Maverick Philosopher

Footnotes to Plato from the foothills of the Superstition Mountains

Singular Existence and Quantification

For Tim M. who wants to discuss this topic with me. ComBox open.

……………………. 

Singular existence is the existence of particular individuals.  It is the existence attributed by a use of a singular sentence such as 'Max exists,' where 'Max' is a proper name.  

A standard way to conceptualize singular existence, deriving from Quine and endorsed by Peter van Inwagen, is in terms of the 'existential' — I prefer to say 'particular' — quantifier of standard predicate logic. Thus,

Max exists =df for some x, x = Max. 

In general,

x exists =df for some y, x = y.

In the standard notation of modern predicate logic with identity, 

x exists =df (∃y)(x = y).

What the latter two formulae express is that an individual exists if and only if it is identical to something. Assuming that there are no nonexistent objects in the domain of quantification, these biconditionals are undoubtedly true, and indeed necessarily true.  Meinongians reject the assumption but it is quite reasonable, so let it stand. Even so, I cannot see that the biconditionals  just listed sanction the reduction of existence to identity-to-something.  

Those of a deflationary bent would welcome such a reduction. For it would allow the elimination of existence as a topic of metaphysical investigation in favor of the sober logic of 'exists.'  You will notice that on the left-hand side of the biconditionals there is the apparently non-logical, content-rich word 'exists' whereas on the right-hand side all the symbols are logical.  If we can get rid of the word 'exists,' then perhaps we can get rid of the temptation to ask about Existence and Being. Aquinas, for example, tells us that God is not an ens among entia, but esse, Being or To Be: Deus est ipsum esse subsistens.  This presupposes that there is such a 'thing' as Being.  If the deflationary account is correct, there isn't.

So my question is this: is the deflationary account adequate? Or is there more to existence than can be captured by the so-called 'existential' quantifier of modern predicate logic?

An Argument Against Reduction

If Max is identical to something, then this thing can only be Max. The upshot is that the existence of Max is his self-identity.  But note that whereas my cat Max, being a contingent being, might not have existed, it is not the case that Max might not have been self-identical. It is true that Max might not have existed, but it is false that Max might not have been Max.  So existence cannot be reduced to self-identity. This holds for all contingent beings. Only a necessary being such as God could be such that existence and self-identity are one and the same. The argument, then, is this:

P1. Every contingent existent is possibly nonexistent
P2. No contingent existent is possibly non-self-identical
————
C1. No contingent existent is such that its possible nonexistence = its possible non-self-identity
————
C2. No contingent existent is such that its existence = its self-identity.

It follows that there is more to existence than what is captured by our Quinean biconditionals.  

An Objection

Is the above argument decisive? A Quinean might respond by denying (P2) and running the argument in reverse.  Insisting that to exist = to be self-identical, he argues that if a thing is contingent (possibly nonexistent), then it is possibly non-self-identical. If Max is contingent, then there is a possible world W in which he doesn't exist. Since Max does not exist in W, he has no properties there. Hence he is neither self-identical nor non-self-identical in W.

Is this objection any good?  

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Bill Vallicella

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15 responses to “Singular Existence and Quantification”

