Everything Exists

According to Quine, the question that ontology asks is: “What is there?”  And the lazy but obvious answer is: “Everything.” 

Now to show my utter ignorance of logic.  I’m trying to rephrase Quine’s answer—i.e., “Everything exists”—in first order logic, but find myself stuck.  Here’s why.

On the one hand, I shouldn’t write: ∀x Exists(x).  For in first order logic, existence is not expressed by means of a predicate, but by means of existential quantification.

On the other hand, this doesn’t seem to work either: ∀xx.  That doesn’t look well formed, since quantified statements have the form ∀x x…, ∃x x…, ∀x y xy…, and so on.

Here’s a third option: ∀x x = x.  Or let me try: ∀x y x = y.  But these are not what I wanted to say.  I want my statement to express the thought that everything exists, not the different thought that everything is identical to itself, or the thought that for anything there’s something identical to it.

So the question is, how do I state “Everything exists” in first order logic?  And while we are at it, how about also: “Something exists”?


10 thoughts on “Everything Exists

  1. After posting this I googled on the topic, and came across one textbook which asserts that “Everything exists” and “Something exists” cannot be expressed in first order logic (link to text):

    …More precisely, is the work done by “exists” exhausted by the existential quantifier? The answer is “No.” For there is no way of saying “Something exists” or “Everything exists” unless there is a predicate—either primitive or defined—available to do the work of “exists”: it will make no sense simply to use the quantifiers.

    [Fitting and Mendelsohn, First-order Modal Logic, p.172]

    If it’s true that “Everything exists” and “Something exists” cannot be expressed in first order logic, then it seems to me that we have at least two options:

    (1) Insist that we do not express anything coherent or meaningful in saying “Everything exists” and “Something exists”. Then we will not need to introduce a non-quantificational notion of ‘exist’ as a predicate.

    (2) Insist that we do express something coherent or meaningful (and obviously true, as Quine himself admits) in saying “Everything exists” and “Something exists”. So, introduce ‘exist’ as a predicate, primitive or defined. (Note that we will then have to reject Quine’s dictum: “To be is to be the value of a bound variable”!)

    Frankly I prefer option (2). I believe ‘exist’ can be defined as a relational predicate, but I will blog about that another time.

  2. There’s two options that I find palatable.

    One is to go with your suggestion \forall x x=x. Since self-identity is supposed to be the entirely trivial property that each thing stands in to itself, this statement will apply to exactly every existing thing. So the standard rationale would have it that this is equivalent to Quine’s answer.

    The other option, as you say, is to introduce an existence predicate. Free logicians do this and think it is fine. In particular, one way to understand the existence predicate is by means of what are called “inner” and “outer” domains which have a sort of Meinongian flavor. The outer domain includes all of the stuff it is possible to quantify over, the inner domain includes just the stuff that exists. We use the inner domain as the extension of our existence predicate EX and then we can say ‘everything’ = \forall y EX y.

  3. Colin, thanks for your answer. Both of your options are interesting, and you make a plausible case for the equivalence of “Everything exists” and “Everything is self-identical”. This seems unlike “…is a creature with a heart” and “…is a creature with a kidney”, which are extensionally equivalent only in our actual world. In contrast, “…exists” and “…is self-identical” seem extensionally equivalent in all possible worlds.

    But what I find esp. interesting is that your second option seems to exclude your first option, since the outer domain of free logic will include self-identical things that don’t exist. So we may choose either your first or second option, but not both. (Also, “Everything exists” will be false, if the quantifier ranges over both the outer and inner domains.)

    I like the free logic option better than the self-identity option, though for largely inarticulate reasons. To make the free logic option metaphysically more palatable, perhaps we can have a free logic that is not Meinongian, if ‘∃’ = “some” and ‘∀’ = “all”, and nothing more…. That is, if they can be used to make quantified statements in a given domain of discourse, without committing us to the existence of the subjects of discourse (or to their subsistence, or some other shady mode of existence).

    Clearly in meta-fictional discourse (by that I mean talk about fictional characters) we often want to make quantified statements without thereby committing ourselves to the existence of the characters we quantify over. E.g., ∃x such that x is a faithful friend and biographer of Sherlock Holmes.

    So my suggestion is that we can distinguish between quantificational and ontological commitment. We assign to quantifiers the job of committing us to how many objects, in a coarse-grained way, satisfy this or that predicate in the intended domain of discourse: zero, at least one, all, not-all. We assign to the predicate ‘exist’ the job of committing us to the existence of a given type of object.

