Conceivability and perfect being theology

Abstract According to Jeff Speaks, one sort of perfect being theology takes its rise from a description of God as the greatest conceivable being. Speaks offers two problems for this project. One is finding a suitable sense of ‘conceivable’. The other concerns what he calls ‘troublemakers’, attributes that seem to show that God cannot be the greatest conceivable being. Speaks brings these together in a dilemma.

Philosophers have two large ambitions when it comes to God.They would like to give good arguments for or against His existence.They would also like to work out what attributes a God would have, ideally without drawing from any claimed revelation.One of philosophy's chief ways to pursue the second ambition has been perfect being theology (hereafter PBT).This begins from a claim that God would be perfect.It then tries to work out what a perfect being would be like, in hope of learning what God would be like.
The idea of being perfect can be developed in two ways.For Leibniz, to be perfect is to be the greatest possible.Anselm calls a perfect being 'that than which no greater can be thought'.Jeff Speaks glosses this as being the greatest conceivable being (GCB) (Speaks (2018), 51). 1 Each way to understand being perfect, Speaks thinks, begets its own kind of perfect being theology.Speaks offers two problems for GCB theology.One is finding a suitable sense of 'conceivable'.The other concerns what he calls 'troublemakers', attributes that seem to show that God cannot be the GCB.Speaks brings these together in a dilemma.
To set up the first horn, I note that as Speaks anatomizes them, both GCB theology and greatest possible being (GPB) theology derive divine attributes from two basic premises.One is a 'modal principle', that God is actually greater than any other being in any possible (conceivable) world (ibid., 16). 2 The other is a 'greatness condition' (ibid., 17).Speaks considers many candidates for this role.But every greatness condition for GCB theology has a perfect parallel in GPB theology, the two differing only in whether they quantify over possible or over conceivable worlds, particulars, attributes, and states of affairs. 3o suppose that these items are conceivable if and only if they are possible.Then to quantify over conceivable items is just to quantify over possible items.But only the quantifier domains were supposed to distinguish the two projects.So GCB theology turns out to be just a 'notational variant' on GPB theology (ibid., 59, 69).The two differ only verbally.
To avoid thisto be a genuinely different project -GCB theology needs a sense of 'conceivable' in which it is not the case that items are conceivable if and only if they are possible.Suppose that it has one.Then in this sense of 'conceivable', some states of affairs are conceivable but not possiblefor if any world, particular, or attribute is, so is some state of affairs.If some states of affairs are conceivable but not possible, Speaks argues, some of these are troublemakers, for reasons that emerge below.Eliminating the troublemakers would shrink the space between the conceivable and the possible.Speaks thinks this might well shrink it to zero.If it all disappears, then items are conceivable if and only if they are possible.So GCB theology collapses into GPB theology (ibid., 71-72).Thus the dilemma: either troublemakers, and so God is not the GCB, or GCB theology is just GPB theology in disguise. 4 now argue that neither horn of Speaks's dilemma gores.I set out a sense of 'conceivable' that works for GCB theology and keeps it distinct from GPB theology.I then argue that the conceivable (in this sense) can outstrip the possible, but not generate troublemakers.

Conceiving
I first develop an account of conceivability for GCB theology.Being conceivable, I take it, is being possibly conceived.Since GCB theology is a project for current humans, that's being possibly conceived by the likes of us, with our current cognitive equipment.We are obviously non-ideal conceivers.So the greatest we can conceivethe GCBmight be less great than the greatest there can be.But it might also be greater, that is, impossibly great.We might well conceive and ascribe to it perfections which are not genuinely possible.We might, for example, find a coherent definition of omnipotence, GCB theology might ascribe it to Godand yet the realm of the possible might include nothing omnipotent.
As being conceivable is being possibly conceived, I start from the relevant sort of conceiving.One sort is sensory imagination.That can't apply in God's case.The sort that can apply is intellectualfilling out and understanding descriptions of individuals, attributes, or states of affairs.For present purposes, a description of an individual is a bit of language expressing how it is, and a description of an attribute or state of affairs is a bit of language expressing its content, as 'Lassie runs' expresses the content of the state of affairs Lassie's running.Understanding a description of an item conceives it intellectually to some degree.
Consider a state of affairs, that Lassie runs through the woods.'Lassie runs through the woods' perfectly expresses its content.Such sentences as 'a dog runs', 'a collie is in the woods', and 'Lassie exists' partly, inadequately, or abstractly express its content.They give descriptions we can improve on.Let us think in terms of improvable descriptions.Imagine trying to express the content of a state of affairs, and gradually building up better and better descriptions of that content.We more fully conceive the state of affairs the more of its content we describe, the more adequate and accurate the description, and the better we understand the description.Improvable intellectual conceiving comes in stages.Adding detail to a state of affairs' description, or improving its accuracy, or coming to understand part of it better, constitute new stages in conceiving it.Improvable intellectual conceiving also has degrees.As we fill out, improve, or better understand a description, we conceive the state of affairs more fully, and so to a higher degree.This is so even if the state of affairs is infinitely detailed.If it is, any finite progress in filling out its content is in a certain sense infinitesimal.But infinitesimal progress is still progress.
Fully or perfectly conceiving a state of affairs would be fully understanding a description that captured all of it with complete accuracy and ideal adequacy.There are two ways to think of fully conceiving Fido's being a dog.In one, all it takes is fully understanding 'Fido is a dog'.In the other, it includes fully understanding a way of realizing Fido's being a doga particular concrete, detailed way he might be a dog.That is, it includes giving him a particular breed, size, colour, etc.I call this realization-conceivability.

I now define a sort of conceivability:
Person A conceives 1 state of affairs S (attribute F) = df.A to some degree understands S's (F's) content.Person A conceives 1 individual B = df.A to some degree understands some attribute B has, and B's having it.S, F, or B is conceivable 1 for A = df.possibly A as A actually is, in A's actual circumstances, conceives 1 S, F, or B.
What we do not understand, we do not 'have in mind' in the intellectual sense, and so have not conceived 1 , even if we are perceptually confronted with it.Referring to an individual does not suffice for conceiving 1 it: I can refer to Euclid's favourite theorem -I have just done sobut I have no conception of what it was.I only begin to have a conception of it when I begin to understand it.'Item' ranges over individuals, attributes, and states of affairs.In line with the definitions, an item is the more conceivable 1 the more adequately, accurately, or completely we can understand it or its content.These definitions do not imply that S could be actualized, F could be exemplified, or A could exist.They do imply that there are items there to understand.Where there is none, there is nothing to conceive 1 , and so nothing to be conceivable 1 , and so nothing is conceivable 1 . 5Here is a second sort of conceivability: Person A conceives 2 item I = df.A conceives 1 I to some degree, and A's a priori evidence favours the claim that possibly I is actualized, is exemplified, or exists.I is conceivable 2 for A = df.possibly A as A actually is, is in A's actual circumstances, with A's actual a priori evidence, and conceives 2 I.
Where I competes with other candidates for possibility, the evidence favours I's being possible just if it does not more favour another candidate: if competitors are tied in first place, I mean to call both rather than neither conceivable 2 .
Both conceivability 1 and conceivability 2 could in principle power versions of GCB theology.But conceivability 2 is a better notion to use, because as we see below, it helps with the problem of troublemakers.Speaks suggests that for GCB theology not to collapse into GPB theology, its sense of 'conceivable' must not imply being absolutely possible.Conceivability 2 does not.That the a priori modal evidence favours that possibly P does not imply that possibly P.Even being favoured at the point of full conceiving 1 does not imply this.So even full conceivability 2 does not imply possibility.So GCB theology employing conceivability 2 -GC 2 B theologydoes not meet the only sufficient condition Speaks gives for being a 'notational variant' of GPB theology.Later I consider another candidate sufficient condition.
The definition of conceivability 2 speaks of a priori evidence for possibility.This comes from many sources.I now describe one that is particularly relevant here.

Evidence for possibility
Eliminating sources of impossibilityimpossibility-makersprovides one sort of evidence for possibility.We understand a grammatical sentence 'P'.We see that 'P' doesn't state a contradiction or contra-mathematical, or contradict a conceptual truth or something we take to express the content of a nature. 6We consider salient entailments and don't see any that make it impossible that P. Doing all this is eliminating sources of impossibility for the claim that P. Eliminating them is reason for greater confidence that nothing does make it impossible that P. It is evidence that nothing makes this impossible.Here's why.
Suppose that the Dictator has thrown me into the arena.I face three doors.He commands, 'pick a door or die'.I must choose one, or he'll shoot me.He might have put tigers behind any.He likes to do that.But I do not know whether there are any tigers behind any doors.I know the Dictator.I know that there are eight possible distributions of tigers over doors, and only one has no tigers in it.I know nothing that would favour any one distribution.I open 1 and 2: no tigers.This does not change the chance that door 3 hides a tiger.But I have eliminated all but two candidate tiger-distributions, and one is 'no tigers'.So I should be more confident in a no-tiger distribution than I was at the start. 7This would be true even if the Dictator had told me that there might be more doors round the bend, though the confidence boost in that case should be less.There being no tigers behind 1 and 2 is evidence that there were no tigers in the distribution, even though it is not evidence that there is no tiger behind 3, or further on.So too, eliminating door 1 and door 2 as sources of impossibility does not change the chance that an impossibility-maker lurks behind door 3, but it is evidence that there are no impossibility-makers in the distribution.
If there are no impossibility-makers in the distribution, it's possible that P.So we at this point have some evidence that possibly P.There is of course a difference between not seeing it to be impossible that P and seeing that it is not impossible that P. My point is that not seeing it to be impossible that P sometimes constitutes evidence that it is not impossible that P.Not seeing an elephant in the room is evidence that there's no elephant in the room, if the lights are on, my eyes are open, etc.
On the other hand, not seeing a neutrino in the room is no evidence that no neutrinos are there.I am suggesting that not seeing impossibility-makers is at least sometimes more like not seeing an elephant when the lights are on etc. than like not seeing a neutrino.Whether not seeing does provide evidence, and how strong I should think the evidence, depend on how confident I should be in a given case that SEE.If there were an impossibility-maker, I would see it. 8is varies by subject-matter.I myself should be confident of (SEE) for simple sentences like 'the cat is on the mat', but not for complex higher-mathematical sentences.

