Thursday, 29 June 2017

Deformed Epistemology

For some reason, there are some christians who find calling Plantinga’s “Reformed Epistemology” by another name, namely “Deformed Epistemology”, insulting.  They called it name-calling.  I think that’s a little unfair, so I’m going to put a little effort into defending the use of the term “deformed epistemology”.

First, we need to look at what is being either reformed or deformed – epistemology.  This is what Wikipedia has to say (I’ve tidied up the list though):

Epistemology is the branch of philosophy concerned with the theory of knowledge. Epistemology studies the nature of knowledge, justification, and the rationality of belief. Much of the debate in epistemology centers on four areas: the philosophical analysis of the nature of knowledge and how it relates to such concepts as truth, belief, and justification; various problems of scepticism; the sources and scope of knowledge and justified belief; and the criteria for knowledge and justification.

Alternatively, there is the Stanford Encyclopedia of Philosophy entry:

Defined narrowly, epistemology is the study of knowledge and justified belief. As the study of knowledge, epistemology is concerned with the following questions: What are the necessary and sufficient conditions of knowledge? What are its sources? What is its structure, and what are its limits? As the study of justified belief, epistemology aims to answer questions such as: How we are to understand the concept of justification? What makes justified beliefs justified? Is justification internal or external to one's own mind? Understood more broadly, epistemology is about issues having to do with the creation and dissemination of knowledge in particular areas of inquiry.

the study or a theory of the nature and grounds of knowledge especially with reference to its limits and validity

Now I could go on adding text from the IEP (“Epistemology is the study of knowledge”), other dictionaries (like “(epistemology is) a branch of philosophy that studies the origin, nature, methods, and limits of human knowledge”), other encyclopedias (like Britannica: “Epistemology (is) the study of the nature, origin and limits of human knowledge”) and papers on epistemology (JT Tennis in Epistemology, Theory, and Methodology in Knowledge Organization: “Epistemology is how we know”), but I think the point has been adequately made already – epistemology is about knowledge.

If we are going to reform (or deform) epistemology, then we are going to reform (or deform) something about how we interact with or think about knowledge. 

So, let’s look at Plantinga's deformed epistemology and specifically his "proper functionalist version of epistemic externalism" which he summarises in Warranted Christian Belief (p133):

Put in a nutshell, then, a belief has warrant for a person S only if that belief is produced in S by cognitive faculties functioning properly (subject to no dysfunction) in a cognitive environment that is appropriate for S’s kind of cognitive faculties, according to a design plan that is successfully aimed at truth. We must add, furthermore, that when a belief meets these conditions and does enjoy warrant, the degree of warrant it enjoys depends on the strength of the belief, the firmness with which S holds it

Note that, according to Plantinga, knowledge is warranted true belief rather than justified true belief:

That may not come as much of a surprise, given that this book is a sequel to Warrant: The Current Debate and Warrant and Proper Function. In the first of those books I introduced the term ‘warrant’ as a name for that property—or better, quantity—enough of which is what makes the difference between knowledge and mere true belief.

So ... is there a problem with Plantinga's deformed epistemology?

This might not come as a surprise, but I think there is.  Right off the bat, there is the assumption of a "design plan".  I understand that theists believe that there is, in some sense, a design plan to humans.  I do think it's possible to work backwards from a belief in a god, including a belief that one's belief in a god is a true belief and reach a belief that one has knowledge about the existence of a god.  Breaking it down a bit:

Is there a god?
 Do you believe there is a god?

Do you consider your belief to be a true belief?

Is there a truth (or god) detecting design plan in the human brain?

Is your brain working in accordance with the design plan (or is properly functioning)?

Does your belief with respect to god constitute knowledge?

Hopefully it can be seen that the only situation in which Plantinga's deformed epistemology both matters and works is one in which:

1) there is actually a god of the right sort,

2) there is actually a design plan for brains such that, when functioning properly, they seek and find truth (or detect the being that created the design plan),

3) the brain of the theist is actually functioning properly in accordance with that design plan, and

4) the believer does actually have the correct sort of belief about the god that exists.

If there is no god (or no god that fiddles with brains the way that it would have to in order to ensure that humans accurately and truthfully detect it), then Plantinga's argument falls in a big heap.  Remember that Plantinga’s
“reformed epistemology” involves warrant, that warrant is that which “makes the difference between knowledge and mere true belief” and that a key element of warrant is that it involves “cognitive faculties functioning properly … according to a design plan”.

