also false of him. The current objection flags not just the departure concerning the full status of Christ’s divinity; the same can be raised concerning Christ’s humanity. Christology is not only the guiding light into trinitarian theory but, as above, is likewise fed, so to speak, by trinitarian reality. As above, the divinity of Christ looks different when its essential tie to trinitarian reality is bracketed out. Contrary to the position affirmed out of caution and in the absence of trinitarian details, the overall theology is one in which ‘Christ is divine’ is both true and false given that ‘Christ is God’ is true and false – and principle M demands that they be intersubstitutable. But what now of Christ’s humanity? Is ‘Christ is human’ true and 12 By ‘negative’ is only meant the logical negations of target truths already at hand. The use of ‘negative’ here is not normative in any fashion.

94 divine contradiction false too? Since theological entailment doesn’t contrapose, the falsity of Christ’s ignorance, the falsity of Christ’s mutability, the falsity of Christ’s peccability, and so on (see Chapter 4 of 2021 for full discussion) does not force that humanity is false of Christ.13 What, then, to say? While acknowledging the possibility of an asymmetrical treatment of ‘Christ is human’ and ‘Christ is divine’ (e.g., as just-true and glutty, respectively), the simple and natural route is to treat them symmetrically – at least without special reason not to do so. And that’s hereby the official account in the overall – albeit herein revised – contradictory theology: namely, that Christ’s humanity, like Christ’s divinity, is both true and false of Christ.

5.6 Heresy! Objection. It is heretical not only to conflate the divine persons, which the contradictory account avoids; it’s heretical for theology to advance the falsity of Father’s divinity, of Son’s divinity, or of Spirit’s divinity. The contradictory account of trinitarian reality, as herein advanced, centrally claims the falsity of Father’s divinity, Son’s divinity, and Spirit’s divinity. The cost of accepting the contradictory account of divine reality is therefore heresy. Reply. Per 2021 (specifically, Chapter 4 therein), there are two central forms of heresy: one by commission, one by omission. The distinction is straightforward. H1.

Presence of Negation: the theory contains the given negation (e.g., ‘It’s false that Father is divine’). H2. Absence of Nullation: the theory fails to contain the given ‘nullation’ (e.g., ‘It is true that Father is divine’). The current objection charges an H1 heresy, not an H2 heresy. The question concerns the relevant or principal sense of heresy in the context 13 For convenience I say ‘humanity is false of Christ’ or the like, which is shorthand for ‘human’ is false of Christ and so on.

seven objections 95 of divine reality. When only consistent (i.e., contradiction-free) accounts of divine reality are on the table, H2 is the principal heresy because any H1 heresy (in such cases) is equivalent to an H2 heresy – a flag that the principal H2 heresy is active. When contradictory accounts are on the table – as they should be in the face of perennial appearance of divine contradiction – H2 remains the principal heresy; it’s just that any H1 heresy is no flag that the principal heresy of omission is active. H1 is hardly a heresy if the target negations are true. The substantial heresies, at least by my lights, involve an outright rejection of the orthodox claims, a rejection reflected by the conspicuous absence of such claims from the target theology. The substance of serious heresy is in H2: namely, having a theory that omits the given truth (e.g., ‘It’s true that Father is divine’). If, for no good reason that I know, one insists that divine reality must be contradiction-free, then any theology that entails the falsity of Father’s divinity, of Son’s divinity, or of Spirit’s divinity is thereby a theology that one rejects. But that’s a loss to said insister; the theology is thereby at best conspicuously incomplete. The full truth of God contains falsehoods; for the full truth is contradictory. Being identical to each of three pairwise non-identical persons is but one reflection of divine contradiction. That the full truth of divine reality demands the negations of some divine truths makes plain that serious heresy is along H2 lines. The idea that H1 ‘heresies’ are the central ones is an idea based on an inadequate diet of merely consistent theories.

5.7 Theology is at most analogical or mere model building! Objection. Your entire project presupposes that theological inquiry is a truth-seeking discipline that aims after the full, true theory of divine reality. But God – divine reality in general – transcends not only our understanding but our language. The apparent contradictions of trinitarian reality are the result of our language falling short of the transcendent reality of God. The apparent contradictions are reminders that when we try to speak both truly and fully of God we fail – and the apparent contradiction is a vivid reminder of the failure. At best, we speak truly of God

96 divine contradiction only through analogies (Eschenauer Chow, 2018) or mere models (Crisp, 2019). But we would be incorrect in thinking that we have done anything more than that. In this way, theology is not on par with physics, biology, mathematics, or any other standard truth-seeking discipline; its transcendent subject matter blocks true theories at central turns – notably, for example, the incarnation and trinitarian reality (Cotnoir, 2019). Accordingly, your entire theory must be rejected if taken to be true. At best, your contradictory account of the trinitarian reality is a reminder that our claims ‘about God’ are at best mere models, mere analogies, and crash into fatal contradiction when taken to be more than that. Reply. There is a lot to say but I’ll again say just a little of the lot. First, there is no question that our theories – regardless of subject matter – offer models of reality. But that doesn’t mean that the aim isn’t the true and complete description of the given portion of reality. This is true of physics, biology, and mathematics just as much as it is true of theology. The aim is to fully and truly describe the target reality, regardless of the challenges that reality inexorably poses, be it physical reality or divine reality. Meeting said aim is achieved in part via a model-building process: one rarely jumps to the true and complete theory all at once; one instead works in a start-stop, model-building fashion. But diminishing the principal aim of true and complete description is close – if not right upon – a dereliction of responsible inquiry.1⁴ The appeal to analogical language as a response to apparent divine contradiction remains both active and common in theology. I confess that I understand neither its motivation nor its details. On the former, I should think that one must know a great, great deal about transcendent reality to know that one’s language is unfit to truly describe it. But if one’s language is unfit to describe it, how is it that one knows so much 1⁴ Sometimes I think that theologians and/or philosophers and/or scientists who describe ‘truth-seeking inquiry’ as nothing more nor less than model-building (regardless of whether the models are full and accurate with respect to their target portion of reality) do so out of an otherwise proper humility. Finding the true theory is very difficult. Anything but proper humility during the process is generally counterproductive. In the face of transcendent divine reality, the humility may be even more important – but, one should hope, all the more natural in the face of divinity. Still, allowing humility to divert one’s efforts towards the ultimate aim of the true and complete theory is a misstep, one that may border on false humility. Clearly, not every theorist who speaks chiefly in terms of model building is thereby engaged in the potential misstep just described, but it’s a misstep worth flagging.

seven objections 97 about it? I present this not as a dilemma but as a genuine confession of ignorance.1⁵ Moreover, if anything, the special revelation of God’s incarnation looks to be at odds with the claim that our language doesn’t truly (or falsely) apply to divine reality. After all, our language applies to Christ, and Christ is God. Moreover, Christ truly spoke of the triune god (by revealing himself, Father, and Spirit), and we speak the same language (on translation) as Christ. Set motivation aside and ask after details. Does analogical language involve a degree-theoretic account of truth, so that something is ‘analogically true’ of God if and only if the given predicate is true-todegree-n or the like? If not truth, what about exemplification itself? Does analogical predication involve degree-theoretic predication, so that God exemplifies-to-degree-n some (analogically predicated) feature? And, of course, there’s a related question: to what degree, if any, does something ‘analogically true of ’ God entail other non-analogical predications of God? All of these (and many more) questions are ones the details of which remain in question, at least as far as I know. If such details are available then – but only then – can a full ‘analogical’ response to apparent divine contradiction be evaluated against the contradictory account. Analogical language is common and useful, but its use isn’t in truthseeking theories, at least not in any central fashion. Analogical language is useful for doing things, specifically, for getting one to think of an object in new terms. If the new terms are true of the object then the analogy is inessential to the object’s true and complete description. If the new terms are not true of the object, they may still be useful for arriving at the truth of the object; it’s just that they themselves don’t do the job. Nothing in this reply purports to undermine attempts to rely on analogical language or model building; it’s just that neither project undermines the importance of truth-seeking theology. If there were some strong argument for proscribing contradictory theology then perhaps the strength of apparent divine contradiction would ultimately undermine truth-seeking theology. But, for all I can see, there’s not only no strong

1⁵ I note that Eschenauer Chow (2018) allows for negative truths of God but not positive truths, though it’s hard to see how the negation of a ‘positive’ claim A can be true without the language in A directly applying to God – just falsely attributing something to God.

98 divine contradiction argument against contradictory theology; the strength of apparent divine contradiction cries out for contradictory theology. *** This chapter responds to seven pressing objections to the contradictory account of theology. Many other objections, as said above, are answered in Chapter 4 (and some in Chapter 5) of 2021. And, of course, other objections peculiar to trinitarian reality or to the overall contradictory theology herein advanced are likely to arise in due course. Such is the truth-seeking process.

6 Measuring Some Non-Contradictory Accounts This chapter is not a substantive review or rehearsal of either traditional or recent work on either the ‘logical’ or the 3-1-ness problem of trinitarian reality. Such reviews are available in many places.1 This chapter compares the contradictory account of trinitarian reality with relevant mainstream and non-mainstream competitors, where ‘relevant’ requires at least the natural option of robust monotheism (as expressed, e.g., via M and, relatedly, TG in §1) and the fundamental trinitarian entailments (per §3). There are highly interesting and very wellknown deviant accounts – one such being the family of social-trinitarian accounts (briefly but only briefly mentioned below). Deviance, in this context, is not necessarily a mark of untruth; however, it does reflect a focus on something other than standard theological theory (where, e.g., heresies mark deviations from the standard). Except, again, for a very brief note on social-trinitarian accounts in §6.1, my focus is on nondeviant accounts.

6.1 Social-trinitarian accounts Despite its ongoing influence, so-called social-trinitarian accounts of trinitarian reality are engaged in a different project from one that takes robust monotheism to demand identity to God (as expressed via principle M in §1). Such accounts reject not only the axiomatic full divinity of Father, Son, and Spirit (i.e., reject that each divine person is identical to

1 Beyond the survey by Tuggy (2021) is the accessible discussion by McCall (2010), and another by Hasker (2013). A recent classic survey is by Rea (2009), including the contribution by Craig (2009). (And see citations in all three works for more.) Divine Contradiction. Jc Beall, Oxford University Press. © Jc Beall 2023. DOI: 10.1093/oso/9780192845436.003.0006

100 divine contradiction God, since God on such accounts is a social group of sorts); they reject one direction or other of the fundamental trinitarian entailments (see §3), a rejection that accentuates the departure from robust monotheism and standard axioms.2 I have little to add to the many (and widely accessible) problems confronting core social-trinitarian accounts (Coakley, 1999; Leftow, 1999; McCall, 2010, 2015; Rea, 2006; Tuggy, 2004).3 But three points are worth flagging in addition to the many other criticisms elsewhere advanced: • the under-motivated, all-out rejection of (even the logical possibility of) divine contradiction fuels the social-trinitarian account (§2); • the under-motivated thought that there is exactly one identity relation shared by all true theories (of the trinity or otherwise) also appears to fuel social-trinitarian accounts (§3);⁴ • a natural unified account of ‘logical’ problems in both the incarnation and trinitarian reality is achievable on the social-trinitarian account only if the incarnation is likewise treated as a ‘social unity’ of (what else is there?) two persons – which is heretical.⁵ When combined with its rejection of robust monotheism (i.e., the rejection of the identity of Father and God, Son and God, and Spirit and God), such points serve as further reasons to reject the social-trinitarian account, at least in comparison with the contradictory account.

2 It would enormously help evaluation of social-trinitarian theories if they precisely defined the central entailment relations governing trinitarian reality. Saying that while three persons are divine (in some sense) but God is ‘the group’ (in some sense) doesn’t answer critical questions about entailment patterns under which the theories are constrained. Are socialtrinitarian theories (on any variety) constrained by T+ and T− ? I don’t see how they can be so constrained without inconsistency, but extant such theories, as far as I see, are entirely silent on such fundamental, pressing entailment questions. 3 And for many, many more discussions, consult the bibliographies of the given list. ⁴ Witness Swinburne (1994) on trinitarian axioms: ‘no person and no Council affirming something which they intend to be read with utter seriousness can be read as affirming an evident contradiction’ (p. 180). Charitably read (given the state of logical studies at the time of the claim), the idea must be that trinitarian theses involve classical identity (or some suitably similar equivalence relation) if identity (versus a unique predication relation) is involved at all. ⁵ This assumes that being divine is not an empty property. Thanks to Tom McCall for flagging this point.

measuring some non-contradictory accounts 101

6.2 Pure relative-identity accounts The basic idea with ‘pure relative-identity’ accounts is that all (nonderived) identity relations (and the respective predicates that express them) are relative to some sort of predicate or ‘sortal’ term. Example: the claim that the number 1 is identical to the number 123-122 is not expressing some non-relative identity relation between 1 and 123-122; rather, it’s expressing the relative-identity claim that, say, 1 and 123-122 are identical-qua-number or, more colloquially, the-same-number.⁶ In response to the apparent contradiction(s) in athanasian axioms, including, for example, from the explicit identity axioms, namely, • 𝔣 ∼ 𝔤. • 𝔰 ∼ 𝔤. • 𝔥 ∼ 𝔤. together with the explicit non-identity axioms (where x ≁ y is just ¬x ∼ y, where ¬, per §2, is logical negation) • 𝔣 ≁ 𝔰. • 𝔰 ≁ 𝔥. • 𝔥 ≁ 𝔣. the relative-identity theorist (whether pure or, as in §6.3, ‘impure’) posits unmarked equivocation: ‘∼’ is not a single identity predicate in both the identity and non-identity axioms; there are two different relative-identity relations going on. In particular, the given axioms, when spelled out explicitly, are as follows, where ∼𝔾 and ∼ℙ are the identical-qua-god and identical-qua-person (relative) identity relations: • 𝔣 ∼𝔾 𝔤. • 𝔰 ∼𝔾 𝔤. • 𝔥 ∼𝔾 𝔤.

