Must theists reject the modal ontological argument?
June 9, 2010 — 18:59

Author: Michael Almeida  Category: Uncategorized  Comments: 45

**N.B. This post belongs to Stephen Maitzen**
I take it that the modal ontological argument (for instance, in Plantinga’s version) requires a principle at least as strong as
B: If possibly necessarily P, then P,
where the modal operators are read in the metaphysical (or “broadly logical”) sense. That is, the argument from “It’s possible that a maximally great being exists” to “A maximally great being exists” sooner or later requires B. Indeed, Plantinga’s version seems to invoke the stronger S5 even if all it needs is B.
Here’s the problem. B can be shown to imply the Barcan formula (BF), and BF implies that the only individuals that could have existed are the individuals (past, present, or future) that actually exist. For BF implies this: “If there could have been an individual that satisfies a given predicate, then there actually exists an individual that could have satisfied that predicate.” But surely God could have made an X instantiating a given individual essence even though God didn’t actually make X (and X doesn’t otherwise exist). Creation could have had X although in fact creation lacks X, and God wouldn’t have been constrained to “make X out of some actual individual” or “turn some actual individual into X.” Those quoted phrases don’t even make sense, given that X would have its own individual essence. Must the friend of the modal ontological argument defend BF by saying that this world, down to its every particle, is the only world God could create or actualize?
If God could have made X even though X never actually exists, then BF is false, in which case B is false, in which case the modal ontological argument is unsound even if we grant what most folks say is the truly controversial premise, namely, “It’s possible that a maximally great being exists.”
What have I missed? Has this objection been replied to somewhere? If you’d rather reply off-list, please email me at smaitzen [shift+2] acadiau.ca
.

Comments:
  • Mike Almeida

    Stephen,
    Plantinga’s argument does not require B, as far as I can tell. Plantinga’s argument goes from,
    1. Possibly, God is necessarily F.
    to the conclusion,
    2. God is F.
    (B) does not tell us that anything that in any world has the essential property P, actually exists with P.
    But aside from this, let’s suppose that S5 commits us to the Barcan Formula, BF, (as it seems to do under the assumption of the lower predicate calculus). Does this lead is to any untoward consequences? Not that I can see. BF says this,
    BF. (∀x)􀂅α ⊃ 􀂅(∀x)α
    One interesting way BF is true is if ‘(∀x)ô€‚…α’ is necessarily false. It is necessarily false that everything has some property essentially. Plantinga might claim (I would, in any case) that it’s necessarily false that ‘(∀x)ô€‚…α’, (except perhaps for trivial essential properties that every possible thing exemplifies). So all you need to validate BF is that every world have at least two things that do not exemplify the same non-trivial essential properties.

    June 9, 2010 — 19:34
  • Mike,
    Thanks very much for publishing my post. If Plantinga’s argument doesn’t require B, then how do you account for his invoking, in the middle of that argument, “the principle that what is necessary or impossible does not vary from world to world” (Nature of Necessity, p. 216)? That principle is in fact stronger than B; it’s S5. Furthermore, if the argument doesn’t require B, I’d like to see it laid out in formal terms without the use of B or anything stronger.
    The version of BF you gave above straightforwardly implies this: “If possibly there exists an x such that Fx, then there exists an x such that possibly Fx.” (Recall that I emailed you the short proof of this implication.) So let F be the full specification of some individual essence. According to BF, then, there could have been an individual with that individual essence only if there actually exists an individual that could have had that individual essence. For reasons I gave in the post, I think this implies the heterodox view that God can’t create any individual he doesn’t in fact create.

    June 9, 2010 — 19:52
  • Mike Almeida

    The version of BF you gave above straightforwardly implies this: “If possibly there exists an x such that Fx, then there exists an x such that possibly Fx.”
    That’s mistaken, as far as I can tell. BF says that if EVERYTHING (everything that exists, everything in the actual world) has some property F necessarily, then it is true in every world that everything in those worlds (everything that exists in those worlds, everything there) has F necessarily. It says this,
    BF. (∀x)􀂅α ⊃ 􀂅(∀x)α
    But you are reading it as though it said this,
    BF*. (∀x)(􀂅α ⊃ 􀂅(∀x)α).
    This is the difference between saying on the one hand, ‘everything that is red is square’,
    (∀x)(Rx ⊃ Sx)
    and saying, on the other hand, that if everything is red, then everything is square.
    (∀x)Rx ⊃ (∀x)Sx.
    These are obviously not the same. BF is analogous to this latter claim. That’s why we have the trivial validation of BF that I mentioned above.
    Make sense?

    June 9, 2010 — 20:13
  • Mike: I’m not misreading BF. I emailed you the derivation as a PDF, so I won’t try to reproduce it here (symbols and all). See in particular steps (6)-(9), where I derive my version of BF from your version of BF. Where do steps (6)-(9) go wrong? (I can email the same PDF to any others who might be interested.)

