Joshua says the GAE is still integral to the Mission

This example can be easily distinguished, in that the definition of “A married bachelor” does not contain an assertion as to the (necessary) existence of “A married bachelor”, so agreeing to its definition does not also mean that you are agreeing to its existence.

You forgot omnibenevolent – and that puts considerable limits. It compels God to create the best possible world – and so entails (i) that it is impossible to create a world better than this one, and (ii) that creating any world worse than this one would be against God’s nature.

This means that although the names of the monsters might change, and the horrors they commit slightly different, but we would still, on average, be left with the same statistical average of major horrors and petty cruelties per century.

This is further accentuated by the fact that Immutability is one of God’s attributes under Classical Theism. Which means that God would always create the world he created.

Yes it is, because rejecting its existence also rejects the definition, when that definition includes an assertion of existence.

I would agree that it is a low bar – but would not agree that Plantinga’s argument demonstrates it is rational. The argument is blatantly Begging the Question, and so a blatantly illogical reason for accepting God’s existence. This does not mean that there may not be rational reasons for believing in God – just that this isn’t one of them.

I’ve been enjoying the discussion and thanks to all participants for their contributions, but (yes, there’s always a but) I am finding it rather esoteric.

I am reminded of many ultimately fruitless discussions at Uncommon Descent where “A or not A” cropped up. Whilst I’m no mathematician or philosopher, I think I grasp the basics of set theory but the “logic” of applying it to anything in the physical world, especially living organisms, seems a bit strained. Take a tree. Any tree. Where are the boundaries of a tree? Can we definitively decide to put a tree in the category of all trees? Can we count them?

And talking of other worlds that we can conceive of or are possible? I mean it’s fun (I eagerly await further episodes of the Three Body Problem) but what practical use is such speculation?

1 Like

Strictly logic applies only to propositions. You wouldn’t directly apply it to a tree but to statements about the tree. Formal logic is a lot like algebra but it’s about propositions and truth or falsehood rather than numbers and values,

Simple deductive logic simply shows that if the premises are true then the conclusion must also be true. If done correctly. But you really have to be very clear and precise about the meaning of the statements for it to work reliably.

The issue of showing the truth of the premises is ultimately outside logic. Plantinga’s Modal Ontological Argument uses deceptive phrasing to trick people into thinking that it’’s single premise is very likely true without further argument - even though more closely considered, it is not at all.

1 Like

Alright, seeing as we agree that the symbolic representations are not the point, we can drop this and get back to the issue, which, I suppose, I have been struggling to articulate. I shall try once again, in a different manner:

For binary attributes, such as having or not having a certain feature, one might prima facie try and say that they are pairs of attributes, rather than just one attribute and its negation. Since there are exactly two distinct truth-values we are allowing ourselves to assign (within classical logic), there exist 2² combinations of truth-values one can assign to a being regarding these two attributes: It could have both attributes, neither attribute, or either one to the exclusion of the other.

Now comes a matter of intuition: To me, it feels like the attributes H = “does have a horn” and \bar H = “does not have a horn” should be somehow linked inherently, by their very own definitions, and independently of what object we happen to investigate for having them. If I were constructing a formal language / logical framework, I would try and put some effort towards rendering statements like the following true in my language:

\mathop\forall x:\left(\left(Hx\rightarrow\neg\bar Hx\right)\wedge\left(\neg Hx\rightarrow\bar Hx\right)\right)

In other words, it should, by my intuition, be that when we say something has no horn, that’s the same as saying that it is false that it has a horn, and vice versa. Requiring existence for attribute assignment, however, breaks that. The equivalence only holds for existing things (if that), but certainly not for non-existing ones. Having no horn is not the same as not having a horn anymore (at least not automatically / for all cases). And while I appreciate that this can be made consistent, and that with a charitable framing it is in its own right somewhat intuitive, my point is that it is at the very least not obviously a more correct, better, or more intuitive formalization of the relationships between predicates and existence.