⁶ For terminology, full discussion and standard references, see Rea 2003. Also, I use ‘∼’, as throughout this book, for whatever identity relation (or predicate) is under discussion. Per §3 I use ‘≃’ for trinitarian identity as I’ve defined it.

102 divine contradiction and the axiomatic non-identities: • 𝔣 ≁ℙ 𝔰. • 𝔰 ≁ℙ 𝔥. • 𝔥 ≁ℙ 𝔣. And so there’s not even a hint of contradiction, at least in the given axioms. Contradiction arises only when the subject matter isn’t shifting. Details of different relative-identity accounts offer different virtues and vices, but the core strategy is the same: posit (unmarked) equivocation! That Father is identical-qua-god to God and that Son is identical-qua-god to God in no way contradicts the falsity of Father’s being-identical-qua-person to Son. As far as relative-identity theorists have spelled them out (which, alas, isn’t far), the relations are ultimately unrelated to each other. (E.g., relative-identity theorists haven’t spelled out the entailment patterns over the various relations. If you and I are identical-qua-person to each other, are we thereby also identical-quamind to each other? identical-qua-desires? identical-qua-moral-agent? And so on.) Pending further details of what the relations are and how they’re entailment-wise related to each other, it’s difficult to evaluate relativeidentity strategies. For present purposes, I highlight three issues for ‘pure’ theories (though some apply equally to ‘impure’ theories discussed below):⁷ • ‘pure’ relative-identity theories carry an arbitrary or nonviable rejection of non-relative identity relations; • an unmarked-equivocation strategy is unmotivated in general; and • the aim of avoiding contradiction fails given a simple ‘revenge problem’ that confronts extant relative-identity theories. Each in turn.

⁷ With a possible exception, per §6.2.4, the bulleted points in §6.1 are significant problems with many non-contradictory theories, including relative-identity ones (pure or otherwise), but I focus on additional problems peculiar to given strategies or accounts.

measuring some non-contradictory accounts 103

6.2.1 Identity relations and their many inevitable relatives Relative-identity theorists do not define relative-identity relations so much as point to examples (e.g., ‘is the same dog as’, where this is supposed to be primitive and not involving some non-relative identity relation). This is fine as far as it goes, but when it comes to responding to the target ‘logical’ problem and defining the relations involved in trinitarian theory, more is required. Per §3, identity relations, in general, are usually understood per the leibnizian recipe: indiscernibility with respect to relevant properties (or satisfaction of respective relevant predicates). Again, per §3 and specifically Definition 3.3.1, a target identity relation ∼ is defined via a sentential schema 𝜑(⋅) ⇔ 𝜑(⋅), namely (where T is the target theory in question), and where 𝜑 ranges over the set 𝒫 of T’s (non-identity) predicates: • t ∼ t′ is true according to T iff every instance of 𝜑(t) ⇔ 𝜑(t′ ) is true according to T; • t ∼ t′ is false according to T iff some instance of 𝜑(t) ⇔ 𝜑(t′ ) is false according to T. The entailment behavior of ∼, so defined, turns on the ingredients used in the schema, specifically, both the range of the variable 𝜑 (i.e., the set of relevant predicates over which 𝜑 ranges) and the entailment properties of the biconditional ⇔. Relative-identity theorists tend to cling to the so-called classical account of logical consequence, and so tend to rely on the classicallogic features of logic’s material biconditional. Since, per the classicallogic account, the material biconditional is reflexive, symmetric, and transitive, identity relations so defined are one and all equivalence relations (i.e., relations that are reflexive, symmetric, and transitive; they differ only in the range of the defining schema’s variables (the set of properties thereby covered). With no other clear candidate for defining the relative-identity relations, the leibnizian recipe, using the classical-logic material biconditional, is the default.⁸ ⁸ To repeat from §2 and §3, even if relative-identity theorists accept a subclassical (say, FDE) account of logical consequence, they can (and do) still constrain their theories in such

104 divine contradiction This is where debate over relative-identity relations becomes difficult to understand. After all, given the leibnizian recipe (and no other recipe has been explicitly given), two facts jump out: F1. F2.

Non-relative identity relations are inevitable. Relative-identity relations are inevitable.

Each in turn.⁹

6.2.1.1 F1: non-relative identity relations are inevitable Let T be a theory. Let ∼ be an identity relation defined per usual via the scheme 𝜑(x) ≡ 𝜑(y), where, according to T’s consequence (entailment) relation, ≡ is reflexive, symmetric, and transitive over the sentences of the language of T. (Again, taking ≡ to be logic’s biconditional, and assuming that, according to T’s entailment relation, ≡ exhibits all classical-logic entailment behavior in T, ≡ is thereby reflexive, symmetric, and transitive in T.) In this context, ∼, so understood, is non-relative if and only if the range of its defining schema (i.e., the range of 𝜑 in the schema) is the set of all of T’s predicates (i.e., all predicates in the language of T). There’s simply no notable relativity to the relation. None.1⁰ And such a relation is a way that logic’s material biconditional is per the classical-logic account, and so leibnizianrecipe-generated identity relations are one and all equivalence relations in such theories. (The constraints, per §2 and §3, wind up excluding the logical possibilities of gluts and gaps from the space of possibilities relevant to the theory in play.) ⁹ One might think that the debate over relative/non-relative identity relations is fueled by the thought that if you have both sorts of identity relations expressible in the theory then both sorts must be used in all axioms or central theses of the theory. But that is plainly incorrect. Many true theories have a variety of conditionals in them (i.e., in the language of the theory). Example: if you try to give the true account of dispositions – or, for that matter, entailment – using, say, logic’s material conditional, your theory would most definitely be false or irresponsibly incomplete. Each axiom or the like has the resources that fall out of the language of the theory. Just because you have, say, classical identity expressible in the theory doesn’t mean that it’s the identity relation involved in expressing all axioms of the theory. 1⁰ One might say, as Rea (2003) might be saying, that such a relation is ‘relative’ in the sense of being relative to the theory T in/for which it is precisely and fully defined. True, but also true of every fully and precisely defined identity relation. To suggest that this is a bad – or potentially ‘antirealist’ – relativity is comparable to suggesting that truth itself is badly relative because, as Tarski (1956) long ago established, no true, classically closed theory (i.e., closed under classicallogic entailment) contains a truth predicate in the language of the theory for the theory itself. This doesn’t make truth relative or ‘antirealist’. Truth is real; it’s just that it is precisely and fully definable in and for a true theory (or its language) only if the true theory is closed under a weaker-than-classical consequence relation. (And recall: most – if not all – relative-identity theorists accept, for reasons that I don’t understand, that the classical-logic account of logical consequence is correct, and hence, according to such theorists, all true theories are classically

measuring some non-contradictory accounts 105 as ‘fundamental’ or non-derivative as any fundamental or non-derivative relation. Accordingly, that there is no non-derivative identity relation given a well-defined family of relative-identity relations is not only untrue; it’s fairly clearly so – at least on the leibnizian-recipe account of identity relations.

6.2.1.2 F2: relative-identity relations are inevitable Let L be the language of theory T. Let 𝒫 contain L’s one-place predicates (including derivative ones such as ‘x stands beside t’, where t is a singular term in the language). Finally, let an identity relation be defined in terms of the usual schema 𝜑(x) ≡ 𝜑(y) but – note well – let the range of 𝜑 in the schema be restricted to a unique singleton {𝜑} from L’s set 𝒫 of one-place predicates. For clarity, let the identity relation, so defined, be ∼{𝜑} . The identity relation ∼{𝜑} , so understood, is simply a standard relativeidentity predicate: namely, is-identical-qua-𝜑 or, same, is-the-same-𝜑-as. Example: in the theory of canines, 𝜑 might be ‘canine’ (say, ℂ), in which case the relation ∼{ℂ} is exemplified by x and y (in that order) just if ℂx ≡ ℂy is true of them. (One needn’t check other predicates in the schema 𝜑(x) ≡ 𝜑(y) because the range of 𝜑, in the current context, just is {ℂ}.) Relativizing an identity relation to a single predicate, as above, is one thing, but what of setting the range to {𝜑, 𝜓}, where 𝜓, like 𝜑, is in L’s set 𝒫 of unary predicates? Here, the natural thought is that ∼{𝜑,𝜓} is likewise a relative-identity predicate but defined ‘conjunctively’ along so-called supervaluational lines (van Fraassen, 1966): ∼{𝜑,𝜓} is simply is-identical-qua-𝜑-and-qua-𝜓 or, same, is the same with respect to 𝜑 and 𝜓.11 Importantly, this goes for the case where the relevant subset of predicates – the relevant ‘comparison class’ or ‘sortal class’ (so to speak) – is 𝒫 itself, the non-relative case in §6.2.1.1 above wherein, again, 𝒫 contains all unary predicates in L (versus just one, just two, etc.).

closed, and hence, perhaps unbeknownst to such theorists, do not contain a precisely and fully defined truth predicate in and for such theories themselves.) 11 Consider too a ‘dual’ relation along so-called subvaluational lines (Hyde, 1997; Varzi, 1999), treating ∼{𝜑,𝜓} in a ‘disjunctive’ style to get different entailment patterns. And other options exist.

106 divine contradiction

6.2.1.3 Upshot of F1 and F2 One notable upshot concerning the debates over relative ‘versus’ nonrelative identity predicates, at least if they are precisely defined via the leibnizian recipe, is that either the non-relative predicate (defined over all predicates in the language, per F1) or the relative ones (defined over subsets of all predicates in the language, per F2) may be taken as ‘fundamental’ or ‘derivative’. When you’ve got the one, you get the other; when you’ve got the other, you get the one. If labeling one of them as ‘fundamental’ or ‘derivative’ is important then let the labels be arbitrarily applied. By my lights, I see no fruit in seeing one as fundamental (or derivative). So long as the relations are precisely defined, the relations can do work and be evaluated on their jobs. Another notable upshot is that it’s very difficult to maintain ‘pure’ relative-identity theory, at least per the standard description thereof (Rea, 2003); for such theorists reject the existence of a non-relative identity predicate – unless it’s derivative. But, again, since relative-identity relations deliver non-relative ones, as above, and vice versa, as above, the would-be distinction between derivative and non-derivative such relations promises little value, at least for all I see. * * Parenthetical remark. One might charge that the foregoing reflects at best an inadequate grasp of ‘real absolute identity’, which, some might say, is ‘theory-independent’. Well, I fear that I must agree: I have no idea what is being talked about.12 My first question to such a charge is to ask about other would-be ‘absolute’ relations. Is there an ‘absolute truth’ predicate in and for the language of all true theories? What of ‘absolute validity’? If the answer is affirmative then I want details!13 But once we realize that identity is not logical (ergo, not in the language of every true theory), and 12 I am assuming that the target relation is not supposed to be just the union of all identity relations expressed in all true theories. If it is just said union then that’s well and good but it’s also completely in line with the foregoing remarks on, for example, F1 and F2. So, I assume that that’s not what is under discussion with ‘real absolute identity’. 13 Tarksi’s Theorem, as above (see fn. 10 on p. 104), is obviously directly relevant. If the charge claims that Tarski’s theory of logical consequence is incorrect, and that some subclassical account is correct, I’ll turn to another question:  OK, that’s fine, but is there an ‘absolute validity’ (predicate) in all true theories? Here, results due to Haskell Curry are directly relevant (Curry, 1942; Church and Curry, 1942). Eventually, the pusher of ‘absolute truth/validity’ will realize that the utility of these notions is achieved only when they’re theory-relative in the very banal sense that they’re precisely, fully defined only in the confines of the language of a given true theory, etc.

measuring some non-contradictory accounts 107 that each true theory needs to define its own identity relation, it’s far from clear what the debate over relative-vs-absolute identity is all about.  Just to be clear: truth is real! Truth is absolute! But, like identity (and validity, etc.), truth is precisely defined only in the confines of true theories. This is not in any way ‘antirealist’. It’s the real, cold, hard truth. Not bad; not good; just what it is. End remark. * *

6.2.2 Unmarked equivocation Regardless of details, the relative-identity account of trinitarian identity – qua ‘the’ relation in the flat-footed, univocal, trinitarian axiomatic identities and non-identities – is not only that there are many different relative-identity relations (same-person, same-god, same-creator, samecounselor, same-savior, etc.) but, and this is the rub, all of these critically important and fundamentally distinct relations are entirely unmarked in the trinitarian axioms. In short, the conspicuous ‘logical’ problem of trinitarian reality arises because athanasian axioms equivocate. Such equivocation has caused a lot of confusion but, once flagged (i.e., written out in an unequivocal fashion), there is simply no ‘logical’ problem at all. (There may still be a 3-1-ness problem, but that’s not a threat of contradiction; it’s simply a problem of expressing the trinitarian counting convention.) The ‘logical’ problem, per §1, arises from the flat-footed – univocal – reading of the core trinitarian axioms. It’s no surprise that if the axioms are equivocating then the apparent contradictions quickly vanish. But, per §1, a univocal reading of the core axioms is better than otherwise, at least if there’s a natural and simple one available. There is: the contradictory account. The reason that relative-identity theories avoid the simpler, univocal, contradictory account is that, for no good reason that I’ve seen, they are married to the unwarranted dogma of the classical-logic account of logical consequence. Alas. One final comment on the charge of unmarked equivocation. Per the very beginning of the target creed (trans. Philip-Neri Reese, O.P.): Whoever wants to be saved should, before anything else, hold the Catholic faith – [the faith] which, if someone does not preserve it

108 divine contradiction whole and inviolate, [that person] will, without doubt, perish eternally. And the Catholic faith is this: . . . [herein enters a parade of allegedly unmarked equivocations].

