Incompatible and Incomparable Perfections: A New Argument

Penultimate draft (corrected). Please cite the published version or contact the Author.

Incompatible and Incomparable Perfections: A New

Argument Against Perfect Being Theism

Jashiel Resto Quiñones

Purdue University

jrestoqu@purdue.edu

Abstract. Perfect being theism is the view that the perfect being exists and the property

being-perfect is the property being-God. According to the strong analysis of perfection, a

being is perfect just in case it exemplifies all perfections. On the other hand, the weak

analysis of perfection says that a being is perfect just in case it exemplifies the best

possible combination of compatible perfections. Strong perfect being theism accepts the

former analysis while weak perfect being theism accepts the latter. In this paper, I argue

that there are good reasons to reject both versions of perfect being theism. On the one

hand, strong perfect being theism is false if there are incompatible perfections; I argue

that there are. On the other hand, if either no comparison can be made between sets of

perfections, or they are equally good, then there is no best possible set of perfections. I

argue for the antecedent of this conditional statement, concluding that weak perfect being

theism is false. In the absence of other analyses of perfection, I conclude that we have

reason to reject perfect being theism.

Introduction

Many theistic philosophers in the Abrahamic tradition are perfect being theists. Perfect being

theism is the view that a unique perfect being exists and that being (uniquely) perfect is a

necessary and sufficient condition of being God. It follows logically that God exists. In an

attempt to describe what a perfect being would be like, if one existed, perfect being theists argue

that a being is perfect only if such a being instantiates certain properties (often called perfections

or great-making properties) that increase their possessor’s intrinsic value. However, what perfect

being theists want to achieve is giving necessary and jointly sufficient conditions for perfection.

Thus, a complete analysis of perfection would inform us what conditions are needed for a being

to be perfect.

There are two standard analyses of perfection that perfect being theists employ. On one

analysis, being perfect requires instantiating all perfections there are. Call this the strong

analysis. If correct, then perfect being theists would be committed to accept the proposition that

being perfect entails having all perfections. Recent developments in philosophical theology (see

Morris 1987; Nagasawa 2017; Muphy 2017; Leftow 2011) have cast doubt upon such an

analysis. The heart of the worry is that there might, for all we know, be incompatible perfections

For the main accounts of PBT see references in the Introduction. See also Leftow (2004) and Wierenga (2011) for

historical perspectives. For objections to PBT see Oppy (2011), Speaks (2014, 2016).

or, worse, there is evidence that there are such perfections. If so and the strong analysis is

correct, then it is impossible that there is a perfect being.

Considering this, another analysis of perfection has been developed. According to this

analysis, being perfect just requires instantiating the best possible combination of compatible

perfections (Nagasawa 2017; Morris 1987). Call this the weak analysis of perfection. Not only

does it have interesting consequences for the ontological argument, but also deals with the

previous worry generated by the strong analysis of perfection. That there are incompatible

perfections does not entail that there cannot be a perfect being because what is required is just

that a being instantiates perfections that are compatible and that those are the best among other

combinations. This way, perfect being theists dismantle the problem of incompatibility.

I do not, however, think that the weak version of perfect being theism is free of problems.

In fact, I show in this paper that it does not even completely avoid the problem of

incompatibility. More generally, I argue that we have good reason to believe that either no

comparison relation holds between sets of compatible perfections, or (if some holds) that they

are equally good. On both alternatives, it is false that there is a best combination of perfections.

Since the weak version of perfect being theism entails that there is such a combination, then by

modus tollens, that version is false.

The Problem of Incompatibility

A perfection is a kind of great-making property. A great-making property is a property the

exemplification of which increases the value of its exemplifier—i.e., increases the intrinsic value

of any thing that exemplifies it. In other words, a great-making property makes a positive

contribution to the intrinsic value of anything that exemplifies it. What kind of great-making

properties are perfections? They are absolute great-making properties, which means that they

increase the intrinsic value of its exemplifier (not its extrinsic value). Perfections are perfectly

exemplified absolute great-making properties, which means that they are exemplified to their

optimal degree. The optimal degree need not be its maximum degree but often is. An example of

optimality without maximality is perfect freedom. A being may be free to do irrational things but

this does not mean that such freedom makes this being better (Draper 2019). Plausibly, it is

maximal freedom consistent with rationality that is best. An example of optimality as maximality

is omniscience.

With this concept of perfection, let’s review the definition of PBT for ease of exposition.

PBT is the proposition that the perfect being exists and that (the property) being perfect is (the

property) being God. All strands of PBT claim this. We can identify two strands of PBT which

enrich our current definition by analyzing what being perfect is. There are two main analyses of

perfection that are the focus of this paper, namely, the strong and weak analyses of perfection.

On the one hand, the strong analysis of perfection says that a being, b, is perfect b iff b

exemplifies all perfections there are. On the other hand, there is the weak analysis of perfection

according to which b is perfect iff b exemplifies the best possible combination of compatible

perfections. Taking each analysis of perfection, we can define two strands of PBT accordingly:

(SPBT): the perfect being exists, being perfect is being God, and , b is perfect iff b

exemplifies all perfections.

(WPBT): the perfect being exists, being perfect is being God, and , b is perfect iff b

exemplifies the best possible combination of compatible perfections.