With confidence in (SEE)
Any item expressed in a sentence or predicate that isn't jabberwocky-style nonsense is at least initially conceivable 1 .Hereafter, take 'or predicate' as read; to smooth exposition, I will mention only sentences.If and only if the subject-matter should yield enough confidence in (SEE), what is expressed is also initially conceivable 2 .It takes some confidence in (SEE) even to think that if a sentence expressed a contradiction, we would see it: many of us should not have that confidence when confronting many mathematical sentences.If we should not have that confidence in a given case, then our not seeing an impossibility-maker is not genuinely evidence for possibility, even if we think it is.If we do not have enough confidence in (SEE) in a given case, we do not hold that seeing no impossibility-makers is evidence that there are none.If we should have had confidence in (SEE) in this case, we possess evidence (our not seeing) that we do not believe is evidence.If we should have that confidence for P, then if 'P' makes initial sense to us and does not include a facially obvious contradiction, that is initial weak evidence that possibly Pone impossibility-maker down.Further, confronting simple sentences that seem within our competence, our reaction to any that don't immediately strike us as saying something impossible is an initial 'well, that could be.Why not?' We do this because of how it seems to us intellectually.This seeming is a sort of intuition, and is also initial evidence that possibly P. At this point in our reflection, we have yet to bring further evidence to bear that might shift the balance against the claim that possibly P.So at this point in the process, the balance of our evidence favours its being possible that P: that P is then conceivable 2 .
We then test further.In effect, we ask why it wouldn't be possible that P, seeking other impossibility-makers.We flesh out our picture of a state of affairs in which P. This is a way to check for impossibility-makers, even if we don't frame it to ourselves that way.It amounts to putting the claim that P to the a priori tests mentioned above.If the claim passes tests, our a priori evidence still favours the claim that possibly P. It favours it more strongly because it passed the tests.Suppose that after all tests, our a priori evidence still favours its being possible that P. Suppose too that it seems likely that we could continue fleshing out our description of a world in which P without ever bringing an impossibility-maker into view.Then we may judge it fully conceivable 2 that P, even if we have not fully conceived 1 that P.An item is the more conceivable 2 to us the fuller or higher-degree the conception 1 of it that could survive tests, or the more tests it would survive.Thus Item I is fully conceivable 2 to A = df.if A as actually is, in A's actual circumstances, with A's actual a priori evidence, fully conceived 1 I and gave this conception 1 of I all a priori tests, A's a priori evidence would favour I's possibility (that is, S's being actualizable, F's being exemplifiable, or B's possibly existing).
A priori tests range from simple fleshing-out to checking consistency with the most sophisticated a priori truths.For those who buy into it, PBT provides another a priori test.But in most contexts, PBT is not relevant.So it is possible to make sense of conceivability 2 , and of this facet of modal epistemology, without bringing PBT into the mix.Suppose now that at some point, our testing leads us to believe that that some item, Item, is not possible.That may be because failure of a test is facially obvious in our very first grasp of a sentenceas with evident self-contradictions.In other cases, this is not facially obvious, but as we more fully conceive 1 Item, it becomes clear a priori at some point that Item is impossible.Then Item is conceivable 2 only up to that point.Item ceases being conceivable 2 there, and so is not more fully conceivable 2 .Up to that point, Item is conceivable 1 and our evidence favours a possibility-claim.At that point and beyond, Item may be more fully conceivable 1 -for example, we can see entailments of what makes Item impossiblebut our evidence definitely goes against them.How conceivable 2 Item is to us depends on whether and where the balance of evidence would shift.

Without confidence in (SEE)
So far, I have discussed only cases in which we should be confident in (SEE).In considering complex higher-mathematical sentences, most of us should not be confident in (SEE).For most of us will be aware that we do not have access, or easy access, to relevant impossibility-makers, or suspect that there might be kinds of impossibility-maker we don't even know about, or know we lack needed background knowledge.A complex higher-mathematical sentence could be mathematical nonsense, or imply a contradiction in a way that would be facially obvious to a mathematician, and I probably would not see it at all.As I know my own deficiencies, I know that I cannot really inspect impossibilitymakers here.Thus where we should lack confidence in (SEE), the impossibility-maker approach cannot provide evidence for possibility or impossibility.

An alternative approach
I base my account on (SEE) and impossibility-makers.One could also seek possibilitymakers, and take failing to find any as defeasible evidence for impossibility if it seems plausible enough that SEE*.If there were a sufficient condition for possibility, I would see it. 9th (SEE) and (SEE*) have their uses.If we use both, we will (obviously) find at most a possibility-maker or an impossibility-maker, not both.But we may not find either.In that case, use of the two yields equally defeasible evidence for both possibility and impossibility, and if we have no other relevant evidence, we must suspend judgement.If we think we see both, then if the candidates seem equally plausible, we must again suspend judgement.If one candidate is more plausible, then we get some evidence in that directionbut less than we would if there were no candidate pointing the other way.
(SEE) may be more useful than (SEE*), particularly in PBT.Being actual and following from something actual are sufficient conditions for possibilitythat is, they imply it.Not much other evidence for possibility doesfor example, lack of obvious a priori falsehood does not.By contrast, we have good access to many kinds of impossibility-maker.Further, (SEE*) is dubious in the case of PBT.One sufficient condition of possibility here would be the exemplifiability of a perfect being's essence.But most philosophers and theologians who have thought about it have held that we don't have access to that essence; Leibniz held PBT delivers it (Leibniz (2016), 71), but his is very much a minority view.If we have no access to that essence's content, then it is not the case that if it were exemplifiable, we would see that it is.Further, if a perfect being's essence is hidden, then we cannot 'see' that any possible world contains itand so a second main sufficient condition of possibility is unavailable.Another sufficient condition for possibility is actuality.But even if there actually is a perfect being, many don't see this, and so it's not the case that they wouldand nothing makes them relevantly different from any other human.Further, even if we don't assume that there is a perfect being, it's not clear why we should think anyone would.

Problems for conceivability 2 in GCB theology
I want to use conceivability 2 to defend the project of GCB theology.But applying it to GCB theology to produce GC 2 B theology faces at least four problems.
Anselm argued that its perfection puts a perfect being beyond full understanding (Anselm (1965), 137).Maimonides argued that it puts a perfect being's intrinsic nature so far from our grasp that we can only understand what it is not (Maimonides (1956), 49f., 67ff.).Many theists would claim that God is not fully conceivable 1 to us: finite minds just can't grasp all of Him.If God is not fully conceivable 1 to us, then if the perfect being is God, perhaps it can't be fully conceivable 2 to us.So perhaps a GC 2 B theology that hopes to describe God cannot work in terms of being fully conceivable 2 . 10f it can't, the next question is what degree of conceivability 2 it should aim for.For conceivability 2 is degreed.There are continuum-many degrees between inconceivability 2 and full conceivability 2 .Further, any one such degree might be constituted in more than one way (e.g.full comprehension of an inadequate description vs partial comprehension of an adequate one).So we seem threatened with an infinity of possible GC 2 B theological projects.If there are all these, it may be hard even to say which one(s) we mean to pursue, and why it (they) deserve(s) more attention than any others.
Again, a less-conceivable 2 being might be conceived 1 as greater than a more-conceivable 2 being.It might fail a test precisely because its conception includes great-makers that fail.This helps generate the problem of troublemakers.I discuss that problem below.
Finally, how confident we should be in (SEE) for a subject-matter affects our modal evidence.It may block any 'testing' evidence, and can also be an undercutting defeater for modal intuition.So if there is to be a GC 2 B theology, it matters whether not seeing something impossible about a purported description of a GC 2 B is too like my not seeing something impossible in a mathematical sentence that is beyond me.