Now, it cannot be that a belief in a god (or a particular sort of god) is a “true belief” if there is no such god.  However, it is conceptually possible that that which “makes the difference between knowledge and mere true belief” could exist even in the absence of a true belief so that a (false) belief that nevertheless has this quality (“warrant”) could be described as a “rational belief”.

Note that when you look for descriptions of “Reformed Epistemology”, you invariably get referred to Plantinga and his attempts to argue that religious belief may be rational.  IEP:

Reformed epistemology is a thesis about the rationality of religious belief.

A section on reformed epistemology appears in the SEP, but within the article on the Epistemology of Religion, an article which makes it clear that it is concentrating on questions of the justification of religious belief and ignoring questions as to whether “these beliefs count as knowledge or whether these beliefs are scientific”.

So, it seems, “reformed epistemology”, despite Plantinga’s protestations, isn’t about knowledge after all, but about the justification or rationality of belief.  It doesn’t really qualify as epistemology at all, or rather it would qualify as epistemology if and only if Plantinga’s god existed, which means he’s seriously begging the question with his terminology.

Note that a not uncommon defence by theists is that reformed epistemology is no more than the position that “belief in God, like belief in other persons, does not require the support of evidence or argument for it to be rational” (Kelly Clark, Without Evidence or Argument: A Defense of Reformed Epistemology).  So, when challenged, they are more than willing to step back from the appearance of any knowledge claim.  But what they are not stepping back from, and this is tricky bit, is the implication of a truth claim.

Reformed epistemology might claim that a belief in a god may be rational despite not having any evidence or argument in its support, but this is entirely contingent on there being a god and the belief in that god being true.  This is why I call this point of view deformed epistemology, it’s not really about knowledge at all, just about justifying (or warranting) belief in a god in the absence of evidence or convincing argument.  And, to the extent that it is an argument about knowledge (because Plantinga doesn’t step away from implying that warranted true belief is knowledge), it’s both begging the question and special pleading (because you can’t use the same approach on other things you might want to believe and claim as knowledge).

Is there anything about Plantinga's “deformed epistemology” that a person who is not already a committed believer should take seriously?  Does it do anything more than provide a fig leaf of rationality to someone who believes something that otherwise should not be believed without very good evidence and/or argument?

Monday, 26 June 2017

WLC - A Hole (Through which Theological Zombies Emerge)

The following sequence is from about minute 54 in the debate between Raymond Bradley and William Lane Craig, although the text is taken from the transcript at Reasonably Fallacious - so be aware that this is the version sanctioned by WLC:

Dr. Bradley:
I'll return to this microphone, if I may, okay? Now let me try to get my point across on this because this is crucial. And we both agree that Dr. Craig's strategy has been to try to prove that propositions (1) and (2) are compatible on the grounds that there is a third proposition which is consistent with (1) and with together with (1) implies (3). I've said that (3) . . .
Dr. Craig:
Entails (2).
Dr. Bradley:
Entails (2). Sorry! I've said that his (3) is all by itself inconsistent with (1).

Let's look at it like this [draws circle on the overhead]. Let that circle represent the set of all possible persons that God could have created with free will and of whom He knows in advance from the very beginning, in the words of the Bible, what the outcome of their choices would be. Okay? Now that set of possible persons, possibilia as you call them in one of your works, can be subdivided more or less arbitrarily, we'll say, into those who would be saved if He were to create them and those who would be damned as a consequence of their free choices [divides the circle into two halves labeled "sacred" and "damned" respectively]. All right?
Dr. Craig:
I don't want to be difficult, but I think that's too simplistic because some people might be damned if created in some circumstances but saved if they were created in other circumstances. So you can't just divide the line down the middle and put people on either half. It depends on what world these possible persons are put into.
Dr. Bradley:
So we can shift the line wherever you wish according to which actual world God chooses to create. All right?
Dr. Craig:
All right.
Dr. Bradley:
Now He creates an actual world [designates a segment of the circle overlapping both halves]. These are just possible individuals up here, the whole domain of possible individuals with free will [designates individuals outside the segment]. Here we've got the actual ones [designates individuals in the segment]. And, as you can see, some of them on this have been assigned to heaven because God knew in advance that if He were to create them in these circumstances, they would be saved. And, here we have those He's consigned to hell. In fact, I've been generous to God and you here because I've created equal parts. And, of course, Jesus says it's going to be pretty rough on most of them.

