I Necessarily Exist

In several earlier posts I introduced and defended what I call Frigid Stipulation as an alternative to Rigid Designation (Introducing Frigidity, Applying Frigidity, What Kripke Really Thinks). The basic claim is that in natural languages (as opposed to in thoughts)there are no logically proper names at all, no singular terms what so ever. Every sentence with what looks grammatically like a singular term is really a disguised quantifier at the level of logical form and truth conditions and so can be analysed via Russell’s theory of descriptions.

Aside for the argument that I gave for frigidity from the fact that the truth conditions of sentences with so-called rigid designators in them change depending on who the speaker had in mind, there are also all kinds of well-known problems with construing linguistic names as logical constants, in fact with the whole idea of logical constants in the first place. These problems range from the normal ones about identity and existence statements, and belief attributions involving co-referential terms, to the bizarre logical result that we can prove that any given individual exists as a matter of first-order logic. The proof is actually quite simple and takes the form of a reductio of the assumption that the individual in question does not exist.

Here is a version of the proof that Rosenthal once presented in a Quine class I had with him.

(1) Proof by reductio that Saul Kripke exists: ((Ex) (x=SK))

            1. –(Ex) (x=SK)                     assumption for reductio

            2. (x) –(x=SK)                       equivalent to 1.

            3. (x) (x=x)                            axiom of identity

            4. (SK=SK)                             UI of 3.

            5. –(SK=SK)                           UI of 2.

            6. (SK=SK) & -(SK=SK)       4., 5.

As you can see, we derive the contradiction that Saul Kripke is both self-identical and not self-identical from the assumption that he does not exist and the axiom of identity with just two uses of universal instantiation. So we can prove that any given object exists as a matter of first-order logic with identity. But surely that is absurd! We may be able to live with the result that some object or other exists (Ex (Fx)), which naturally follows in standard first-order logic, but we cannot live with the fact that we can prove that any given particular object exists.

Even worse it seems to me that we can give an analogous proof that the object in question necessarily exists!

(2) Proof that Saul Kripke necessarily exists: □Ex (x=SK))
                1. ◊ –Ex (x=SK)          assumption for reductio

                2. ◊ (x) –(x=SK)         equivalent to 1.

                3. (x)□ (x=x)               modal axiom of identity

                4. □ (SK=SK)               UI of 3.

                5. ◊ -(SK=SK)              UI of 2.

                6. –□ (SK=SK)              equivalent to 5.

                7. □ (SK=SK) & -□ (SK=SK)           4,6

But surely this is even more absurd than the last! How can I necessarily exist? These kinds of results offer good reason to adopt frigidity.

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32 thoughts on “I Necessarily Exist

  1. My logic is not very good, but isn’t (talking about first argument) going from step 3 to step 4 by universal instantiation valid? I mean, SK was already assumed not to exist, so it won’t be part of “all x” (which I take is required for universal instantiation).

  2. Hi Tanasije,

    No, it is a legitimate move…(3) just says that everything is self identical, so any given thing will be self identical, hence the UI. UI just says that when you have (x) (Fx) we can write Fa…

  3. Yeah, I understand that UI application in that case is that – if everything is self identical, than any particular thing (which is in this everything) is self identical. But I wonder if “everything” there picks out just existing things, and SK is not one of those.

  4. 1. in standard first-ordder logic there is no restriction on the quantifier, so you don’t get to say ‘any particular thing (which is in this everything)’ because everything is in there

    2. SK has to pick something out, this isn’t free logic!!! The whole point is that logical constants MUST refer…

    We start off by stipulating ‘let “SK” name a particular object’ and then proceed to prove that it is a theorem of first-order logic that it exists, and that it must exist…

  5. So ‘that introduction of SK as rigid reference of a particular object, already assume that it exists? Then it seems that the proof is unnecessary. The contradiction is already in the introduction of SK, and then saying that -(Ex) (x=SK).

    Why wouldn’t Kripkean instead talk about Fx, where F means x is referent of name SK, and formulate the non-existence claims as standard -(Ex) Fx ?