That unmarked equivocation is critically involved in core axioms is at least prima facie awkward. Were there no natural or simple way of avoiding the charge, so be it. But there is: the contradictory account.

6.2.3 The revenge problem Per §6.2.2 relative-identity responses to the target ‘logical’ problem invoke unmarked equivocation over a multitude of relative-identity relations – or at least the few (e.g., ‘is-identical-qua-god’ and ‘isidentical-qua-person’) that appear to be necessary for avoiding the target contradiction. Motivated or not, invoking unmarked equivocation does appear to get around some apparent trinitarian contradiction. But blocking just one contradiction does nothing more than that: blocks that one. But there are what might be called ‘revenge’ cases: cases in which contradiction reemerges via the relative-identity relations that were supposed to keep contradiction at the door. A simple example of the target revenge problem concerns the truth that God is a (divine) person:1⁴ 1. ℙ𝔤.

1⁴ This follows immediately from the principal definitions of divine person (i.e., is identical to either Father, Son, or Spirit) and divine (i.e., is identical to God), together with the fact that each of Father, Son, and Spirit is divine. (See §3.) Moreover, if some other non-identity-involving account of ‘person’ is given (i.e., different from being identical to Father or to Son or to Spirit) then God’s being a person follows from 3-1-ness entailment patterns, namely, T+ (per §3). Furthermore, assume that van Inwagen (1988) is correct that all target relative-identity relations ∼𝜑 entail that the given predicate 𝜑 is true of the given relata: that a ∼𝜑 b is true entails that 𝜑 is true of a and true of b. In this case reflexivity of ∼𝜑 entails 𝔤 ∼𝜑 𝔤 which, given said condition, entails 𝜑(𝔤), which, in the case of ‘person’, entails the truth above. [Technical Note: The relative-identity relations defined merely out of the leibnizian recipe in §6.2.1 do not deliver the entailment above (i.e., that the relata exemplify the given property); the definition would require a minor tweak to give the given condition. For example, where ∆ is a set of the theory’s predicates and 𝜑 is in ∆, define the truth condition for a ∼∆ 𝜑 b-type sentences as follows (thinking of ∆ as the relevant set of ‘difference makers’), where ⇔ is the relevant biconditional in the given leibnizian schema:

measuring some non-contradictory accounts 109 But, now, what of the other truths involving the central trinitarian relation is-identical-qua-person or is-same-person-as relation (herein, for convenience, ∼ℙ )? Of special relevance is the following trio. 2. 𝔣 ∼ℙ 𝔤. 3. 𝔰 ∼ℙ 𝔤. 4. 𝔥 ∼ℙ 𝔤. Who is God if not Father? . . . if not Son? . . . if not Spirit? There’s no evident answer given the truth of (1), and so no evident way to reject all of (2)–(4) in the face of (1). Rejecting instead exactly one of (2)–(4) is a form – even if not an exact traditional form – of (heretical) ‘subordinationism’, at least in that whoever is not the same person as God is in that respect subordinate to whomever God stands in the given relation.1⁵ For such reasons I assume that relative-identity theories (of any level of ‘purity’) accept not only (1) but also all of (2)–(4). There is one more relevant tenet of target relative-identity theories: namely, that all relevant relative-identity relations are equivalence relations (i.e., reflexive, symmetric, transitive). One good explanation for the equivalence-relation features of such identity relations is that if, as extant relative-identity theorists hold, logical consequence is per the classicallogic account and the biconditional in the leibnizian-recipe schema is logic’s material conditional, then, per §6.2.1, all such identity relations thereby defined are equivalence relations. An especially relevant equivalence-relation feature of target relativeidentity relations is transitivity.1⁶ Indeed, the simple example of revenge turns on transitivity (and, strictly, symmetry). Witness: without loss of generality, focus on (2) and (3). Transitivity (and, strictly, symmetry) of ∼ℙ immediately delivers a ∼∆ 𝜑 b is true iff (𝜑(a) ∧ 𝜑(b)) ∧ (𝜓(a) ⇔ 𝜓(b)) is true for all 𝜓 in ∆. This is the usual leibnizian-recipe relation except for the additional feature concerning the target predicate 𝜑 being true of both relata.] 1⁵ To repeat: this is not the exact historical instantiation of subordinationism; it’s the generalized form. 1⁶ As per §1.6, the non-transitivity of trinitarian identity – at least on the flat-footed, univocal reading of the axioms – is conspicuous. One oddity of relative-identity accounts is the transitivity of target relations despite initial appearances. (This is in no way a stand-alone objection. One stand-alone objection is the revenge problem that involves transitivity.)

110 divine contradiction 5. 𝔣 ∼ℙ 𝔰. In short: Father is-the-same-person-as Son. The appearance of heresy is not the revenge problem. The revenge problem is the reappearance of contradiction that is not only not avoided by relevant relative-identity relations; it turns on them. In particular, (5) directly contradicts one of the axiomatic non-identities, namely, 6. 𝔣 ≁ℙ 𝔰. And, to make it explicit, (5) and (6) jointly – logically – entail 7. 𝔣 ∼ℙ 𝔰 ∧ 𝔣 ≁ℙ 𝔰. And this – given the classical-logic account of logical consequence (under which extant relative-identity theories are closed) – is the ticket to triviality: namely, the unique trivial theology according to which all sentences in the language of theology are true.1⁷ Pending some supplemental story that resolves the given revenge problem (and this is but one example of target revenge), extant relativeidentity theories are themselves one in number: the trivial theological theory. * * Parenthetical note. The revenge problem was independently discovered by Jedwab (2015), though in a framework that is slightly narrower than my discussion. Additionally, van Inwagen (1988), who attempts to offer an outline of sorts of a ‘pure’ relative-identity theory (without committing to the ‘purity’), notes something similar. What is clear from my discussion of ‘revenge’, but also apparent in said Jedwab and van Inwagen works, is that all relative-identity theories wherein the identity relations are equivalence relations (regardless of whether they are all-properties-identical or ‘Leibniz Law’-satisfying relations) must go further than merely positing unmarked equivocation over the relativeidentity relations; that alone won’t avoid simple contradiction. What’s 1⁷ Recall: every sentence is logically entailed by any contradiction according to the classicallogic account of logical consequence.

measuring some non-contradictory accounts 111 also required is a non-standard semantics for relevant singular terms involved – a semantics that, if done right, may independently avoid contradiction at the price of potential vacuity (as illustrated, e.g., in my suggested ‘variable-like’ semantics in §6.3.2 below). End parenthetical. * *

6.2.4 Summary evaluation: pure relative identity For reasons above, a ‘pure relative-identity’ account of the target ‘logical’ problem is less promising than the simple contradictory account. For one, the ‘purity’ – rejecting any non-relative identity relation as at most derivative – is impossible to sustain given the leibnizian recipe for identity relations. If the leibnizian recipe is not the defining recipe for the ‘pure’ approach to relative-identity relations then a precise definition needs to be given before such promises can turn into genuine theories. As things stand, the leibnizian recipe remains the only precise definition through which to evaluate such theories. The charge of unmarked equivocation, per §6.2.2, versus the flatfooted univocal account of the target axioms, should be accepted only if there’s no natural univocal account. Since the contradictory account gives a simple, natural, and univocal account, the charge of unmarked equivocation should be rejected.1⁸ This is especially so given that the driving motivation towards the charge of unmarked equivocation is contradiction. (As repeated before: if there are no markings of equivocation, the axioms are read univocally until a contradiction forces the charge of unmarked equivocation.) Why is contradiction a sufficient reason, according to extant relative-identity strategies, for retreating from a flat-footed univocal account of axioms to the charge of unmarked equivocation? Alas, so far as I can tell, the answer points to the classicallogic account of logical consequence. But that’s a bad answer. As in §2, the classical-logic account was never formulated as an account of logical consequence – qua universal, basement-level entailment relation involved in all true theories (be it theology, physics, algebra, what have you) – but rather as an account of the entailment behavior of logical vocabulary in true mathematical theories. Accepting the classical-logic 1⁸ This applies to the ‘impure’ relative-identity accounts too.

112 divine contradiction account over alternatives (especially subclassical accounts) requires good argument, of which, as far as I see (and I’ve looked far), there is none.1⁹ In this light, retreating to a charge of unmarked equivocation is in palpable need of motivation. One conspicuous virtue of pure relative-identity strategies is that, unless essentially tied to a particular metaphysics, they can share the metaphysical (and epistemological) neutrality of the contradictory account. Moreover, they promise to offer a unified solution to the target ‘logical’ problem of trinitarian reality and the ‘logical’ (so-called fundamental) problem of christology – though the promise fades fast in the face of the revenge problem. Bracketing the revenge problem, the ‘logical’ problem in the case of christology and trinitarian theory is the apparent contradiction involved. As van Inwagen (1994) sketched, there are avenues for each target problem to be naturally unified under a single solution: a multitude of relative-identity relations over which the target axioms equivocate in an unmarked-equivocation fashion.2⁰ Despite said virtues, extant relative-identity theories, be they pure or impure, are inconsistent given the revenge problem §6.2.3 or involve a semantics for singular terms that has yet to be precisely stated. Given the tie to classical logic, such theories are one and the same: the trivial theology. Pending some supplemental account that gets around the revenge problem, such theories should be rejected.

6.3 Impure relative-identity accounts: constitution So-called impure relative-identity accounts, so called by Rea (2003), are ‘impure’ in two ways: they tie themselves to a particular metaphysical theory as an essential part of the relative-identity account of trinitarian reality; they accept the (inevitable) existence of non-relative identity relations, pace ‘pure’ cousins. (Such relations are inevitable per 1⁹ When the candidate accounts of logical consequence are classical logic versus FDE per §2, there is a simple argument for the latter (2018): the subclassical (FDE) account loses no true theories (including no classically closed true theories) and gains live and strongly viable theories for persistently peculiar – apparently contradictory or incomplete – phenomena. 2⁰ Rea (2011) offers work towards a unified impure relative-identity account of both trinitarian reality and the incarnation. To my uninitiated metaphysical eyes, the account looks to be fairly metaphysically complicated, but that is a matter for genuine debate elsewhere.

measuring some non-contradictory accounts 113 §6.2.1.1–§6.2.1.2.) The central impure account is the so-called constitution account (Rea, 2003; Brower and Rea, 2005; Rea, 2009), which is a systematic account of both the target ‘logical’ problem and the related 3-1-ness (counting) problem. Take each in turn.

6.3.1 Impure relative identity: the logical problem The constitution account addresses the ‘logical’ problem along relativeidentity lines: namely, unmarked equivocation over a multitude of relative-identity relations. While the constitution account, in particular, wraps the relative-identity relations in a robust metaphysical theory (and thereby appears to lose the virtue of metaphysical neutrality vis-à-vis the target ‘logical’ problem), the metaphysical story does not play a role in the explicit resolution of the target ‘logical’ problem (and hence said appearance of losing said virtue is merely apparent). (The metaphysical story plays a key role only in the 3-1-ness problem, as discussed in §6.3.3.) Accordingly, the impure account faces exactly the same problems as its pure cousins (see §6.2.4), including the inconsistency arising from the revenge problem (§6.2.3). Ultimately, then, without some supplemental story that avoids the revenge problem, the central impure theory doesn’t get beyond scratch, at least on explaining how trinitarian reality is supposedly devoid of contradiction.

6.3.2 Supplemental semantics-of-‘God’ story There is a supplemental story that might be given: a supplemental semantic story (of sorts).21 The supplemental semantic story concerns the name ‘God’. The aim of the supplemental story is to maintain five desiderata:

21 While the following is not to be officially attributed to either Rea or Brower, the following is suggested by comments about the singular term ‘God’ in the target works (Rea, 2003; Brower and Rea, 2005; Rea, 2009). Moreover, in correspondence (January 2022), at least Rea was inclined to accept the gist of the following supplemental semantic story, though not necessarily in exactly the terms that I give it. (Note again: this footnote is in no way the claim that Rea’s account in fact involves the following supplemental story as a way of responding to the target revenge problem.)

114 divine contradiction • • • • •

T+ and T− as central trinitarian entailments [see §3]; the truth that God is a (divine) person; all axiomatic identity and non-identity axioms; all of the same-person truths (2)–(4); and the equivalence-relation features of all relevant relative-identity relations.

