Divine Contradiction: replies to critics

Abstract

This is a collection of replies to critics of Divine Contradiction, each critic a symposiast in the Religious Studies symposium on said book.

Introduction

It is both humbling and exciting to have five very accomplished thinkers, strongly representing both philosophy and theology, critically engage with Divine Contradiction and also, indirectly, thereby engage with its essential predecessor (‘Book 1’, so to speak) The Contradictory Christ. The work of each symposiast does all that any reasonable thinker can hope: the work highlights promise and potential problems with the current contradictory theology, thereby opening doors for progress. To each symposiast I hereby express my sincere gratitude: truly, thank you.

In what follows I skip a great many issues raised by symposiasts, and each such skip is done with a firm frown as every skipped issue demands a thorough reply. The skipping is not for aerobic exercise but rather imposed by severe page limitations. My hope is that my replies address at least the principal objections or questions raised by each work. Familiarity with each article is assumed, though in cases where it's useful I briefly rehearse my understanding of the relevant objection, question, or issue at hand. Replies are alphabetically ordered by symposiasts’ surnames.

Response to James N. Anderson

James N. Anderson examines two proffered virtues of the contradictory theology developed in Divine Contradiction and its essential predecessor The Contradictory Christ, namely, epistemic and metaphysical neutrality. Anderson argues that contradictory theology, as I've developed it in target works, achieves neutrality by silence; said works commit no epistemic or metaphysical transgressions by omitting the topics altogether. But that route towards ‘achieving’ virtues is cheap; it's on par with never singing off key by never singing at all.

The difference, Anderson suggests, is that while singing might be indefinitely postponed, epistemic and metaphysical questions are not. At some point, epistemic and metaphysical questions have to be answered, including ones arising from theology. When that time comes – and Anderson urges that that time is nigh – contradictory theology may well (if not probably will) involve not only some substantive epistemology and metaphysics, but quite possibly an unhealthy dose of the very elaborate metaphysical or epistemological scaffolding that allegedly plagues non-contradictory theologies. In that case, contradictory theology loses its glimmer of simplicity and viability, and possibly also gains unexpected complexities.

In this reply, I first clarify the sense (and somewhat limited context) in which epistemological and metaphysical neutrality is a virtue. In turn, I give a general response to Anderson's discussion of ‘metaphysical creeping’ (my word) into contradictory theology; in turn, I give a general response to Anderson's direct epistemological challenge – ‘the nub of the [epistemology] matter’, as Anderson puts it.

On the would-be virtue of target neutrality

Clarifying the would-be virtue of target neutrality demands more space than it receives in my target books and definitely more space than is allowed in this symposium. The short but very important clarification is twofold.

First fold: epistemic (ditto metaphysical etc.) neutrality is a virtue in response to an apparent contradiction if the apparent contradiction is in fact neutral with respect to epistemic features – if it arises independently of epistemic features. Ditto for metaphysics. Accordingly, epistemic neutrality of a theory of phenomenon X is a virtue if apparent contradictions of X are explained without invoking special epistemic phenomena. Ditto metaphysical neutrality of a theory. Of course, sometimes surprises happen, and no natural and simple theory of X sufficiently explains the apparent contradictions of X.¹ In that case, the surprise is that epistemic scaffolding of the theory (or, ditto, metaphysical or, for that matter, semantical, etc.) might be required. In general, the search for true theories continues to be guided by a standard trio of methodological rules of thumb (discussion of which is in Divine Contradiction, chapter 7).

The second (and, for present purposes, final) fold: epistemic and metaphysical neutrality are especially important when it comes to the target contradictions of the most central doctrines of christian theology (including the incarnation and trinity). Why especially? Christian theology is a revealed theology, and the doctrines (axioms) are ultimately founded on revelation. In the absence of a revealed epistemology or metaphysics, responses to apparent contradictions on the very surface of core doctrines (axioms) that essentially invoke epistemology or metaphysics have a knock against them. The knock against them is not that there is no true epistemology or true metaphysics; there is. The knock is that, at least as yet, there is no revealed epistemology or metaphysics, and so invoking such resources to explain apparent contradiction at the core of christian theology carries a prima facie flag. Anderson is correct that, all things equal, epistemic and metaphysical neutrality are virtues of any would-be theory, at least along the first fold above; however, such neutrality carries more weight when it comes to revealed theories – namely, theories (e.g. theologies) the truth of which is revealed. There is a lot more to say, some of which arises from what (little) is above. But there is no space to say more if Anderson's explicit problems/challenges are to be even briefly addressed.

Metaphysics creep: propositions and gluts

A glut theorist – and, so, anyone endorsing a contradictory theology – accepts that some truths are also falsehoods: the logical negation of some truth is true (and false), and the logical negation of some falsehood is false (and true). Logical negation is simply logic's falsity connective, dual to logic's usually ignored (because logically redundant) truth connective – ‘logical nullation’ (or the invisible null connective). By way of example, Anderson notes that metaphysical questions quickly creep into an otherwise metaphysically neutral theory right from the start. After all, one and the same truth (e.g. that some people are bald) is expressible via different sentences in one language, and also in different languages altogether. But, then, it's natural to infer that ‘the primary truths’ – the ‘primary bearers of truth’ – are not sentences but rather something else, say, ‘propositions’. While I use the English sentence ‘Some people are bald’ to speak a truth, someone else might use a non-English sentence to speak exactly the same truth. On this (fairly standard) picture, the truth we express is a proposition; the vehicle we use to do so is some sentence or other (which, as Anderson says, is derivatively true by expressing a truth).

Assume, just for present purposes, that the foregoing account of truths qua propositions is true. The question at hand isn't whether such an account is true; the question that Anderson raises is whether the account raises problems for contradictory theology (or glut theory, generally). Anderson thinks so. In particular, Anderson notes that:

[I]t seems that propositions characteristically play a representational role. A proposition is true if it accurately represents how things are. A proposition is false if it misrepresents (i.e. fails to accurately represent) how things are. If this relatively modest metaphysical theory is correct so far – the theory that says propositions are real entities with representational features or capacities such that they can bear truth-values – then it already seems bizarre that a proposition could be both true and false, as Beall's proposal entails (Beall [2023], 41). How could a proposition both accurately represent how things are yet also misrepresent how things are?

What to make of Anderson's alleged problem? The would-be problem is indeterminate or otherwise underspecified. To see this, note that, on pain of circularity, ‘accurately represents’, if it defines truth of a proposition without circularity, must mean something other than the usual truly represents. (Ditto for falsely represents and a proposition's thereby being false.) But if it doesn't just mean truly represents (or, dually, falsely represents), what does ‘accurately represents’ or ‘inaccurately represents’ or ‘misrepresents’ mean here? They are clearly technical terms that may or may not carry heavy metaphysics (or more). The trouble is that they remain entirely undefined.