  6. No, what we are trying to do is to prove that there is a contradiction by assuming that the thing does not exist…

    A kripkean could not take that route as that is to give up on there being singular terms and to opt for a Quinian strategy of taking names to be descriptions…

  7. How about using SK just when we know that SK exists, and reformulating the talk about existence of SK in the terms of “if the name ‘SK’ rigidly refers or not to some object”?
    In such case any usage of SK as a logical constant will be just in cases where we know that name ‘SK’ designates some object, and when we talk about issue of existence, we use the Fx formulation where Fx means “x is designated by name ‘SK’ ” in case where name ‘SK’ doesn’t designate anything, and using SK just in case if it does rigidly designates some object?

    It seems to me in Kripkean spirit, to reformulate the question “Does SK exists?”, to the question “Does ‘SK’ have referent?”.

  8. Hi Guys,

    I’m not sure if this is in the spirit of what Tanasije is getting at, but one concern I have is as follows.

    Since, as Richard points out, he’s not working with a free logic and assumes that all constants refer, we can prove the existence of SK directly from the axiom of identity.

    1. (x)(x = x) (Axiom of identitiy)
    2. SK = SK (from 1 by UI)
    3. (Ex)(x = SK) (from 2 by EG)

    To mobilize this for a criticism of Richard’s point, when he writes

    So we can prove that any given object exists as a matter of first-order logic with identity. But surely that is absurd!

    It turns out there isn’t anything absurd about it. This is because since the logic isn’t free and all constants are referring, then “any given object” means “any given existing object”. And so, given any existing object, it follows that it exists. That’s not absurd. It’s trivial.

  9. Hi Richard,

    I’ve recently been thinking about similar issues myself. Concerning your proof of Kripke’s necessary existence, is “◊ –Ex (x=SK)” really the only way of spelling out the intuition that Kripke might not have existed? As I said on my own blog:

    “Perhaps we could cash out the intuition some individuals, such as myself, do not necessarily exist as follows:

    “It is possibly the case that a, b, c…; a_1, b_1, c_1…; a_n, b_n, c_n… exist and there is no x such that x is not identical to a and x is not identical to b and x is not identical to c…” (where, a, b, c, (etc.) are constants denoting everything existent with the sole exception of me. )

    If the foregoing is true then, while the proposition “Jason Zarri does not exist” cannot be true, it does not follow that I necessarily exist. For it seems perfectly possible that everything besides me could have existed while it was also the case that these things were the only things which existed. It is not necessary for this to be the case that there exist some proposition which truly asserts my nonexistence. If a given possible world does not contain me, it fails to contain the proposition that I exist as well as the proposition that I do not exist, so Williams’[sic] argument cannot go through. Though it is not possible that I lack existence, it is still true that there are possible worlds which do not contain me.”

    Of course I know very little about predicate logic, so the above may simply be misguided. But if it does work do we still need to adopt frigidity to account for the fact that Kripke doesn’t necessarily exist?

  10. Hi Tanasijie

    I think you are right that itis in the Kripkean spirit to reformulate existence claims into claims about the reference of singular terms…in fact in Kripke official formulation of rigid designation he stipulates that a rigid designator is one that designates the same object in all possible wordls where that object esist…but I don’t see how your proposal will get you out of the difficulty that the argument presents…For, even if we restrict the use of singular terms to cases where we know that the object exists, we will still be able to run the proof and get the results that the object’s existence is a theorem of first-order logic, and that it must necessarily exist…The problem arises if there are ANY singular terms…

  11. Pete,

    I think the argument you present is invalid because the conscusion should be

    3*. Ex (x=x)

    which is fine, as it is, as I said in the post, just a consequence of working in a logic that isn’t free…we have to live with the fact that we can prove that some object or other exists (what 3* says) but we can’t live with the fact that it is a tautology that I exist!

  12. Hi Jason,

    Thanks for the comment!