Full conceivability?
I first address full conceivability 2 .I stated above that an item I is fully conceivable 2 to A just if it is the case that were A as actually is, in A's actual circumstances, with A's actual a priori evidence, to fully conceive 1 I and give this conception 1 of I all a priori tests, A's a priori evidence would favour I's possibility.This can be true even if A cannot gain a full conception 1 of it.Even if we cannot fully conceive 1 God, there can be substantive truths about whether God's having a certain attribute is fully conceivable 2 , and we can have a basis for judgements about them.
Imagine a novel so long that with only my current human natural endowment, I cannot live long enough to finish it.Even though I cannot finish it, it might be a substantive truth that were I to finish it, I would like it.The content of the book and my tastes might jointly make this true.Similarly, even if no-one with only the current human natural endowment could fully understand a perfect being's nature, it could be a substantive truth that GRASP.If anyone with only the current human natural endowment fully conceived 1 a perfect being, his/her a priori evidence would favour the claim that possibly it is F.
If (GRASP) is true, it is fully conceivable 2 that a perfect being be F.According to many theists, (GRASP) has an impossible antecedentit is a 'counterpossible'.On the standard approach, counterpossibles are all trivially true for semantic reasons (Lewis (1973)).But truth can be overdetermined.I think that some counterpossibles' truth is overdetermined.They are true for substantive as well as trivial semantic reasons.The one about the toolong novel could be one of these, and (GRASP) could be another.The content of a perfect being's nature and of the current human natural endowment might make it true.So surprisingly, it could be a substantive truth that the nature of a perfect being is fully conceivable 2 even if it is not fully conceivable

Many projects
I next note two particular GC 2 B projects that are likely to attract interest.If any did GC 2 B theology, many of them would not do it solely for its own intrinsic interest.For a good proportion of perfect being thinkers hope that PBT will describe (or support an already-going description of) a God they believe to exist.To those with that motivation, GC 2 B theology can be interesting only to the extent that it seems to remain within the bounds of the possible.For the more doubt about that, the more doubt that it really does describe an existing God.So a degree of conceivability 2 in GC 2 B theology would more interest these thinkers the greater the evidence of possibility it embodies or provides, ceteris paribus.One can also seek from PBT evidence for the possibility premise of an 'ontological' argument for God's existence.For those with either interest, a degree of conceivability 2 will be more worth having the more support it seems to give a possibility claim.Full conceivability 2 would be the best evidence for possibility.So GC 2 B theology in terms of full conceivability 2 would interest many philosophers.
The interest of another project arises from earlier points.For Anselm, Maimonides, and other theists who think that we cannot fully conceive 1 a perfect being, the degree of conceivability 2 best for GC 2 B theology might be whatever degree is the maximum consistent with perfection's impact on conceivability 1 .In other words, they might most want to know what the modal evidence would favour at the point of the fullest available human conception 1 of God, rather than what it would favour if humans fully conceived 1 God.Thus we see two degrees of conceivabity 2 for GC 2 B theology that are likely to interest philosophers.There may be other degrees worth noting, but hereafter, when I discuss GC 2 B theology, I take the degrees involved to be either or both of these.

Perfect beings and (SEE)
I now turn to how confident GC 2 B theology can be in (SEE).Impossibility-makers specific to PBT might be (a) the content of the attributes it considers, (b) the content of relevant attributes beyond our ken which are not the divine nature, (c) the nature of God, (d) incompatibilities between items in (a)-(c), or (e) brute lack of relevant possibility.It might just be a brute fact that however well we can conceive 1 exalted perfections, Marcus Aurelius (say) was the greatest being absolutely possible.That is, there might just be a lot of brute impossibility out where PBT wants to work.
I discuss these in order.I start with (a).Most PBT finds the live candidates for ascription to a perfect being in the vicinity of divine attributes traditional in Abrahamic theologies.This is not surprising.The Abrahamic God is a personal being.If we think in terms of greatness, we do not see anything greater to be than personal; my treatment of (b)-attributes, noted below, gives reason not to worry much about better things we're unable to see; and we have a good sense of what the perfections for persons would be.Now philosophers debate how to define the standard divine attributes.Some offer definitions.Others object, and often consensus arises about whether the objections are decisive.False candidate definitions are necessarily false.So any fact that falsifies a definition makes it impossible.That Fido is a dog and has four legs makes it impossible that to be a dog is to be a three-legged animal that. .., and does so just as truly as the content of the attribute of caninity does.The attribute of caninity may be that definition's minimal impossibility-maker.But just as minimal truthmakers are not the only truthmakers, 11 minimal false-and impossibilitymakers are not the only ones.
The debate over divine attributes shows that many philosophers have some skill in spotting and winnowing out candidate impossibility-makers in this subject-matter.So there is reason for at least some confidence in (SEE) for divine attributes.However, often, no clear consensus arises.Thus philosophers as a whole are not very good at spotting divine attribute impossibility-makers and knowing that they have done so.So the divine-attribute debate shows that while we're entitled to some confidence in (SEE) for (a)'s divine attributes, it should not be very high and perhaps should not be high.
Again, pace Kant and other critics, we are not less competent in philosophical theology, or in metaphysics more generally, than in other areas of philosophy.Kant et al.'s evidence that we are metaphysical incompetents is largely persistent metaphysical disagreement, and the same sorts of disagreement dog epistemology, ethics, etc.So we should have as much confidence in (SEE) for purported definitions of divine attributes as we do for other philosophical hypotheses.Philosophers across all subfields do tend to see surviving philosophical scrutiny as evidence for truth, and therefore for possibility.This evinces moderate confidence in (SEE) for philosophy.We have no reason to be less confident in the particular sort of philosophy PBT is, and in particular in our competence to spot impossibility-makers in (a)-attributes.
I argue elsewhere (Leftow (forthcoming b)) that (b) and the chance of incompatibilities between (b)-attributes and others should not much diminish our confidence in PBT.The same arguments, slightly redirected, would show that they should not much affect our confidence in (SEE) for GC 2 B theology.Turning to (c), 'the nature of God' might refer to: (i) a conjunction of (a)-attributes, (ii) a further property 'behind' that, which (i)'s conjunction partly or inadequately expresses (Aquinas's view), which might be one we cannot fully conceive 1 (so Aquinas) or cannot conceive 1 at all (so Maimonides), (iii) a further property 'behind' that, which (i)'s conjunction does not even partly or inadequately express (so Maimonides), which might be one we cannot fully conceive 1 , or cannot conceive 1 at all, or (iv) a conjunction of (i) and (ii) or of (i) and (iii).
So with (c), the question becomes whether we have enough grip on candidate items in (i)-(iv) to spot impossibility-makers in them, or incompatibilities between them and other attributes.
I have said enough to handle (i) already.Turning to (ii), suppose that the further property is one nothing can havedue to its content, rather than as a brute lack of possibility.Then having it entails having (i)'s conjunction, since it entails everything.So the conjunction is in that sense compatible with the further property.In another sense, the further property is incompatible with all properties, since in no possible world is it co-exemplified with any other property.We see this clearly.So we're quite good at 'seeing' the facts about compatibility here.But the real issue for (SEE) is not the divine nature's compatibility with anything else, but whether we could pick up on that about its content which makes it unexemplifiable if we have limited or no conception 1 of its content.I now discuss Maimonides' sort of view; what I say will make separate discussion of Aquinas' sort unnecessary.
For Maimonides and others, we can only grasp what God is not by nature.We have literally no positive, contentful idea of what's in God's nature.What it is not will not tell us that it's exemplifiable or that it is not, and if I have no idea what's in there, I can have no idea whether something in there makes it unexemplifiable.Even if I somehow could survey all possible worlds and 'see' that nothing had this property, that would not tell me whether the problem lay in the property's content, or was instead just a brute lack of possibility.However, as I note below, Maimonides' and similar completely negative-theological views of the divine nature have little argumentative support.So till we see a better case for such a view, we can fairly set this aside.However, if we do, such views remain undefeated hypotheses.Even though we have no reason to believe them, the fact that they are undefeated must lower our confidence in the possibility of a perfect being despite any evidence for possibility that PBT could provide.For then it is an undefeated hypothesis that something, we know not what, makes it impossible that there be a perfect being.But I cannot see that more than a slight lowering is warranted, as Maimonidean views have only slight support, and do not favour the unexemplifiability of a perfect being's nature over its being exemplifiable.
Further, Maimonides' view's being undefeated is no reason to lower our confidence in (SEE) for PBTand even knowing that Maimonides was right would not be.For we are assessing how confident we should be about (SEE) in the case of attributes we can feed into perfect being reasoning.If Maimonides is right, the divine nature is not one of these.PBT is a search for the right thing to say about God: Anselm introduces the project as a search for 'what of all the things that can be said of something . . . is appropriate to this . . .nature substantially' (Schmitt (1946), 28, my translation).What cannot be 'said of something' is thus not a candidate input to the process.So it should not affect our confidence in (SEE) for things which are candidate inputs.Note further that even impossible attributes have definitions: being a round square is being round and square.The right thing to say about what it is to be a round square is just that, not any of the things that follow from being a round square (being triangular, being octagonal, etc.).PBT is an effort to articulate what it is to be a perfect being, and God.Suppose that a perfect being's nature is inconceivable 1 to us.Then even if its content makes it unexemplifiable, there are right and wrong answers in PBT, and PBT might still be able to articulate its content in conceivable 1 or conceivable 2 terms ('round' and 'square' are both).Further, the attributes PBT chooses to do so might all be possible ones, as long as it's the case that they can at best articulate it only partly and inadequately.
Suppose on the other hand that the divine nature is exemplifiable.Then if (i)'s conjunction also is exemplifiable and expresses the divine nature's content, even if only partly or inadequately, it is compatible with the divine nature's content.So (ii) poses no problems beyond (i)'s.Nor does the first conjunction in (iv).Only complete negative theologians about the divine nature -Plotinus, Ps.-Dionysius, Maimonideswould maintain (iii).All three rest their view about the divine nature on a doctrine of divine simplicity and ontologies of predication which are now very widely rejected.The lack of appealing arguments for complete negative theology about the divine nature, and the difficulties it entails, constitutes a strong case that there is little here to detain usnor then with (iv)'s other conjunction.Aquinas and others let the 'way of eminence' speak of the divine naturebut since we apply this to (a)-attributes, this turns out to be a version of (ii), not (iii).Note finally that a general sense that some aspects of God may be humanly inexpressible is not sufficient to motivate (iii).For the aspects need not be in or of God's nature.Even if they are, that we can't get our heads around these aspects of God's nature doesn't entail that we can't get our heads around things like them, and the fact that they are like them, and use our grasp of all that to speak positively, indirectly, and inadequately about a nature we can't directly and adequately express.
My treatment of (d) would just repurpose things I've said about (a)-(c).As to (e), suppose that it's not possible to be wiser than Aurelius.We can conceive 2 of much wiser beings.So it's natural to wonder why it isn't possible to be wiser.(e) is a bruteness hypothesis.On (e), the answer is 'no reason.It just isn't.There is no difficulty about the property of being wiser than Aurelius.It's just that brutely, when we survey all the possible beings, nothing has it'.Brute impossibility is precisely impossibility without impossibility-makersimpossibility simply due to the absence of a possibility.So while brute lacks are a worry for a priori modal epistemology quite generally, they are not a worry for (SEE) specifically.For on (e), if I don't see any impossibility-maker for being wiser, I'm getting it right.There is none.There is nothing to 'see', and so nothing that should affect our confidence in (SEE).
Nor should the thought of lacks of possibility without any impossibility-maker affect our confidence in the impossibility-maker approach to possibility-claims.All it implies is that there can be evidence (the lack of an impossibility-maker) for a false claim (that the state of affairs is possible), and that is true quite generally.I've already noted that favouring by the a priori modal evidence doesn't entail possibility, and it nonetheless remains the case that it is reason to endorse a possibility claim.How good a reason it is depends on the proportion of cases in which the evidence is for a claim that is in fact false.Our view on that will depend at least in part on how plausible we find it that there is a lot of (e)-type brute impossibility.
(e) itself, and also the proportion question just noted, raise the question of why we should believe in all this brute impossibility/necessity.A good answer must motivate a metaphysics of modality that restricts the possiblesay, to what actual natural causes actually can naturally cause.It will also have to persuade us to think less of the fact that far greater things appear to be conceivable 2 , for instance by arguing that our relevant faculties are reliable, or knowably reliable, only in dealing with the kind of hypotheses they evolved to handle (about contingent natural states of affairs, not supernatural degrees of anything).Discussing such moves demands more space than I have here.So I settle for a dialectical move.Most readers of this article will not hold a metaphysics of modality that generates such a restricted domain for possibility, and probably most do accept some sort and degree of conceivability as at least some evidence for possibility on other metaphysical questions.To those readers, at least, I simply say: if you accept conceivability evidence elsewhere, it takes argument to make an exception for PBT.I do not think such an argument can be had.(e), then, leaves those my dialectical move spoke of entitled to moderate confidence in (SEE) for PBT: as much as we can have in any other philosophical domain.I think I could make a persuasive case for others against (e), but that task must wait.
I have had to paint with a broad brush.But I submit that philosophers should have enough confidence in (SEE) for PBT to operate with impossibility-makers in PBT.