For a taste of how underspecified the matter stands, ask yourself whether standard logical behaviour involving logic's falsity connective (viz. logical negation) is immediately validated by the details of the target technical terms ‘accurately represents’ and ‘misrepresents’. Does every proposition either ‘accurately represent’ or ‘misrepresent’ – and do so out of relevant necessity? Are there relevant possibilities in which a proposition neither accurately represents nor misrepresents something? (Think excluded middle.) Moreover, at every relevant possibility, is a proposition's misrepresentation of a proposition's misrepresentation of something thereby the ‘accurate representation’ of the thing (and/or vice versa)? (Think double negation.) And so on. Pending details on these apparently primitive technical terms, Anderson's alleged problem remains an alleged but as yet underspecified problem. One can hardly assess whether a proposition can be true and false on the given account when the terms remain undefined.

One other effect of the relevant underspecification is that the extent to which the account promises to be a ‘relatively modest metaphysical theory’ is likewise very wide open. Filling out primitive relations like ‘accurately represents’ and ‘misrepresents’, if they're to (non-circularly) define truth/falsity of propositions, is often a long and wobbly project of heavier and heavier scaffolding. That's not a bad thing in itself; it's just that Anderson's chosen example of potential metaphysical weight gain might ultimately be a programme to be avoided entirely. This doesn't mean that there's no metaphysics in the true theory of propositions (unless, of course, there are no propositions, but for present purposes we're assuming otherwise); it just means that the level of metaphysical weight depends very, very much on the details – not only of propositions, but of truth itself.²

I conclude that Anderson's example of a potential problem for contradictory theology (and glut theory in general) is not a problem as it stands. Should the true metaphysics in general preclude gluts (or gaps), all contradictory theories (and gappy prime theories) should be rejected. But, of course, should the true theology be contradictory (or gappy), as my target work has argued, any metaphysics that precludes gluts (or gaps) should be rejected. One thing is clear: we repair our rafts while trying to use them in an often turbulent sea. (Worn-out but still apt metaphor.) Such is our epistemic condition. We do the best we can with an eye firmly towards the truth. (See chapter 7 of Divine Contradiction for a sketch of methodological rules of thumb in the pursuit of true theories.)

Before turning explicitly to a few remarks on Anderson's epistemological challenge, I should comment on Anderson's question about glut theory and realism/antirealism in metaphysics. (See Anderson's ultimate paragraph in his ‘Metaphysical neutrality’ section.) In short, Anderson suggests that glut theorists (perhaps also gap theorists, though Anderson doesn't say) are likely (Anderson says that he suspects ‘that a theological glut theorist would’) to adopt an ‘anti-realist view of propositions’. There are big and many issues here, but space limitations force the shortest of replies. I have never found the ‘realist vs antirealist’ terminology in any fashion fruitful. If there are propositions, they're real – ‘really there’ etc. – and, so, there's then nothing worth being ‘anti’ over. I don't have firm views of propositions, but I see no reason to think that there can't be such things. (Indeed, there certainly are propositions on a standard ‘Lewis/Stalnaker model’, appropriately and explicitly filled out – as in my reply in this symposium to Peter van Inwagen's invocation of them.) But the critical point is this: I'm ‘realist’ (for whatever that's supposed to mean) about there being a true theory of propositions, and, as above, whatever that theory claims, metaphysically or otherwise, such is the truth. The question is whether the truth about propositions either precludes or even makes unlikely the truth of contradictory theology (or glut theory generally). Anderson suggests that the answer will ultimately be affirmative; however, as above, the suggestion remains too underdeveloped to be telling – at least as things stand at the time of writing.

Epistemological bills

Anderson correctly notes that I've compared his formidable (epistemic-mystery) theology with my target contradictory one, finding his to be unnecessarily burdened by a heavy epistemology. Anderson also correctly checks the playing field, noting that his account was driven by important epistemological challenges that my own account, as things stand, has yet to meet. Fair point. What's required is a careful, extended, full discussion of these issues; however, given space limitations, that's impossible to carry out in this symposium. What shouldn't happen but, given said constraints, is the only available option is a bullet-ish sketch of the direction of replies to Anderson's explicit epistemological challenge. (Readers would be sorely wrong to infer that the brevity reflects anything more than necessity; it doesn't reflect priority or a measure of importance.)

For efficiency I quote Anderson's well-stated challenge and, in turn, offer the briefest sketch of (the direction of) replies. According to Anderson, contradictory theology, as I've so far developed it, ultimately requires

the support of a plausible epistemic theory according to which all of the following hold: (1) Christians are warranted in believing the claims expressed in the relevant Christian creeds; (2) Christians are warranted in believing that those claims are contradictory; (3) Christians are warranted in believing that there can be true contradictions; and (4) any intuitions or ‘seemings’ they have regarding the impossibility of contradictions do not constitute defeaters for any of the beliefs in (1)–(3). Consider this, then, a friendly invitation to fill in the blanks. Whether that requires an ‘elaborate epistemological story’ remains to be seen.

By way of too-brief (sketch of the direction of) reply to each of the four given conditions:

1. 1. Yes, they're warranted in the sense of being epistemically justified. (General principle is the reverse of Descartes: you're epistemically justified in not rejecting what you accept unless you have some good reason to do so.)

2. 2. Yes, same. (Good reason to reject that christian theology is contradiction-free: both The Contradictory Christ and Divine Contradiction.)

3. 3. Yes, same. [Among others, good reason to reject the logical impossibility of gluts/gaps are the well-known phenomena that perennially push against ‘classical’ boundaries, together with long-standing methodological rules of thumb that guide the pursuit of true theories (see Divine Contradiction chapter 7), together with the fact that the ‘classical’ truth and falsity conditions – the ‘meanings’ – of logical vocabulary obviously don't preclude the logical possibility of gluts/gaps.]

4. 4. Yes, for sure. (See third item in current list, but also: be very, very aware of one's limited diet and the ‘hangry’ behaviour towards real nutrition outside such limits!)