  4. Yeah the story with free logics is a little complicated. In one type of free logic, positive free logic, you would have it come out as logical truth that each thing is self-identical even the non-existent things. But there are also negative free logics which are based around the principle that non-existent things have no properties, not even the property of self-identity. In the case of such a free logic the two statements — of all things with the ‘existence’ property and of all self-identical things — would be equivalent. The semantics of the inner and outer domain doesn’t really have to be interpreted in any heavy-handedly Meinongian way. It is after all just a formal semantics. I mentioned Meinong just because that would be one sort of interpretation of the inner and outer domains, along the lines of the distinction between existent and subsistent entities. But one surely needn’t take on such metaphysical views just in virtue of using a free logic. So I would have to agree with your suggestion that we can ‘quantify’ without making any ontological commitments in the process. Uh oh, did I just contradict Quine?

  5. Mmmm… meta-ontology. :-)

    My reading of Quine there, in “On What There Is,” is that you can express an ontology in a correct but trivial way: “Everything.” So I think Quine would agree that you can’t put that in FOL, and that, rather, one does make ontological commitments in statements translatable to FOL.

    I don’t see the option (x)(x=x) as saying “everything exists”; maybe it entails that, given an ontology in some nonlogical form. Rather, it seems to me you’d have to go with option (2) and some kind of existence predicate, of which Quine would certainly not approve; this would be in keeping with his apparently not thinking the ontological answer “Everything” to be of logical interest.

    [Colin] “…one way to understand the existence predicate is by means of what are called “inner” and “outer” domains which have a sort of Meinongian flavor.”

    Or Carnapian?

  6. Well I’m not sure inner and outer domains have a Carnapian flavor to the extent that Carnapian external questions are supposed to be meaningless from the internal perspective, whereas on the free logic approach questions about non-existents in the outer domain are perfectly meaningful even though we are talking from the perspective of some existent agent within the inner domain.

  7. Greetings from the first night of our huge-ass road trip to Connecticut!

    Yeah, that makes sense that the free logic approach wouldn’t really be Carnapian. But maybe, statements like “everything exists” would be a Carnapian external question that, while we know what it’s supposed to mean, from the approach of Quinean meta-ontology it’s otiose owing to the fact that you can make all your ontological commitments in FOL. Since, on this view, “exists” isn’t equivocal, existence is a straightforward concept, and there’s no call to inflating one’s ontology willy-nilly. I think that’s what Quine’s getting at in “On What There Is.”

  8. The way I interpret Carnap is this: existence simpliciter questions/claims are cognitive nonsense (from any perspective). Either that, or claims of the form “X exists (simpliciter)” have to be construed as linguistic proposals of the form: “Adopt X-talk!”

    Any meaningful existence claim is not an existence simpliciter claim, but a claim about existence-in-some-restricted-domain-of-discourse (or in other words, a claim about domain-membership) where linguistic rules for formulating and verifying such claims are in effect. But there are no linguistic rules in place for formulating existence simpliciter questions and verifying answers to them. E.g., mathematics provides rules for formulating questions such as “Is there a prime number between 2 and 10?”, as well as admissible answers to such questions, but in mathematics itself we cannot meaningfully ask or answer the question “Are there numbers?”; much less can we meaningfully ask it in any other discipline that offers linguistic rule-guidance in formulating and verifying claims.

    I agree with Carnap’s distinction between internal and external questions of existence. But I disagree with him on external questions of existence (what I’m calling existence simpliciter questions), and my reasons for thinking this I will save for a later blog topic.

    As with most interpretations of Carnap out there I’ve probably put my own spin on it, so here’s the classic Carnap on internal and external questions of existence:

    Empiricism, Semantics, and Ontology

  9. Micah,

    I hope you are enjoying the road trip! As to how Quine would put “Everything exists”, “Something exists” in FOL, if at all…. I’ve just read his “Existence and Quantification” (in Ontological Relativity and Other Essays), where he explicates singluar existence claims of the form “a exists” as:

    (1) ∃x x = a

    This suggests to me that Quine might explicate “Something exists” as the existential generalization of (1):

    (2) ∃x ∃y x = y

    …and “Everything exists” as:

    (3) ∀x ∃y x = y

    In plainer words, you can read (1) as “Something is a“, (2) as “Something is something”, and (3) as “Everything is something”.

    This is quite speculative, I’m trying to guess how Quine would explicate “Something exists” and “Everything exists”, but what he says about “a exists” might not be reliable evidence about how he would explicate these two claims.

    I vaguely seem to recall van Inwagen (a self-avowed Quinean in meta-ontological matters) briefly (in passing) explicating “Everything exists” as “∀x x = x”. I tried to locate it but unfortunately I can’t.

  10. Everything exists we know something about,

    at this monument,

    at this moment,

    at this monument,

    at this moment,

    ad infinitum …….

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