Now the point is this. Why did God have to create just this subset of possible individuals with free will? [designates segment of actual individuals] He could have sliced the pie a very different way. He could have sliced the pie so that there weren't any in this segment at all, the segment of hell [shades sub-segment of actual, damned individuals]. He could have chosen to create a world in which no actual individuals like you or me were existent [draws another segment outside the segment of actual individuals].
Dr. Craig:
Dr. Bradley:
After all, there's nothing all that great about us, is there?
Dr. Craig:
Dr. Bradley:
So He could have created all these possible individuals . . . [ticks new segment]

Dr. Craig:
And my point is He wouldn't be able to guarantee--so long as those people have free will--that they would FREELY RESPOND TO HIS OFFER OF SALVATION AND BE SAVED.
Dr. Bradley:
But if He knows in advance that these will in those circumstances be saved by virtue of freely accepting God's offer of salvation through the blood of Jesus, then why not?
Dr. Craig:
Because there may not be a compossible set of individuals such that if you put all of them together in a world, all of them freely receive God's salvation and are saved. It may be that individual "S" would only be saved in a world if in that world individual "S-prime" were lost …

So that it's impossible for God to … or infeasible for God to create a world in which all are saved …
Dr. Bradley:
I understand quite well about them having to be compossible. And, let's just say that out of the set of all possible inhabitants of this world that God is going to choose to create, only some are compossible. So let's make it a subset. We now have a subset of compossible individuals all of whom would be saved.
Dr. Craig:
But, see, my point is that you don't know that such a set is not the empty set. It could be the empty set.
Dr. Bradley:
Well, look, you play with possibilities. You talk about it's possible that this, it's possible that that. . . I'm asking you to confront some actual examples of possibilities. Heaven is allegedly a state of affairs in which God exists and the only other persons to exist are people who either have been saved because they believed in Jesus' name or would have believed in Jesus' name and have been saved or you could throw in a few of those who get there by general revelation.
Dr. Craig:
But that in itself presupposes there was an antecedent pre-mortem world …
Dr. Bradley:
It doesn't logically presuppose it. Causally perhaps. But you understand the distinction between causal ties and logical ones as well as I do. [long pause]
Dr. Craig:
You've still got three minutes.


There's a lot of evasion from Craig in this sequence, and many appeals to ignorance.  The debate has been discussed previously by others such as Jonathan Pearce (who argues that WLC's argument violates the ontological argument, because we can conceive of a better god than WLC's god, one that is not obliged to create a world in which a significant proportion of the inhabitants would be consigned to hell) and Luke Muehlhauser (who applauds Bradley for being prepared for the debate, unlike many other atheists who meet WLC on the debating podium), but I want to take a slightly different tack.

My ears pricked up when WLC opened up this hole in his world view:

I don't want to be difficult, but I think that's too simplistic because some people might be damned if created in some circumstances but saved if they were created in other circumstances. So you can't just divide the line down the middle and put people on either half. It depends on what world these possible persons are put into.

Now, note that WLC is not arguing for divine ignorance, by which I mean the idea that free will might preclude a god from knowing whether a given person will be saved or not.  Bradley gave him the opportunity (a few times) to make such an argument and instead of doing so, WLC gave a "compossibility" argument.

Dr. Bradley:
But if He knows in advance that these will in those circumstances be saved by virtue of freely accepting God's offer of salvation through the blood of Jesus, then why not?
Dr. Craig:
Because there may not be a compossible set of individuals such that if you put all of them together in a world, all of them freely receive God's salvation and are saved. It may be that individual "S" would only be saved in a world if in that world individual "S-prime" were lost …

So that it's impossible for God to … or infeasible for God to create a world in which all are saved …

I'm willing to accept that WLC misspoke and that he was not suggesting that his god could find certain acts impossible (as opposed to infeasible) and instead I'll focus on what he is saying about the possible lack of compossible sets.

What is being suggested here, by WLC himself, is that there is a possible world in which S-prime is freely saved although in that world, S would be lost, as well as a possible world in which S is freely saved while S-prime is lost.  Conceivably there would be another world in which both would be lost.  Think about that for a moment.