Speaks's argument against GCB theology
My account of conceivability 2 instances a general kind that Speaks discusses.Let's call this notion conceivability 3 .It is conceivable 3 that P, says Speaks, iff it is not a priori false that P and the claim that P meets some further condition 'weaker than possibility' (Speaks (2018), 70).My further condition is that the a priori modal evidence favour its being possible.This is a condition beyond not being a priori false because competing, mutually exclusive candidates for possibility may all not be a priori false without all being equally favoured by a priori modal evidence.
Speaks argues that GCB theology cannot decide between competing conceivable 3 theological claims.He asks us to consider a pair of mutually exclusive, jointly exhaustive conceivable 3 theological hypotheses.As both are conceivable 3 , neither is a priori false.If neither is a priori false, neither is knowable a priori.For if one were knowable a priori, the other would be a priori false.So both are conceivable 3 only if neither is knowable a priori.But that God is the GCB is knowable only a priori.Arguments of GCB theology from this axiom are also a priori.So if GCB theology can in fact back one hypothesis in a way that renders it knowable, that hypothesis is knowable a priori.So the other is a priori false.So it is not conceivable 3 after all (ibid.).Thus GCB theology cannot decide between conceivable 3 hypotheses: 'a successful (GCB) theology would deliver a priori knowledge of claims about God which must be assumed not to be knowable a priori for the method to get off the ground in the first place' (ibid., 70-71).But GCB theology is precisely for the purpose of deciding between conceivable theological claims.So if Speaks is right, GCB theology is useless.
Given the backdrop I've laid out, I reply as follows.GC 2 B theology begins with initially conceivable 2 theological hypotheses.It sorts out what remains conceivable 2 on fuller reflection.It starts with theological hypotheses conceivable 2 to some degree.It hopes to sort out which are fully conceivable 2 , or as conceivable 2 as our limits in conceiving 1 allow.Consider a pair of mutually exclusive, jointly exhaustive theological hypotheses.Each is initially conceivable 2 .It does not show up as false a priori at first glance.But if PBT supports one hypothesis, it gives reason to call the other a priori false on fuller reflection.If it is a priori false on fuller reflection, it does not remain conceivable 2 .There comes a point in a priori reflection when the modal evidence tips against it.PBT induces this.There is no inconsistency with its starting point in its doing so.Being initially conceivable 2 is compatible with not being more fully conceivable 2 .Thus Speaks's argument fails.For it ignores the need to specify stage and degree when it comes to conceivability.Thus conceivability 2 is a sort of conceivability 3 that can serve GCB theology.I now take up what Speaks calls troublemakers.

The problem of troublemakers
A troublemaker is a property (i) God would be better with than without, which (ii) God conceivably has and (iii) cannot have.Here's how trouble-makers make trouble (Speaks (2018), 54-55).Suppose that God would be better with than without F. Then F meets (i).Further, if we think that God would be better with than without F, we have already conceived 1 God being F (to some degree).For we have mentally compared God with and without F.So it is to some degree conceivable 1 that God has F. If God's being F is not facially impossible, it is also to some degree conceivable 2 that God has F. So F meets (ii) in GC 2 B theology.
If God is the GC 2 B, conceivably 2 has F, and would be greater with than without it, then seemingly He must have F.For suppose that He does not.Then we can conceive 2 something greater than God, namely God with F. So without F, God is not the GC 2 B. But we've supposed that He is the GC 2 B. Thus if F satisfies (i) and (ii) and God is the GC 2 B, God is F, by reductio.(III) tells us that God can't be F.So if (i)-(iii) are true of some property, either God is not the GC 2 B (Speaks (2018), 55), or He is, but He isn't possible.
Apparent troublemakers are not hard to find.Speaks notes that first-year philosophy students often think that God would be greater if not 'limited by the laws of logic' (ibid., 58).That is, they think it would be greater to have than to lack Cartesian omnipotence, which includes the ability to bring about the logically impossible.If a state of affairs is logically impossible, it cannot come about.So nothing can have the power to bring it about.So if any state of affairs really is logically impossible, and it is to some degree conceivable 2 that God have Cartesian omnipotence, apparently Cartesian omnipotence is a troublemaker.Speaks also mentions being able to make Hesperus not be Phosphorus (ibid., 66).Perhaps that is another contra-logical troublemaker.Perhaps being able to make a circle that isn't perfectly round is yet another (ibid., 57-58).If it is not, mathematical impossibilities get their own class of troublemakers.Speaks also brings up God's changing His nature, that is, making Himself have necessarily an attribute He now cannot have (ibid., 69-70).This combines a law-of-metaphysics troublemaker (changing one's nature) and a theological troublemaker.So the problem of troublemakers seems quite live.I now try to kill it.