It seems to me that you are assuming that you can speak sensibly about non-existent things without tacitly assuming - for the purpose of the argument - that they exist. I do not accept that, and think it better to simply acknowledge the tacit assumption and move on.

So let’s look at the problem I see with your view. What does “Charlie the unicorn has a horn” mean if Charlie the unicorn does not exist? There is no horn belonging to Charlie the unicorn. So if it means that there is a horn belonging to Charlie the unicorn it is clearly false. There is no such horn.

That’s correct. In my view, “exists” is – at least in the formal logical language – not a predicate. I can have in my language an entity that is Charlie’s horn, but there is no formal proposition that the natural language symbol string “Charlie’s horn exists” corresponds with.

It’s not false that Charlie’s horn exists, or that Charlie himself does. These just aren’t propositions in the first place. “Charlie has a horn” is, but to find out whether it is true I have to review some prior premises. If it is supposed to be a statement about something in nature, the way to decide it is to go out and try and find Charlie the unicorn out in the real world, and only then – if that ever happens – will one have definitively decided the truth-value of the proposition within the real natural world. And if it is a proposition in a purely logical sense, with no ambition to describe a state of affairs in nature, then no amount of physical evidence will settle it, but only agreed upon definitions, axioms, and assumed premises will.

And other worlds? Conceivable or possible? Is conjecture here useful ?

The clarifier there was to distinguish the scope of the statement. A “possible world” in modal logic is not a parallel universe, and the “actual world” is not the physical/natural world around us. Rather both are merely sets of truth-assignments to propositions, and there is in principle no obligation to have any correspondence between that abstract construct and anything outside/beyond formal logic.

In colloquial speech we do not make this distinction, and we feel quite comfortable saying that horses exist and unicorns don’t. In formal logic, I have argued, existence is a more complicated thing that is not the same as just any other property. In my opinion, this is one of the central flaws about ontological arguments for the existence of God: They both equivocate between existence in nature and existence in the abstract, and treat the latter in a way that can with some ease be misused to prove things that many would not find agreeable to translate back into natural language.

That seems odd indeed, since existence predicates are quite normal in mathematics, and the modal logic “possible” and “necessary” are framed in terms of existence.

But it doesn’t really answer my question. What does it mean to say “Charlie the unicorn has a horn”?

But doesn’t that essentially boil down to my position? If it’s said to be a statement about nature then Charlie the unicorn does not exist and so does not actually have a horn. If you’re speaking about some other context then that context determines the meaning and everything still works out.

Much of science, I believe, is of not much more practical use than abstract logic.

That’s news to me. As far as I was made aware, the closest thing to saying anything about existence in mathematics is the existence quantifier \exists. \exists is not a predicate, though. The symbol string \exists x is not something that can be mapped to true or false, nor is the symbol string \exists4. “\in\mathbb N” is a predicate, and one can say that 4\in\mathbb N i.e. that predicate is true of the object symbolized by 4. Logicians would probably rather use something like N\left(4\right), or N4 and say that the predicate N means “is an element of the natural integers”.

As far as I know, there is no mathematical theorem that states that “There exists the object S[1]. Existence only ever occurs in the form "There exists an object x such that predicate P is true of x, or, more commonly, “There exists an object x such that if the predicate P is true of x, then so is the predicate Q”. Like one could say \mathop\exists x:\left(x\in\mathbb N\right), meaning that \mathbb N is a non-empty set. Or one could say that \mathop\exists x\in\mathbb N:x<7, which would be a common shorthand for the stricter notation employed by logicians, wherein that should rather look as \mathop\exists x:\left(\left(x\in\mathbb N\right)\to\left(x<7\right)\right).

As for modal logic, that only adds a new operator (a “predicate” for propositions only; one might want to say there is two, but really \Box=\neg\Diamond\neg and \Diamond=\neg\Box\neg, so one of the two would suffice[2]), explained most conveniently and commonly by the possible world semantics I briefly sketched in post #81. If an existence predicate was not already explained within the predicate logic we construct our modal logic atop of, then neither is it explained in the resulting modal logic, if introducing the unary operator(s) is all we did to get it.