But how can this aim be achieved given that the last two bullets jointly entail the revenge problem? After all, the (relative-) identities (2) and (3) jointly entail the problematic (5) given the transitivity (and symmetry) of the relative-identity relation, and (5) contradicts the axiomatic nonidentity (6). Given the equivalence-relation features of relevant relative-identity relations, there’s no straightforward way of avoiding the revenge contradiction, but there is a way. In particular, give a semantics of ‘God’ such that ‘God’ is a variable-like term: given a context, ‘God’ picks out (denotes) either Father, Son, or Spirit or – functioning as a sort of pluralvariable-like term – any two or all three of said persons. Details are (very) pressing but the general idea is clear enough: the consistency of, say, the (relative-) identities (2)–(4) with the axiomatic non-identities is ensured via taking the transitivity of ∼ℙ off the table in the context of (2)–(4). How so? The point may be seen by reflecting on the effect of the axiomatic non-identities on determining the value (so to speak) of ‘God’ in (2)–(4). Given, for example, the axiomatic non-identity (6), the only available ‘values’ for the variable-like term ‘God’ in (2) and (3) are ‘Father’ and ‘Son’, respectively. Hence, (2) and (3) are – in effect – open sentences (i.e., contain a variable-like term) whose values, when properly given, result in the true sentences 8. 𝔣 ∼ℙ 𝔣. 9. 𝔰 ∼ℙ 𝔰. And, obviously, the transitivity of ∼ℙ is simply irrelevant to (8) and (9), at least in that (6) clearly doesn’t follow from them via transitivity. This is all for the good: given the supplemental semantic story the impure relative-identity theory can avoid the revenge contradiction(s), and thereby avoid the simple path to the trivial theology.

measuring some non-contradictory accounts 115 The conspicuous problem with the given supplemental semantic route towards avoiding the trivial theology is that it comes at the cost of a different sort of ‘triviality’ – a sort of vacuity. Recall that the ‘logical’ problem emerges with axiomatic identities and axiomatic non-identities, such as 10. 𝔣 ∼ 𝔤. [axiom] 11. 𝔰 ∼ 𝔤. [axiom] 12. 𝔥 ∼ 𝔤. [axiom] 13. 𝔣 ≁ 𝔰. [axiom] 14. 𝔰 ≁ 𝔥. [axiom] 15. 𝔥 ≁ 𝔣. [axiom] The relative-identity theory posits unmarked equivocation: the trio of explicit axiomatic identities involves one (unmarked) relative-identity relation (viz., ∼𝔾 , same-god-as); the trio of explicit axiomatic nonidentities involves another (viz., ∼ℙ , same-person-as). This avoids the standard ‘logical’ problem but not the revenge problem concerning, for example, (2), (3), and (5). This problem, under the supplementalsemantics strategy, is resolved by would-be variable-like semantics for ‘𝔤’ (i.e., ‘God’). The strategy works by reducing (2)–(4) to self-identity claims. Given that ∼ℙ is an equivalence relation (and therefore reflexive), the given claims are vacuously true – and hence true, but entirely uninformative. The vacuity problem goes slightly further. Given the supplemental semantic story, the fundamental ‘logical’ problem that fueled the target impure (similarly, pure) relative-identity account(s) is a conspicuous non-starter. After all, if ‘𝔤’ is variable-like per above, how are the axiomatic identities (10)–(12) in any fashion whatsoever in apparent contradiction with the axiomatic non-identities (13)–(15) even given that ∼ is a single equivalence relation with no equivocation going on? Answer: the given trios aren’t apparently contradictory, not even at first glance (given the supplemental semantic story that avoids the revenge problem). Indeed, even if ∼ is classical identity (i.e., the identity relation defined by the leibnizian recipe using the material conditional in a theory closed under classical logic), there’s simply no apparent contradiction provided that the variable-like nature of ‘𝔤’ is out front. The axioms are curious

116 divine contradiction but not at all apparently contradictory (given said semantic story); each occurrence of ‘𝔤’ in the (classical-) identity axioms is resolved as a selfidentity claim – true but entirely uninformative. In the foregoing respects, all of the talk of relative identity, like the essential metaphysics of the impure theory, is doing little (to nothing) on the would-be ‘logical’ problem. The supplemental semantic story resolves the revenge problem and the original ‘logical’ problem, and does so in a way of ridding the theory of unmarked equivocation (and the need for relative-identity relations). The only requirement for avoiding the ‘logical’ problem is a variable-like semantics for ‘God’ – pressing details to be given. In short: the supplemental semantic story appears promising with respect to arresting the theory’s march towards the trivial theology but does so with a very blunt instrument that makes the traditionally problematic axioms vacuously true. (There may be many so-called conversational implicatures that ‘follow from’ the axioms, but that doesn’t change their vacuity given said supplemental story.) This is not a virtue of the supplemented theory.22

6.3.3 Impure relative identity: the 3-1-ness problem The 3-1-ness problem is the problem of specifying a counting convention for trinitarian reality, normally one tied to the identity relation(s) invoked in response to the ‘logical’ problem. Assume that, contrary to §6.2.3–§6.2.4, the ‘logical’ problem is ultimately resolved not by relativeidentity relations but by something along the supplemental-semantic route (§6.3.1). The 3-1-ness problem still remains: the vanishing of the ‘logical’ problem (the apparent contradiction) on such an account doesn’t thereby vanish the basic 3-1-ness question. The basic 3-1-ness question

22 Worth noting too is that while all desiderata (see p. 113) are met via the supplemental story, the route towards satisfaction is peppered with similar vacuity. Take T+ , namely (§3), T+ . 𝜑(𝔣) ∨ 𝜑(𝔰) ∨ 𝜑(𝔥) ⊣⊢𝜃 𝜑(𝔤). Details notwithstanding, this bi-entailment holds just from logic (viz., disjunction), axioms on the range of variable ‘God’ (viz., the range is simply {𝔣, 𝔰, 𝔥, 𝔤}), and the variable-like semantics of ‘God’. (Ditto for T− of §3.)

measuring some non-contradictory accounts 117 is how three non-identical persons are sensibly counted as one god (viz., God).23 By way of comparison, and to repeat from §3, the contradictory account – which is comparatively neutral on metaphysics – points to the following three facts (backed by the precise leibnizian recipe account of identity and the essential entailment patterns in T+ and T− , all per §3). • Fact One: the number of gods (equivalently, triune gods) is 1. (See §3.7.3.1.) • Fact Two: the number of divine persons is 3. (See §3.7.3.2.) • Fact Three: the precise trinitarian counting convention is simply the standard convention, reflected in many counting practices, by which relevant duplicates are not (double-) counted, where relevant duplicates, in trinitarian theory, are defined – unsurprisingly – by trinitarian identity. On the contradictory account, the sense in which the number of gods is 1 (viz., God) is that the number of triune gods is 1 (viz., God). (See again TG in §1.5.) In turn, the sense in which the number of divine persons is 3 falls out of the fact that God is identical to three pairwise non-identical persons.2⁴ The explanation of the correct (viz., church-stamped) counts is that relevant duplicates are counted by trinitarian identity, the relation which is exemplified only by divine reality. How does the central impure relative-identity account – namely, the constitution account (Brower and Rea, 2005) or the semanticssupplemented constitution account – compare on its 3-1-ness answer? Aside from recognizing the (inevitable) existence of non-relative identity relations (e.g., classical identity or such), the ‘impurity’ of the given (constitution) account is that it is decidedly metaphysics-driven.2⁵

23 Thanks to Mike Rea for discussion on this point. 2⁴ Recall the beginning and end of the trinitarian section of the Athanasian Creed: ‘unity in trinity and trinity in unity’, describing – in the middle – the unique triune being to whom three pairwise non-identical persons are identical. 2⁵ There’s nothing inherently impure about metaphysics. The terminology is analogous to ‘abstract’ versus ‘applied’, where the latter is tied to ‘impure’, along the analogy of pure/impure mathematics.

118 divine contradiction Unlike the contradictory account, the constitution account attempts to answer the 3-1-ness problem by, in effect, three facts:2⁶ • C-Fact One: Aristotelian metaphysics is true at least in that all objects (material and immaterial) are either ‘hylomorphic compounds’ of matter and form or ‘hylomorphic compounds’ of something ‘analogous to’ matter and form – where, in either case, the central unifying relation in any ‘compound’ is constitution. • C-Fact Two: there is exactly one ‘divine matter’ (viz., the ‘divine essence’) but three distinct ‘forms’ (viz., being a Father, being a Son, being a Spirit) which, when ‘compounded’ via constitution with ‘divine matter’, results in three triune gods: ∘ the number of triune gods is 3; ∘ the number of divine persons is 3; ∘ the number of ‘divine matter’ is 1. • C-Fact Three: the trinitarian counting convention, whatever its precise details may be, involves ‘numerical sameness’ and, in particular, its species ‘numerical sameness without (classical) identity’. On the constitution account, there’s exactly one god in the sense that there’s exactly one ‘matter’ (viz., divine essence) that ‘constitutes’ any divine being, and, towards the second fact, there are three non-identical persons each of whom is a ‘form’ of the given matter and thereby thesame-god-as each other (i.e., identical-qua-god-to each other) but not the-same-person-as (i.e., non-identical-qua-person-to each other). The counts – as with the contradictory convention – are thereby governed by a relation other than classical identity, even if, when details are eventually (precisely) spelled out, classical identity plays a role somewhere. Worth emphasizing is that the critical relations ‘numerical sameness’ and, in particular, ‘numerical sameness without (classical) identity’ are undefined but they are illustrated. And the illustration drives the constitution account. In particular, familiar cases of ‘material constitution’ are driving examples of the metaphysics of constitution generally and, 2⁶ The following are simplified. The simplifications do not affect my comments or evaluations; they’re in place for readers. There is absolutely no substitute for reading the now-classic target paper on constitution (Brower and Rea, 2005) or subsequent debate (McCall and Rea, 2009; McCall, 2010, 2015).

measuring some non-contradictory accounts 119 of special relevance, the theory’s counting convention tied to the given examples. Consider a common example of material constitution. Let S be a statue entirely made of hunk H of clay. Fix a time period – for example, right now – when S exists, H exists, and S is actually ‘constituted by’ H. (There is some such period for most, if not all, statues and hunks of stuff of which they’re composed.) Now, S is a piece of art, which, supposedly, can be severely damaged by changing its shape (e.g., squashing it to remove all of its intended artistic features); however, H needn’t be thereby severely damaged by changing its shape. Hence, S and H differ with respect to the given modal features: it’s possible to severely damage S by squashing it without thereby damaging H. Hence, S and H are not all-relevantproperties-the-same, and hence not identical simpliciter. It’s true that S and H, so described, are two non-identical objects, at least when evaluated by classical identity or some other all-propertiesconsidered identity relation.2⁷ But how does this relate to a common counting convention that figures into the motivation behind trinitarian counts? Answer: on the given constitution view, there’s a ‘conceptual truth’ concerning material objects: MO. No two (non-identical) material objects can occupy the exact same region of spacetime. Given MO, the only conclusion to draw about S and H is twofold: • S and H count as exactly one piece of matter. • S and H count as exactly one material object. • S and H count as different objects – non-identical objects, measured by all-relevant-properties identity. And the case of God and trinitarian counting not only is analogous but involves exactly the same, very common relation of constitution and the associated counting convention: namely, that there’s exactly one ‘matter’

2⁷ Important to note is that restricting the range of relevant properties to actual properties concerning intrinsic features in the given region of spacetime motivates a count of one (not two), at least if relevant duplicate is measured only in terms of the restricted range of properties.

120 divine contradiction involved (see C-Fact Two on p. 118), even though, similar to S and H, there are exactly three ‘forms’ which/who are the divine persons (see C-Fact Two as above).2⁸ There’s no question that material constitution is a common relation, regardless of its full details. There’s also no question that counting conventions governing common cases of material constitution might serve to refute the thought that ‘duplicates’ (or double counts) are always counted by classical identity. Whether in fact common cases of material constitution (like S and H) serve to make the case is not of principal importance;2⁹ such common cases certainly illustrate how different counts – and different measures of ‘duplicates’ in such counts – are possible (if the possibility was somehow in doubt).

6.3.4 Summary evaluation: impure relative identity and constitution In its original form, whereby the ‘logical’ problem is to be solved by unmarked equivocation over a multitude of relative-identity relations, the constitution account faces exactly the same evaluation as other extant relative-identity theories (see §6.2.4).3⁰ But on the proffered supplemental semantics-of-‘God’ account, the theory has a response to the logical problem, and the metaphysical wrapping (Aristotelian-hylomorphic metaphysics), while independent of the ‘logical’ solution, offers motivation towards a trinitarian counting convention.