The second stage
The road to troublemakers above follows Speaks' own presentation of GCB theology.But if anyone really has pursued GCB theologywe see below why I raise this question then historical GCB theology has had a second stage.Its first stage runs the reductio from (i) and (ii) above.If Anselm is doing GCB theology, his first stage is in Proslogion 2 and 5. Proslogion 2 introduces the reductio form of argument.Operating on 'existence in reality', the reductio turns into that chapter's 'ontological argument' for God's existence.Proslogion 5 tells us that since God is a thing 'than which nothing greater can be thought', He is 'that which is greatest of all things, alone existing of itself, that made all other things from nothing' and 'the greatest good, through which every good thing exists' (Schmitt (1946), I, 104).It adds for good measure that 'whatever is not this, is less than can be thought.But this cannot be thought of you' (ibid.).This repeats Proslogion 2's key description, 'thing than which no greater can be thought', and reminds the reader of the move that drives the reductio.The rest of the quoted text gives us the capsule description of God from which Anselm's Monologion derived a wide range of divine attributes.Here Anselm tells us in effect that by using the argument-form Proslogion 2 introduces, he can derive the full description of God the Monologion laboured to produce.This means that he can do what the Proslogion's introduction promised: derive from unum argumentum all the results for which the Monologion used many varied arguments.
That's PBT's first stage.Its second stage takes the outputs of the first-stage reductio and checks their consistency.In Anselm, the second stage occupies Proslogion 6-11.Anselm worries at length, for example, about whether perfect justice and perfect mercy are compatible.He then concludes, in this section's last sentence, 'so, therefore, you truly are . . .whatever it is better to be than not to be' (Schmitt (1946), I, 109).This sentence means, in effect, 'consistency check complete, and all first-stage outputs passed'.For Anselm, then, checking consistency is a larger, longer part of the PBT project than the use of the (i)-(ii) reductio.Only when it's done does he endorse the first stage's outputs and consider it 'case closed' that God has them.For Anselm, then, (i)-(ii) reasoning establishes only a prima facie case that God is F. Cases for other attributes can in principle override that case.If Anselm did not believe this, he would not have bothered checking consistency.
If GCB theology's second stage leaves the GCB in possession of a troublemaker, that's still trouble.But if a second stage is necessary, it's not automatic that the GCB will wind up with any particular first-stage output.For apparent conflicts between first-stage outputs might prove genuine.If they are, then the GCB must lack at least some first-stage outputs.For clear inconsistency tilts the balance of a priori modal evidence against the GCB's possibly having all of the inconsistent attributes at once.So GC 2 B theology must reject some of them.So even if the first stage can have troublemaking outputs, it is not automatic that the GCB wind up with any.It may be automatic that it does not.The second stage's job is to find the best compossible set(s) of first-stage outputs. 12hat is in itself impossible is not compossible with any set of attributes.If so, the second stage automatically jettisons trouble-makers.They never do cause trouble.GC 2 B theology gets rid of them because perceived impossibility changes the balance of modal evidence and makes the attribute or combination of attributes cease to be conceivable 2 .
Speaks might well reply, as he does to a slightly different move, that this is just a way to deny that God is the GCB (Speaks (2018), 58).After allreturning to the reasoning from (i)-(iii) abovewe have conceived that God has the troublemakers, He would be greater if He did, and yet we're saying that He does not have them.I demur.Part of being conceivable 2 is being favoured by the a priori evidence for possibility.In GC 2 B theology, God is the greatest conceivable 1 being whose possibility the a priori evidence favours.That evidence does not favour the possibility of something that could make a contradiction true ( pace Descartes).For it does not favour the possibility of a contradiction's being true.When we deal in conceivability 2 , if an attribute meets (iii), it fails (ii).So troublemakers are impossible.Because we're dealing in conceivability 2 , denying God a troublemaker does not deny that He is the GC 2 B. For nothing with a troublemaker is a C 2 B.
Again, Speaks might say that if we eliminate troublemakers this way, the result is just 'a notational variant' of a PBT that works with absolute possibility, not conceivability (ibid., 59, 69).I think not.The two-stage GC 2 B project begins as Speaks specifies, with 'space between conceivability and possibility' (ibid., 54).It lets some theological propositions that will eventually prove impossible be initially conceivable 2 .If we're feeling charitable to Descartes, perhaps one such proposition is that God can violate classical logic.If we're not, a better candidate might be that God is not omnipotentor if He happens not to be omnipotent, that He is. 13 So too, many non-theological propositions which must be impossible are conceivable 2 to some degree.It is initially conceivable 2 that Goldbach's Conjecture is provable.It is initially conceivable 2 that it is not.Only one is possible.
One reason GC 2 B theology's second stage does not remove the 'space' between the conceivable 2 and the possible is that such non-theological cases remain.Perhaps some must remain.For perhaps some mathematical propositions are provable, but only by greater resources than humans can have.But it may be that even theological cases must remain.Suppose that the GC 2 B is a possible being and there are things about it that we can't fully conceive 1 .Then it is not possibly fully conceived 2 , because fully to conceive 2 it, we must fully conceive 1 it, and so must fully conceive 1 these things.So even in theology, space must remain between what is possibly fully conceived 2 and what is possible, because the possible outstrips what is possibly fully conceived 2 .The possible might also outstrip the conceivable 2 simpliciter, for perhaps the things we can't fully conceive 1 , we can't even partly conceive 1 . 14hen the evidence for possibility ceases to favour a hypothesis about the GC 2 B, the hypothesis ceases to be conceivable 2 , and so the space between the conceivable 2 and the possible shrinks.Suppose that GC 2 B theology leaves no conceivable 2 but not possible hypotheses about God.Suppose too that we somehow fully conceive 1 God, so that no possible but inconceivable 2 ones remain either.Then the only hypotheses about God that remain are conceivable 2 and possible.At most non-theological cases distinguish some conceivable 2 from possible worlds.The theological space between conceivability 2 and possibility is zero.Then one might ask: isn't GC 2 B theology now a 'notational variant' of greatest possible being theology?For now, by comparing a conceivable F-God with a conceivable non-F God, we compare possible states of God, and so we just do GPB theology in another key.
One answer is that if this happens, there is no more PBT to do.We've solved all questions GC 2 B or GPB theology can answer.For I show elsewhere (Leftow (2022), Idem (forthcoming a)) that PBT can only answer questions about how God necessarily is.So if the question whether God is F is really one for PBT, then necessarily, if God is F, necessarily He is F.So if possibly He is F, it follows that He is F, assuming a modal logic containing Brouwer.So if we establish which conceivable 2 hypothesis is the possible one, we answer our perfect being question. 15Thus if we've done this in all cases, there is (again) no PBT left to do.But as long as we are en route to this promised land, there is still theological space between conceivable 2 and possible worlds.So GC 2 B and GPB theology remain distinct projects.As long as they are projects, they are distinct.Once they're complete, their results might well be mere notational variants.But as I argue below, that may not be a bad thing.
What the a priori modal evidence is and what it favours depends on whose cognitive powers are in use.Absolute possibility does not depend on this.So conceivability 2 and possibility are distinct notions.Further, being conceivable 2 doesn't entail being possible.So one may wonder how we could ever infer from God's being the GC 2 B to His being possible.But this is an instance of a general sort of inference we routinely make.That the evidence most favours a claim is good defeasible reason to infer that it is true.We make the inference unless the reason is defeated.Thus evidential favouring is good defeasible reason to infer possibility.So if conceivability 2 includes evidential favouring, it is good defeasible reason to infer that something is not just the GC 2 B, but possible.It's at worst matter for further thought whether something defeats the inference.
The second-stage approach to PBT works against all troublemakers if it works against any.I now turn to one specific sort of troublemaker.I suggest a way to disarm it.I then suggest that the way I do that extends to all other troublemakers, thus providing a second fully general solution to the troublemaker problem.

Logical troublemakers
Beginning philosophy students are indeed drawn, lemming-like, to Cartesian ideas about omnipotenceto the claim that logic can't limit God's power.Speaks seems to think that their intuition is that God's power cannot have any limits at all (Speaks (2018), 58).I disagree.I think that what really drives them is that phrases like 'limited by the laws of logic' make it sound as if logic is independent of God and imposes itself on Him.If it is and does, He does seem less supreme than is to some degree conceivable 2 .He faces an external environment He can do nothing about.He must merely cope with it.These are not marks of greatness.I think the real intuition students typically have is that God must face no externally imposed limits (so Leftow (2012), 119).It's that logic seems external to God that makes logic seem like an unacceptable sort of limit.
I support this by noting a parallel.The Euthyphro dilemma asks whether actions are right because God wills that we do them, or instead He wills that we do them because they are right.First-year students react to one horn of that dilemma as they do to 'limited by logic': it galls many of them to say that He wills it because it's right.This is because if things are right independent of God's will, their being right seems an external condition to which God is merely subject, which controls His actions and 'limits His power' much as logical truths seem to do.I now cite yet another student reaction.I have rarely if ever found a student who wasn't content to deal with the Euthyphro dilemma inter alia by basing goodness on God's natureone who insisted that only a divine command theory of goodness would do.Limits that merely express God's being who and what He is did not seem objectionable to the students I've had.
Here is one way to develop this move. 16Plato believed in an objective standard of goodness, the Form of the Good.To be good, he thought, is to 'participate' in this.(Plato never spelled out just what that meant.)To ask Plato why being good consists in participating in the Form is exactly like asking a realist about universals why being red consists in having the property of redness.In each case, it just does.That's just what the ontology is.That's where explanation runs out.There is no more to say.One way to make the Euthyphro move just noted is to give God the role Plato gave the Form of the Good.We then spell out what relation takes the place of Plato's 'participation'.On one proposal, to be a morally good action is to be one that could express a sufficiently Godlike character in the circumstances.On another, to be a morally good action for humans to do is to be relevantly like an action God might do in relevantly similar circumstances if He were human, and to be a virtue is to be a disposition to such actions.
On this second approach, for Christians, the question 'what would Jesus do?' gains a new metaphysical resonance.Others who think an Incarnation at least possible can approach matters in broadly the same way.Those who think Incarnation impossible can treat some sentences of the form 'if God were human, He would . ..' as expressing substantively true counterpossibles.Either way, God's character provides an objective standard of good action and moral virtue, and for humans, being morally good consists in standing in the right relation to God's character (which determines what He might do, and so plays a vital background role in the second proposal).Asking a theist why God is the standard in this way is just like asking Plato why being good consists in participating in the Form of the Good.In each case, the item just does by its own nature, with no imposition from without, play the central role in the metaphysics of moral good.Thus truths about the morally good express the nature of God.A theory of goodness is of course not directly a theory of rightness.But the goodness of God's nature provides internal limits on what God can make right that help in answering the Euthyphro question: because He is perfectly good, God commands good acts because they are good.They are right because He wills themonly His command makes them right as well as good.But the facts about what is good make His commands non-arbitrary, and for the reasons given, the facts about what is good are no external limit on His commands.
On a broadly parallel move for logic, logical truths also express the nature of God.One story starts with God just naturally thinking in certain ways.As I argue elsewhere, He is the source of all possibilitypossible worlds have the content they do because He gives it to them (Leftow (2012)).The content reflects the way He thinksstories naturally reflect the way their author thinks.If negation is classical rather than intuitionistic, say, that is ultimately because God thinks that way.Because He does, He so 'wrote' possible worlds that across them all, whenever it is not true that P, it is true that not P.So logic is written into the world, and reality necessarily conforms to it, because it expresses the way God's mind naturally works.Thus the 'limit' truths of logic represent is not external to God.It just reflects how His mind workswhich is a facet of His nature.Further, when we get logic right, we are picking up on the way God's mind works.Now God may not actually infer anything.Aquinas and many others deny that He does.They think He simply knows intuitively whatever He knows (Aquinas (1911), Ia 14, 7).But even if this is true, when we get logic right, we are expressing as well as can be expressed in the medium of inference the thought-patterns of a mind too perfect to need to infer.In that sense, a valid inference is the most Godlike action we can perform when called on to make a deduction.This is just a sketch of one way to develop the logical parallel to the Euthyphro move.I can give no more due to limits of space.But I hope it seems at least a bit promising.If it does, what it promises is inter alia a way to deal with logical troublemakers.