To epistemologists (at least so-called traditional ones, though perhaps not so much to so-called formal ones), such replies might appear to be irresponsibly short or weak. (Compare with the pages and pages and pages of ink used by Alvin Plantinga on Anderson's challenge.) In the end, if the reader allows a dose of autobiography, it's difficult not to see Anderson's challenge through the lens of Descartes's challenge – which, in my limited historical view, is precisely the driving challenge in response to which much of Plantinga's work transpired. Descartes's challenge, as I see it, rested on the principle that you should (i.e. ‘should out of epistemic duty’) reject what you accept unless you have good reason to accept it. In a slogan: nothing you believe is justified unless you have (in some sense of ‘have’) good reason to accept it. (Implicit: reject everything that's not justified.) Of course, Descartes gave no good reason for accepting said principle. The flip side of Descartes is in the vein of Thomas Reid, the crude version of which is that you're justified in what you believe unless you have good reason to reject it. In an imperative (flip-side-Descartes) slogan: you shouldn't reject something you accept unless you have good reason to reject it.

My position, to the extent that it's a ‘position’ at all, is along the crude (which mightn't be in letter or, possibly, in the spirit of) Reidian lines: you're epistemically justified in what you believe, theologically or otherwise, unless you have (in some sense of ‘have’) good reason to reject it. All of this is messy, since much of this comes in degrees. Still, so long as you're doing your best both diligently and vigilantly to pursue the truth, worries about justification shouldn't hold you up. Just find the truth!

Anderson, of course, asks not after justification but ‘warrant’, which is sometimes used in a very technical fashion for the secret ingredient that converts true belief into knowledge. What do I say about that? On that question, Anderson is right that I am happy to remain silent. I am pessimistic about coming up with a precise and accurate analysis of warrant in the technical sense, and (as a mere autobiographical note) I find it more fruitful simply to pursue truth – true belief – and let the secret ingredient of warrant fall where it falls. I do know some stuff. Do I know what exactly makes it knowledge? No, but I'm also happy to rephrase when it really matters: I have true beliefs about some stuff.³ Anderson is correct that the true epistemology must be compatible with the true theology, and of course neither one of us doubts that such compatibility will necessarily transpire. But as for Anderson's explicit challenge, the direction of reply is above. If there's a genuine problem with the given direction of reply, I should very much want to know it. Until then (using up one last autobiographical line), I am very happy to have Anderson among the crew while we all jointly attempt to repair our raft.

Response to Karen Kilby

Karen Kilby's article is exemplary in its clarity, conciseness, and clear engagement with the work. Kilby raises two style objections (relative to the stated goals of the book), and raises one substantive objection about the overall account. I respond to each point in turn.

Style I: audience

In the preface to Divine Contradiction I state the hope of reaching not only theologically informed philosophers but also theologians, church leaders, and many non-academic readers. Kilby, explicitly writing from a theologian's perspective, objects that said hope is likely to be frustrated by failure. Kilby points to widespread evidence of (for lack of a better word) logico-cum-maths-English speak that is perfectly explained in the book to those with some familiarity with it (and certainly to those, like Kilby, with serious training in it) but not for those without requisite experience.

By way of reply: alas, fair enough. What Kilby highlights is truly regrettable but a lesson learned. While I didn't intend to write a so-called trade book, I did intend to write in a way that, with suitable interest, anyone in the listed groups could profitably read it. Kilby's remarks at least cast serious doubt on that. The lesson: either do better (and take cues from Kilby's own exemplary writing) or suitably harness my hopes.

Style II: God's ‘place’ in reality and theological audiences

In a line that brought both a burst of laughter and a sharp pang of regret (see Kilby's last line in the following), Kilby's second style comment runs thus:

In short, I don't think many of my non-analytic-theological colleagues will make it far in reading this book. Any who do, however, will encounter another kind of difficulty in the language, and this is in the way that God is rather casually described as a portion or fragment of reality (cf. e.g. 50, 74, 96). According to the Christian tradition, as I understand it, God is not a portion or fragment of anything else whatsoever: God is certainly not something rare which is located ‘in an otherwise consistently described created world’ (75). For me and those with a similar theological training, these sorts of phrases are the intellectual equivalent of fingernails on chalk board.

By way of reply, the uses of ‘portion’ or ‘fragment of reality’ are truly regrettable uses given the fingernail-screeching effects (see Kilby's flagged pages 50, 74, 96). (Moreover, such language also had misleading metaphysical connotations, as Anna Marmodoro's article in this symposium reflect.) Such words weren't meant in mereological senses in any substantive fashion, but instead just to indicate that (as the old metaphor goes) writing the full book of reality has some divine fragments or portions to the story and some non-divine.

More troubling is the deeply regrettable ambiguity – in fact, lamentable sloppiness – in the cited line ‘in an otherwise consistently described created world’ (cited by Kilby pointing to page 75 of Divine Contradiction). This is ambiguous between two salient readings:

• • that God is just part of the created world or the like (a fingernail-screeching sound indeed);

• • that God is ‘part’ (not in a mereological sense, but think full-story sense) of reality that's different in the target way (viz. contradiction) from many otherwise consistently described ‘portions’ (in the full-story sense!) of reality.

I truly regret such a sloppy slip, and – for the record – even I hear the awful screeching on the board.

Enough of Kilby's two style points – both very well taken. On to Kilby's substantive objection.

Substantive: denial, rejection, and heresy

Kilby's objection is put thus:

This proposal – to take so seriously what we learn of the Trinity that we rethink logic itself – is something I find initially very appealing. However, when I attend to what it means a little more specifically, I have to say that the appeal disappears. According to Beall, the doctrine of the Trinity means not only that the Father is God, but also that this is not the case: it is false that the Father is God.

Can this really be said by a Christian believer? Can I maintain, as part of Christian belief, that ‘it's false that the Father is God’: can I deny the Father's divinity?

By way of reply, one must be very careful about ‘denial’ in this context. ‘Denial’ is not tied to logical negation (falsity) except at most in restricted contexts (wherein one's denials are constrained by the behaviour of logical negation in classically closed theories). Denials are speech acts that reflect rejection, just as assertion is the speech act reflecting acceptance. By my lights, and the lights of all who recognize even the logical possibility of gaps or gluts, acceptance and rejection are mutually exclusive but not exhaustive in the sense that one can't both accept and reject a claim but one needn't either accept or reject a claim. Indeed, these are the states many truth-seeking theorists find themselves when reality confronts them with recalcitrant or strikingly different phenomena. As in The Contradictory Christ, perhaps the most strikingly different phenomenon is the GodMan, a god who is a human; a human who is a god; and this is one person, and so God (given monotheism). The force and immediate appearance of the contradiction – the strongly apparent contradiction at the heart of christian doctrine – is something on which many truth-seeking theorists pause, neither accepting nor rejecting; some eventually reject; some – Christians of the ‘orthodox’ or standard sort (my focus in target works) – accept even if they don't fully understand all implications (a common state).