Note that SPBT and WPBT do not entail each other. One might think that SPBT entails WPBT

because, if all perfections are compatible, then the best combinations of perfections is the one

that contains all perfections. But this relies on the assumption that adding more perfections to a

collection makes them better—an assumption that we will get to later in the paper. But without

such an assumption, then entailment from SPBT to WPBT is blocked.

Is there any reason to prefer one strand of PBT over the other? Some philosophers think

there is.

One reason to prefer WPBT over SPBT is what I call here the problem of

incompatibility. To explain the problem, however, I need to define incompatibility. Let us take

that task next.

We say that some property(ies) is(are) incompatible iff they make an inconsistent set. A

set, S, of properties is inconsistent iff it is (broadly) logically impossible to exemplify all the

properties in S. Let ‘Φ’ be the set of all perfections. The problem of incompatibility starts with

the question, is Φ an inconsistent set? If it is, then no being can have all perfections. This seems

to contradict SPBT.

For brevity, let ‘’ be the sentence ‘Φ is an inconsistent set’. Assume for the sake of

the argument that SPBT is the correct strand of PBT, i.e., if PBT is true, then SPBT is true. (I will

use this assumption in 4-9 below.) Then,

1. If , then PBT is false. [premise]

2. . [premise]

 3. PBT is false. [MP 1,2]

We can establish premise 1 easily by formulating a conditional proof (CP).

4. ). [Assumption for CP]

5. , it is logically impossible that b exemplifies all perfections in . [Definition of

Incompatibility]

6. SPBT is false. [5 definition]

7. If PBT is true, then SPBT is true. [Assumption of Strong Analysis]

8. PBT is false. [6, 7 MT]

9. If ) then PBT is false. [4-8 CP]

Cf. Nagasawa (2011, 2017); Murphy (2017).

The crucial question, however, is whether premise 2 is true.

I do not have the space to argue for 2, but two points are worth noting. First, I can direct

the reader to many arguments that try with varying degrees of success to establish 2 by arguing

that a pair of perfections is incompatible. For one, even though Nelson Pike (1969) showed that

moral perfection and omnipotence are compatible, Morriston (2001) argues persuasively that

necessary moral perfection is incompatible with (necessary) omnipotence.

Kretzmann (1966)

argues that omniscience and immutability are incompatible. Another argument challenges the

compatibility between moral perfection and perfect freedom on the grounds that moral perfection

entails moral praiseworthiness, which in turn entails being free (in the libertarian sense) to do

wrong. Yet another argument claims that perfect mercy and perfect justice are incompatible

because the former sometimes requires not giving others what one deserves while the former

requires always giving others what they are due. Still another argument claims that existing

necessarily is incompatible with being a concrete entity.

In other words, there are many

arguments for the conclusion that a pair of perfections is incompatible. If we take the conclusion

of each argument and put them in a disjunction, the likelihood that  increases.

The other point comes from a discussion with Paul Draper on incompatibility arguments.

Note that the main strategy of incompatibility arguments is to identify a pair of divine attributes

and see if they are compatible or not. Thus, incompatibility arguments against PBT can exploit

the tension between God’s so-called metaphysical attributes (immutability, impassibility,

timelessness, simplicity, necessary existence, etc.) and God’s perfect agency. But if one defines

‘God’ as the perfect agent instead of the perfect being, arguments that exploit this tension don’t

work. In other words, it is easier to hit PBT than to hit perfect agent theism. Now, the fact that it

is easier to hit PBT with one of these arguments makes PBT more at risk of being false and,

therefore, less probable than perfect agent theism. Therefore, from an evidential point of view, I

am inclined to think that 2 is more probable than not. Hence, be it through demonstrative or

evidential support, it seems 2 wins against its contradictory.

If what I’ve said is true, theists are justified in worrying about holding SPBT. But assume

that I am wrong. There is another problem for PBT-ists. One might think that 2 is possibly true in

the epistemic sense of ‘possibly’. Assuming that it is, we can formulate a modal-epistemic

version of the incompatibility problem as follows.

1*. Necessarily, if , then PBT is false. [premise]

2*. Possibly, . [premise]

 3*. Possibly, PBT is false. [MP 1,2]

Though this is not at odds with the proposition that God exists necessarily (as many theists

believe), it is an evidential challenge for those who endorse SPBT. I find 2* plausible. For all we

know, there may be perfections too complex for us to understand, let alone grasping the

entailment relations between them (see objection V below for a related concern). At this point,

See Mawson (2002) for an objection and Morriston (2003) for a reply to Mawson.

This was pointed out to me by Paul Draper.

then, it seems that the burden of proof is on the theist who endorses PBT. Since it is the case that

if one has reasons to believe 2* is true, then there is tension between those reasons and the

common theistic belief that PBT is true, I submit that the problem of incompatibility is a problem

those who endorse SPBT should take seriously.

Either way, I think theists find this sufficiently troublesome to reject SPBT. In fact, it is

the assumption that being perfect entails having all perfections (7 above) that generates the case

for 1. And since 2 is sufficiently plausible, 3 goes through. This is just one

reason many theists

reject SPBT and accept WPBT, because endorsing WPBT opens the door to reject 1 through

rejecting that the strong analysis of perfection is the correct one.