The truculent reply
The basic thought in my approach to logical troublemakers is that the only 'limit' on God if He is 'limited by logic' is His being Himself, and that's no limitation.Particularly truculent first-year students might reply that God would be greater if He weren't limited to being Himself, or His kind of being, but could be someone else or some other kind of being instead, or in addition.This sounds absurd.They are certainly not thinking through what they're saying.But it is just Descartes taken to the utmost, and so it won't do simply to ignore it.
It is not an effective reply to say, 'He can't be someone else or some other kind of being'.For the student can then ask why.If what prevents it is external, the initial objection returns.If it is internalwell, the student is precisely questioning whether even internal limits are too much.To deal with the nerve of the truculent response, one must instead argue that having the possibilities the student gestures at does not satisfy (i) and (ii) above.That is, one must argue that these possibilities (or what undergirds them in actuality) might or would not add to greatness, or that God's having them is not really conceivable 2 .Pointing out the latter in effect shows that the truculent have indeed not thought things through.I now pursue the truculent, to show that my approach to logical troublemakers can be sustained.I begin with a way to force suspension on the claim that logic-based troublemakers would make God greater.I then provide an argument by cases, examining ways to interpret what the truculent say would be greater and arguing that in each case, we don't get both (i) and (ii).Inter alia, this shows that such cases don't actually yield a troublemaker.

Forcing suspension
The truculent assert that it is great-making not to be 'limited by logic'.Even if it is, it is also great-making if logic expresses God's nature.For then • logic is not an external limitation for Him.On my earlier story, this is great-making in itself, and part of His having a great-making status as the source of all possibility.• His mind is wholly rational, as we now understand rationality (and what else can we work with than the concept of rationality we have?).Perfect rationality is great-making.• logic helps make Him the source of absolutely everything, which is great-making.
• Every inference we make depends for its validity on His existence and nature.This seems great-making.• He avoids the following problem.Pace dialetheism, to make a contradiction true would make reality fundamentally unintelligible to us.We simply would not be able, even in principle, to come to a clear picture of how reality is.Further, to believe a contradiction is (again, pace dialetheism) irrational, as we now understand rationality.So to make a contradiction true would make something we now consider irrational necessary to grasp how things are.Many say that God would be more perfect if unable to do evil.A parallel seems to apply here.If God shouldn't be able to make the world evil, He shouldn't be able to make the world fundamentally unintelligible, and comprehensible only by what we now consider an act of irrationality.
(Perhaps that would make the world in one respect evil.)• We avoid an invidious parallel.The power to violate classical logic is the power to make false what is necessarily true.For unless that's what it is, either classical logic is only contingently true, just waiting on divine will to 'flip' to falsity, or it is not true at all.If it is either, the ability to violate it is not the big deal it intuitively should be.The ability to make the necessarily true false has a moral parallel (or instance, if there are necessary moral truths).This is the power to make what is necessarily good evil (and vice versa).God's having this power is the defining feature of the least defensible sort of divine command theory.The motivations of the logical and moral views are also parallel.In each case, we're told, if God cannot do this, His power, control, supremacy, etc. are limited.On that sort of divine command theory, God could do anything and declare it good.So on that account, His being perfectly good does not constrain His actions at all.It places nothing morally off-limits.We can for example have no expectation at all that God will be truthful if it's simply up to Him whether to make lying for its own sake a good.He could do that at any point, lie about anything He pleases, and still come out perfectly good.That is surely not an ideally great God.
If the demand to be able to do the impossible is worth heeding anywhere, it should be equally worth heeding in the moral case.If we heed it there, the claim that God is perfectly good loses one key sort of content.It can no longer support any expectation about God's behaviour, however minimal or 'merely in principle'.All this has parallels with violating logic.Just as the moral move leaves nothing morally off-limits, the logical move leaves nothing rationally off-limits.
I think these points leave it at best a toss-up whether candidate logical troublemakers really add greatness.If it's a toss-up, we can't endorse the claim that they do.So we don't get troublemakers, and the first-year student subsides into suspended judgement on the claim that God should be able to violate logic.Now a toss-up is an issue PBT can't settle.But PBT doesn't claim to settle all relevant things on its own.Further, I'm not really sure that it is a toss-up.I think that greatness really tilts the other waythat it's greater to be a supreme reason whose nature all necessary truth expresses, than to be an inscrutable will whose rationality, if any, is not expressed in anything we can understand to be rational.I think I can motivate the intuitive judgements I feed into the PBT machine better than the other side.But that is work for another day.I now take up my argument by cases.

The argument by cases
There are many ways to read the truculent demand.The truculent might be plumping for God (a) possibly being only someone else, (b) possibly being only some other kind of thing, (c) possibly being in addition someone else, or (d) possibly being in addition some other kind of thing.Again, they might insist that (1) He could have been someone else or some other way from all eternity or (if the other kind's instances have beginnings) ab initio, without having done anything to bring this about, or (2) that He can bring any or all of (a)-(d) about.I discuss these in order.
Being only someone else I begin with (a1), the idea that God could have only ever been someone else.This says: God is from all eternity only God.There could have been (say) a gorilla, Koko, who was not God.God is not Koko, but He could have been. 17Now if God is the GC 2 B, it is not conceivable 2 that being only someone else be greater.If God is the greatest, it is only conceivable 2 that others be lesser or incommensurable.It is not even conceivable 2 that they tie God, if God is the greatest.So possibly being only someone else would not be possibly being greater.It's therefore not clear why the possibility would be worth having, if it's not something God could bring about.
Further, (a1) does not assert contingent identity.It does not say that in one possible world, Koko ≠ God, and in another, Koko = God.For if it's contingent identity, then in the latter possible world, one thing is both God and Koko, not only Koko.Once we see this, it's clear that (a1) isn't even something we can coherently describe.If (a1) is what the truculent are on about, they only think they can conceive 1 it because they haven't really thought about it.
A coherent variation on (a1)hereafter (a1*)would ascribe to God such possibilities as contingent identity with Koko.Now contingent identity is controversial.But true contradictions are more so, and so if the truculent are willing to countenance those, they will hardly get squeamish about contingent identity.The truculent might say that such possibilities are great-making because they expand God's possible knowledge -He can know what it is like to be Koko, and to be a gorilla, as well as what it is like to be God.He can know such truths as what Koko would express with 'I am Koko', which might otherwise be inaccessible to Him.Further, they might say, being Koko would cost Him no greatness, since He would also be God.So possibly being Koko would cost no greatness either.
I reply, to begin, that as God is always omniscient, He always knows what it is like to be a gorilla, and to be Koko.So God does not have to be Koko to know this, and one supposed reason for this possibility being great-making goes away.Again, God has made Koko, but was never her.If He did not know what it was to be like to be Koko, He created in ignorance things of which He remains ignorant.That is hardly what one expects of a perfect being.Further, that ignorance may have moral consequencesif bad experiences are part of what it is like to be Koko, or a gorilla, then He inflicted them in ignorance, and that is hardly an act of moral or more generally agential perfection.
Further, as Koko, God either would or would not possess and have access to the resources of the divine nature. 18If He enjoyed His full divine knowledge, He would hardly by being Koko know what it is like to be an ordinary gorilla.So He gains ordinary-gorilla knowledge only if He at least does not occurrently consider some divine knowledge.But on (a1*), it is not the case that God deliberately chooses to lay His knowledge down, in order to save gorillas by an incarnation or to learn something.Rather, it is simply possible that He not have been only divine from all eternity, but instead have eternally been a gorilla and divine, or just a gorilla.So on (a1*), we can't see God's possibly not enjoying divine knowledge as enabling some noble self-sacrifice on behalf of gorillas.It's just a possibility of enjoying less knowledge.It's just a possibility of living a lesser life.That does not seem like a good thing.Parallel points apply about limits of power, and lack of even the possibility of moral goodness if gorillas are not moral agents.Further, if it is less-making to be Koko, it's not clear why it would be great-making to know what only Koko can know ('I am Koko').
Finally, on (a1*), God can coexist with things that He might be identical with, without being identical with them.Otherwise there would be perfectly good contingent states of affairs He could not bring about, and so He would not be omnipotent.If Koko exists, and God can be Koko, but is not, then possibly being Koko seems to lessen God's greatness: there actually are truths He could know ('I am Koko'), but does not.Thus I submit that (a1*) is not clearly great-making.