2⁸ One might find it at least somewhat surprising that the unifying relation in trinitarian reality is the same relation that unifies one’s garbage bin and the material that constitutes it. Of course, the divine ‘matter’ is no less holy or perfect for standing in said relation, but the commonness is surprising. (On the other hand, the fuller story of God in the incarnation revealed beyond question the gobsmacking commonness of God. This surprise in no way undermines the truth of God’s utter uncommonness – at least on the contradictory account.) 2⁹ In fact, I take S and H to refute the alleged ‘conceptual truth’ MO. Presumably, science is our best account of the material world, but while science itself might accept MO the given acceptance is not telling, since science almost certainly brackets out modal-involving properties such as ‘can be damaged as a statue’ or the like. In the end, S and H are two material objects distinguished by the given modal properties. 3⁰ And maybe worse, as the Aristotelian-metaphysics-driven account complicates a simple unified account of both ‘logical’ problems in christology and trinitarian theory. But set this aside – and, again, see the work of Rea (2011) towards potential unification.

measuring some non-contradictory accounts 121 Problems aside, few accounts have been as fruitful in moving trinitarian theory forward as the given constitution account. Given the revenge problem (§6.2.3) and the problems with the supplemental semanticsof-‘God’ story, the progress, as I see it, is not so much towards the truth (which involves contradiction) but rather in forcing theology to recognize that counts are via counting conventions, and that counting conventions neither need nor always do involve merely (if at all) the classical-identity relation. Few lessons, when it comes to the 3-1-ness problem, are more important. Perhaps none. Does the constitution account provide reason to reject the contradictory account? No. Should Aristotelian-hylomorphic metaphysics turn out to be true, this would only augment the contradictory account, not diminish it. The metaphysics is unnecessary for the trinitarian counting convention (see §3 wherein metaphysics is missing); what the metaphysics provides, as above, is a possible example of how identity relations other than, say, classical identity might be involved in various counting conventions. But is Aristotelian-hylomorphic metaphysics true? Maybe. Maybe not. But being driven to such a metaphysics in an attempt to avoid divine contradiction is motivated only if a ban on divine contradiction is itself well-motivated. It isn’t, so far as I see. (The usual argument points to logical consequence and its alleged constraints against any would-be true contradiction. But that argument is only as strong as the very weak arguments for the given account of logical consequence.)

6.4 Epistemic-mystery accounts The epistemic-mystery response to the ‘logical’ problem (not necessarily to the 3-1-ness counting problem) is advanced most clearly by Anderson (2007). The bare bones are three in number.31 • The apparent contradictions of trinitarian reality (and, note well, the incarnation) are unavoidable from our epistemic position; they are

31 I should flag that the bare bones are bare indeed. Anderson 2007 is a robust and systematic work (one of the rare Ph.D. theses that makes a pioneering contribution). My aim is to say just enough for comparison with the contradictory account.

122 divine contradiction due to equivocations that, given our epistemic position, we cannot detect. • The classical-logic account of logical consequence is the true account. • The true theology is contradiction-free. Unlike the metaphysics-driven constitution account (§6.3.3) the epistemic-mystery account is robust in its epistemology – not surprisingly. Like many consistency-questing accounts, the epistemic-mystery account, per the second bullet above, toes the classical-logic line. Because I have discussed the epistemic-mystery account elsewhere (2021), I limit comments to just a few.

6.4.1 Virtue: unified account A big virtue of the epistemic-mystery approach is that it offers a natural, unified account of the ‘logical’ problems in both trinitarian reality and the incarnation. Indeed, were it not for the contradictory account, the epistemic-mystery account would be the hands-down leading route towards a unified account of both ‘logical’ problems. For reasons given below and elsewhere (2021), the epistemic-mystery account does not provide reason to reject the contradictory account; however, compared only with all extant consistency-clinging accounts, the epistemic-mystery is the leader on the unified-solutions front. One might object that the ‘logical’ problem of trinitarian reality demands a solution to the 3-1-ness problem too. I agree. The problems are independent but tightly related. But even here, the epistemic-mystery account has a unified response: the precise counting convention – just like the hidden equivocation – is beyond our ken. The church demands suchn-so counts explicitly; however, the precise convention governing such counts is something beyond what our epistemic position can handle, at least without further divine revelation.32

32 Note that Anderson (2007) doesn’t distinguish the ‘logical’ and 3-1-ness problems as I have, but I think that the given response is in keeping with the account he advances.

measuring some non-contradictory accounts 123

6.4.2 From a subclassical point of view Were there good reason to accept the classical-logic account of logical consequence, the epistemic-mystery account would be a strong candidate for the true theory of divine reality (whatever, per §6.4.3, the theory might be). But, per §2, while the classical-logic account is the right account of the behavior of logical vocabulary in many true theories, there remains no good argument for it as the true account of logical consequence in general – qua universal, basement-level entailment relation governing logical vocabulary in all true theories. From a subclassical point view (i.e., a view according to which a subclassical account of logical consequence is the true one), it’s very difficult to see how the epistemic-mystery account is the right one. The contradictions are strongly apparent. They are unavoidable. And – from the subclassical perspective – they are at least logical possibilities. Why, then, not accept them as theological possibilities that are in fact realized by divine reality? The uniqueness or even rareness of something does not imply non-existence. (Witness: the incarnation.)

6.4.3 What is the theory? There is a glaring problem with the epistemic-mystery account. Question: what exactly is the theory? We have it that the classical-logic account of logical entailment is true. Accordingly, if the theory’s axioms (e.g., trinitarian identity and non-identity axioms) have the inexorable appearance of entailing a contradiction (as they do, according to the epistemicmystery account) then the contradiction is in the theory or it isn’t. If it is, the theory is trivial – that is, it’s the trivial theology that contains every sentence in the language of the theology. What, then, is the extralogical – theology-specific – entailment relation that blocks the contradiction that inexorably appears to follow? Without an answer to this question, the would-be theory of divine reality is far from the systematic theory to which epistemic-mystery theorists aspire – or, at least, the leading advocate of such accounts aspires (viz., Anderson). The rub goes deeper than merely acknowledging that ultimately we know not what either the theological entailment relation or the

124 divine contradiction theological axioms amount to. What motivates the epistemic-mystery account is the unavoidable persistence of apparent contradiction in systematic attempts to truly describe divine reality as much as possible. But why think that the appearance of contradiction is unavoidable if we don’t even know the target entailment relation? (Knowing logical entailment is one thing, but it alone won’t deliver the theological contradictions.) Perhaps we know just enough of the target entailment relation to know that the apparent contradictions are inevitable. But, then, if we know (do we?) that the axioms are true, and that they entail (by a relation that we know) contradiction, wherein lies the epistemic mystery? Do we not just have knowledge that the true theory, whatever its other details, is contradictory?

6.4.4 Evaluation The epistemic-mystery account has strong virtues but also conspicuous problems. Ultimately, it is unclear what the theory – what the true theology – is supposed to be. Were it not for commitment to the mainstream story of logic the epistemic-mystery account would be a robustly epistemic glut-theoretic (i.e., contradictory) account of divine reality. With the given commitment the account appears to falter at just the points that it cannot: namely, inexorable derivation of contradiction and epistemic mystery around the derivation (or the axioms). By my lights, said commitment to the standard story of logic is without good reason. * * Long parenthetical. Anderson (2007) gives an account that, on one hand, is clearly truth-seeking and systematic, and appears to be exactly the project in which most truth-seeking theorists are engaged when addressing the ‘logical’ problem of trinitarian (or divine-incarnate) reality: namely, formulating the true and complete theory as far and as precisely as possible. This is the vein along which the epistemic-mystery account is discussed above. On the other hand, one might charge that my discussion is thereby misplaced because ultimately there are critical invocations of ‘analogical language’ in Anderson’s account, and such invocations clearly reflect a different project. By my lights, if the account is ultimately – even just at critical points – expressible only via ‘analogical

measuring some non-contradictory accounts 125 language’ then systematic, truth-seeking theorists must specify exactly what such language amounts to. Is it a degree-theoretic account of predication or exemplification? If so, what are the details? Is ‘true by analogy’ a new kind of truth? If so, how does it work? What are the entailment patterns that govern it? Is the language itself one that we can never speak? If so, how do we know so much about it? And so on. To be sure, there is a long and lively tradition of so-called analogical talk in theology. But what does it come to? Using analogies in everyday language is convenient and fruitful, but not for expressing the cold, hard truth about a phenomenon. Such talk usually helps one to do something such as, for example, think or imagine along different lines from which one is currently thinking or imagining. This is all good and perhaps necessary for truth-seeking progress; however, it is simply a different activity from the project of precisely, systematically advancing a true theory. Analogical talk is a wonderful tool in which activities such as worship or sharing a sense of sacredness often occur; it is not talk that serves true theories that describe reality. End parenthetical. * *

6.5 Gap-theoretic accounts One – to my knowledge uninhabited – candidate for the ‘right’ (I don’t say ‘true’) theory of trinitarian reality is a gap-theoretic one supplemented by a revision of assertion conditions for sentences. The account is worth noting as a sort of middle position between the contradictory account (i.e., the glut-theoretic account) and the epistemic-mystery account. The account is aired elsewhere (2021); I mention only the barest of details.33

6.5.1 Logical consequence Take logical consequence to be FDE, per §2. Hence, logic recognizes the possibility of gluts (true and false) and gaps (neither true nor false).

33 The account is related to responses to an otherwise unrelated problem in the philosophy of language of logic (Beall and Ripley, 2004; Maudlin, 2004).

126 divine contradiction Recall that, given gaps, there is a difference between falsity and untruth: the former demands the truth of logical negation while the latter doesn’t. Example: if sentence A is gappy (i.e., neither true nor false) then each of A and its logical negation ¬A is gappy; both are untrue; neither is false.

6.5.2 Assertion or acceptance conditions Consider the standard norms of assertion or acceptance: • a sentence is assertible or acceptable only if true; • a theory (as a closed set of sentences) is assertible or acceptable only if all elements are true. Instead of the standards, so understood, the target account offers minimal revisions: • a sentence is assertible or acceptable only if it is not false; • a theory (as a closed set of sentences) is assertible or acceptable only if no elements are false. Note that neither the standard nor the substitute nonstandard norm mandates assertion or acceptance; they give necessary conditions only. Sufficient conditions for assertion or acceptance are difficult to come by. (That’s the hard nut of serious truth-seeking inquiry.) But, for purposes of the current account, at least two sufficient conditions can be given: • a sentence is assertible or acceptable if it is true or demanded by reliable authority; • a theory (as a closed set of sentences) is assertible or acceptable if all elements are true or demanded by reliable authority. To be clear: the given sufficient condition is problematic through and through.3⁴ The given condition is only for illustration of the target account – not that the account requires the given condition. 3⁴ If the many problems aren’t evident, some useful discussion is available by Harman (1986, 1999).

measuring some non-contradictory accounts 127

6.5.3 On trinitarian untruths The basic idea is now straightforward: take trinitarian identity to be per §3 except reject that any of the contradictions (i.e., sentences of the form A ∧ ¬A) in the ‘right’ theory are true, and hence – given the truth and falsity conditions for logical negation – reject that any of them are false. What one gets is a simple, ‘acceptable’ theory of divine reality, one which can be unified with the same gap-theoretic christological account. And yet there are no true contradictions. The theory is acceptable – and assertible – not on the grounds that everything in it is true; it’s acceptable because, for example, relevant, reliable authority demands as much. The given gaps in divine reality are essential gaps, but they’re the only language that ‘rightly’ describes the reality.

6.5.4 Evaluation The given theory is a serious challenge to the glut-theoretic theory. Sentence by sentence, the theories are duplicates. The difference is that all sentences in the theory are true according to the glut-theoretic account while some (many?) of them are untrue (not false, but untrue) according to the gap-theoretic account. But the gaps-only account leaves a conspicuous problem. Once gluttiness of contradictions is traded for (assertible/acceptable) gaps, the nontransitivity of trinitarian identity remains entirely unexplained. What exactly about trinitarian reality explains the non-transitivity? In the contradictory (i.e., glut-theoretic) account, it’s essentially divine contradiction that explains the non-transitivity of trinitarian identity. Without true divine contradiction – with only gappy contradictions in the unfalse theory – there’s no explanation for such non-transitivity.3⁵ This is not a light defect, at least weighed with the glut-theoretic alternative. Serious challenge though it may be, I see no reason whatsoever to accept the given theory if I were to reject that it’s entirely true (i.e., sentence-by-sentence true). Church authority is sufficient for one sort of acceptance: accepting in a faith-seeking-understanding fashion. Yes. 3⁵ Note that, per §3, were one to kick out gluts one thereby gets a theory in which the logical vocabulary – without further constraints imposed on the theory – behaves per so-called K3, wherein the material biconditional is transitive.

128 divine contradiction But if one is convinced that the to-be-accepted claims are gappy, one is thereby convinced that the given authority is either knowingly or ignorantly pushing untruths. That’s not an absurd position by any stretch, but it’s not a position that sings out as well-motivated.3⁶ Finally, why exactly should the truth of the theory’s contradictions (i.e., sentences of the form A ∧ ¬A) be rejected? Presumably, this has to do with some absolute proscription from principles of rational truth-seeking endeavors. I know of no solid such principles.

6.6 Piecemeal theology: losing logic – one more time Finally, there’s an approach to the ‘logical’ problem that is strict in its diagnosis of the problem. The problem, on this approach, is to think that the true theology is closed under any entailment relation! It isn’t (on this approach). In a nutshell: • • • •

all axioms are true; there is no equivocation; classical identity is trinitarian identity; there is no entailment relation – neither logical nor extralogical (viz., theological) – under which the set of truths about divine reality is closed.