Also note that endorsing

WPBT deals with the modal version of the incompatibility problem since the mere possibility of

there being incompatible perfections does not entail that, possibly, there is no perfect being.

These are strong reasons to prefer WPBT over SPBT.

The Problem of Incomparability

Theists have good reasons to think that 2 is probably true, thus giving them reasons to

prefer WPBT over SPBT. In this section, I argue that there is a pressing argument against WPBT,

namely, the problem of incomparability. As with the problem of incompatibility, we need to

understand what incomparability is before getting into the problem. Let’s take this task next.

There are three “canonical”, dyadic value comparison relations: better-than, worse-than,

and equally-good-to. According to some, the Trichotomy Thesis (TT) holds:

(TT)  if x is comparable with y, then either x is better than y; y is better than x; or x

and y are equally good.

The consequent in TT is read “exactly one of the relations better-than, worse-than, or equally-

good-to holds between any two items” because the three relations are obviously mutually

exclusive.

Now, as a definition of incomparability, we can say that:

(IP)  x is incomparable with y iff neither x is better than y; y is better than x; nor x

and y are equally good.

Note that TT and IP can both be true. Furthermore, many philosophers (Chang 2002, 2014;

Griffin 1986; Hsieh 2005; Parfit 1984) suggest that there are cases where no canonical

Though not the only nor (perhaps) the main one. Theists (and non-theists) might reject the strong analysis, not

because it leads to the conclusion that God does not exist (that would be bad reasoning), but because they just don’t

think it is true by definition that a perfect being is perfect in every way. Instead, what it means to be perfect is to be

as intrinsically valuable as it is possible to be.

In fact, this was Nagasawa’s (2017) brilliant insight. One can reject 1 on the grounds that being perfect just

requires exemplifying the best possible combination of compatible perfections. This way, Nagasawa, by rejecting 1,

provides an argument against the problem of incompatibility without having to tackle each incompatibility argument

for 2.

I am using Chang’s (2002) version of TT. But TT can be distinct from the Comparability thesis. See Espinoza

(2008). See Steele & Stefánsson (2020) for a different terminology. For a novel defense of TT see Dorr & Nebel &

Zuehl (2022).

comparison relations hold between certain things while arguing that such things are still

comparable because the relations in TT do not exhaust the comparison relations. For example,

Chang would say that (for some people

) no canonical comparison relation holds between being

a musician and being a lawyer, but that does not entail that they are incomparable since there is a

fourth relation, parity, which holds between them. Thus, many philosophers think that IP is not

an adequate definition of incomparability precisely because TT is false.

For our purposes, however, I take two items to be incomparable just in case no canonical

relation between them holds. (I will consider deviations from the traditional view in the

“Objections” section.) Hence, certain things are incomparable when and only when none of these

three relations hold between the things under consideration. For instance, a career as a musician

and a career as a lawyer are incomparable since neither is better than the other nor are they

equally good.

A few other clarifications are required about the concept of (in)comparability. Chang

(2014, 3-5) says that the comparability relations (better-than, worse-than, and equally-good-to)

are three-place functions: x is (in)comparable to y with respect to value V. In other words,

comparisons always require a covering value. Hence, if we compare two things, it must be done

relative to a certain value. I assume this is true. I’m also inclined to believe that it is true only if

relativizing comparison to values does not remove the possibility of comparing two things

relative to all values, i.e., all things considered. Chang (2014, 3) says that covering values can be

specific like the value of being-pleasing-to-my-grandmother. So, it seems that the same goes for

generality; covering values can be generic like being-intrinsically-valuable. Furthermore, I also

assume that one can make generic value comparisons all things considered. This would mean

comparing some things with respect to all values. I think Chang’s account is consistent with

these points and, therefore, it is safe to assume them for the sake of the argument.

Finally, note that we are comparing perfections (properties of some kind), not careers or

objects like tables and chairs. This leads me to note the following. Either perfections are values,

or they are not. If not, then we compare them relative to some value. If they are, however, can we

compare values to other values? Can one value be better than another with respect to some other

value? Is being loving better than being rational? I think these questions are coherent. Being

loving is better than being rational with respect to pleasing-my-grandmother-who-suffers-from-

Alzheimer’s disease. It seems, then, values too are comparable. Therefore, I will allow myself to

extend the conceptual framework of comparability, not just to things like tables or careers, but

also properties that could be considered values themselves (e.g., power, agency, etc.).

Enter incomparability. In addition to assuming that Φ being an inconsistent set, I am also

assuming that Φ has the following property: the members of Φ that are incompatible are

incomparable as well, i.e., no canonical comparative relation holds between them. Suppose 

Chang would also say that for other people being a lawyer is better than being a musician, for instance, those who

dislike or do not have a talent for music, and vice versa!

I think a logical consequence of B(x, y, V) is that it is always true that B(x, y, x) for any y. Anything is better than

any other thing with respect to the first thing. The Eifel Tower is better than New York relative to being-the-Eifel-

Tower. Similarly, pleasure is better than compassion relative to being-pleasurable.

  . Then, not only Φ is inconsistent, but also a is incomparable with b, b with c, and a with

c, and no member is incomparable with itself. This is extremely relevant to the argument so let’s

subscript 



to state that Φ has this incomparability property.