There is a chance we were in part talking past each other, yes. When I’m insisting on caution or an altogether forfeiting of the use of existence as a predicate, this is within formal logic.

I understand that when we speak colloquially, that same symbol has a meaning that we are liberally assigning to or refusing from things, like horses and unicorns, or Santa and Dan Dennett, and that Santa has a white beard, or that Charlie the unicorn has a horn are both false statements, in that due to the non-existence of Charlie and Santa, we are unable to verify them having these attributes.

My suggestion, then, is that this colloquial usage of that symbol is one for a meaning that is rather distinct from its formal meaning in maths and logic. At the risk of sounding somewhat Wittgensteinian here, a lot of the disconnect might well be trivial, boiling down to this common equivocation on the symbol “exists”, and either a misunderstanding of just which meaning it is we were discussing, or at most a disagreement about how strictly or blurry the line between them ought be drawn as a matter of practicality.


  1. One could at most say that the existence of the empty set is a statement like that. It is an axiom, rather than a theorem, though, and I imagine there is probably a way of interpreting it as a definition, i.e. saying that \emptyset is the name of the set that has the property that \mathop\forall x:\left(x\notin\emptyset\right), without any need of talking about scopeless existence as a property mathematical objects can have or lack. ↩︎

  2. For that matter, the quantifiers obey the same structure. We do not need to have both the existence and the universal quantifier, because \forall=\neg\exists\neg and \exists=\neg\forall\neg. ↩︎

You seem to be putting up objections for no reason - in fact that seems to be the main thrust of this conversation. Am I to take it when you say that existence is not a predicate you simply mean that existence taken alone without defining the thing that is meant to exist is meaningless? Surely that’s a triviality.

This seems to evade the point. Surely if they do not exist they cannot actually have any attributes. I really don’t see why you seem so determined to reject that - even to the point of inventing “problems” that clearly don’t exist.

After building some formal arguments I’m beginning to see what you are saying, all in all. What follows is a sketch of some of the thoughts I had pondering this matter and toying around with formalities to grasp it.

Suppose we formalize existence as a predicate, and denote it with E. In Zermelo-Fraenkel set theory, the axiom of comprehension allows us, given any predicate P, to subdivide any set S into two subsets: the elements of S that satisfy P, and those that do not. Now, let S be some non-empty set. The axiom tells us that there exists then the set

S_{\neg E}:=\left\{x\in S\,\middle|\,\neg Ex\right\}\rlap{\quad,}

i.e. the set of elements of S that do not satisfy the existence predicate E. The only question now is whether this set empty or not. If it is, then universal quantification over it gives trivially true propositions, while existential quantification gives trivially false ones. I am tempted to agree that this would be the more intuitive answer, and it is consistent with the stipulation you suggest, that only items that exist get to have properties. Since the elements of S_{\neg E} by definition do not exist, this rule can only work if there are no such elements.

Without this stipulation, however this does not seem to follow. Neither the introduction of existence as a predicate, nor any of the rules of propositional or predicate logic or set theory seem to forbid this set from having elements as a matter of logical consequence (I’m open to seeing a proof otherwise). If it is non-empty, though, then the statement that existence as a predicate is a prerequisite for all other predicates is just false, since the predicate “is an element of S_{\neg E}” is satisfied for objects that violate the existence predicate. Statements like “There exists an x such that x does not exist” translates to an actually really true proposition “\exists x:\neg Ex”, absurd though it may sound on the face of it.