Hence, the alleged ‘logical’ problem is merely apparent. The target contradiction, for example, 𝔣≠𝔰 ∧ 𝔣=𝔰

3⁶ I flag, only for future exploration, that some positions according to which our language cannot truly or falsely describe God might best be characterized along the given gap-theoretic lines. (This is neither directly related nor irrelevant to §5.7.) The idea: we only have the language we have; the transcendence of God is beyond what our language can truly or falsely describe; the result of trying to truly describe transcendent reality is contradiction, but such contradiction is the best description available – it’s not false (and not true) because divine reality is beyond as much, but it’s the ‘best’ or ‘acceptable’ account nonetheless. Again, this is flagged only for potential future exploration.

measuring some non-contradictory accounts 129 does indeed follow from the true axioms (and the transitivity of = together with axioms ‘𝔣 = 𝔤’ and ‘𝔰 = 𝔤’); however, the mistake behind the ‘logical’ problem is to think that the theology is closed under logic or any entailment relation.

6.6.1 Unified ‘solution’ The piecemeal-theology approach affords a unified ‘solution’ to both of the ‘logical’ problems (christology, trinitarian theory). Again, that’s a genuine plus, at least as far as it goes.

6.6.2 No equivocation whatsoever The piecemeal approach is a perfectly simple one that avoids the charge of equivocation despite clinging to the classical-logic account of logical consequence, and despite identifying trinitarian identity with the classicalidentity relation.

6.6.3 Consistency Logical consistency – absence of contradiction – is achieved by not closing the target set of sentences under any entailment relation (ergo, not under logical entailment). Even though the true axioms do entail a contradiction the contradiction is in no way in the theory, since the theory isn’t closed under its entailments.

6.6.4 Evaluation My evaluation in short: the piecemeal approach is an option, but why do it? In some ways, the piecemeal approach is perfectly forthright and simple. The piecemeal approach acknowledges that the apparent contradiction is strongly apparent for a simple reason: namely, it follows from the axioms given the (extralogical, theological) entailment relation

130 divine contradiction that otherwise governs the axioms. But the governing relation is cut off. The would-be truths are in no way in the true theory; for the theory rides free from any entailment relation that would otherwise close it. Why pursue such an approach towards the true ‘systematic’ theology? I see no reason, except an unfounded rejection of would-be divine contradiction.

6.7 On the consistency-questing field In the end, the consistency-seeking accounts do not offer strong reason to reject the contradictory account. Each of them pushes trinitarian theorizing further but, whether it’s the ‘logical’ or 3-1-ness problem, none of them offers as much as the contradictory account, at least so far that I see. Ultimately, each is driven by the thought that logic precludes contradictory possibilities. With such a thought firmly in mind, it may be difficult to see divine contradiction; for one sees the appearance but runs full-steam away. Sometimes, one needs to seriously, systematically explore whether one’s reasons for so running are good ones. I do not see that they are, but if they are, that changes everything. Until then, the contradictory account appears to be not only the simplest account that avoids the charge of unmarked equivocation and affords a natural unified account of the ‘logical’ problem in christological and trinitarian theories; it appears to be true. Trinitarian reality was revealed via the contradictory Christ; it is not surprising that trinitarian reality, like God enfleshed, is contradictory.

7 Towards Future Contradictory Theology Trinitarian reality, revealed via the contradictory incarnation of God, is contradictory. Are there theological contradictions beyond trinitarian reality and the incarnation? I remain neutral pending good reason to recognize other theological contradictions. The aim of this chapter is not to argue the case one way or another but rather to simply give considerations behind my currently neutral position on a number of relevant issues. In many ways, the exploration of both theological gluts and theological gaps (i.e., theological claims that are neither true nor false) remains incipient. The process of systematic, truth-seeking exploration of theological gluts and gaps must proceed before an overall evaluation can responsibly be given.

7.1 On omni-property problems So-called omnigod problems – not peculiar to christian theism – require no review.1 A trio of such problems is standard: • omnipotence: too-heavy stones; • omniscience: knowing what God cannot know; • omnibenevolence, omnipotence, and omniscience: suffering, horrors, evil. The first two are often thought to be ‘logical’ problems (i.e., involve entailment-relation issues); the third goes beyond apparent ‘logical’ problems.

1 Nagasawa (2008, 2017) provides classic positions and recent modified positions.

Divine Contradiction. Jc Beall, Oxford University Press. © Jc Beall 2023. DOI: 10.1093/oso/9780192845436.003.0007

132 divine contradiction Were one to explore either gluts or gaps beyond the contradictions of Christ and trinitarian reality, a natural first step would turn to the given trio (and related ‘omnigod problems’).

7.1.1 Towards gluts Why, pending further exploration, do I remain neutral on ‘omnigod contradictions’ given that the fundamental doctrines of christian theism (viz., incarnation and trinity) are contradictory truths? As always, the matter is a difficult one of balancing three methodological rules of thumb that jointly guide systematic truth-seeking (2021, Chapter 6): • Completeness: try to give the whole truth of the target phenomenon. Precisely: try to put either A or ¬A into the theory, for each sentence A in the language of the theory. (In other words: completeness shoots to avoid gaps in the theory, deciding at least the truth or falsity of every sentence in the given language.) • Consistency: try to avoid falsehoods of the target phenomenon. Precisely: try to avoid putting both A and ¬A into the theory, for any sentence A in the language of the theory. (In other words: consistency shoots to avoid gluts in the theory, deciding at most the truth or falsity of every sentence in the given language.) • Simplicity/Naturalness: try to give a simple and natural theory of the target phenomenon. (There is no precise formulation of this rule of thumb.) When reality resists the satisfaction of all three rules of thumb, at least one of them must be transgressed. The strength of apparent contradiction with respect to God enfleshed and trinitarian reality, together with fundamental axioms that cannot be treated as gappy without rejecting the standard (orthodox) theory, calls for transgression of the second methodological rule of thumb, as I have argued herein and in The Contradictory Christ (2021). But what of the apparent ‘omnigod’ contradictions? Cotnoir (2018) argues not that such apparent contradictions are true, but that their truth must be explored in the pursuit of the true theology. On that point,

towards future contradictory theology 133 I wholeheartedly agree. But the question is: why not accept that they’re true given that God is contradictory? The short answer to why I remain neutral on would-be omni-problem gluts is twofold. First, each such problem appears to rely on some sort of ‘no gaps’ principle, perhaps something along the lines of DG (see §3). Such a principle, of course, is in no way required by logic itself. The question is whether, as assumed in §3, the absence of gaps in divine reality is theologically required. Perhaps the would-be ‘omnigod’ problems ultimately demand gaps in a way that the consistency rule of thumb is naturally preserved. (See §7.1.2 for brief discussion along such lines.) Second, the apparent omni-problem contradictions appear to be of a single kind.2 If so, a principle of unified solution requires a uniform solution to all such problems. If such a solution be glutty then the problem concerning suffering/horror/evil involves a true contradiction. But what would that be? Nothing natural jumps out.3 I do not have any knockdown arguments against a glut-theoretic response to the family of apparent omni-property contradictions. (If I did, I would not be neutral as things stand.) As yet, I also do not have 2 Nagasawa (2008) distinguishes different ‘types’ of omni-property problems. Such types do cut distinctions (e.g., one type arises from a given property on its own; another requires two or more such omni properties; the third requires omni properties in concert with non-divine, largely contingent matters); however, all such types are more like different flavors of a single phenomenon than different phenomena. 3 Of course, there are obvious glut-theoretic responses; it’s just that they don’t appear to be natural. One conspicuously unviable example: • It’s true that ∘ God can rid the world of evil; ∘ God knows how to rid the world of evil; and ∘ God wishes the world to be rid of evil. • It’s true that God rids the world of evil (from above). • It’s false that God rids the world of evil (from conspicuous empirical observation). Another less conspicuously unviable example: • It’s just true that God wishes the world to be rid of evil. • It’s true and false that ∘ God can rid the world of evil; and ∘ God knows how to rid the world of evil. • It’s just false that God rids the world of evil. • Explanation: the falsity of God’s omniscience (God’s relevant know-how) or the falsity of God’s omnipotence (God’s relevant can-do) upends the expected entailments from omniscience, omnipotence, and omnibenevolence in such a way that evil exists. As said, it’s not as if such options are unavailable (though none have been seriously explored, to my knowledge); it’s that none jumps out as natural or viable; the obvious options buck the simplicity-cum-naturalness methodological rule of thumb, in addition to bucking against a lot of emotional weight tied to such problems.

134 divine contradiction any strong arguments for such a treatment. The truth about such wouldbe gluts is a matter for future debate.

7.1.2 Towards gaps Dual to a glut-theoretic account of apparent omni-property contradictions is a gap-theoretic account, wherein some central gap explains the absence of the otherwise apparent contradiction. Some initial work has been done along such lines (Beall and Cotnoir, 2017), at least on the omnipotence problem. Claims like ‘either God can create a stone too heavy to lift or it’s false that God can create a stone too heavy to lift’ are treated as gappy on the grounds that any world in which they are true is a world that limits God. Obviously parallel responses to omniscience problems (e.g., ‘It’s false that God can know that this sentence is true’) are viable if the gap-theoretic response to otherwise apparent omnipotence contradictions is viable. Moreover, inasmuch as the ‘problem of evil’ involves instances of excluded middle (e.g., ‘either God can rid the world of evil or it’s false that God can rid the world of evil’), the promise of a unified solution to all such apparent omniproperty contradictions is at least open. As in the glut-theoretic direction, I have no knockdown argument against such gap-theoretic accounts.⁴ As in the glut-theoretic direction, I have no strong arguments for such a treatment. The truth about such would-be gaps is a matter for future debate. * * Parenthetical note. Were such problems to demand gaps, the fundamental trinitarian entailment pattern DG (see §3) would thereby be a restricted pattern. Nothing in any such restriction would undermine the general glut-theoretic account of trinitarian reality; it would just admit both gaps and gluts, and thereby neither transitive nor reflexive according to the given theological entailment relation. End note. * *

⁴ Tedder and Badia (2018) raise an issue for such accounts, at least as they currently stand, and the issue would require resolution before the given gap-theoretic direction is viable. There are directions for resolution that can be explored but have as yet not been explored.

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7.2 Free will and determinism A glut-theoretic account of omnipotence and omniscience, while (as above) in no way obviously correct, is at least natural. Given such an account, what of contradictions beyond the standard trio of omnigod contradictions? In particular, what of foreknowledge and freedom? Some philosophers argue that the familiar freedom-determinism ‘dilemma’ involves gluts (DeVito, 2021). The idea, in short, is that the truth of human freedom arises via the falsity of God’s foreknowledge, just as some philosophers have thought; however, the truth of God’s foreknowledge arises via the falsity of human freedom, just as other philosophers have thought. On the proposed glut-theoretic picture, freedom and foreknowledge actually coexist in one way: via contradiction. The longstanding ‘dilemma’ is better seen as a sound argument for more divine contradiction. While the picture is in some ways natural, at least on a large-scale viewing, details very much matter but remain open. Does each free human action involve a contradiction? If so, what exactly is the contradiction? Is it that one both freely acts and doesn’t freely act? And does the falsity of freely acting entail the falsity of acting simpliciter? (Example: does the falsity of driving a tractor follow from the falsity of freely driving a tractor? Etc.) The strength of the apparent contradiction involved in the longstanding freedom-foreknowledge ‘dilemma’ is close to (if not equal to) the strength of the apparent contradictions in familiar omnigod problems. But, as with such problems, there looks to be an implicit or explicit appeal to an absence-of-gaps principle. If such a principle rests entirely on the would-be logical validity of excluded middle (i.e., the would-be logical validity of all instances of A ∨ ¬A) then a pause is in order, and an exploration of gap-theoretic options is in order. Of course, if, per the assumptions in §3, DG governs divine reality, the absence of gaps is largely ensured by theology, in which case, perhaps the gaps fall on the side of human reality. In the end, details are important. Pending details, I see no reason to accept that the freedom-foreknowledge dilemma is another theological contradiction.

136 divine contradiction

7.3 God’s transcendence Another direction of apparent contradiction comes from the transcendence of God. Many a theologian has pointed to the transcendence of God as something that knocks up against apparent contradiction. If it were true that our language and thoughts cannot express or comprehend God in any substantial fashion then apparent contradiction rises up with a fury: our language and thoughts truly describe God’s transcendence but thereby falsely describe God’s transcendence. I see no reason, as yet, to accept that God’s transcendence in and of itself is contradictory. As with all such ‘potential gluts’ the details must be given. If God’s transcendence demands contradiction, the relevant ingredients of transcendence are. . . what? Details matter. As in other cases, I remain neutral on would-be ‘gluts of transcendence’ pending the details.

7.4 God’s love Another direction concerns God’s love. The love exemplified by divine persons is perfect and sufficient. Love that extends beyond perfect, sufficient love either is imperfect or belies the sufficiency of love exemplified by divine persons alone. Such issues may raise apparent contradiction, but not in a straightforward fashion that falls out of core theological axioms. While logic certainly involves possibilities in which love restricted to divine persons is perfect, sufficient and yet is also imperfect and insufficient, theology’s space of possibilities doesn’t obviously involve as much. Might it? Yes, but as yet there’s no strong reason to accept as much.

7.5 God’s creation Yet another direction, intimately related to §7.4, is creation itself. This apparent source of contradiction is not peculiar to christian theology, but it arises there all the same. In particular, God’s unchanging perfection appears to be in contradiction with God’s creation of imperfect, changing beings.

towards future contradictory theology 137 I admit that such issues revolving about God’s creation are candidates for contradiction. But that God’s creative action involves contradiction is far from clear. The apparent contradiction, perhaps similar to those of love or transcendence, involves a host of variables around the details of God’s creative act and the relevant relations of it to time and ‘the created order’ – none of which enjoys simple, stark axioms in the way that, for example, core doctrines of divine-incarnate or trinitarian reality do.