    I don’t think that your strategy will work because I can simply take one of the constants that you have listed (a, a_1…a_N) and run the proof on that…the problem is that if there are ANY logical constants then we can prove that it is a tauttology that they exists and that they must exist…

    finally, you say “Though it is not possible that I lack existence, it is still true that there are possible worlds which do not contain me.” I don’t understand this because to say that it is possible that you do not exist is JUST to say that there are possible worlds that do not contain you, so this looks like a contradiction to me…but maybe I misunderstood your point?

  13. A third thought…

    OK, so I think I may be wrong about EG, Bessie and Glennan misled me!!! They didn’t say that it was ‘one or more’ of the constants that could be replaced. My bad.

    In that case I don’t think that you are offering a critisism of the argument that I am making, but rather just a simpler proof of the point I was trying to make. And it further illustrates the problems with assuming that there are logical constants by illustrating the problem I mentioned above. I can prove that unicorns exist and that they necessarily exist…this doesn’t seem to be trivial…

  14. Perhaps you’re right about my strategy not working, and I agree my conclusion looks contradictory at first sight. Hopefully this will clarify what I was trying to say: my idea was to transform statements about the non-existence of certain individuals into statements which are not about them at all, but which give the intuitively correct result that they do not necessarily exist (which in my view is not to say that they *fail* to exist in some possible world). So assume there are exactly three objects, denoted by ‘a’, ‘b’, and ‘c’. Suppose, then, we wish to say that a does not necessarily exist. Instead of expressing this as “◊ –Ex (x=a)”, we can say “◊ Ex & Ey ((x=b) & (y=c)) & (z) (z=b) v (z=c)” As a is nowhere mentioned in my paraphrase, I don’t see how you can take the paraphrase and prove that it (necessarily) exists. If it is possible for b and c to have existed while at the same time everything was identical to one or the other, doesn’t this vindicate the idea that a is not necessarily existent? If a were necessarily existent, wouldn’t my paraphrase be necessarily false? And if the problem arises of asserting that b or c are not necessarily existent, couldn’t we paraphrase those statements in the same way?

    As for the distinction between the lack of existence of an individual in a world and the failure of that world to contain that individual: Suppose we take an ersatzist view of possible worlds as sets (or classes, or fusions, or whatever) of propositions; Russellian propositions, to be precise. Then we can say that an individual is contained in a world if they are a constituent of at least one proposition which is a member of that world. If they are not a constituent of at least one proposition which is a member of that world, that world does not contain them. We can say that an individual exists in a world if that world has as a member a proposition which asserts their existence, and that an individual does not exist in a world if that world has as a member a proposition which asserts their nonexistence. To take myself as an example, any world which does not contain me fails to have as members any propositions which have me as a constituent, and consequently any propositions which assert my existence or nonexistence will not be members of such a world. Worlds which have as members propositions asserting my nonexistence are impossible since I am a constituent of such propositions, and hence would have to exist in order for such worlds to be actual. Yet I see no reason why worlds which do not contain me could not be actual. Of course this view entais that not all possible worlds are “maximal” in that there are some propostions such that neither they nor their negations are a member of them; but I do not find this consequence absurd, and at any rate it seems far more plausible to reject this assumption than to hold I necessarily exist.

  15. Hi Jason,

    Thanks for the response. That helps me see where you are comming from.

    Onequestion I have right off thebat is if you are going to be using quantifiers and diamonds anyway, then what is to preven us from using ◊ –Ex (x=a)? If the answer is nothing, then your strategy seems ad hoc…after all we only write it that way because we are trying to prove □Ex (x=SK)

    Also, I am not trying to prove that SK necessarily exists, I think that is clearly absurd. I am trying to show that this is an undesired result of accepting logical constants. So your paraphrase is not problematic…

    As for the second part of your comment, first, I don’t see how itis supposed to explain what you said, namely “Though it is not possible that I lack existence, it is still true that there are possible worlds which do not contain me.” What you argued is that worlds in which you don’t exist don’t contain any propositions about you, not that it is impossible that you lack existence, which you seem to accept, but again I might be missing something.