The problem of incomparability can be stated as follows. Assume for the sake of the

argument that WPBT is the correct strand of PBT, i.e., if PBT is true, then WPBT is true. Then,

10. If 



 then PBT is false. [premise]

11. 



.

[premise]

12. Therefore, PBT is false. [MP 10, 11]

10 is true. To see this, let us take some concrete examples. Assume, for the sake of illustration,

that   



 



     . Here p

and p

, along with 1, 2…, n are perfections since Φ is the

set of all perfections. Assume Φ is an inconsistent set because p

and p

are incompatible. Then

   members of Φ entails the negation of the remainder. Therefore, p

(along with 1, 2, …, n)

entails ~p

, and p

(along with1, 2, …, n) entails ~p

. This implies that a being cannot exemplify

all the perfections. But note that there are other perfections that are compatible (thus co-

exemplifiable) with only one of the elements of {p

, p

}, namely, {1, 2, …, n}. Let’s continue to

denote all perfections that are not in {p

, p

} with natural numbers. In this case, there would be

two (maximal) sets of perfections that a being could exemplify:

Set 1

Set 2

Since p

and p

are incomparable, then no comparison relation between the sets containing them

(i.e., Set 1 and Set 2) holds. And since Set 1 and Set 2 have more perfections than    ,

Recall, I(Φ) abbreviates the sentence ‘the set Φ of all perfections is inconsistent’. By adding the subscript, I intend

‘



’ to be read as ‘the set Φ of all perfections is inconsistent and those members of Φ that are incompatible with

each other are incomparable’.

Note that this is true because of the monotonicity of classical logic (which is the one I employ here). What this

means is that a valid argument cannot be made invalid by adding new premises. If p entails q, then p and r and s

entail q also. In our case, if a perfection p

entails ~p

, then p

and q

and… entail ~p

also, where q

is a

perfection.

they are better than    ,

so one cannot take p

and p

out (so to speak) because then a

being exemplifying     would not exemplify the best set of perfections.

Suppose now that   



 



 



    . Let’s go through the motions again. All

members of Φ are perfections since Φ is the set of all perfections. Since Φ is an inconsistent set,

   members of Φ entail the opposite of the reminder. Suppose (as before) that p

1-3

are

incompatible with each other. Therefore, p

and p

(along with 1 through n) entail ~p

; p

and p

(along with 1 through n) entail ~p

; and p

and p

(along with 1 through n) entail ~p

. This

implies that the perfect being cannot exemplify all three. Again, note that there are other

perfections that are compatible (thus co-exemplifiable) with at most two of the elements of {p

, p

} that are incompatible with each other. Denoting these other perfections with natural

numbers, these are the sets (maximal) of perfections that the perfect being could exemplify:

Set 1

Set 2

Set 3

Now, p

, p

, and p

are incomparable, which means that p

is incomparable with p

and p

, and p

is incomparable with p

. Since this is so, then no comparison relation holds between them. It

follows that Sets 1-3 are incomparable as well. Moreover, since Sets 1-3 have more perfections

than     they are better than    . Thus, the perfect being cannot exemplify

    since it is not the best combination possible nor it could exemplify Sets 1-3 since

none is the best set of perfections to exemplify because they are incomparable.

We could go on and on, but these examples suffice to prove two important things,

namely, that (i) whenever







  (i.e., the cardinality or size of Φ is n) there are n subsets of Φ

that have the greatest possible number of compatible perfections, and (ii) that the greatest

possible number of compatible perfections is all-minus-one. Let’s explain each in turn.

(ii) is obvious. Since Φ is an inconsistent set, then no being can exemplify all perfections

in Φ. But a being exemplifying less than all-minus-one perfections would not exemplify the

greatest possible number of perfections that can be co-exemplified. Since an intuitive

aggregation principle

says that a set of intrinsically good things is better than another set if the

An anonymous referee points out correctly that this claim is dubious. I address this worry in the Objections

section below (see objection II).

See Rubio (forthcoming, 8).

former has more things than the latter; and since a being is perfect (according to WPBT) iff that

being exemplifies the best possible combination of compatible perfections, it follows that a

perfect being cannot exemplify less than all-minus-one perfections.

Since it is impossible to

exemplify all perfections, and it is impossible for a perfect being to exemplify less than all-

minus-one perfections, it follows that a perfect being exemplifies all-minus-one perfections.

In the examples above, this is captured by the ps. Since the ps are incompatible, then

there is always one of them that we must “take out” so to speak. Note then that whenever









, the new compatible set we form by taking out one of the ps will have all the ps possible plus

all the others (those denoted by natural numbers). But this entails that this the greatest possible

number of perfections in this new set is all-minus-one. Therefore, (ii) is true.

(i) follows from Φ being an inconsistent set. Recall that if Φ is inconsistent, then   

elements of Φ entail the negation of the remainder, so no being can exemplify all perfections in

Φ. A being can exemplify at most    perfections in Φ. The question is, how many consistent

subsets of Φ with the greatest possible number of elements can we form? The answer is we can

form n compatible subsets of Φ. The way to prove this is using the equation 





 













What this equation allows us to do is to count how many sets with k elements we can form by

taking elements from another set with n elements. Since a consistent combination of perfections

with the greatest possible number of elements must have    members (as shown in the

previous two paragraphs), we substitute k for    and get the following result.