7.6 Denominationally distinct doctrines The foregoing directions towards theological gluts or gaps are common to standard christian theology, not turning especially on unfortunate divisions of the christian church. But once divisions are enforced, denomination-specific theories can sometimes invite apparent contradiction. Witness for but just one (of so very many): predestination. The apparent contradictions of such a thesis, like those of all canvassed cases above, require no rehearsal. (In some ways, the apparent contradictions of varieties of ‘predestination’ are entwined with elements of freedomforeknowledge apparent contradictions in general, as in §7.2.) Whether a particular denominational apparent contradiction is true depends not only on the details; it depends on the broader (denomination-specific) theory. My own view, while governed by the usual trio of methodological rules of thumb (§7.1.1), leans towards conservatism. In particular, theological gluts concerning anything but central, core, and simple axioms of christian theology – versus specific denominational variations – are one thing; going beyond is something very different. The conservatism, in the end, is a restriction of gluts (and gaps) to the few simple core axioms of christian theology unless, in the process of following standard rules of thumb, the truth requires more.

7.7 . . . apart from divine-incarnate and trinitarian reality? While so much more work must first be done, I remain, as above, neutral with respect to potential gluts and gaps in divine reality except for those

138 divine contradiction perennially apparent in the foundational core doctrines of christian theology: God enfleshed and the triune god therein revealed. Why, in the end, are these two apparent contradictions true while others – including but also beyond those canvassed above – not equally obviously true? Slivers of explanation are given above, but a simple, flatfooted one is most telling. In short, the obvious appearance of contradiction in the axioms governing divine-incarnate reality, such as • Christ is divine (and therefore God). • Christ is human (and therefore human as we are human). and likewise governing trinitarian reality • Each of Father, Son, and Spirit is identical to God. • Father, Son, and Spirit are pairwise non-identical. are not appearances that are buried deeply or require some clever argument to churn up. Christian thinkers and non-christian thinkers (be they theists or otherwise) do not require a subtle argument to see the apparent contradiction in such fundamental axioms of true theology. The axioms themselves, once seen, invite the same reaction over and over again: What?! ...But that’s contradictory! In the case of the omniproperty problems, the problems do not arise from simple, core axioms or doctrines. The apparent omni-property (or foreknowledge-freedom, or transcendence, or beyond) are genuine problems, but they involve a broader family of issues than the core trinitarian and incarnation axioms involve. One ought not seek contradiction any more than one ought seek incompleteness. Likewise, one ought not run from contradiction any more than one ought run from incompleteness. Truth, in the end, is discovered by careful, sturdy, case-by-case steps. But in that rare case wherein the few core axioms blink brightly of apparent contradiction – over and over and over again – one ought to consider the rare but real possibility: the reality itself is contradictory. This is especially so with divine reality wherein contradiction is the Alpha and the Omega, the First and the Last, the Beginning and the End . . . the only truly holy contradiction known to creation.

APPENDIX A

Athanasian Creed (tr. Philip Neri Reese, O.P.) Whoever wants to be saved should, before anything else, hold the Catholic faith – [the faith] which, if someone does not preserve it whole and inviolate, [that person] will, without doubt, perish eternally. And the Catholic faith is this: that we venerate one god in trinity and trinity in unity, neither confusing the persons nor separating the substance. For the person of the father is other than the son, [and] other than the holy spirit. But of the father, and the son, and the holy spirit, there is one divinity, equal glory, and coeternal majesty. As the father [is], so the son [is], and so the holy spirit [is]. The father [is] uncreated, the son [is] uncreated, and the holy spirit [is] uncreated. The father [is] boundless, the son [is] boundless, and the holy spirit [is] boundless. The father [is] eternal, the son [is] eternal, and the holy spirit [is] eternal. And yet [there are] not three eternals, but one eternal – just as [there are] not three uncreateds, nor three boundlesses, but one uncreated and one boundless. Likewise, the father [is] omnipotent, the son [is] omnipotent, and the holy spirit [is] omnipotent. And yet [there are] not three omnipotents, but one omnipotent. So too, the father [is] god, the son [is] god, and the holy spirit [is] god. And yet [there are] not three gods, but [rather], there is one god. So too the father [is] lord, the son [is] lord, and the holy spirit [is] lord. And yet [there are] not three lords, but one lord. [This is] because just as Christian truth compels us to confess each person singly [to be] god and lord, so too are we prohibited by the Catholic religion from speaking of three gods or lords. The father is in no way made, or created, or begotten. The son is from the father alone – neither made, nor created, but begotten. The holy spirit is from the father and the son – neither made, nor created, nor begotten, but proceeding. Therefore, [there is] one father, not three fathers, one son, not three sons, one holy spirit, not three holy spirits. And in this trinity nothing [is] prior or posterior, nothing [is] more or less. Rather, all three persons are coeternal and coequal to each other, such that – as was said above – in all things the unity in trinity and trinity in unity is to be venerated. Whoever wants to be saved should think thus of the trinity. But it is [also] necessary for eternal salvation that one faithfully believe in the incarnation of our lord Jesus Christ. Therefore, it is the correct faith that we should believe and confess that our lord Jesus Christ, the son of god, is god and man. [He is] god from the substance of the father, before time began, and he is man from the substance of [his] mother, born in time. [He is] perfect god and perfect man, subsisting from a rational soul and human flesh – equal to the father according to divinity, less than the father according to humanity – who, although he is god and man, nevertheless [is] not two, but is one Christ. [He is] one not by a conversion of divinity into flesh, but by an assumption of humanity into god. [He is] entirely one, not by a confusion of substance, but by a unity of person. For just as one man

140 appendix a is rational soul and flesh, so one Christ is god and man – who suffered for our salvation, descended into hell, [and] resurrected from the dead on the third day. He ascended to the heavens [and] sits at the right hand of the father, from whence he shall come to judge the living and the dead. At his coming all men shall arise with their bodies and render an account of their deeds. Those who did good will go to eternal life, [while] those who [did] bad [will go] to eternal fire. This is the Catholic faith – [the faith] which, unless someone believes it faithfully and firmly, [that person] cannot be saved.

APPENDIX B

§2 Appendix: Formal Sketch of FDE This appendix sticks closely to but deviates slightly from the presentation in Chapter 2 of 2021, where, among other places (2017, Omori and Wansing 2019), a fuller discussion is offered. Except for recognizing all four semantic statuses (viz., true, false, both, neither) the language, like the definition of consequence, is the same as the standard language of the so-called classical account of first-order logic.

Syntax of the language of FDE The syntax of the language is standard first-order syntax (without identity, which, per further discussion in §3, is beyond the topic-neutral, universal logical vocabulary).

Basic alphabet The basic alphabet or set of building blocks is standard: a set 𝒫 of extralogical predicates (viz., P, Q, R with or without natural-number subscripts), each with an ‘arity’ (viz., the number of ‘slots’ for individual terms required to make a sentence); a set of alogical punctuation marks (viz., ‘)’ and ‘(’ ); a set 𝒯 of so-called (extralogical) individual terms, where 𝒯 is the union of a set of variables (viz., x, y, z with or without natural-number subscripts) and names (viz., a, b, c with or without numerical subscripts);1 two primitive logical unary sentential connectives (viz., † and ¬); two primitive logical binary sentential connectives (viz., ∧ and ∨); and two primitive logical quantifiers (viz., ∀ and ∃);2 and one derivative binary logical connective (viz., logic’s material conditional →, where, for any sentences A and B, A → B is defined to be ¬A ∨ B).

Well-formed formulæ The set of formulæ (whether well-formed or otherwise) is simply the set of all strings of items from the alphabet. The important set 𝒲ℱ of well-formed formulæ is defined as follows. Definition B.0.1 (𝒲ℱ) Where an atomic wff (atomic wooff) is any n-ary predicate followed by n many terms (i.e., elements of 𝒯), the set 𝒲ℱ is defined: 1 Function symbols can be added in the usual way but this is up to individual theories (much like the names, predicates, and variables are). 2 Note for experts: this is very much a redundant list of ‘primitive’ connectives, all of which enjoy their usual definitions in terms of others. But for simplicity and for readers unfamiliar with such standard definitions, all are taken to be primitive.

142 appendix b 1. Every atomic wff is in 𝒲ℱ. 2. Let 𝜎 and 𝜎 ′ be in 𝒲ℱ. Then so too are †𝜎, ¬𝜎, (𝜎 ∧ 𝜎 ′ ), (𝜎 ∨ 𝜎 ′ ), (𝜎 → 𝜎 ′ ), and, where v is an individual variable, also ∀v𝜎 and ∃v𝜎.3 3. Nothing else is in 𝒲ℱ (i.e., no other well-formed formulæ).

The sentences The sentences of the language are likewise exactly per the standard account: defined as a proper subset of 𝒲ℱ. And, as usual, the definition invokes the distinction between open wff – that is, open well-formed formulæ – and closed wff, where the former involve so-called free occurrences of at least one variable and the latter involve only so-called bound occurrences of variables (if any variables at all). The definition is slightly tedious but here it is: Definition B.0.2 (Free variable) Let u and v be individual variables. Then v is free (or ‘occurs free’ in, or ‘is unbound’) in a well-formed formula (wff) 𝜎 iff

1. 𝜎 is an atomic formula and v occurs in 𝜎, or 2. 𝜎 is a negation ¬𝜎 ′ and v occurs free in 𝜎 ′ , or 3. 𝜎 is a nullation †𝜎 ′ and v occurs free in 𝜎 ′ , or 4. 𝜎 is a conjunction 𝜎 ′ ∧ 𝜎 ″ and v occurs free in either 𝜎 ′ or 𝜎 ″ , or 5. 𝜎 is a disjunction 𝜎 ′ ∨ 𝜎 ″ and v occurs free in either 𝜎 ′ or 𝜎 ″ , or 6. 𝜎 is a universal ∀u𝜎 ′ and v is not u and v occurs free in 𝜎 ′ , or 7. 𝜎 is an existential ∃u𝜎 ′ and v is not u and v occurs free in 𝜎 ′ . With the notion of free variable defined, the set of sentences is just this: Definition B.0.3 (Set of sentences: 𝒮) A well-formed formula 𝜎 is in the set 𝒮 of sentences iff there are no free variables occurring in 𝜎.

Semantics of FDE The semantics for the language of FDE is the usual ‘classical’ semantics except for allowing the otherwise ignored two of four semantic statuses. In particular, the semantics is given in terms of FDE models or FDE interpretations (henceforth, models or interpretations unless context demands more), which are triples M = ⟨D, 𝛿, 𝜌⟩, where D is a non-empty domain (consisting, so to speak, of all individual objects that exist according to the model), 𝛿 an interpretation function that assigns to each n-ary predicate 𝜋 a pair ⟨𝜋 + , 𝜋 − ⟩, where 𝜋 + and 𝜋 − contain all n-tuples of which 𝜋 is true and, respectively, false according to the model, and 𝛿 assigns to each name t an

3 Note: for ease-on-the-eyes reasons outer parentheses are dropped from well-formed formulæ provided that context clarifies.

appendix b 143 element of D (viz., 𝛿(t)); and 𝜌 is a so-called variable assignment, which is a function from all individual variables into D. A semantics must deliver, generally in a systematic (and compositional) fashion, semantic values (or ‘semantic statuses’) for all sentences of the language. The semantic values are subsets of {1, 0}, which is the set of fundamental values (viz., truth and falsity). In particular, the set 𝒱 of semantic values is {{1}, {0}, {1, 0}, ∅}. In turn, the values are assigned to elements of 𝒮 (viz., sentences) by a model-relative function: | ⋅ |M ∶ 𝒮 ↦ 𝒱, a function from every sentence into the semantic values. The modelrelative function | ⋅ |M is partially defined via a function 𝛿M from all terms (i.e., names and variables) into the domain D, a function that piggybacks on M’s denotation function and variable assignment:

𝛿M (t) = {

𝛿(t) 𝜌(t)

if t is a name; if t is a variable.

The relevant FDE functions are all and only such functions that ‘obey’ (or satisfy) the following clauses, where, as a heuristic, one can think of ‘1 ∈ |A|M ’ and ‘0 ∈ |A|M ’ as sentence A contains truth (respectively, falsity) according to the given M-relative function (Dunn, 1976; Omori and Wansing, 2019).