No matter how you want to paint it or buff it, this is predestination.  The claim entails a situation in which a god chooses which world to implement, notionally allowing either S or S-prime to freely choose to satisfy the requirements for salvation (although another optimum solution could have both S and S-prime being lost so that T, U and V together with their primes might be saved).  And this is despite there being possible worlds in which one or another of S and S-prime might be saved.  Therefore, the salvation of S and damnation of S-prime is purely the result of god's decision.  From the very moment of creation, S-prime is excluded, despite being able to act freely.  And therefore, god consciously places S-prime into the (future) eternal flames of hell.

Bradley's argument, which I agree with completely, is that an omnibeneficient god would not and could not consciously choose to create an intelligent being knowing (via omniscience) that that being would be tormented forever.  WLC also agrees (at least where bunnies are concerned).

Remember that we are talking about a maximally excellent being (MEB) here, which incorporates omnipotence - unlimited in power, unlimited by time, unlimited by space, although apparently limited by logic (the logically impossible is not required of an omnipotent god, according to WLC and his ilk, which is strange, since they could bypass such objections as the problem of evil in one fell swoop if they only allowed their god to be immune to logic).  An omnipotent MEB is not restricted to creating a single world in which all its playthings interact in such a way as to enable the salvation of one at the expense of another.

Remember that this MEB knows the outcome of all your freely made decisions, and knew it before you were created.  It already knew all your choices, all your interactions, all your innermost thoughts and the same applies to every being around you.  So, as far as you are concerned, from the perspective of the MEB, it doesn’t matter whether you interact with actual free acting lesser beings or theological zombies (soulless automatons which merely act out the behaviour of those who would otherwise be damned).  The priority of this MEB (according to WLC) is to ensure that the maximum number of humans come to freely choose to know and love it.  It's not to minimise the number of worlds created, or to maximise the actual interaction between lesser beings, or to not create theological zombies.

Therefore, an MEB of the sort that WLC believes in would choose to create a possible world in which S would be saved (largely inhabited by theological zombies, including a theological zombie playing the role of S-prime), a possible world in which S-prime would be saved (in which the damned S would be portrayed by a theological zombie) and so on such that each and every created soul would be saved.

That's a rather terrifying idea.  For each and every believer, there is a possibility that this universe is created just for them and people like them, because in any other configuration they would be damned.  While it might be comforting to know that there might be a number of actual, predestined-to-be-saved people in amongst all the theological zombies - you could never know which is which.  Your mother, zombie or real person?  Your child, zombie or real person?  Your partner, zombie or real person?

But it gets worse.  This world would be the one, presumably the best one in which these people would be saved.  Everything that goes on around them would be contributing to that salvation and every bad thing that happens would (due to the thorough goodness of the MEB) be necessary.  Therefore, for these people to be saved, it is necessary that three quarter million people die each year from dysentery.  It was necessary for journalists to be beheaded in Syria.  The tsunamis in Japan and Indonesia were necessary.  Droughts and floods and fires and earthquakes and mudslides and mental illness and drug addiction, the election of Trump and all sorts of social chaos - all are necessary to save these people.

These are not good people if they could not possibly have been saved in a better world.  Apparently WLC is one of these people (although I freely admit that, within this premise, it's much more likely that he is a theological zombie).

What does the theist have to do to escape this conclusion?  I see a few options:

Reject maximal excellence (although WLC argues that a less than maximally excellent being is not god)

Reject the arguments of WLC and people like him (a very good start on the road to reason and intellectual freedom)

Appeal to ignorance (the standard fall-back option)

Argue that the theological zombie is logically impossible (this would have to be a valid argument, of course, otherwise it's just another appeal to ignorance hidden behind a veil of rhetoric and hand-waving - of the sort that I'd expect WLC to embark upon)

Finally, I would ask any theist who is accidentally convinced by my argument to treat any and all theological zombies with the utmost care and respect.  It seems that we're here - dying and suffering, putting up with you, or at least simulating it - just to help you get into heaven because you failed to manage it in any better world.


More zombies and the logic thereof.