 

























 .

This shows that whenever







 , there are n subsets of Φ with the greatest possible number of

compatible perfections. Adding all remaining perfections (represented by natural numbers), we

get that there are n compatible proper subsets of Φ with the greatest possible number of

elements. In the examples above, this is captured by the number of columns. A generalized

illustration looks like this. Let







 .

See footnote 12.

n! is the product of all the integers   . In other words,    



  







  



    .

Set

n-2

Set

n-1

Set

omit 





















omit 





















omit 















omit 



…





omit 















omit 















omit 















The number of columns is n corresponding to the n consistent subsets of Φ while the number of

rows (obviously excluding the first row) is    corresponds to the cardinality of each subset of

Φ which is the greatest possible number of compatible perfections.

Since all the consistent proper subsets of Φ are incomparable, then there is no best

possible set of compatible perfections. If so, then WBT is false. But since we are assuming that if

PBT is true then WPBT is true, it follows that PBT is false. Because we arrive at our conclusion

from the assumption that 



, we conclude that if 



 then PBT is false.

The challenge is to prove 11. The strategy here is to take those perfections that are

thought to be incompatible and argue that they are also incomparable. Since I already motivated

the case for 2—specifically, that Φ is an inconsistent set—I need not repeat myself here. Hence,

what I need to show is that those properties are incomparable. To this end, I offer two reasons to

support 14, namely, that the pairs of perfections thought to be incompatible are also

incomparable.

The most cited argument to establish incomparability is the so-called small improvements

argument (SIA).

The argument goes like this. Suppose that two value-bearers, a and b, are such

that neither is better than the other. If a and b were equally good, then a small improvement in

either one would make the improved one better than the other (thus making one prefer the

improved over the non-improved). Thus, suppose we improve a, having as a result a+. If a and b

were equally good, then a+ would be better than b. However, it is possible—indeed, plausible—

that even when a+ is clearly better than a, it is false that a+ is better than b. This shows that a

See Broome (1997), Espinoza (2008), Gustafsson & Espinoza (2009), Gustafsson (2013), and Anderson (2015)

for arguments claiming that SIA fails to establish incomparability. See Carlson (2011) for criticism of Gustafsson &

Espinoza.

and b are not equally good. Combined with the starting assumption—that neither is better than

the other—then we have a case of incomparability: a and b are such that neither is better than the

other nor they are equally good.

We can clearly see the validity of this argument by stating it formally. I take Espinoza’s

(2008) formulation with some notation changes.

13.    [Assumption]

14. 



 [Truth]

15.   



  



 [Indifference Principle]

16. 



 [Assumption]

17.   



 [from 15, 16]

18.  [from 14, 17]

19.      [from 13, 18]

13 and 16 are assumptions made in cases under consideration. For instance, if we take a and b

above to be a career as a professional musician and a career as a lawyer, respectively, it is clear

that 16 and 19 come out true. I take 14 to be an analytic truth. If the career as a musician has n

amount of value, an improvement to the career (say, better pay) makes the improved career better

than the unimproved one. It is from the meaning of the word ‘improve’ that this follows. The

indifference principle states that if there are two things that are equal in value, adding value or

improving one “will tip the scale in favor of the improved option” (Spinoza 2008, 130). I take

this to be extremely plausible, if not (necessarily) true. Therefore, the question is whether the

assumptions—16 and 19—are true when applied to perfections.

The Appeal to Radical Difference. One way to show that 16 and 19 are true with respect

to the pairs of perfections considered above (necessary moral perfection/necessary omnipotence,

perfect mercy/perfect justice, perfect freedom/moral perfection, omniscience/immutability, etc.)

is to argue that they are radically different value-bearers. True, they are all perfections. But when

confronted with the question “is it better for God to be knowledgeable or unchanging?” one

might justifiably answer that those two perfections are too different to say that one is better than

the other. Think about the case of careers in music and law. One the one hand, a career in music

gives one artistic liberty and promotes creativity. On the other hand, a career in law gives one

economic stability and promotes critical thinking skills. One is better with respect to some

things, and the other is better with respect to others. But all things considered, it is extremely

plausible that neither is better than the other precisely because what they offer is radically

different. The same applies to perfections. Immutability, perhaps, implies metaphysical stability

and uniformity while knowledge provides rationality and other virtues. But it seems to me that

these perfections (and what they provide) are sufficiently different to say that one is not better

than another and to say that it is also false that an improvement won’t make one better than the

other.

The Appeal to Optimality. Another way to motivate 16 and 19 with respect to perfections

is by noting that since we are dealing with perfections God would exemplify if he existed, we are

dealing with perfections exemplified perfectly (see “The Problem of Incompatibility”). Put

differently, the great-making properties God would exemplify would not admit further

improvements. Thus, one cannot provide reasons against 16 and 19 on the grounds that one

perfection is exemplified to its optimal degree while the other is not. For example, one cannot

say that exemplifying power is better than exemplifying freedom because power is exemplified

to the optimal degree while freedom is not. Moreover, since all perfections we are considering

are absolute perfections—properties the possession of which makes its bearer intrinsically more

valuable—it is extremely difficult to affirm that possession of one perfection is better than

another since not possessing either one would result in a decrease of intrinsic value. But that they

all make their possessor more intrinsically valuable does not imply that they are equally good.