That is to say, in order to introduce an existence predicate, we must stipulate that predicates are only true for individuals that satisfy the existence predicate. It looks like I am coming to agree with you more and more here. Now, naively, we would say that the stipulation should look something like this:

\forall P:\left(\forall x:\left(Px\to Ex\right)\right)

Of course, any negation of a predicate is itself a predicate. To have no horn is just as much of a property as to have a horn is. Let \mathbb P be the set of all predicates. Then

\forall P:\left(\left(P\in\mathbb P\right)\to\left(\neg P\in\mathbb P\right)\right)\rlap{\quad.}

In particular, this means that since E is a predicate, non-existence \neg E must also be a predicate. And now we get some silliness: Since existence and non-existence are predicates, we can instantiate our stipulation with the non-existence predicate \neg E substituted for P, and logically so conclude:

\forall x:\left(\neg Ex\to Ex\right)

In other words, because in order to satisfy a predicate something must exist, the predicate of not-existing is only satisfied by things that exist. All things that do not exist, do exist.

This is, of course, utterly absurd. Again one is tempted to think of the empty set, and of equating being an element of that with non-existence. That would resolve the absurdity, but we already adopted a stipulation to get rid of it, and here we are again, in need of a new stipulation?

The other idea I’ve had is to restrict the stipulation. Rather than saying that for all predicates P, E must be satisfied in order that P can be, we might want to make a special exception to a few. After all, the stipulation itself already marks E as a special predicate: No other predicate needs to be satisfied in order that all the others can be. So why not go one step further and say that all predicates except \neg E are satisfied only by things that satisfy E? Alternatively, we could just stipulate that the rule about negations of predicates being predicates holds for all predicates except existence. That’d be another rule that naively should apply to all predicates, but would have to suffer an exception just to allow for existence to be a predicate.

So, it would appear existence is rather quite special. Naively introducing it invites all manner of either absurd sounding consequences or outright contradictions. It is a “predicate” with consistency only when packaged together with a new rule, and that rule, that should naively apply to all predicates and their negations, must be formulated so as to specifically not apply to the negation of existence itself, or some other universal rule for predicates must be made to not work for existence. Existence seems to just naturally not want to be a predicate, and ways to force it into being one all require building up special facilities just for it, enough to make one question just in what sense it even is a predicate anymore, outside of enjoying the notation “Ex” that we grant other predicates.

Of course, all along we had the existence quantifier \exists: Propositions wrapped in it map to \texttt{FALSE} when ever there is no logically consistent way of satisfying what ever properties the object so quantified is assigned within the scope. It plays perfectly well with the universal quantifier, and with the empty set. The only thing it is lacking, really, is the ability to be put by an individual without a scope and yield a proposition. If I let c be a symbol for Charlie the unicorn, the formula “Ec” is a proposition, but the formula “\exists c” is not. The existence quantifier seems to do just about everything we wanted from the existence predicate, except for being a predicate. And even then, to say that E was a predicate by the end feels (to me, at least) somewhat suspect, considering how categorically unlike every other predicate we had to make it, just to get to define it coherently.

1 Like

I really think you are overcomplicating things. How about dealing with the basic question of what it means to say that “Charlie the unicorn has a horn: if Charlie the unicorn does not exist?

Well, I don’t find the basic question all that interesting. On a basic, colloquial, intuitive level we probably agree. I was much more curious to see as to the far-reaching logical consequences of existence as a predicate, especially given the context of ontological arguments for the existence of God. That does get technical, and I for one do not mind it. I think I have a better appreciation and understanding of some of the subtleties of this topic than I had before engaging with it through challenges like yours.

Of course I could have also not engaged with it in this manner, but I would not have had the same learning experiences I did, had I stuck to “the basic question”. Whether you ever intended to lead anyone down that rabbit hole – and by the looks of it you do not much care to follow, and I say this without judgement – I feel richer for the journey, and I wish to thank you for sparking it.

1 Like

But without an answer to that question it certainly can’t be said to be true or false. You might declare the statement meaningless in which case it is neither.

However if the statement is considered meaningful it must be false if it refers to the ordinary understanding of the statement when it refers to beings that do exist e.g “Frederic the Indian rhinoceros has a horn on his head”

And I do think that this is relevant to ontological arguments which seek to infer existence from properties that the proposed entity is defined as having.

1 Like

There are few greater rewards to a moderator than seeing that - nearly 100 comments after I last checked - a deep discussion has broken out. :slight_smile:

3 Likes