Kilby's driving question is whether one can (perhaps ought?) assert the falsity of Father's divinity – assert that it's false that Father is divine. Of course, given conversational implicatures (i.e. not actual entailments or deductive implications of what's said but commonly inferred ‘implicatures’ of what's said), any Christian would be irresponsibly misleading to assert only the falsity of Father's divinity and not the critical truth of Father's divinity. This is one important respect in which Kilby's question needs to be understood – namely, that responsibility of affirmation remains even if, as I've argued, God is a contradictory being. As Divine Contradiction (and, likewise, its essential predecessor The Contradictory Christ) makes plain, the contradictions of God explain the appearance of contradiction; they don't undermine the truth or dictate different practices in the pulpit or pew.

Perhaps the most important response to Kilby's question is that, as christian theology in general – and trinitarian studies in particular – makes plain, questions of Father cannot be answered in isolation from those of Son or Spirit; indeed, questions of God – triune God who is Father, and who is Son, and who is Spirit – cannot be answered in isolation from questions of each person who is God. With this common caveat in mind, one returns to Kilby's question not as it stands but rather in concert with the fuller picture: the question is one of explaining how, despite apparent contradiction, it's true that Father is God, true that Son is God, true that Spirit is God but, in the same breath (so to speak), it's false that Father is Son, false that Son is Spirit, false that Spirit is Father, and yet the number of gods is 1. This is the so-called logical problem of trinitarian reality; it demands an explanation of how trinitarian axioms (doctrines) are true despite the glaring appearance of contradiction. I have argued that the truth of the given axioms involves contradiction: it's true that Son is God but, since it's false that Son is Father and it's true that Father is God, it's also false that Son is God. The truth of God, I've argued, is indeed ‘radically other’ as some have said (though I'm not hereby endorsing any particular account along those lines); it's certainly other than common, especially so since it essentially involves contradiction.

The falsity involved in trinitarian axioms in no way diminishes the truth. Too many thinkers (and perhaps readers) think of falsity as a ‘taking away’ feature, but this is so only in the restricted cases of classically closed theories.⁴ The ‘good news’ is that each of Father, Son, and Spirit is divine; if christian practice, from the pulpit or pew or anywhere in between or beyond, motivates only telling one half of the full story (viz. only that each is divine), then so be it. Sometimes, what's best to say is not the full truth.

After observing that, in her view of the history of doctrines, ‘it is almost always easier to identify what has been ruled out as a result of doctrinal controversy than [affirmed]’ (Kilby 2024), Kilby asks, more or less in closing: How then can one be persuaded by Beall's announcement that certain forms of statements are not really heresies even though they give that appearance? My response is that, though the relevant recorders of target doctrines didn't fully understand (all implications of) what they wrote, they still, under divine inspiration (one may hope), wrote what had to be recorded in order to get at core truths (even if, as is plainly the case, they appear to be contradictory). If, as may be likely, they – like as yet still many today – were thinking within a restricted space of possibilities, their ‘ruling out’ was an effect of not seeing the radical truth on the table (e.g. not so much one or the other exclusively, but in fact both). When it comes down to it, the situation is this: regardless of their understanding, fears, hopes, intentions, and so on, the doctrinal writers wrote what they wrote, and the result has struck one and all (and, I dare say, even such writers) as apparently contradictory or difficult to understand. Theologians and philosophers (and many lay people) have for a long time wrestled to understand how on earth (or in heaven) such doctrines can be true despite the glaring appearance of contradiction. For that same long time explanations have been offered, explanations that run the gamut from complicated metaphysics to sophisticated epistemology to new-fangled semantics to unmarked equivocation and so on.⁵ The explanation I offer is the most straightforward: the appearance of contradiction is explained by the reality of contradiction – that is, by a contradictory god. Pending a better explanation, for reasons given in the target work, the contradictory account stands, and it immediately (provably) yields distinctions among different sorts of heresy – ones outlined in target works.

What Kilby claims is that my explanation of the apparent contradiction of trinitarian axioms (each of Father, Son, and Spirit being identical to God while all being pairwise non-identical) is ‘ultimately unpersuasive’. I take Kilby's reservations very (very) seriously. The problem remains: that Kilby has neither pointed to substantive problems with the proposal nor, perhaps more importantly, advanced an explanation that purports to be better than the contradictory one. As things stand, the contradictory account, for reasons given in the target book, remains the frontrunner.

Response to Anna Marmodoro

Anna Marmodoro, in her typically clear and concise fashion, raises two substantive objections to Divine Contradiction, each revolving around (two different) relevant senses of ‘simplicity’, and each discussed below. Before turning to the objections Marmodoro raises an overriding question about the would-be ‘special status’ of gluts/gaps in reality and the relation of the ‘divine fragment’ of reality vis-à-vis the rest of reality. In what follows, I first address said question and then reply to Marmodoro's two objections.⁶

Gluts, gaps, divine reality, and the rest

Misled by my regrettable lack of clarity in Divine Contradiction, Marmodoro infers that, on my view, glutty and/or gappy fragments of reality (i.e. fragments the full truth of which involves gluts and/or gaps) involve a ‘special status’ that is ontologically significant. Marmodoro writes (and I quote at length):

A key metaphysical assumption Beall makes is that most of reality has no gaps or gluts: we can make claims about it that are either [just] true or [just] false. There are nevertheless what he calls ‘recalcitrant data’: parts of reality that are ‘gappy’, namely such that what we can say about them is neither true nor false; and even more surprisingly perhaps, parts of reality that are ‘glutty’, of which what we can say is both true and false. In other words, reality harbours some contradictions. In this book Beall considers ‘trinitarian reality . . . a rare fragment of reality’ (p. 84) which has glutty status [note a typo as Marmodoro cited page 74].

The emerging view is very intriguing for the reader, who would want to hear more about the overall metaphysical theory Beall endorses. He holds that reality as a whole has fragments of special status (gappy or glutty). If the Trinity is a rare but not unique fragment of this kind, what other suchlike fragments are there? And if there are others, do they all pertain to the realm of the divine, or are some of them natural phenomena? Either way, what does this entail about the Trinity? In other words, is the Trinity an ordinary or extra-ordinary fragment of reality? If it is an ordinary albeit ‘recalcitrant’ fragment of reality, does this diminish in any sense what's extra-ordinary about the divine? If on the other hand it is an extra-ordinary fragment, why take it to reveal what reality in general is like? My point is that it would be good to know more about how the author's understanding of the divine sits within his understanding of reality as a whole.