    Second, on the view you sketched can I assert that Socrates lived in 400 BCE? He no longer exists, so cannot be a part of a Russellian proposition…

    Thirdly, if the answer to 2 is that Socrates used to exist or something, then we can say the same sort of thing modaly and say that the individual must be contained in the word or a world that is accessable from one that contains it

    Lastly, if the actual world is one that conains propositions about you and these propositionas are expressed via singular terms, then we can prove that all possible worlds contain you, which is the problem! Now, as I said, I agree that we should reject the conclusion that you, or I, necessarily exist, which is why we should reject the view that there are singular terms (in language…it is, according to me, a different story when it comes to thoughts)…

  16. Thanks Richard, you’ve given me a lot to think about, so for now I’ll confine myself to a few quick comments.

    First, as to your question, “…what is to preven us from using ◊ –Ex (x=a)?” my answer is, indeed, “Nothing.” It’s just that, if you say that, you’re saying something false. I think the reductio you gave of it is sound (even though it seems you don’t), and it is for similar reasons that I tried to paraphrase negative existentials: so we could say something that, while different, is true and acheives the same purpose. (As for “□Ex (x=SK)”, it also seems to be something my view commits me to reject, though I see now I’d have to do some work to make good on that rejection.)

    Secondly, you say: “What you argued is that worlds in which you don’t exist don’t contain any propositions about you, not that it is impossible that you lack existence, which you seem to accept, but again I might be missing something.” I don’t think I ever said that worlds in which I don’t exist do not contain propositions about me. I think ‘worlds in which I don’t exist’ are impossible worlds, because I *am* contained in such worlds in virtue of the fact that I am a constituent of propositions which assert my nonexistence. Worlds which don’t contain me don’t have any singular propositions about me, so it is not correct either to say that I exist in such worlds or that I fail to exist in them: I have no status in them whatsoever.

    Finally, since I am an eternalist, I *do* think Socrates exists (in an unrestricted sense), it’s just that his existence is earlier than now.

    I’ll have to do some more thinking before responding to your other points. One again, thanks for your comments.

  17. First of all, i think that you should be more clear in your math effort.
    You should note what all of your signs mean. For instance, i don’t have a clue what “□Ex (x=SK)” means. With x=SK you probably explicitly tell the readers what x means…
    A tip: whenever you try to explain something, consider all the readers as if It’s the first time in their life that they have thought about the words that you are writing.

    For instance, what does the tilted square mean?
    What does minus the existence of SK mean? It means non existence?
    Then, when the minus affects existence, does it really say that an existence is transformed into non existence? It’s like saying that the “minus” is equal to non existence. So, when you put a minus in front of a non existence, you get… existence? Non existence of non existence means existence?

  18. Hi goran123,

    First of all, it’s not math, it’s logic.

    As for your tip, if I abided by it then I would never get anything done! Although I certainly don’t mind explaining things to people who ask nicely…

    As for you questions, the ’tilted square’ (also known as a diamond) means ‘it is possible that’, the box means ‘it is necessary that’ the ‘-‘ means ‘it is not the case that’ and ‘x’ is a variable; so “□Ex (x=SK)” says ‘it is necessary that there exists some object (x) that is identical to Saul Kripke, or as I said ‘Saul Kripke necessarily exists’. “[M]inus the existence of SK” means ‘it is not the case that Saul Kripke exists’ or ‘Saul Kripke does not exist’

  19. Hi Jason,

    Thanks for the response, sorry for the delay in getting back to you…

    So it seems we agree, at least in part, where the problems lie for your view, so I will only respond to your second comment…

    You say “I don’t think I ever said that worlds in which I don’t exist do not contain propositions about me.” but in your earlier comment you said “To take myself as an example, any world which does not contain me fails to have as members any propositions which have me as a constituent, and consequently any propositions which assert my existence or nonexistence will not be members of such a world.” So…it looks to me like you did say that, though nothing much hangs on it I suppose…

    Also, you say

    I think ‘worlds in which I don’t exist’ are impossible worlds, because I *am* contained in such worlds in virtue of the fact that I am a constituent of propositions which assert my nonexistence. Worlds which don’t contain me don’t have any singular propositions about me, so it is not correct either to say that I exist in such worlds or that I fail to exist in them: I have no status in them whatsoever.