  1. Thomas E. Vaughan Avatar
    Thomas E. Vaughan

    I wonder whether you might find interesting a peculiar idea that is related to identity and perhaps even to the matter of self-identity. Your post reminds me specifically of an idea from the interface, in physical theory, between quantum mechanics and statistical mechanics: That two physical things might be so radically indistinguishable as for their being swapped not to count as a different state of the system of which they are part.
    For example, suppose that in a certain system there are two marbles and three locations, each of which can hold a single marble. The system is such that, in the absence of intentional interference, each marble will quickly fall into one of the three locations and linger nowhere else. How many stable states are possible for this system? Let the marbles be M1 and M2, and let the locations be L1, L2, and L3. Then we have the possible states S1, S2, S3, S4, S5, and S6.
    S1 = M1 at L1 and M2 at L2.
    S2 = M1 at L1 and M2 at L3.
    S3 = M1 at L2 and M2 at L1.
    S4 = M1 at L2 and M2 at L3.
    S5 = M1 at L3 and M2 at L1.
    S6 = M1 at L3 and M2 at L2.
    In classical mechanics, even if M1 and M2 were intrinsically indistinguishable, so that nothing intrinsic to M1 or to M2 could be used to tell them apart, there would still be six possible states. After all, one could distinguish M1 and M2 by something extrinsic, such as their initial locations. In this case we might arrange the marbles so that one is at L1 and the other at L2. Then we distinguish between S1 and S3 by reaching in a hand, grabbing the marble at L1, reaching in another hand, grabbing the marble at L2, and then manually swapping their positions. By keeping an eye on the system as it continuously changes configuration during this process, one knows that S1 and S3 are distinct states.
    However, treating this system as it would be in quantum mechanics, one finds that, if there be particles that are intrinsically indistinguishable, then there is also no extrinsic way in which to distinguish them: There are only three different states. What classically are the distinct states S1 and S3 are indistinguishable in quantum mechanics. So, too, for S2 and S5; and for S4 and S6. Instead, quantum mechanically, there are only three states, QS1, QS2, and QS3.
    QS1 = M at L1 and M at L2.
    QS2 = M at L1 and M at L3.
    QS3 = M at L2 and M at L3.
    If M were an electron instead of a marble, then in a quantum-mechanical description of a system, every electron is so absolutely identical to every other electron as that exchanging the location of any pair whatsoever does not count as a different configuration of the system.
    Now one might be tempted to ask, Why should one care about this obscure detail of quantum mechanics? An answer is that the observational results of thermodynamics can be predicted by way of statistical mechanics only if one assume that identical fundamental particles are radically identical. The number of states available to a system is a key concept upon which many a result from statistical mechanics rests; in fact the theory agrees with real-world measurements only under the assumption that fundamental particles like electrons are radically indistinguishable.
    Maybe I’m hung up on nothing, but, to this day, I am still struck by the apparent insanity of the quantum-mechanical idea. I still remember my sudden realization of this, when I was a second-year graduate student. I was sitting at a desk in the peace and quiet of the physics library in the Spring of 1992. I was reading a passage in which the author was discussing the history of statistical mechanics. When I came to the part about how the theory’s success hinged on the assumption that the exchange of two (“identical”) particles did not count as a different state of the system, my jaw dropped. It’s as though there must really be only one electron, somehow mysteriously multiplied to occupy different locations in the system. On the one hand, there seems really a numeric plurality, for, according to the theory, the total electric charge in the system due to the electrons is as it would be if there were many, but, on the other hand, there seems also to be the lack of a fundamental sense of plurality, as if in a sense there is really not a distinction between one electron and another in the system.

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  2. The Dark Ostrich Avatar
    The Dark Ostrich

    “Assuming that there are no nonexistent objects in the domain of quantification, these biconditionals are undoubtedly true, and indeed necessarily true.”
    Actually, even if there are nonexistent objects in the domain of quantification, i.e. assuming that ‘existent’ means something other than as you have defined it, and given the standard meaning of the ‘=’ sign in predicate calculus, the biconditional is true. The standard meaning of ‘=’ is a relation that each thing bears to itself, and no other thing.
    I.e. we don’t need the assumption in question. ‘For all x, x=x’ is necessarily true however we define ‘exists’. If we define it as ‘self identical’ that is a definition, rather than an assumption, and given that definition it is necessarily true that everything exists.

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  3. BV Avatar
    BV

    A dark saying, Ostrich Man. The RHS of the following biconditional is satisfied by Meinong’s golden mountain which does not exist. The LHS is not. So the biconditional is true only on the anti-Meinongian assumption.
    x exists =df (∃y)(x = y)

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  4. Josh Avatar
    Josh

    I am inclined to agree with the Quinean objector here, although the ease with which I am so inclined suggests (to me, at least) that I do not understand the problem fully.
    A question: your objector says, >>[Max] is neither self-identical nor non-self-identical in W<<. It seems right to me to say that Max is not self-identical in W. Yet I don't understand why he is not *non*-self-identical. What is wrong with simply saying that Max is non-self-identical in W on account of his not existing in W? Another point: as far as I can tell, one would not need to adopt the Quinean "reduction" in order to agree with your objector, since he could say that "exists" and "is self-identical" are distinct intensionally but not extensionally. Right?