By way of reply, Marmodoro is right that, on my view, most of reality is free of gaps/gluts in the sense that the true theories of most phenomena are gap-free and glut-free – indeed, are ‘classically closed’ (i.e. closed under so-called classical logic). But on my view of reality, being glutty/gappy in the given sense is not ontologically significant; it's just what it is. For example, the number two is completely and truly describable without gluts or gaps; however, a sentence that says of itself only that it's true (or, similarly, another sentence that says of itself only that it's false), is at best incompletely and truly describable without gluts or gaps. The former entity (viz. number two) is of a familiar sort, truly and fully describable without gaps/gluts; the latter entity (viz. spandrels of truth) is an unfamiliar sort, truly and fully describable only with gaps/gluts. In the sense of ‘ordinary’ that's in effect equivalent to ‘familiar’, entities like the number two (or, if you don't like numbers, the tree in the quad) are ordinary while entities like spandrels of ‘true’ (e.g. ‘This sentence is true’ or ‘It's false that this sentence is true’, etc.) are extra-ordinary. In this sense, the theology advanced in Divine Contradiction puts ‘the divine fragment of reality’ (more on ‘fragment’ below) in the extra-ordinary camp; however – and this is critical for clarification – there is nothing ontologically significant that's implied (or intended to be implied) by this sort of extraordinariness.

By way of clarifying ‘fragment’ talk, I should clarify that the term is not intended in a metaphysically (let alone ontologically) significant way. (As above, I regret not making this clearer, as the terminology, in retrospect, obviously invites meaty readings by metaphysicians and theologians.) While I take metaphysics and ontology very seriously, the theology advanced in Divine Contradiction was intended to be as metaphysically neutral as possible (compatible with the theological doctrines or axioms that constrained and fuelled the project). Accordingly, my talk of ‘the divine fragment of reality’ was on par with ‘aspect’, ‘part’ (in a non-mereological sense), ‘bit’ (as in ‘that bit of reality’), and so on. The term is used in an ontologically light way. Of course, the theological constraints (viz. more or less standard christian theology as reflected up through the familiar creeds and doctrines up through Chalcedon 451) have long reflected an ontologically significant distinction in reality between created reality and uncreated reality. Such a distinction, while not discussed in any significant way in Divine Contradiction, is assumed as part of the theology; however, the distinction in itself is neutral with respect to gluts and/or gaps, at least on my view.

Alas, my regrettable lack of clarity (or, perhaps better, regrettable choice of terminology), clarified above, may have partly fuelled Marmodoro's two principal objections; however, the objections are nonetheless important to address.

Simplicity: the ontological cost of explanatory parsimony

A virtue of the contradictory account (of both the incarnation and trinitarian reality) is its simplicity – or so I argue in Divine Contradiction (and its essential predecessor The Contradictory Christ). Marmodoro correctly clarifies that at least one aspect of the given simplicity is explanatory parsimony. In particular, the strong and perennial appearance of trinitarian contradiction – pretty much right on the surface of the core trinitarian axioms – is very simply explained if such appearance is veridical, that is, if such apparent contradictions are true (i.e. they're gluts). Marmodoro agrees that explanatory parsimony is a feature of the advanced contradictory account; however, she alleges that the explanatory parsimony is purchased at the price of ontological complexity. Marmodoro: Here Beall's stance [is] that accepting contradictions is explanatorily parsimonious; but even if so, it isn't ontologically parsimonious, because it achieves its target explanation by positing that bits of reality have a special status. So the methodological simplicity comes at a cost. As above (see section ‘Gluts, gaps, divine reality and the rest’), the alleged special status in reality is a false allegation (based, regrettably, on my own poorly chosen terminology). The only special ontological status attributed to the divine versus the rest is whatever such status is attributed by the theological constraints assumed from the start – and so constraints shared by all accounts of the target ‘logical’ problem of trinitarian reality.

Divine simplicity against gluts

In her most trenchant objection Marmodoro cuts to the very heart of the account (which is in chapter 3 of Divine Contradiction). Indeed, Marmodoro, alone among symposiasts, explicitly challenges one of the main contributions (viz. chapter 3) which concerns trinitarian identity and the fundamental trinitarian-identity entailment patterns. Marmodoro's challenge is this:

The reader would benefit from further explication by the author here. On a classical conception of identity, God, being identical to Himself, is Simple [my emphasis and my upper case to note a technical term]; however, God for Beall is [identical] to Himself [via contradiction]: each Person is identical to God, but not (pairwise) identical to the other Persons. The challenge is to understand how trinitarian identity, so explained, impacts on the simplicity of God.

Section 2 of chapter 5 of Divine Contradiction is devoted to the topic of divine simplicity, a feature traditionally attributed to the god of traditional monotheism. Alas, as I explicitly note in said section, there is no one notion of divine simplicity but rather many, and theologians (and theologically inclined philosophers) continue to debate the right account of divine simplicity. On the matter, Divine Contradiction makes three explicit claims:

• • Contradictory theology, as advanced in the target work, is committed only to a negative characterization of divine simplicity.

• • The negative characterization to which the target theology is committed: there are no mereological proper parts in the triune god.

• • There is a positive characterization that's promising, new, and worthy of exploration; it falls out of the definition of trinitarian identity that sits at the heart of the contradictory account of trinitarian reality. The account: all properties of God (either directly or via ‘divine-simplified predicates’) entail identity to God. This is discussed in some detail in said part of Divine Contradiction.

Marmodoro's challenge purports to cut through the foregoing trio of claims to the challenge above: namely, that contradictory (equivalently, glutty) identity claims are incompatible with Simplicity.

I note that Marmodoro does not object to the negative characterization of divine simplicity explicitly given in Divine Contradiction. Moreover, Marmodoro does not object to the new positive characterization of divine simplicity in terms of trinitarian identity, at least not directly. Marmodoro, as above, objects to any account of divine simplicity that requires false identity claims, and a fortiori objects to any account of divine simplicity that requires contradiction. In particular, on Marmodoro's view (as I understand her challenge), divine simplicity is perfectly compatible with the self-identity but precludes the falsity of self-identity – in which case, the target contradictory account should be rejected, as trinitarian identity, involved in divine simplicity, essentially involves contradiction.

My reply to the challenge is that the falsity of a glut does not in any fashion take away from the truth. That's just what gluts are: they're truths that are also false – they're true and their respective logical negations (logic's falsity connective) are also true. As Divine Contradiction (likewise its essential predecessor) emphasizes: any true and classically closed theory (of which there are many) collapses the truth of logical negation with absence from the given theory. In other words, if T is a classically closed (and non-trivial) theory,⁷ and A is a sentence in the language of T (with ¬A its logical negation, which is just it is false that . . .), then ¬A is in the theory T (i.e. true according to T) if and only if A is absent from T. But, of course, not all true theories are classically closed (at least as I've argued); some contain gluts (and some gaps), in which case ¬A's being in the theory implies only that A is false according to the theory, even if A is also true (and, so, both ¬A and A are in the theory) – and this is the case, Divine Contradiction argues, for trinitarian theology. Accordingly, Marmodoro's challenge is to explain how divine simplicity can allow, for example, the falsity of the following central trinitarian axioms:

• • Father is (identical to) God.