For all we know, they might make its possessor more intrinsically valuable in diverse ways. (In

fact, they all make their possessor more valuable with respect to themselves.) Therefore, it is

plausible that 16 and 19 are true with respect to perfections.

These two factors give evidence in favor of 13 and 16 as applied to perfections. But even

if they fail, one might defend the argument by affirming that the comparable perfections are

equally good to each other. In other words, even if they are comparable, one might plausibly

think that the most plausible relation that holds between them is that they are equally good. But

since the WPBT-ist is committed to say that it is the best combination of perfections (which is

equivalent to say that such combination is better than any other), their being equally good

suffices to show that WPBT is false. I do not claim to have proven that WPBT is false, but I

think that the WPBT-ists must show that there is a best combination of compatible perfections.

And what this requires is explaining how one perfection can be better than another, something

that has not been offered yet.

Objections

This is a long and somewhat complicated argument, which is open to an enormous number of

objections. In this section, I address what I take are the main objections to my argument, namely,

(I) further weakening WPBT, (II) the appeal to organic unities, (III) the appeal to a fourth

comparison relation, (IV) the appeal to indetermination, and (V) skeptical theism.

(I) We saw that theists have reason to move from SPBT to WPBT. But now consider a

weaker formulation of PBT than WPBT.

(EWPBT) God has the best compatible set of comparable perfections.

If theists are prepared to countenance a from SPBT to WPBT, it is not clear why theists should

not also be prepared to countenance a move from WPBT to EWPBT. And even if we have

reasons to believe that there is no uniquely best set of compatible and comparable perfections,

this is a problem that was already present in WPBT. Staying with the problem but solving

That is: if we consider all of the compatible sets of comparable perfections, God has the best of those. More

precisely, suppose that there is a "largest" set of comparable perfections. Consider all of the compatible subsets of

that set. Whichever of those is best is what God has.

another (i.e., the problem of incomparability) is still a move toward a solution. Furthermore, if

there were something that instantiated the best set of compatible and comparable perfections, it is

not obvious that it would not be a good candidate for God. Specifically, it does not seem that this

being would not be worthy of worship.

Thus, moving from WPBT to EWPBT suffices to

undermine my argument.

This is a multifaceted objection; let’s respond to the claims in reverse order. (I.1) The

problem with the claim that plausibly a being having the best set of compatible and comparable

perfections is worthy of worship is that for PBT-ists worthiness of worship is neither necessary

not sufficient for being God (see Murphy 2017 for one example). (I.2) The claim that if there

were something that instantiated the best set of compatible and comparable perfections it would

be God seems incorrect for two reasons. First, it assumes that there is such a set. But this is far

from obvious. The PBT-ists would need a way to show that perfections are comparable, but until

they do, the burden of proof is on their shoulders and prospects of finding such a way to

determine comparability among perfections seem dim. Second, even if there is a best set of

compatible and comparable perfections, a being exemplifying the perfections in such set would

not be God. For surely a being that has the best set of compatible and comparable perfections

wouldn't count as God according to PBT if some other being were better because it had all those

perfections plus some incomparable ones.

A last word on there being—for all we know—no uniquely best set of compatible and

comparable perfections. If PBT by itself of with the aid of argument (e.g., the ontological

argument (Nagasawa 2017)) we might end up concluding that multiple entities that are as

valuable as it is possible exist. In other words, once we entertain that there are distinct equally

valuable sets of perfections, polytheism becomes a possibility. Of course, this may not be a

problem for some. But since the audience toward which this argument is directed are perfect

being monotheists, this surely is a problem for them. Thus, even if appeal to this claim solves

some problems, it might create other equally difficult problems.

(II) One problem with my argument is that I assume that sets of perfections are just that,

mere sets or collections of perfections. By assuming this, I am implicitly claiming that there is no

more to these collections other than their trivial and logical relations. But this is false. Since

these are the sets of perfections God would exemplify if he existed, we need to consider them

organic unities, that is, sets of properties that exhibit non-trivial, non-logical relations that affect

the value of such collections. If we consider them as such, then it is not true that a set with more

intrinsically good things than another is better. In fact, it may worsen it. But since this is a crucial

step in my argument against WPBT, by rejecting the aggregation principle one is entitled to

reject the conclusion of such argument.

One attempt to answer this objection is to claim that organic unities have value in virtue

of the extrinsic properties they have. For instance, one can claim that a property that makes

something an organic unity is that thing’s teleology. A group of people that ought to carry a

heavy rock is an organic unity. The non-trivial, non-logical relations matter in this case. For

My gratitude to an anonymous reviewer for raising this objection.

example, adding one more person to the group might not make the group better since the rock

may have certain dimensions such that it would be a hassle for six people to carry. Five people

would be better since it would be easier for them to carry. But what makes this so is the purpose

or end the group has, namely, carrying the rock, which is an extrinsic factor to the group. And

even if what makes an organic unity is not solely the thing’s teleology, it is plausible that what

makes it so are other extrinsic factors that contribute to the value of the thing. In the case of

WPBT, however, we are not considering any extrinsic factors, let alone any teleology, relative to

God. We are considering properties the possession of which makes their possessor intrinsically

more valuable, i.e., independent of any extrinsic factors.