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  5. BV Avatar
    BV

    Josh,
    My thought was that if Max does not exist in W, then nothing is true of him there. So it is neither true of him that he is self-identical or the opposite.
    >>Another point: as far as I can tell, one would not need to adopt the Quinean “reduction” in order to agree with your objector, since he could say that “exists” and “is self-identical” are distinct intensionally but not extensionally. Right?<< There is no question about extensional equaivalence: whatever exists is self-identical, and whatever is self-identical exists. The question, however, is whether the 'property' of existence can reduced to the 'property' of self-identity. I say it can't. Max might not have existed; but it is not the case that he might not have been self-identical. Existence is no ordinary property; it is not the intension of 'exists.' Existence is what makes a thing be in the first place. Suppose the existence of Max is his being created by God. That would illustrate how the existence of Max could not be reduced to his self-identity.

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  6. The Dark Ostrich Avatar
    The Dark Ostrich

    I meant that you seem to be confusing two separate assumptions
    (Ass 1) there are no non-existent objects in the domain
    (Ass 2) ‘a is existent’ meansdf for some y, a = y.
    The first relies on ‘existent’ already having an understood meaning. The second assigns a meaning. Clearly the assumptions cannot be the same, for the first relies on a meaning, the second assigns a meaning. So, having already assigned the meaning (ass 2) it follows that ass 1 is necessarily true, given that necessarily for all x, x=x. So you didn’t need to write ‘assuming there are no non-existent objects in the domain’.

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  7. The Dark Ostrich Avatar
    The Dark Ostrich

    For example, I can assume (1) that cats have wings or (2) that the word ‘cats’ means birds. These are entirely different assumptions, don’t you agree?

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  8. BV Avatar
    BV

    It is not about meaning unless you think meaning can be accounted in wholly extensional terms.
    a exists =df for some y, a = y
    merely states an equivalence.
    The point I am making is very simple and strikes me as self-evident. If there are Meinongian objects, then the biconditional has counterexamples. That is why the biconditional is true only if the Meinongian theory is rejected.
    Hung up on a red herring, you ignore my main argument.

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  9. The Dark Ostrich Avatar
    The Dark Ostrich

    >>’a exists =df for some y, a = y’ merely states an equivalence.
    Why the ‘df’ sign? You are defining the term ‘exists’.
    I don’t agree that this is a red herring.
    >>If there are Meinongian objects, then the biconditional has counterexamples.
    But there can’t be counterexamples if your ‘df’ defines the meaning of ‘exists’. If its meaning involves self-identity, then there can be no exception.
    You need to be clearer about what ‘df’ means.

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  10. Josh Avatar
    Josh

    >>My thought was that if Max does not exist in W, then nothing is true of him there. So it is neither true of him that he is self-identical or the opposite.<< But aren't the two propositions, (1) "Max is self-identical in W"; and (2) "Max is not self-identical in W" contradictory opposites? If so, this would mean that one of them has to be true (2, in this case). >>There is no question about extensional equaivalence: whatever exists is self-identical, and whatever is self-identical exists.<< Good. Agreed. I suppose I have the same question as Ostrich, then, albeit from an entirely different angle: what is implied in your "df"? Surely it is more than just a statement of material equivalence, since this is admitted by all, Vallicella and Quine alike. I have been assuming (perhaps erroneously) that by "df" you meant to say something stronger, i.e., that "exists" and "is self-identical" are synonymous. I take this to be the Quinean position.
    So there are at least two ways to agree with the objector (or so I argue):
    1) “Exists” and “is self-identical” are synonyms (Quine’s position), and so the following is trivially true: If Max does not exist in W, then Max is not self-identical in W.
    2) “Exists” and “is self-identical” are not synonyms, and yet it is still true that if Max does not exist in W, then Max is not self-identical in W.
    Sorry if this is convoluted.