1. Atomic: Let 𝛼 be any n-ary atomic wff 𝜋t1 , . . . , tn in 𝒲ℱ. (a) 1 ∈ |𝜋t1 , . . . , tn |M iff ⟨𝛿M (t1 ), . . . , 𝛿M (tn )⟩ ∈ 𝜋 + . (b) 0 ∈ |𝜋t1 , . . . , tn |M iff ⟨𝛿M (t1 ), . . . , 𝛿M (tn )⟩ ∈ 𝜋 − . 2. Molecular: where A and B are any elements of 𝒲ℱ, and v a variable: (a) 1 ∈ |†A|M iff 1 ∈ |A|M . (b) 0 ∈ |†A|M iff 0 ∈ |A|M . (c) 1 ∈ |¬A|M iff 0 ∈ |A|M . (d) 0 ∈ |¬A|M iff 1 ∈ |A|M . (e) 1 ∈ |A ∧ B|M iff 1 ∈ |A|M and 1 ∈ |B|M . (f) 0 ∈ |A ∧ B|M iff 0 ∈ |A|M or 0 ∈ |B|M . (g) 1 ∈ |A ∨ B|M iff 1 ∈ |A|M or 1 ∈ |B|M . (h) 0 ∈ |A ∨ B|M iff 0 ∈ |A|M and 0 ∈ |B|M . (i) 1 ∈ |A → B|M iff 0 ∈ |A|M or 1 ∈ |B|M . (j) 0 ∈ |A → B|M iff 1 ∈ |A|M and 0 ∈ |B|M . 𝜌v

(k) 1 ∈ |∀vA|M iff 1 ∈ |A|M , for all so-called v-variants 𝜌v that differ at most from M’s given 𝜌 only in its assignment to v. 𝜌v

(l) 0 ∈ |∀vA|M iff 0 ∈ |A|M , for some so-called v-variant 𝜌v that differs at most from M’s given 𝜌 only in its assignment to v. 𝜌v

(m) 1 ∈ |∃vA|M iff 1 ∈ |A|M , for some so-called v-variant 𝜌v that differs at most from M’s given 𝜌 only in its assignment to v.

144 appendix b 𝜌v

(n) 0 ∈ |∃vA|M iff 0 ∈ |A|M , for all so-called v-variants 𝜌v that differ at most from M’s given 𝜌 only in its assignment to v.

FDE: logical consequence Let ⊢ (unsubscripted) be logical consequence. Then the FDE account of ⊢ is defined exactly standardly in terms of absence of counterexample; the relation is defined only over set-sentence pairs where the relevant sets are sets of sentences. Definition B.0.4 (FDE Counterexample) Let X ⊆ 𝒮 and A ∈ 𝒮. An FDE counterexample to ⟨X, A⟩ is any FDE function | ⋅ |M , as above, such that 1 ∈ |B|M for all B ∈ X but 1 ∉ |A|M . Note that every so-called classical counterexample (i.e., any FDE model in which 𝜋 + ∩ 𝜋 − = ∅ for all predicates 𝜋, and also 𝜋 + ∪ 𝜋 − = Dn for all predicates 𝜋) is an FDE counterexample, though the converse fails. Definition B.0.5 (FDE Consequence) X ⊢ A iff there’s no FDE counterexample to ⟨X, A⟩. (Terminology: ‘X ⊢ A’ may be read: A is a logical consequence of X; A logically follows from X; X logically entails A; etc.)

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Index 3-1-ness 35, 52, 53, 55, 56, 77, 87, 90, 91, 107, 108, 116, 117, 122 3-1-ness problem vii, viii, xii, 1, 10, 11, 16, 36, 56, 57, 60, 63, 64, 66, 68, 77, 78, 99, 113, 116, 118, 121, 122, 130 analogical 96, 97, 124 Anatolios, Khaled 1, 2 Anderson, Alan Ross 28 Anderson, James N. xv, 78, 87, 121–124 Aristotelian 118, 120, 121 Asenjo, F. G. 32, 50 athanasian axioms x, 10, 11, 13, 14, 36, 37, 43, 63, 66, 72–75, 101, 107 Athanasian Creed vii, x, xix, 1, 2, 9, 10, 13, 38, 56, 74, 86, 88, 117 tr. Philip-Neri Reese, O.P. 36, 86, 139, 140 Augustinian 9 Ayres, Lewis 1 Badia, Guillermo 134 Barnes, Michel René 1 Belikov, Alexander 28 Belnap, Nuel D. 28 Berto, Francesco xv bi-entailment 11, 52–54, 88, 116 Bimbó, Katalin 28 Bray, Dennis xvi Brewster Stevenson, Rebecca xv Brower, Jeffrey E. 2, 11, 16, 84, 113, 117, 118 Byzantine patristics vii, viii Callahan, Laura Frances xv Chowdhury, Safaruk Zaman vii, xv christology 2–5, 42, 69–71, 82, 92, 93, 112, 120, 127, 129, 130 Church, Alonzo 106

Coakley, Sarah 1, 2, 5, 100 completeness 132, 138 Concha, Abby xvi conjunction 5, 17, 24, 25, 28, 29, 44, 46, 48, 52, 85, 87, 142 consequence 5–8, 23, 26, 28, 31, 33–35, 41, 50, 52 consequence relation 5–7, 32, 43, 49, 52, 92, 104 consistency 11, 76, 78, 80, 100, 113, 114, 122, 129, 130, 132, 133 constitution account 113, 117, 118, 120–122 contradictory christology 3–5, 82 contradictory theology 82, 131–138 contradictory christology, see contradictory christology contradictory theory contradictory account 100 contrapose 92, 94 Cotnoir, A. J. xv, 96, 132, 134 Council of Chalcedon vii, ix, 4 counting convention 9, 17, 19, 36, 38, 52, 56–64, 67, 73, 88, 107, 116–122 Craig, William Lane 99 Crisp, Oliver xv, 2, 96 Cross, Richard xv, 1, 3 Curry, Haskell B. 106 Cutter, Brian xv D’Agostini, Franca xv, 92 Dahms, John V. 8 DeVito, Michael xv, 7, 31, 84, 135 DG 54, 55, 133–135 disjunction 24, 25, 28, 29, 48, 116, 142 distribution of gaps 54 DG, see DG Dunn, Michael J. 28

152 index entailment bi-entailment, see bi-entailment extra-logical entailment, see extra-logical consequence logical entailment, see logical consequence theological entailment, see theological entailment trinitarian entailment, see trinitarian entailment entailment pattern 12, 52–55, 62, 87, 90, 91, 100, 102, 105, 108, 117, 125, 134 entailment relation 5, 6, 26–28, 31–33, 43, 49, 52–54, 100, 102, 104, 123, 128, 129, 131, 134 equivalence 29, 42, 88 equivalence relation 47, 48, 100, 103, 104, 109, 110, 114, 115 Eschenauer Chow, Dawn 96, 97 Estrada González, Luis xv exclusion 20, 29–32, 50 exhaustion 20, 29–32 explosion 29, 30, 32, 42, 91 extension 22, 23, 51, 58–62, 65–68, 86, 87 antiextension 22, 23 fairdinkum x FDE 25, 28, 49–51, 70, 103, 112, 125, 142–144 Ficara, Elena xv, 92 fundamental problem 3, 112, 115 gap 20, 21, 27, 28, 30, 34, 35, 50–56, 70, 104, 125–128, 131–135, 137 General Catechetical Directory 77, 78, 80 glut 20, 21, 27, 28, 30, 34, 50, 51, 55, 59, 60, 64, 70, 82, 92–94, 104, 124, 125, 127, 131–137 Harman, Gilbert 126 Hasker, William 99 heresy 2–4, 75, 94, 95, 110 absence of nullation (H2) 94, 95 presence of negation (H1) 94, 95 Hofweber, Thomas 92

Holmes, Stephen 2 Hughes, Michael 32 Hyde, Dominic 105 identity viii, 9, 10, 30–32, 36–39, 42–51, 53, 56–65, 68, 70–75, 78, 83–86, 88, 91–93, 99–121, 128, 129, 141 leibnizian recipe, see leibnizian recipe non-identity, see non-identity incarnation vii, 2–5, 11, 18, 21, 35, 39, 41, 54, 56, 69–71, 75, 76, 79, 82, 96, 97, 100, 112, 120–123, 131, 132, 138 indiscernibility 18, 45, 103 intersubstitutability rule 12 intersubstitutability 92, 93 Jaeger, Andrew xv Jedwab, Joseph xv, 110 Joaquin, Jeremiah Joven xv Kelly, J. N. D. 9 Kelly, Jon xvi Kleene, S. C. 50 Leftow, Brian 100 leibnizian recipe 17, 18, 37, 44–51, 62, 64, 73, 74, 103–106, 108, 109, 111, 115, 117 Lincicum, David xv Logan, Shay 50, 51, 141 logical consequence xiii, 6–8, 16, 18, 20, 22, 26, 28, 31, 34, 35, 49–51, 70, 74, 84, 103, 104, 106, 107, 109–111, 121–123, 125, 129, 144 extra-logical consequence 92 logical entailment 5, 7, 22, 26–29, 54, 104, 123, 124, 129 extra-logical entailment 32, 33, 129 logical equivalence 29 logical problem vii–ix, 1, 13, 15, 113, 120, 122, 124, 131 incarnation, see incarnation trinitarian reality, see trinitarian reality

index 153 logical vocabulary 5, 6, 14, 18, 19, 23, 24, 26, 28, 32–34, 43, 45, 57, 60, 71, 73, 111, 123, 127, 141 extra-logical vocabulary 33 Lourié, Basil vii, xv

Pelikan, Jaroslav 9 Pleitz, Martin xv, 16 Positive Uniqueness 10 Priest, Graham xv, 16, 50 principle M 38, 63, 65, 92, 93, 99

MacFarlane, Dylan xv Martínez-Ordaz, M. d. R. xv Martinich, A. P. 2 material biconditional 17, 18, 44, 48–51, 73, 103, 104, 127 material conditional 14, 19, 24, 30, 37, 48, 73, 104, 109, 115, 141 material conditional identity 30 material constitution 118–120 Maudlin, Tim 125 McCall, Thomas xv, xvi, 1, 2, 84, 99, 100, 118 methodological principle 4 methodological rule 4, 55, 132, 133, 137 methodological rule zero 4 M0 4 modus ponens 30 monotheism 2, 10–13, 38, 39, 42, 63, 82, 88, 90, 93, 99, 100 Morris, Thomas V. 83 Mullins, R. T. 84

Rea, Michael C. xii, xv, 2, 11, 16, 92, 99–101, 104, 106, 112, 113, 117, 118, 120 Reese, Philip-Neri, O.P. xv, 36, 86, 107, 139 reflexivity 46–51, 103, 104, 108, 109, 115, 134 non-reflexive-(bi)conditional 48 non-reflexivity 48, 49 reflexive-(bi)conditional 46–51 Restall, Greg x, xv, xvi Ripley, David xv, 125 Russell, Gillian xv Rutledge, Jonathan xv

Nagasawa, Yujin 131, 133 negation 6, 23, 24, 28, 29, 39, 42, 48, 52, 67, 78, 79, 91–95, 97, 101, 126, 127, 142 Newlands, Sam 92 Nicene Creed 2 Niceno-Constantinopolitan Creed 9 Nolan, Daniel xv non-identity 10, 14, 36, 37, 41, 42, 45, 52–54, 56, 64, 66, 70, 72, 74, 75, 77, 84, 89–91, 95, 101, 103, 108, 114, 115, 117, 123 nullation 23, 24, 52, 94, 142 Ogden, Stephen xv Omori, Hitoshi xv, 28, 141 Paoli, Francesco 29 Pawl, Timothy xv, 92

Sanders, Fred 2 Shramko, Yaroslav 28 shriek 32, 33, 54 shrug 32, 33 Snodgrass, Jace xvi Southwell, Sharon xv style conventions explicit ix–xiii subordination 2, 109 subvaluation 105 supervaluation 105 Swineburne, Richard 100 symmetry 14, 47, 50, 51, 94, 103, 104, 109, 114 symmetric-(bi)conditional 46–48 T+ 52, 53, 55, 62, 87, 90, 100, 108, 114, 116, 117 T− 52, 53, 55, 62, 90, 100, 114, 116, 117 Tamburino, J. 50 Tarski, Alfred 45, 104, 106 Tedder, Andrew xv, 134 TG 11, 12, 42, 63, 77, 88, 99, 117 theological entailment 92, 94, 123, 129, 134 Thom, Paul 16 Thompson, Rachael Elizabeth xv

154 index Torrance, Andrew xv, xvi transitivity 14–16, 18, 43, 47–51, 64, 72–74, 103, 104, 109, 114, 127, 129, 134 non-transitive-(bi)conditional 48 non-transitivity 15–18, 36, 37, 48, 49, 51, 73, 74, 109, 127 transitive-(bi)conditional 46, 47 trinitarian axioms 14, 15, 70, 91, 100, 107 trinitarian counting convention 60–63, 67, 88, 89, 107, 117–121 trinitarian entailment 12, 38, 51–53, 56, 66, 70, 90, 91, 99, 100, 114, 134 DG, see DG T+, see T+ T−, see T− trinitarian entailment patterns 52 trinitarian identity 1, 15–19, 35, 37–40, 43, 51–57, 60, 62–66, 69–71, 73, 77, 78, 83–85, 90, 91, 101, 107, 109, 117, 123, 127–129 trinitarian reality vii, xii, 1, 36–40, 42, 43, 52, 53, 55, 59–61, 63, 64, 66, 70–78, 80, 82, 90, 91, 93–95,

98–100, 107, 112, 113, 116, 120–122, 124, 125, 127, 130–132, 134, 137, 138 trinitarian theory viii trivial theology 30, 91, 110, 112, 114–116, 123 trivial theory 7, 29, 110, 123 Tuggy, Dale vii, xv, 2, 99, 100 Uckelman, Sara L. 16 van Fraassen, Bas C. 105 van Inwagen, Peter 108, 110, 112 Vandergrift, Ross 32 Varzi, Achille 105 Wansing, Heinrich 28, 141 Ware, Timothy 9 Weber, Zach xv Wolfson, H. A. 1 Wood, William 2, 84 Yang, Eric xv Zaitsev, Dmitry 28 ZFC 61