• • Son is (identical to) God.

• • Spirit is (identical to) God.

The answer is that all such claims are true – not true in some special or diluted or blinking or degree-theoretic fashion, but true simpliciter. That each of the given claims is also false in no way takes away from or diminishes the truth of them (as emphasized in Divine Contradiction). Indeed, it is precisely via contradiction that the truth of the triune god is expressed. If that's correct, the explanation for how relevant self-identity claims can be false of a divinely simple being is ultimately the simple one: namely, that the being is contradictory.

Response to Thomas McCall

After providing a valuable backdrop to the target so-called logical problem, together with a big-picture account of the target contradictory response, Thomas McCall focuses on a very fundamental concern. The concern:

• • core trinitarian and other theological axioms find their formal motivation from arguments that appear to be invalid according to contradictory theology (as advanced in Divine Contradiction);

• • in the absence of clear and valid arguments, target axioms (doctrines) stand on loose footing at best.

Note that matters of ‘epistemic justification’ needn't be the prime issue here. (I don't think that that's the target issue.) McCall explicitly wonders – and quite reasonably so – whether the target doctrines would've been accepted as axioms (doctrines) by the Church were the invalidity of target arguments known. In the end, the issue on which McCall's concern rests is whether the target arguments enjoy valid forms even if their standardly represented forms are invalid.

Examples of historical arguments

McCall points to various arguments around divine simplicity (‘Simplicity’ in some sense that true theology must spell out) from the history of doctrine. Either of the examples suffices for present purposes but here is Gregory of Nyssa's (represented as follows by McCall, herein keeping McCall's enumeration):

(4) If the doctrine of simplicity is true, then there can be no gradations of divinity or divisions within the divine nature.

(5) The doctrine of simplicity is true.

(6) Therefore, there can be no gradations of divinity or divisions within the divine nature.

McCall acknowledges that, on the contradictory account, the argument could be valid via ‘shrieking’ if that were an option. [For present purposes, I leave the details of shrieking to Divine Contradiction and sources cited therein, except to note that McCall overlooks the fact that while, as he notes, a theory is shrieked if and only if all of its predicates are shrieked – something that, as McCall notes, is not the situation with true theology on my account – a theory can still shriek some predicates and not others, thereby being ‘partially shrieked’ for a lack of better terminology. Indeed, given my account of ‘divine simplicity’ in §5.2 of Divine Contradiction, which I believe is novel and worth further exploration, being divinely simple is a matter of entailing identity to God. Even though the latter predicate is not shrieked – on pain of otherwise trivializing the true theology – the predicate ‘entails identity to God’ is shrieked (and shrugged), since the entailment relation itself is glut-free (and gap-free). But I set all of this aside for present purposes.] If shrieking (in a technical sense) doesn't afford validity to McCall's sample argument, what, if anything, does?

A different representation of target arguments

For present purposes, assume that the modus-ponens form of Gregory of Nyssa's argument is indeed invalid according to the true theology. (As above, it is not clear that this is so, as it depends on whether ‘Simple’ is a predicate shrieked in the theology, or indeed whether ‘if’ is logic's material conditional versus some other ‘detachable’ conditional, etc. But for present purposes, just assume that the form presented by McCall is in fact invalid.)

My primary reply to McCall's concern is to note that the proffered form of argument is not the only candidate in the vicinity. Indeed, since the validity of any theologically important argument is going to involve a language that goes well beyond the sparse bag of topic-neutral logical vocabulary (which, as McCall rightly notes, is devoid of predicates), relying on extra-logical entailments from other common vocabulary provides an immediate candidate. One example, in which an it is impossible that connective is in the (object) language, is natural (one may add ‘Divided’ too below):

4*. It's relevantly impossible for divine nature to be both Simple and Gradated (uppercase for technical terms in the theology).

5*. Divine nature is Simple. (Ergo, it's relevantly possible – because actual – that divine nature is Simple.)

6*. Therefore, it's false that divine nature is Gradated.

Similar x-valid arguments are available for other relevant examples of logically invalid argument patterns, where x-validity is whatever extra-logical relation is in play in the target theory.⁸ What this relies upon is a basic entailment from it is impossible that such-n-so to the falsity (or, perhaps, depending on the semantics, gappiness) of such-n-so, which, technically, relies on ‘shrieking’ the impossibility connective.⁹

Perhaps an even likelier candidate ‘form’ of argument, and intimately tied to that above, cuts directly to a primitive predicate entailment pattern between ‘Simple’ and ‘Gradated’ (and/or ‘Divided’ etc.). In this case, there's no (object-language) ‘impossibility’ connective doing the work; instead, the predicate entailments are achieved directly via the relevant extra-logical entailment relation (which is defined in the so-called metalanguage of the target theology):

• Simplicity Entailment: the truth of Simplicity x-entails the falsity (or untruth) of Gradation and Division, where (again) x-entails is whatever extra-logical entailment relation governs the target theology.

More directly but blunter, the idea is a direct ‘shrieking’ enforced by the extra-logical entailment relation (all defined in the so-called metalanguage):

• Where y is any object: that y is Simple and Gradated (Divided, etc.) x-entails B, for any sentence B in the language of the theory (theology).

Regardless of the preferred form, there are close cousins of the sample arguments that McCall discusses.

I conclude that McCall's principal concern, while a clear and pressing one, has a fairly straightforward answer. In particular, details aside, the concern arises from the fear that contradictory theology invalidates all forms of historically doctrine-motivating arguments. As above, that's just not true; it should be rejected. There are nearby candidate ‘forms’ that are valid (further details notwithstanding).

McCall's closing question

McCall's article closes with an important question:

[S]hould theological developments that have occurred ‘downstream’ of orthodox Trinitarianism be vacated if those developments are based upon arguments made via classical logic (or, alternatively, via theory-specific shrieked accounts)?

This question is a fundamental one that goes well beyond theology. A full answer demands a different, less constrained venue. I close with a short gesture towards the direction of an answer.