This reply is inadequate. For it assumes that an organic unity’s value rests upon its

extrinsic properties only. But this is clearly false. Take the set Φ of all perfections and let Δ and Γ

be small proper subsets of Φ. (By ‘small’ I mean proper subsets of Φ with less than the biggest

number of compatible perfections a subset of Φ can have. Recall that we are assuming that Φ is

an inconsistent set.) Suppose that Δ and Γ have no members in common (i.e., their intersection is

the null set)

and that there is a perfection, p, in Φ such that    and    are consistent

sets. It might be the case that p makes a different contribution to the intrinsic value of an object

that exemplifies the perfections in    from the contribution p makes to an object that

exemplifies the perfections in   . In other words, p’s co-instantiation with distinct

(collections of) perfections could (and plausibly does) make different contributions to the

intrinsic value of the being that exemplifies them simply because what p contributes to intrinsic

value in the presence of the perfections in Γ is different from what p contributes to intrinsic value

in the presence of the perfections in Δ. Note, moreover, that we may safely assume that all

properties are intrinsic, and that the value of an object exemplifying the collections herein are not

dependent upon extrinsic factors. This shows that the value of some organic unities does not

depend on extrinsic factors. Therefore, the response above does not work.

Here is an adequate response to the problem. That organic unities can be made only of

intrinsic properties (as shown above) does not affect this argument. In fact, it makes the problem

of incomparability worse. Why? Because if a perfection can make different contributions to

different packages of perfections (and thus to the objects that exemplify them), it seems we have

strong reason to believe that no comparative relation holds between the packages of perfections.

In other words, we can appeal to the radical difference between, not only the perfections, but

between the different combinations of perfections to claim that they are incomparable. In the

case above, p could make different contributions to the intrinsic value of an object depending on

what other perfections are co-instantiated with p by that object. Not only the contribution that p

makes to these combinations of perfections is different (perhaps radically different), but also the

I take this case for ease of exposition, but the same applies with other cases (e.g., Δ is a subset of Γ or vice versa;

Δ and Γ have some (but not all) members in common but neither is a subset of the other).

I am very thankful to an anonymous reviewer for pointing this out to me. The reviewer also rightly points out that

“it might be said that this objection overlooks that appeal to optimality: a perfection is the optimal degree of a great-

making property (see p.2). But whether there is such a thing as “the optimal degree of a great-making property”

depends upon whether what a given perfection contributes to intrinsic value depends upon what other perfections it

is coinstantiated with.

relations between perfections themselves and between combinations of perfections are

(metaphysically) very complex so that it would not be surprising at all if these combinations

were incomparable or that their value were indeterminate or on a par. Thus, if we allow that some

perfections, p

…p

, make different contributions on the intrinsic value of an object depending on

which perfections p

…p

are co-instantiated with, then the problem of incomparability becomes a

more difficult problem of the PBT-ist.

In fact, the assumption that collections of perfections are not organic unities works better

for the PBT-ist. The perfect being theologian has a (more or less) clearer way to say which

combination of perfections God exemplifies—namely, the more the better. But if they reject the

“the more the better” claim, we not only have a reason to believe that those combinations are

incomparable (as argued above), but we are also left in the dark to how that combination looks

like in terms of the value it confers to the object that exemplifies it. Therefore, not only this is a

problem for PBT, but it is also a problem for perfect being theology as a method to know how

God is like. If the contributions perfections make in the intrinsic value of an object depend upon

their co-instantiation with other perfections, we seem to undercut perfect being theology as a

(more or less) clear method to know how God is like, a method PBT-ist would like to hold on to.

(III) It is well-known that SIA is used, not to prove that there are things that are

incomparable, but as an intermediate step to argue for a fourth comparison relation like parity

(Chang 2002, 2014).

I focus on Chang’s parity relation. An objection to my argument is that

one can appeal to parity and say that incompatible perfections are not incomparable but are on a

par or roughly equal. If this is the case, then one can affirm that perfections are neither

incomparable nor equally good but are on a par, thus rejecting 14 and making my argument

unsound.

I do not think this strategy works since I can formulate 14 in terms of parity in the

following way:

11*. 



.

If the incompatible perfections of Φ are on a par, then there is no best possible set of compatible

perfections. The resulting subsets of Φ would be on a par. In debates about deliberation, if n

options are on a par with each other, it is rationally permissible to choose one over another even

though it is not the case that one is better than another. (Chang (2002, 2014) has a story to tell

about how this is the case.) We are not talking about rational deliberation, however. We are

talking about perfections. If the incompatible perfections of Φ are on a par with each other, then

one (imperfectly) analogous claim to that of being permissive to choose among options that are

on a par would be that, if God existed, he could exemplify any on a par subset of Φ.

The problem with this claim is that it would be a brute fact. We cannot explain that God

exemplifies this set of perfections because that is the best possible set since there is no such set.