    This is confusing to me because you do think that worlds in which you don’t exist are possible, though it is impossible (on your view) to say anything about you in those worlds…which brings me to a final remark.

    Your view about propositions and modality strike me as being more akin to an Armstongian kind of view than a Lewisian kind of view…if this is so, and you endorse a c=kind of actualist combinotorial view of modality (you kinda have to, since in this actual world you cannot assert anything about things that do not exist) then to really get the paraphrase right you will have to include a list of the things that actually exist as a conjunctive clause in the paraphrase and then your strategy collapses ionto the regular one since you have to assert that you exist…so to take the three object world you mentioned earlier to say that it is possible that A doesn’t exist we will have to say that A, B, and C do exists and that it is possible that there is a world where only B and C exists. This gets you right back into hot water!

  20. Hi Richard,

    As for the failure to contain / lack of existence distinction: I don’t like to say that I don’t exist in worlds which fail to contain me because propositions of the form “Jason Zarri does not exist” seem to be *about* me in a way they could not be if they were true, since if they were true there wouldn’t be anything for them to be about. But worlds which don’t contain me don’t have any singular propositions about me. Thus I prefer to say that only worlds which contain singular propositions attributing a lack of existence to me are worlds in which I don’t exist. These (latter) worlds are impossible worlds because if they were actual I would have to exist in order to *lack* existence, just as a blind person has to exist in order to *lack* sight. But while some people do lack sight, no one does and no one could lack *existence*. So on my preferred way of speaking, worlds which don’t contain me don’t have any singular propositions about me, and worlds in which I don’t exist do have singular propositions about me, namely, singular propositions to the effect that I don’t exist. The point is more terminological than substantive; you can say that I don’t exist in worlds which fail to contain me if you like, but I think this engenders confusion. I hope my view is clearer now.

    I’m not sure I understand your final argument, though you’re probably right that it sinks my position.

    After reading your more recent post on the subject, I think I understand the views you’re concerned with better. The argument I was originally reacting against was metaphysical rather than logical; if you want to understand better where I was coming from, you can check out Timothy Williamson’s paper “Necessary Existents” here:

    http://www.philosophy.ox.ac.uk/members/twilliamson/index.htm

  21. Hey thanks forthe link to the Williamson paper…I read through it and while I think that his argument depends on a lot of very questionable assuptions (my mind is not one of the open ones that he has written the paper for :)) I now better see where you were comming from…

    A couple of thoughts…It seems to me that Kripke’s version of modal logic avoids the argument that he gives in the begining of the paper. So let’s look at it real fast

    (1) Necessarily, if I do not exist then the proposition that I do not exist is true.
    (2) Necessarily, if the proposition that I do not exist is true then the proposition that I do
    not exist exists.
    (3) Necessarily, if the proposition that I do not exist exists then I exist.
    (4) Necessarily, if I do not exist then I exist.

    Switching from proof-theoretic talk lto model-theoretic talk, we can see that there is a model where (1)-(3) are true but do not entail (4) and so do not entail that I necessarily exist…Let the domain be the actual world and let ‘JZ” name you. So (2), and (3) are true at the actual world (in virtue of your acceptance of Russellian propositions and I am not challenging that (though I might) since the actual world contains you and so contains propositions about you.

    The question then is ‘how is the proposition that JZ does not exist get to be true’? How is (1) true at the actual world? It is true just in case there is a possible world where you don’t exist (that is we adopt the Kripkean notion that a proposition is true at a world if there is some world where it is true. That would make (1) true in the actual world just in case there is some possible world where you do not exist, and we agreed earlier that there is such a possible world (trivially, the empty world). So (1)-(3) are true but in a model where we do not get the result that ‘if you exist then you don’t exist’