The question is ultimately whether a doctrine or axiom should be rejected if the only known reason for accepting it is an invalid argument (or family of invalid arguments). Again, the important issue here isn't one of rational justification or the like; it's a question of whether the target doctrine is true. As with McCall's sample argument from the history of doctrine, it's likely that there are very nearby valid arguments in the vicinity. If there are, and those arguments are sound (i.e. not only relevantly valid but all premises true), the question at hand may be put aside: all is well. On the other hand, if there are no valid (sound) arguments for the target doctrine or axiom, and there's also no other sort of good (truth-directed) evidence that delivers the target claim, the question at hand might linger. Exactly what one should (in some sort of epistemic-ought or truth-seeking-ought sense of ‘should’) do is the same difficult question that every truth-seeking theorist faces every time she returns to the search. If the answer had clear-cut rules, truth-seeking would be easier. Alas, there are no such clear-cut rules that apply across the board. The only thing to do is to keep on keeping on in one's search, doing the best one can. (Again, see chapter 7 of Divine Contradiction for common methodological rules of thumb in the search for true theories.)

Response to Peter van Inwagen

Peter van Inwagen does not directly engage the target theology; instead, he targets the very idea of gluts (and, dually, gaps, though he doesn't explicitly discuss them). Van Inwagen states the scope and limits of his principal aim clearly:

[G]lut theory [and, so, a glut-theoretic or contradictory theology] can be right only if the idea of a proposition that is both true and false makes sense.

It seems to me . . . that this idea doesn't make sense. I have to admit, however, that that's not an argument – strongly tempted to think that it is though I am. I have to admit that I have no argument. All I have are, well, [three related] considerations. I'll do my best to attempt to articulate them, to try to frame an explicit statement of what lies behind my conviction that the idea of something that is true being also false makes no sense.

One way to do this is to present a plausible, intuitive semantics that confers validity on the same inference-forms as the Received Semantics (the logic-text truth-tabular semantics), but doesn't proceed by assigning single, ‘exclusive’ truth-values to sentence letters – and therefore, at least, doesn't begin [emphasis original] by contradicting the glut theory. I'll present such a semantics, which I'll call the Received Semantics Variant . . .

Worth noting, before directly responding to van Inwagen's ‘variant semantics’, is that there are many, many, many different ways to define validity relations on any given language. Indeed, as so-called ‘adequacy results’ (e.g. so-called soundness and completeness results, etc.) for standard languages show, one validity relation on a given language can be defined in all sorts of different ways – some without any mention of ‘truth values’ or ‘semantic values’ (or ‘models’ or etc.), some with very different sets of semantic values, and so on. The fact that there are very different ways of defining a validity relation on a language is not in question, at least for the languages of relevance to current discussion. The issue (on logic, not theology), as discussed in Divine Contradiction and especially its predecessor (viz. The Contradictory Christ), is that the standard truth and falsity conditions (i.e. the standard ‘semantics’) for logical vocabulary – the truth of which, as emphasized in said works, is in fact not in dispute – preclude neither the logical possibility of gluts nor (dually) that of gaps. Excluding those logical possibilities requires argument. In the absence of argument, van Inwagen offers his trio of considerations.

Consideration I: received semantics variant

Van Inwagen's first consideration against gluts/gaps is what he calls a ‘variant semantics’ for the classical-logic validity (entailment etc.) relation. As emphasized in both Divine Contradiction (pp. 26f.) and its essential predecessor The Contradictory Christ (pp. 21f.), the classical-logic entailment relation is achieved by restriction: its space of would-be counterexamples is restricted to a proper subspace of the full ‘subclassical’ space of logical possibilities; the classical-logic relation restricts to a space that omits gappy and glutty possibilities. Both accounts (viz. classical and subclassical) are in full agreement on the truth and falsity conditions for logical vocabulary; the former simply restricts itself to a smaller space of relevant possibilities. Critical to observe is that nothing by way of said restrictions changes in van Inwagen's target ‘variant’ semantics: his target semantics is just a restriction of the more general account.¹⁰ The general semantic framework, of which van Inwagen's ‘variant semantics’ is a restriction, has three principal ingredients:¹¹

• • a non-empty universe or domain S of ‘points’ (objects of some sort);

• • subsets of S;

• • assignments (total functions) from sentences of the relevant language to ordered pairs ⟨s ⁺, s ⁻⟩ of subsets of S (where, note well, the superscripts are unnecessary except as a suggestive way to distinguish roles of the subsets in the semantics, as illustrated below).

Being a compositional semantics, the ‘semantic values’ of the whole are determined/defined via the semantic values of the component parts, ultimately down to the values of the atomic sentences. All sentences – atomic and compound – are assigned an ordered pair of subsets. All relevant assignments (‘models’ in van Inwagen's terminology) conform to – or, if you prefer, are defined by – the following conditions for each type of sentence:

• • Atomic sentences (atoms): every atom A is assigned a pair ⟨A ⁺, A ⁻⟩.

• • Molecular/Compound: where A and B are any sentences:

  • – Nullation (†): †A is assigned to ⟨A ⁺, A ⁻⟩ iff A is assigned to ⟨A ⁺, A ⁻⟩.

  • – Negation (¬): ¬A is assigned to ⟨A ⁻, A ⁺⟩ iff A is assigned to ⟨A ⁺, A ⁻⟩.

  • – Conjunction (∧): A ∧ B is assigned to ⟨C ⁺, C ⁻⟩, where C ⁺ = A ⁺ ∩ B ⁺ and C ⁻ = A ⁻∪ B ⁻.

  • – Disjunction (∨): A ∨ B is assigned to ⟨D ⁺, D ⁻⟩, where D ⁺ = A ⁺ ∪ B ⁺ and C ⁻ = A ⁻∩ B ⁻.

The target validity (entailment, consequence) relation is defined as absence of counterexample:

• • A counterexample (or ‘countermodel’ in van Inwagen's terminology) to a set-sentence ‘form’ or ‘pattern’ {A ₁,. . .,A_n} ∴ B is any such assignment wherein the intersection of all the given A ⁺_i is not a subset of B ⁺.¹²

• • Validity (entailment, consequence): the pattern {A ₁,. . .,A_n} ∴ B is valid (or the given set entails B, or B is a consequence of the set) iff there's no counterexample to the given pattern.

That the semantics above is the relevant general case of van Inwagen's restricted case is clear. To get van Inwagen's restriction (viz. classical propositional validity), simply define c-restricted assignments to be any such assignment (in the general case above) such that A ⁻ in ⟨A ⁺, A ⁻⟩ is the classical complement of A ⁺ (relative to the given universe S), namely, every element in the universe S that isn't in A ⁺. Using standard relative-complement notation wherein, for any sets X and Y, the set X \ Y is the complement of Y relative to X (i.e. everything remaining in X once you've removed any Y members, if any, from X), the definition of van Inwagen's c-restricted assignments is exactly as above, but with the requirement that every (c-restricted) assignment is such that, for all sentences A and universe (‘domain’) of points S,

⟨A ⁺, A ⁻⟩ = ⟨A ⁺, S \ A ⁺⟩.