This would undercut the whole purpose of perfect being theology, namely, figuring out how God

Rough equality (Griffin 1986), rough comparability (Parfit 1984), or clumpiness (Hsieh 2005) could be seen as a

fourth relation.

would be if he existed. If it is a brute fact that God exemplifies a particular subset of Φ then there

is no further explanation as to why God exemplifies such perfections. But then, perfect being

theology would be bankrupt as a method to discover and explain how God would be if he

existed. In fact, PBT would be uninformative since, if we didn’t know what set of perfections

God exemplified, we would not know what being perfect consisted of in the first place.

Therefore, I submit, appealing to parity does not work to undermine the argument against WPBT

(and PBT generally).

(IV) A claim put forward against IP is that the fact that no canonical comparative relation

holds between two things does not entail that they are incomparable since it can be

indeterminate. It is indeterminate that p iff it is neither true nor false that p. Let ‘D’ stand for a

determination operator and ‘I’ for the indetermination operator. This way,  iff   

and  iff   .

Espinoza (2008, 134-135) shows that if we assume vagueness anti-

symmetry (if , then 







 







), the following sets are consistent.

s.1 







 







 









s.2 







 







 









s.3 







 







 









s.4 







 







 









s.5 







 







 









s.6 







 







 









s.7 







 







 









Therefore, one could object that even if no canonical relation holds between a pair of things, that

does not entail that those things are incomparable. What it entails is that either they are

incomparable, or it is indeterminate that they are equally good. One could, then, argue that s.7

holds, granting that no perfection is better than the other while claiming that it is indeterminate

that they are equally good. Since I have not shown that , then I cannot conclude that

they are incomparable. Therefore, my argument fails.

Again, I note that I have stated my argument in such a way that it works if the perfections

are equally good. I have defended the claim that they are incomparable since it seems to me more

plausible than the claim that perfections are equally good. Still, if this objection goes through, it

will supply a reason to be skeptical of my argument. I do not think, however, that this objection

affects my argument negatively. For if s.7 holds, i.e., it being neither true nor false that the

perfections are equally good does more damage to the WPBT-ist than to my argument. If it is

determinately true that no perfection is better than the other; and if it is indeterminate whether

they are equally good, the WPBT-ists are in trouble since now there is no fact about whether the

subsets of Φ are equally good, which makes an appeal to bruteness (as with parity above) much

harder to swallow. If p is a brute fact, then . So, under indetermination, there is no best

See Fine (1975) for an elaborate trivalent logic.

possible combination of perfections, and there is no brute fact about whether the subsets of Φ are

equally good.

Therefore, even if this affects the validity of SIA, it takes WPBT with it down the

abyss.

(V) The last objection I want to consider is skeptical theism. Skeptical theists affirm that

we have no good reason to believe that our knowledge of actual goods/evils and their entailment

relations is not representative of all the possible goods/evils and their entailment relations

(Bergmann 2008). This is closely related to judgments of value. Thus, I imagine a skeptical theist

affirming that we are in the dark about what relations other than entailment hold between

perfections. Nevertheless, if we are in the dark with respect to what comparison relations hold

between perfections, we cannot even affirm that they are equally good, incomparable, or even

indeterminate. We should withhold judgment about such things. If so, then there is an

undercutting defeater for my claims and my argument does not go through.

I do not think WPBT-ists want to go this route since it is a double-edge sword. If we are

really in the dark about such matters, then we are also (at least partially) in the dark about what

God is like. God could exemplify any subset of Φ, but we would not be able to discern which

one. In fact, this kind of skepticism may even make us unable to know whether a certain property

is a perfection. If so, we would not just be in the dark about which subsets of Φ God would

exemplify, we would also be in the dark about what properties would count to be elements of Φ.

This is the reason I think skeptical theism does more damage than not. Therefore, if one endorses

skeptical theism, one provides an undercutting defeater to my argument at the cost of disabling

our capacity to know what God is like. A remarkably high price to pay.

Conclusion

Two versions of PBT have been analyzed. According to SPBT, the perfect being must exemplify

all perfections. If there are incompatible perfections, it is impossible for the perfect being to

exist. There are plausible arguments for the claim that there are incompatible perfections, and

even the possibility of there being such perfections puts the cherished belief that God exists

necessarily at jeopardy. This is sufficient for the PBT-ist to reject SPBT and adopt another

version of PBT.

According to the other, now most accepted version of PBT, i.e., WPBT, what is required

for a being to be perfect is that it exemplifies the best possible combination of compatible and

compossible perfections there is. We saw that if WPBT is true, then there must be a comparison

relation that holds between the combinations of perfections. After all, that there is a best

combination of perfections is to say that one combination is better than all the others. However, I

argued that if there are incompatible and incomparable perfections, then WPBT is false precisely

because, by virtue of incompatibility and incomparability, either no comparison relation holds

between combinations, or if one comparison holds, it is that they are equally good. Either way,

we can conclude that there is no best combination of perfections.

Note that this is true even if any s.1-7 holds.

An important consequence of my argument is that PBT raises a host of issues about

value, comparability, consistency, and adequacy conditions for a definition of ‘God’ seldom

addressed in discussions on PBT. The argument presented here is one attempt to get those issues

out to the light and advance philosophical inquiry on these topics among philosophers and

theologians.

Acknowledgements

I am grateful to Paul Draper and Troy Seagraves for helpful discussions and reading earlier drafts

of the paper, and to an anonymous reviewer for helpful and insightful comments.

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