    Now though Williams doesn’t mention Kripke by name, I take it that he is addressing this sort of move when he says the following,
    It is sometimes said that a proposition can be true of a possible world without being true in that world. We can express propositions in one world about another world. Thus a proposition might be true of a possible world without existing in that world. But this idea does not address the case for (2+) [‘Necessarily, if the proposition that P is true then the proposition that P exists’], for (2+) does not say that the proposition exists in any possible world of which it is true. We could paraphrase (2+) thus: for any possible world w, if the proposition that P would have been true if w had obtained, then the proposition that P would have existed if w had obtained. We can abbreviate that by saying that for any possible world w, if the proposition is true in w then the proposition exists in w. The antecedent concerns truth in w, not truth of w, so the distinction poses no threat to (2+).
    but this doesn’t really address the Kripke move. Kripke says that a proposition is true in a world if it is true at some possible world. So the proposition that you do not exist is true at the actual world because it is true at (or ‘of’ if you prefer) some world. So it is true in this world and you exist in this world. We can agree with everything that he says except the bit about how the distinction poses no threat. It is true in w (the actual world) in virtue of it being true of some possible world.

    He goes on to consider whether this distinction poses a problem for premise (1), which I will skip since the above seems enough to get by the argument (you may argue that I assume a negative answer to the next challenge, but if so, then I will give the argument against it)…what about his general problem with the distinction? The argument from failure to grasp contingency? He says,

    Consider the contingently true proposition that Blair was Prime Minister in 2000. It is supposed to be true of the actual world @ and false of some other possible world w. On the model, the sentence contains a tacit variable; if @ is assigned to the variable, a truth results, if w is assigned, a falsehood. But that does not make the resulting propositions contingent. There is genuine contingency in how things are only if, once values have been assigned to all variables, the resulting proposition could still have differed in truth-value. It is not contingent that Blair was Prime Minister in 2000 in @ and that he was not Prime Minister in 2000 in w. What is contingent is simply that Blair was Prime Minister in 2000. Its contingency requires it not to have a variable waiting to be assigned a world. The reply ‘But contingency just is variation in truth-value with variation in the value of the world variable’ betrays a failure to grasp what contingency is.

    Well right off the bat he has misrepresented the position since he says that the proposition is true of the actual world. Rather it is that the proposition is true in the actual world because it is true at/of some possible world…so I of course agree that what is contingent is the proposition that Blaire was Prime Minister in 2000…that proposition is true in the actual world and contingent because it is possible that it could have been false (though not expressible on your and Williams’ view) in the actual world…

    Finally the business about the ‘illusion’ of a distinction doesn’t seem to me to work either. He says,
    There is the illusion of a distinction between truth in a world and truth of a world for propositions because we appear to be able to model such a distinction on a corresponding distinction for utterances, forgetting that the presence of the latter depends on the absence of the former.
    What he means by that last bit is that the intuitive claim about utterances (i.e. that the utterance ‘there are no utterances’ is true in this world because it is true of some wold) depends on our having the notion of truth in a world for propositons. As he says,

    The utterance need not exist in that world in order to be true of it because the proposition which it expresses in this world exists in that one. We need not carry the utterance across from this world to that one precisely because we can carry the proposition across instead.

    But this is nothing more than question begging…why isn’t the proposition that the utterance expresses true in this world because it is true of some world, just like the utterance? So, anyways, all of this is along winded way of saying that you don’t need to adopt your paraphrase as a way of avoiding the argument if one accepts Kripke semantics for modal languages like I do…

    But of course the argument of the post, and the one you mention, is meant to show that this is not good enough, one also needs to jettison the idea that our semantics inculedes singular terms. Williamson explicity appeals to singular terms in his defense of premise (3)…it is that defense which my argument is aimed at…

    …except there is this very odd phenomenon that for me showing that logic leads to the claim that I necessarily exist is reason to think that we have got something wrong, while for him it shows that he necessarily exists!!

    Finally, finally, I was going to try and expand on that last bit about Armstrong, but since you already think it ‘sinks’ your view, I think I’ll have breakfast instead 🙂

  22. Dang!!! Looks like I got some of the code wrong! There are a couple of quotes that are not indented…they follow “He says,” and “he says the following,”

    **sorry**

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