Wednesday, 21 June 2017

WLC's Mistake in Formulating the Moral Argument

Actually, there is more than one mistake that WLC made in his formulation of the moral argument and there is more than one mistake that WLC made in his Q&A response, Formulating the Moral Argument.  I've addressed the moral argument a few times already; in The Modified William Lane Craig Moral Proof, The Logic of an Apologist, WLC3: When Morality Arguments are Bad, The Moral Argument in Graphical Form and WLC Being a Duffer.

In the first, I show how the same argument can be used to argue for the existence of a god via the existence of Kim Kardashian (if there is a better way to highlight how trivial an argument is, I'd like to know it).  I'll get to the second in a moment.  In the third, I summarise the first and second arguments and go into a little detail as to why I think WLC's formulation of the moral argument is questionable. In the fourth I demonstrated, in a diagram, the circularity of the moral argument.

The fifth is more about his Q&A response to a question about … well, it should be reasonably clear already -  it's the formulation of the moral argument, which I too questioned in The Logic of an Apologist.

Now, The Logic of an Apologist got some attention recently at Craig-Land, in a thread dedicated to lampooning my arguments, eventually focussing in on the moral argument and my assertion that WLC presents the moral argument oddly and, as I interpreted it in WLC3: When Morality Arguments are Bad, deceptively.

There's a bit of noise in the discussion, but through it is an interaction between me and the thread's originator, λ-calculus, that I want to use as my launch-pad to discussing WLC's errors.  Reasonably late in the thread (post #86) I challenged λ-calculus to answer the question I effectively posed in The Logic of an Apologist: why did WLC formulate the moral argument the way he did, if not to deceive?  I also reframed the question, to make it easier for him: Why did he word it in the form (~A->~B;B;∴A) rather than in the form (A->B;A;∴B)?

Someone else immediately chimed in and referred to WLC's Q&A response, Formulating the Moral Argument.  And this led me to take another look at the piece that was the inspiration for WLC Being a Duffer.

What a treasure trove!

My original comment on it was that WLC effectively torpedoes his own argument, but that is only one of many problems.  Here are the ones that I have identified:

Problem 1 - Equivocation on the term "objective"

This is a problem with the content of the argument.  By "objective" WLC really means "absolute", as opposed to "relative".  But he equivocates, perhaps unintentionally, when he tries to persuade atheists to accept that there is such a thing as "objective moral values and duties".  We can use metrics, generally agreed principles and cool reason to work out this action is better or worse than that action - that would be objective - rather than letting ourselves be swayed by personal opinion, hyperbole and how we felt on the day - which would be subjective.  When WLC gives examples of "objective values" saying this or that is "simply wrong", you can tell that he's really talking about absolute morality, rather than objective morality - or he wouldn't be raising emotive topics like rape or torturing children for fun.

Problem 2 - Equivocation on the term "conditional"

This is a problem with his explanation for formulating the argument the way he did.  He writes "Since, as our truth table reveals, the whole conditional statement comes out true if the consequent is true, then it doesn’t matter whether the antecedent clause is true or false."

Here he is mixing up what is known as a material conditional with what is known as a conditional sentence.  He's helped in this in that the terms both contain the word "conditional" and they can be symbolised the same way (A->B).  Wikipedia will explain the potential for confusion better than I:

The material conditional is used to form statements of the form p->q (termed a conditional statement) which is read as "if p then q" or "p only if q". It is conventionally compared to the English construction "If...then...". However, unlike the English construction, the material conditional statement p->q does not specify a causal relationship between p and q. It is merely to be understood to mean "if p is true, then q is also true" such that the statement p->q is false only when p is true and q is false.

These are quite different understandings of the construction "if … then …".

Problem 3 - Equivocation on the term "true"

When, in general parlance, we say that something is "true", we do not mean that it is "trivially true" or "vacuously true" - we mean that it is actually true such that its negation is not true.  WLC is equivocating by presenting as "true" something which is only vacuously true.

Problem 4 - Lack of specificity to morality (if Problem 2 is ignored)

This argument works (to the extent that it works) with anything, including the existence of Kim Kardashian.  It's not really a moral argument at all.

Problem 5 - Triviality due to negation (if Problems 1, 2 and 3 are ignored)

WLC himself points out that, from his point of view, his first premise is trivially true - because he believes that his god does exist.  But this isn't actually just from his (or any other theist's) point of view.  In terms of the argument, if it were to succeed, it makes its major premise trivial.