And that's it. Everything else is the same. Using c-restricted assignments and only c-restricted assignments delivers the classical propositional validity relation; using all relevant assignments (including c-restricted ones but not only them) delivers what I take to be logical validity (viz. FDE, as presented in chapter 2 of both Divine Contradiction and The Contradictory Christ).

Also important to note is that the same ‘intuitive idea’ to which van Inwagen appeals in discussing his target ‘variant semantics’ remains intact, and remains equally ‘simple, and, to [van Inwagen], compelling’ (van Inwagen 2024). Propositions in general are ‘sets of worlds’ but, in particular, pairs of sets of worlds: the proposition assigned to (or, if you want, ‘expressed by’) sentence A is the pair consisting of, first, the set of worlds at which A is true and, second, the set of worlds at which A is false. Moreover, as van Inwagen rehearses, a proposition ⟨A ⁺, A ⁻⟩ is said to be true just if the actual world is in A ⁺; the proposition is said to be false if the actual world is in A ⁻.¹³ Important to note is that this is completely standard and uncontroversial even if often only implicit: namely, propositions, in the general case, are pairs, the first coordinate tied to the proposition's would-be truth, the second to its would-be falsity.

The reason that the falsity component of propositions – the second coordinate of propositions, so to speak – is often ignored when defining classical-logic validity is that, as above, logical negation, in the restricted (classical) case, is assigned to classical (relative) complement. But that's still the coordinate filling the ‘falsity’ role. Van Inwagen claims that only the glut/gap theorist's definition of validity essentially involves a notion of falsity (or, in model-theoretic terms, falsity-in-a-model etc.), and that the classical-logic definition doesn't. But that claim is incorrect; both accounts implicitly involve a notion of falsity; indeed, both are such that the logical negation (i.e. logic's unary falsity connective!) of a sentence A, namely, ¬A, is true on an assignment (model) iff the actual world (of the model) is in the assignment's ‘negative’ or ‘complement’ coordinate A ⁻. In van Inwagen's restricted case, A ⁻, on every relevant (c-) assignment, is just the classical complement S \ A ⁺, that is, the classical complement of the ‘positive’ or ‘nullation’ coordinate of A's ‘proposition’. What van Inwagen overlooks is that restricting logical negation to classical (relative) complement doesn't get rid of falsity; it simply assigns it to classical (relative) complement. In short, assigning the logical negation of A to the classical complement of A (relative to universe S) is nothing more nor less than assigning said negation to the set of all points in S at which A is false. That's just the classical-logic restriction. It's perfectly good as far as it goes. It's just that, per Divine Contradiction (and its predecessor), it doesn't go the whole way to logical validity itself, as it precludes various logical possibilities.

The first consideration, I conclude, is not a consideration against gluts or gaps, and a fortiori not a consideration against the target contradictory theology.

Consideration II: Martians [and classically closed theories]

Given (severe) space restrictions, my reply to van Inwagen's second consideration against gluts/gaps is very brief, consisting of two remarks.

The first remark is that the Martian thought experiment is premised, in his target discussion, on van Inwagen's claim that, in his restricted semantics, ‘falsity is unnecessary’ (or, strictly, falsity-in-a-model is unnecessary). That claim has been refuted above (see above in ‘Consideration I’).

The second though critically important remark is that, as emphasized in Divine Contradiction (chapter 2 section 7), true and classically closed theories (i.e. closed under the classical-logic validity relation) are not only common; they're nearly ubiquitous. Indeed, many – many, many, many – true theories are classically closed. If, either by choice or by cognitive limitation, one focuses entirely on and only on true and classically closed theories one will probably find oneself in the shoes (?) of van Inwagen's Martians.

In classically closed theories, the falsity of a sentence (and so the truth of the sentence's logical negation) is equivalent to the classical (relative) complement of the sentence. In a slogan: a logical negation's membership in a true classically closed theory is equivalent to its negatum's absence from the theory – that is, ¬A's being in a true and classically closed theory is equivalent to A's being absent from the theory. (Presumably, this is what van Inwagen's ‘falsity is simply untruth’ is supposed to mean.) And if true and classically closed theories exhaust what you choose to (or, for whatever reason, all you can) understand, you'll be puzzled indeed by true theories in which such an equivalence is absent. In my view, as emphasized in Divine Contradiction (and its predecessor), I think that very, very, very few true theories are glutty or gappy; but few – even very, very, very few – is still not none. That Martians, by choice or some other limitation, are forever stuck in van Inwagen's c-restricted region of reality is what it is – but it certainly isn't a consideration against either gluts/gaps.

The second consideration, I conclude, is not a consideration against gluts or gaps, and a fortiori not a consideration against the target contradictory theology.

Consideration III: pointless semantics

Space restrictions, at this point, are unbearably fierce. Fortunately, van Inwagen's third consideration, I believe, is the lightest, even by his own lights. In any event, the third consideration is that there are often many different ways to define a given validity (or entailment or consequence) relation, some having nothing ‘in them’ involving truth or falsity or any other semantic-sounding notion. Indeed, even setting aside purely syntactic (i.e. meaningless-symbols-involving) definitions of common validity relations, there are ways of precisely defining a validity relation that, to use van Inwagen's terminology, seem pointless. Whether they are pointless depends on whether they're useful ways to define a relation. I suspect that pointless-appearing is often a subject-relative matter, but the fact that some precise definitions of a validity relation appear to be pointless is (or ought to be) uncontroversial – certainly not controverted by me.

The stated point of van Inwagen's ‘pointless semantics’ is to show that just because a validity (entailment, consequence) relation is precisely defined (whether by pointless-appearing definition or otherwise) doesn't thereby show that that validity relation is logical validity. In the case at van Inwagen's hand, the fact that a validity relation that ‘allows for’ gluts/gaps is precisely defined (by a pointless-appearing definition or otherwise) doesn't show that that relation is in fact logical validity – or, of course, that there are logical possibilities of gluts or gaps. Again, I agree without reservation. In general, there are many validity relations, even on a single language, that are precisely defined but are not logical validity. One extreme case:¹⁴ define the null relation over any language via absence of counterexample: a counterexample to any form is a point (possibility, model, whatever) in which each sentence in the form has some semantic status or other, where, note well, having a semantic status is merely being in the grammar of a language. This precisely defines the null ‘validity’ relation, but that doesn't make the relation logical validity – or, indeed, useful in any fashion.

The third consideration, I conclude, is not a consideration against gluts or gaps, and a fortiori not a consideration against the target contradictory theology.