Bringing this about?
If (a1) is incoherent, so is (a2), the idea that God can bring it about that He is only ever someone else.But suppose that my earlier argument is wrong, and there is a coherent thought here to discuss; in any case, (a1*) is at least coherent.Then (a2) still runs into trouble for another reason.If (a2) is bringing it about that God is only ever (say) Koko, it has God bringing it about that He was never the individual He 'started out' as, which seems just a variant on the 'killing your grandparents' paradox of time-travel.If (a2) is bringing it about that He is only ever contingently identical with Koko, it has God bringing it about that He was never just identical with the individual He 'started out' just identical with, and so involves changing His own past.But the best models of changing the past I've seen (Hudson (2014); Lebens and Goldschmidt (2017)) stop short of making sense of God's changing His own past.So it's not clear that even (a2) built on (a1*) rather than (a1)hereafter (a2*)is really coherent.
Further, if it is not conceivable 2 that anyone else be greater than or equal to the GCB, (a2)'s ability to make Himself only someone else would be an ability to make Himself lesser than or incommensurable with what He actually is.But one of PBT's oldest intuitions is that it is greater to be unable than to be able to make oneself worse.This was the nerve of the first recorded PBT argument, in Plato's Republic II (381b-d).Again, this intuition is the core of the PBT case for divine necessary moral perfection, something perfect being thinkers also almost uniformly endorse.If this intuition is correct, then it would not make God greater to be able to turn Himself into someone else who is lesser.This leaves only the thought that it might be greater to be able to turn Himself into someone else of incommensurable value.To follow the thought out, we must ask why it would make Him greater to be able to do this.The only reply I see is that it would be greater to have a wider range of possible action.
However, not every widening of one's range of action clearly makes one greater overall.Suppose that a psychological block keeps me from committing murder.After a few years' therapy, the block dissolves.I now can murder.So I have a wider range of action.It is not clear that this makes me greater overall.Again, a long Christian tradition holds that the blessed in heaven are no longer able to sin.In a straightforward sense, they thus can do less than they could on earth.Christian tradition holds that this is a better state than our current one, a state to which we should aspire.There is enough to the latter thought, I think, at least to create significant doubt that being able to do more must be an improvement.Again, many writers argue that omnipotence and necessary moral perfection are incompatible (e.g.Pike (1969)).Almost all of them agree that if they really do conflict, it's omnipotence that should go.They think, in other words, that if we have to choose, God would be greater overall with a narrower but morally better range of action.So the mere fact that the range of action the truculent prefer is wider does not show that it is a greater one to have.But I do not see what else could show that it is.And I shortly show that the idea of a wider range of action really isn't a good motivation in this context anyway.
We can deal in related ways with some conjunctive troublemakers.Consider being both necessarily perfectly good and able to do wrong. 19Per my argument, it is not clear that this would make God greater overall.This suggests an epicycle: consider being the GC 2 B by being necessarily perfectly good, able to do wrong, and greater by being able to do wrong.The free will defence suggests that humans are greater by being able to do wrongso why not God?Note first that what makes one kind of thing greater need not make another kind greater: a giraffe-length neck improves a giraffe, but would not improve me.I then answer with another epicycle: if adding murder to a range of actions does not clearly make greater, nor does adding what generalizes on thisbeing able to do wrong.Further, if it does add greatness, there could be greatness it might be greater for some kinds of thing to foregosuch as greatness having which costs more greatness to that kind of thing than it brings.(Consider my giraffe-length neck, if you thought that would improve me.)Further, this conjunctive troublemaker and the prior one cease to be conceivable 2 upon only a little reflection.That's all it takes to see that being necessarily perfectly good is incompatible with being able to do wrong.So having either attribute would not add to conceivable 2 greatness, with conceivability 2 specified as either of the interesting degrees above.

Why wideness won't work
What really moves the truculent may be the idea that it is greater to have a wider range of power, and God would if He weren't 'limited to' the classical-logically possible.This (they may say) is so even if the limits of logic are internal to Him.Perhaps (they may continue) these limits are merely the silhouette of His power.Perhaps contradictions are impossible simply because God has neither the power to actualize them nor the power to give that power to others.Regardless, less-constricted limits are in some sense conceivable (they can insist).Add to omnipotence 'within the bounds of logic' the power to actualize contradictions, and God would be able to do more.
I now reply as follows.It is not conceivable that God does not have a definite range of power, one 'limited to' just certain things.For not to have a definite range is not to have a range at all.(Take 'definite' here weakly enough to allow the range vague borders, if you like.)Now for any definite range that is in some sense or to some degree conceivable, a wider definite range is also in some sense or to some degree conceivable.So it's not conceivable 2 that a conception of God not be subject to the 'complaint' that it would be greater for God's range to be wider.I now show this.
Suppose that classical logic does not bind Goddoes not 'limit' God's ability to combine particulars and attributes into states of affairs, then actualize them.Suppose that He can not only actualize contradictions, but make it the case that Hitler exemplifies Stalin, or a sneeze.In other words, suppose that He can even make reality nonsensical.Still, the available 'vocabulary' of particulars and attributes limits what He can do.It is not even initially conceivable 2 that there is no such thing as roundness, and yet God makes something both round and not-round.Maybe a 'truly unlimited' Cartesian God would always be able to add more 'vocabulary'.But if He did, still, at any point, He would always have added only certain items.It would always be in His power to add more and so have a wider range.To have the power to give oneself more power is not itself to have the further power.Thus our desire to give God an unlimited range is not conceivably satisfied.So it cannot rightly motivate any particular change to our concept of God.For no change will satisfy the desire.'To get more power' can't be a reason to put the limit of God's power in one place rather than another if it is not conceivable 2 that the result not continue to face that demand.'To get more power than this'indicating God's previous limitmay seem like a reason to put the limit of God's power somewhere further on, but exactly the same kind of demand will arise for the new place.So here too, it's not really a reason to change where the limit falls.
Further, suppose that God could make contradictions true.Even so, some logic would describe the 'limits' of what He could bring about, and He would be 'limited' to what that logic 'allowed'.It's just that the logic would be dialethic rather than classical.It is not conceivable 2 that God's power not have (as it were) a stopping-point, a limit, which it is to some degree conceivable 1 that He be beyond.So all one can do is insist that the stopping point be self-imposed, and look for the best place for it.It should not look arbitrary and unmotivated.The break between what classical logic allows and what it does not fills this bill.It does not seem arbitrary to us at all that possibility stop there, nor then that power stop there.On the contrary, it seems self-evident to many that both do.

Contingent identity
Thus my case against (a2).(a2*) is the claim that God can make Himself contingently identical with something else.This does not imply that the greatness of the God who is also (say) Koko is less than or incommensurable with the greatness of the God who is just God.For if God makes Himself Koko, 'two' things are then one, and the one thing they are might have any combination of properties the two things previously hadand so perhaps not just lesser or incommensurable combinations.The truculent might even insist that it could have inconsistent combinations of properties, and so have all properties God and Koko have if not identical.Perhaps one truculent thought here would be: suppose God were both God and Koko, with all the great-makers of both.God is great.So is Koko, a little.So something with all the greatness of both must be greater.But let's think of greatness in terms of greatness units.If we do, God has an infinity of these just for being God.Koko has only a finite number.Cantor tells us that ℵ 0 plus any finite number is just ℵ 0 .So this doesn't pan out.If adding all the gorilla properties and subtracting no divine properties doesn't make God greater, then a fortiori adding only some gorilla properties and/or subtracting some divine ones won't do it.I add that I don't see anything inherently great-making about making a contradiction true, or being able to.Earlier argument suggested that it might even be less-making.If there is not more to be said for (a2*), then the idea that being able to bring about contingent identities with other things would make God greater does not seem to have anything beyond a wider-range consideration behind it.If so, I have dealt with (a2*) as well as (a2).
Only some other kind I now take up (b1), the idea that God could have been only some other kind of thing from all eternity or (if being of that kind gives Him a beginning) ab initio.If this is not some kind His essence actually excludessomething He cannot bethen it doesn't generate a troublemaker.If it is, then if we preserve the link between being a particular individual and having that individual's essence, this includes (a1), the idea that possibly God is only a different individual ab initio, and falls with it.If we don't preserve the individual-essence link, we get a 'law of metaphysics' troublemaker, that is, God supposedly being greater by violating the 'law' connecting individual and essence.
If this troublemaker is a possibility for God, He is not essentially divine and perfect.He could have been non-divine or imperfect instead.But nothing can make anything divine or perfect.To be either, one must be it a se.So on (b1), God is divine and perfect as a fortunate contingencycosmic luck.I argue at length elsewhere that this is an imperfection (Leftow (2012), ch. 7).Insisting that it would be more perfect to be both perfect and imperfect would just play with words.Even the truculent should fold their tents at this point.