There may well be other problems, feel free to point them out.

Tuesday, 20 June 2017


I raised the issue of angels in Talk to Me about Life, the title of which is an obscure reference to Douglas Adams (via Marvin, the paranoid android).  Douglas Adams is also known as the originator of puddle theory.  You are all very welcome.

I did a bit of research while putting that article together because, as a non-theist, I am not that familiar with christian thinking vis-à-vis angels.  What I discovered could be described as "confusing", which is a euphemism papered over what appear to be inconsistencies in the bible.  Who would have guessed?

Much like Eve's apple, the possible first appearance of an angel, namely Satan, in the garden of Eden isn't quite what it appears.  Genesis only talks of a serpent, a serpent which, up until that point, didn't crawl about on its belly, and could talk.  The talking bit indicates that perhaps we aren't talking about a simple snake, but the punishment dished out by god doesn't make sense if it's something more.  While it's certainly possible for christian theists to regard the Genesis story as allegorical and thus not literally true, those who do think that Genesis is literally true seem to be united in thinking that either the serpent was Satan in disguise or it was being controlled by Satan.

When the angels were created is a little unclear, but some say it was on Day 4, meaning that they preceded humanity by a couple of days.  However, there are some that argue that angels would have been created "in the beginning", but "in the beginning" the Earth was created, along with the heavens - and the "Sons of God" were apparently already worshipping god when the foundations of the Earth were being laid down.  This would imply that the angels were created before the beginning.  But again, this is all based on taking the bible quite literally.

Personally, I think that christian theists have an issue as soon as they move away from their Genesis story because they move away from the Fall, which is the crucial to their beliefs.  With no fall, there is no original sin, no deliberate turning away from god, from which we need to be saved by the crucifixion of their christ.

But let's assume, for the purposes of the non-literalists, that there was some sort of flaw that we, as a species, introduced into god's carefully created universe, which is merely allegorically referred to in Genesis and which could be redeemed by the orchestrated and temporary death of Jesus.

Angels are still key to the story.  Mary was advised of her pregnancy by an archangel.  Matthew and Luke both refer explicitly to angels (as does Mark, but in what appears to be a later amendment to his gospel, written from the perspective of an omniscient narrator more suited to fiction than fact).  John doesn't, but his gospel is in a different category anyway, due to its inconsistency with the synoptic gospels.

Perhaps we could ditch the angels by ditching the synoptic gospels, leaving us with only John's testimony for the resurrection of Jesus.  This does seem a little like cherry-picking though, and if we buy into this, then we should also buy into a number of major miracles that aren’t described in the other gospels (raising of Lazarus, turning water into wine, appearing in a locked room with the stigmata to show to Thomas, etc).  And if we buy into John gospels, then we should buy into his Revelations as well.

I've always been mildly curious about where angels fit into the scheme of things in christian thinking.  When was this "war in heaven" which led to Satan being expelled along with a band of lesser angels?  Where is it described?  If it was prior to the creation of humans, why was Satan left lurking about in Eden (or wherever)?  Perhaps they rebelled in the 130 years or so prior to Adam and Eve being expelled from Eden, but there's no mention of the events - events that you would think were quite newsworthy - in Genesis.  Although, there were naughty angels around prior to the Great Flood, procreating with humans and causing trouble and this is recorded in Genesis - so it's almost like the upheaval took place before Adam and Eve.  Would that put it on Day 5, the day after some think angels were created?  Or perhaps early on Day 6, in the morning?

However, if you want to read about the war in heaven, you have to wait until the last book of the bible, John's Revelations.  The problem is that this war happens after the four horsemen of the apocalypse are loosed on the world.  Which hasn't happened.  Which indicates that Satan hasn't rebelled yet and hasn't been thrown down yet.

As I mentioned before … it's confusing.

My understanding after discussing this topic with someone who has experience inside the christian church is that various christians focus on different things and not many think deeply about the ramification of angel mythology on their faith.  Some pretty much ignore the whole thing, not focusing on the supernatural at all (up to and including an expectation that prayer isn't likely to work, but should be indulged in anyway because it has been requested of the faithful).

So, while it's entirely possible that I have been overthinking angels, especially as a non-theist, I live in hope that by doing so I am raising the average amount of thinking being done on the topic to make up for the apparent total lack of thinking being done by those who actually do believe in angels.