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  11. The Dark Ostrich Avatar
    The Dark Ostrich

    I looked vain for a clear definition of ‘=def’ online, even though it is commononly used in textbooks. I always use it as a definition of the meaning of a term. Thus I distinguish
    a exists iff Ey y=a
    which is a true biconditional, from
    a exists =def Ey y=a
    I don’t know how to read the second except as giving a meaning. How else would we read it? Why else would we put the ‘definition’ subscript there?
    I think Bill means the first.

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  12. BV Avatar
    BV

    Ostrich,
    Not all definitions are intensional, as you know. Some are extensional. You can supply your own examples. Charitably interpreted, the Quinean biconditionals are extensional. So read, they are true. But if we took them as giving the sense or meaning of ‘exists’ they would be false.
    When I say of a thing that it exists I don’t mean that it is identical to something; I mean that it has being, is not nothing, is there (in a non-locative sense of the term).

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  13. BV Avatar
    BV

    Josh,
    If two contradictory propositions are such that the common subject does not exist, then neither are true. LEM holds only for what exists.
    It is not just material equivalence but strict equivalence, i.e., the necessitation of material equivalence.
    See response to Ostrich. There is no synonymy.
    What Quine actually says (Ontol Rel and Other Essays, p. 94) is that the RHS “explicates” (his word) the LHS, whatever that exactly means. But given Quine’s views about meaning (sense) he can’t mean that the RHS gives the meaning of the LHS.
    But I am not concerned with Quine exegesis.
    Let’s drop the ‘df.’
    We all agree that a exists iff for some y, a = y. (I don’t think anything hinges in this discussion on whether I use the free variable ‘x’ or the arbitrary individual constant ‘a’ as Quine and the Ostrich Man do.)
    And presumably we all hold that this biconditional is necessarily true.
    My question is: does the truth of the biconditional sanction the reduction of existence to identity-with-something? Answer: No way.
    Parallel question. X is morally obligatory iff God commands x.
    Does the truth of the biconditional sanction the reduction of moral obligatoriness to the property of being commanded by God? There is a genuine question here, and the answer is not obvious.
    And please note that no one who understands English would think that the RHS gives the sense of the LHS. And of course it would be knuckleheaded to advance the biconditional as a stipulation of the meaning of the LHS.

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  14. The Dark Ostrich Avatar
    The Dark Ostrich

    >>Let’s drop the ‘df.’
    OK then I broadly agree with you that existence is not self-identity. But I suspect we do not agree on the meaning of ‘exists’. We have discussed this before, and you know my view is that ‘a exists’ means that the singular concept attached to ‘a’ is satisfied. We have had earlier discussions on singular concepts.
    My most recent arguments (separately by earlier email) are that (i) the sense of a proper name is not ‘proper’ in that the sense is attached to the bearer of the name, i.e. I reject direct reference; (ii) I reject sense-theories of proper names that assign a per se or transportable sense to the name, for reasons discussed by email, essentially the transposition argument; (iii) I hold that the sense is text-specific, i.e. so that we cannot understand or communicate a proper name without access to a specific text, e.g. the text that originally introduced the name ‘YHWH’ to us.
    The transposition argument, you will recall, is that the infernce ‘there is a God called ‘YHWH’ and YHWH is all-powerful, so some God is all-powerful’ is valid with the major and minor in that order, but not valid if they are transposed. That seems conclusive proof of context-specific singular concepts. You haven’t so far given me any argument otherwise.

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  15. Josh Avatar
    Josh

    >>My question is: does the truth of the biconditional sanction the reduction of existence to identity-with-something? Answer: No way.<< Agreed. >>And please note that no one who understands English would think that the RHS gives the sense of the LHS. And of course it would be knuckleheaded to advance the biconditional as a stipulation of the meaning of the LHS.<< Absolutely. An important point, pace Van Inwagen especially.

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