Bringing this about
(b2) is the idea that God can make Himself into some kind of thing He actually is not.Again, to yield a troublemaker, the kind must be one He necessarily is not.If we do get a troublemaker from it, it is one of Speaks's theological ones, God's changing His nature (Speaks (2018), 69-70).If this happens, then first God is (say) necessarily moral perfect, and then He is not.This also violates a 'law of metaphysics'.It requires that one and the same being have first one modal essence, then anotherwhile remaining the same being.(We get a further troublemaker if we add that, nonetheless, there remains just the tie we believe in between essence and the identity of the thing having it.)Replies here include that laws of metaphysics are just internal limits on God, and that violating that law isn't conceivable 2 .More important, perhaps, is that the GCB has as such the greatest conceivable nature.If so, any change of nature would have to be to one which is either less or incommensurable.
It could not be a change to a lesser nature if (as PBT suggests) it is part of perfection not to be able to give up the best for something less.Christians might object here based on the Christian doctrine of the incarnation.This might suggest that perfect love led God to become less for others' sake. 20But one needn't parse the Incarnation as God becoming less.The orthodox view is that He adds a human nature while remaining as divine and perfect as ever.At most, once incarnate, He opts not to draw on some resources of His divinity for a while.If God can achieve what the Incarnation is out to achieve without literally giving up some divine attribute, the claim that He has given one up is at best unmotivated.It's no mark of perfection to make oneself worse without good reason.If love doesn't provide sufficient reasons, it's incumbent on the critic to provide others if we are to take this part of the discussion further.
Our last option is that God give up His initial nature for one of incommensurable greatness.This is conceivable 2 only it is conceivable 2 that the divine nature be completely incommensurable with another.But it isn't.For if the divine nature were completely incommensurable with another, God's natural greatness would be so.It is so only if God's nature includes no attributes that create a ranking between Himself and anything else.For suppose that God is F by nature, I am F, and being F contributes greatness.Then either we are equally F-great, or one of us is greater.Suppose further that God beats me in F-greatness, but in all other respects, we are incommensurable -He neither wins, loses nor ties.If this is so, overall, He wins.For He wins in one category, and in the rest, the game is called on account of rain, without a result.Thus, as noted, if God's natural greatness is completely incommensurable with something else's, His nature does not create any ranking with that thing.It is not conceivable 2 that it not do that.For God is by nature alive, and it is inconceivable 2 that any conceivable 2 other thing not be either alive or not.If it is not, God wins, as live things are naturally greater.If it is, God wins, for God's natural way of living is greater than any other thing's.If there is some vague border between being alive and not, then God wins for related reasons.
Being God and someone else (c1) asserts that God is possibly from all eternity in addition someone else.On one reading, it asserts that possibly God is God, and so not (say) me, but also me.That is, all of God is at once me and also not me.This isn't even initially conceivable 2 .The truculent might therefore suggest that God would be greater if He were not the GC 2 B, but instead the greatest being conceivable in some sense that does not let contradiction impede conceivability.But the motive for this is just the idea that logic-based troublemakers would make God greater.So I have already replied: it's not clear that they would.
On another reading, (c1) asserts that possibly God is from all eternity a compound including a divine being and something else.The thought is that actually God includes only a divine being, but in an alternative state of affairs, He includes both a divine being and (say) me.The truculent may be suggesting that God would be greater were this so.I reply that if He includes me, He is partly divine and partly less.This dilutes His greatness with an inferior vintage.A wine so diluted is not a greater wine, and it's not clear that dilution would make God greater.Suppose we think in terms of 'greatness points'.Then presumably including the divine being gives Him an infinity of these.Including me in addition would increase His greatness if my greatness were of a commensurable kind and a higher-order infinity, or the sheer doing of this neat trick would be.Neither is plausible.Including me would also add to God's greatness if His greatness thereby came to consist of two incommensurable and so disjoint sorts, one His native kind and the other what He gets from me.For suppose that the ruble is not a convertible currency.Suppose too that I first have only n dollars, then have n dollars plus m rubles.Even though there is no sum for these two in any one currency, I am richer than I was before.
It takes incommensurability to get the conclusion that God's greatness consists of disjoint parts.If the 'greatness units' are commensurable, they have a single-currency sum.But perhaps nothing can be incommensurable with God.If something is, then God is not greater than every other possible thing.But we can to some degree conceive 2 that He be greater than every other possible thing.This seems like a great-making attribute.So it seems like the GC 2 B should have it.So we have at best a greatness standoff.Even if God would be greater with a disjoint greatness of this sort, He would also be greater if greater than every other possible thing.Having both is not conceivable 2 .It's not clear which is better.So we cannot endorse the truculent claim that He would be greater overall with disjoint greatness.Further, for God's greatness to be completely incommensurable with something else's, God and this other thing must not be rankable in even one respect.This is inconceivable 2 for reasons already given.
If so, then turning to (c2), the idea that God can bring it about that He includes both a divine being and something else, it seems that only a wider-range consideration might favour this.I've argued that wider needn't be better, and that wider isn't a good reason here anyway, but Christians might object at this point.Christians sometimes explicate the Incarnation as God's coming to include something created (e.g.Flint (2011)).Christians might well think that it is better for God to be able to become incarnate than for Him not to be able to, given what they think He did with His incarnation.So (they might conclude) it is better for God to be able to make Himself include another being.A short reply is that there are other models of the Incarnation (e.g.Leftow (2011)).If there are, then even if it is better that God be able to be incarnate in some way or other, it does not follow that it is better that He be able to do it this way.
Turning to (d2), I add that if the Incarnation is possible, then there is a way for God to have both the divine nature and another.For on the Chalcedonian approach, that is exactly what it involves.So for (d2), Christians, at least, might give the truculent what they want.Apart from that, my treatment of (d1) and (d2) would just reapply points already made.I have argued, then, that truculence fails no matter how we interpret what the truculent demand.If it does, my approach to logical troublemakers stands.

Extending the moves
Not all troublemakers involve logical violations.Some violate mathematics.If essences beget necessities, some violate those.Any other sources of necessity (e.g.'laws of metaphysics') create still other sorts of troublemaker.But we can treat all troublemakers as I treat the logical ones.For a tradition going back at least to Augustine bases truths about mathematics and the natures of things in the content of divine ideas.That is, it holds that (for example) 2 + 2 = 4 because that is how God thinks about 2 and 4. That tradition is pretty much unanimous that God's nature dictates His ideas.On this approach, then, mathematics, essences, and 'laws of metaphysics' express God's nature.(The essences here include the divine essence, trivially, so the move applies also to theological troublemakers (Speaks (2018), 69-70).)So can any other source of necessity.This will yield ways to deal with these troublemakers that parallel moves I've made about logic.I submit, then, that the problem of troublemakers can be handled.Thus, as promised, I've argued that neither horn of Speaks's dilemma genuinely gores.

Notational variants redux
For Speaks, GCB theology is a 'notational variant' on greatest possible being (GPB) theology if an item is conceivable just if it is possible (ibid., 59, 69).GC 2 B theology is not in this sense or for this reason a 'notational variant'.However, one might reboot Speaks's 'variant' charge as follows.GPB theology works just like GC 2 B theology.For both, modal evidence justifies premises.The premises are strict parallels.GPB theology too can have two stages and ditch claims that appear possible in the first.So it can start where GC 2 B theology does and make just the same moves.So GC 2 B theology does not differ in practice from GPB theology. 21It is in a second way a 'notational variant' of it.I reply with a reminder.As Speaks sets them up, GCB and GPB theology each start from a modal principle and a greatness condition, and make parallel deductions from them.So they do run in parallel.Speaks builds this in.Further, there is a reason Speaks sets them up this way.His GCB and GPB projects are only ideal types, distinguished for purposes of discussion. 22Actual historical PBT does not distinguish these projects.Anselm, for instance, uses 'that than which no greater can be thought', but close inspection shows that for him, something is thinkable iff it is possible (Leftow (forthcoming c)).Some even argue that his 'can be thought' expresses alethic possibility (e.g.Smith (2014)).I myself suspect that he uses it because he sees 'thinkability' as our access to what's possible.Be that as it may, it is not clear whether he's doing GCB or GPB theology.He never distinguishes the two.Again, Leibniz's PBT works in terms of compatibility and not implying a contradiction (e.g.Leibniz (1965), 55-56).He defines being possible as not implying a contradiction (Leibniz (1985), 345).So he thinks of himself as doing GPB theology.But most philosophers now think that non-contradiction does not suffice for absolute possibility.If it does not, Leibniz was in fact working only in terms of one sort of conceivability.So it is not clear whether to call Leibniz a GPBer or a GCBer.Further, as Leibniz equates possibility with one sort of conceivability, in his own mind, GCB and GPB theology would not be distinct projects.
Speaks's GPB and GCB projects are idealizations.Historical PBT oftenperhaps usuallyisn't clearly doing just one of them.Speaks may mean his idealizations to illuminate what (he thinks) is really going on in historical PBT.If the idealizations are two ways to frame or analyse the same historical material, it is reasonable that they run in parallel.What matters for keeping the idealizations distinct is whether the projects are notational variants in Speaks's specific sensethat is, whether the conceivable collapses into the possible.I have argued that GC 2 B theology and GPB theology are not notational variants in this sense.On the other hand, perhaps Speaks means the idealizations to represent not what historical figures did, but what they should have been clearly doing, or what it would take to do PBT properly.Even so, the mere fact that they are versions of one projectperfect being theologyshould make it reasonable that they run in parallel.
Speaks offers a dilemma for GCB theology.I have argued that GCB theology can escape both horns.I have also suggested that the second way to be a notational variant is not a problem.So I submit that GCB theology remains a viable project.