    June 9, 2010 — 20:20
  • Mike Almeida

    Here is BF.
    BF. (∀x)􀂅α ⊃ 􀂅(∀x)α
    Steve, this should be easy to settle. If you take BF to entail that if Jim is necessarily F then necessarily Jim is F, then you’re reading the notation incorrectly. That’s not what it entails.
    It says that if EVERYTHING is necessarily F, then necessarily, EVERYTHING is F. That’s just what the notation says. So, BP entails that if Jim AND EVERYTHING ELSE is necessarily F, then necessarily Jim (and everything else) is F.
    Again, this is why we have ways of trivially validating BF (i.e. in cases where it is necessarily not the case that everything has some property necessarily).

    June 9, 2010 — 20:34
  • Mike: All right, here’s the same four-step derivation I emailed to you, although in typography that may look terrible. Let “F” be an arbitrary predicate, “(x)” the universal quantifier, “(Ex)” the existential quantifier, “Nec” the necessity operator, and “Poss” the possibility operator:
    (6) (x)(Nec Fx) –> Nec(x)Fx……….[Your version of BF]
    (7) ~Nec(x)Fx –> ~(x)(NecFx)………[From (6), contraposition]
    (8) Poss(Ex)~Fx –> (Ex)(Poss~Fx)……[From (7), interdefinability of ‘Nec’, ‘Poss’, ‘(x)’, ‘(Ex)’]
    (9) Poss(Ex)Fx –> (Ex)PossFx………[My version of BF; from (8), exchanging “Fx” for “~Fx”]
    Don’t let step (9) throw you. If (6) is a valid schema, then (8) is a valid schema, in which case the uniform substitution of “Fx” for “~Fx” in (9) also produces a valid formula. So my version of BF follows from yours, unless there’s a misstep in (6)-(9).

    June 9, 2010 — 21:09
  • I think the focus on B (in the initial post) is a bit of a red herring here. Note that in the same sense that BF can be proved using B, its converse can be proved from plain vanilla K:
    1. If Fx then for some x, Fx.
    2. If possibly Fx then possibly for some x, Fx
    3. If for some x, possibly Fx, then possibly for some x, Fx.
    1 and 3 are applications of the standard quantifier rules, and 2 follows from 1 by K.
    For both BF and CBF, I think the culprit is really the background quantification theory, not the distinctive modal axioms. If you want to avoid those commitments, your quantifiers had better be governed by a free logic, which says the sensible things when you have variables that don’t refer to anything in the evaluation world.

    June 9, 2010 — 22:25
  • Mike Almeida

    Steve,
    I’m not denying the equivalence you note. I was talking about reading the formula. That’s a side issue that is probably all clear.
    Given S5 + LPC, several interesting things happen. It is straightforward to prove all domains (in all worlds) contain the same objects. It also follows that the distinction between de re modality and de dicto modality collapses. I think this is important to observe, since it offers a cheap ontological argument. An argument that Plantinga does not advance (but might have). It might go like this (where ‘OO’ is the predicate ‘is omnipotent, omniscient, omnibenevolent’…).
    1. M(Ex)NOOx
    2. M(Ex)NOOx –> N(Ex)OOx
    3. /:. N(Ex)OOx
    4. /:. (Ex)OOx
    Here we have a proof that necessarily there exists an omnipotent, omniscient, omnibenevolent being from the assumption that possibly something has those properties essentially. Notice that this ontological proof does not need the assumption that God has the property of necessarily existing.
    Why would Plantinga redundantly add the assumption that God has the property of existing in every world? He doesn’t need to make that assumption given S5 + LPC. It’s because he is not running this sort of argument. So, go to a logic in which you have B, but you do not have the derived rule DR4. Maybe KB or TB will do. In those weaker logics, you cannot derive BC or CBC. And Plantinga’s ontological argument will still go through.
    Nonetheless, I think it is really important to note that anyone committed to S5 + LPC will have to make peace with the Barcan formula and its implications. I can’t believe the proof in Cresswell hadn’t been noticed before, since so many people are committed to S5. There is a way Plantinga might manage this, and it echos a way that Williamson suggests (I’ll get the reference). Distinguish between existing at a world and being instantiated in a world. Since all individual essences exist at every world, it is true that all of us necessarily exist. But it is false that all of us is necessarily instantiated. But God is necessarily instantiated.

    June 10, 2010 — 7:46
  • Mike: Thanks for your latest comment. I’ll await the reference to Williamson that you mentioned, but if it’s what I think it is, then it’s a reference to some very weird things Williamson has said in defense of BF. I’ll stay tuned. Meanwhile:
    (1) I reject (as I have in our emails) your parsing of ‘x has F essentially’ as ‘NFx’. I’m essentially human, but I’m not necessarily human since I don’t exist in all worlds. Let ‘m’ denote me. ‘Maitzen is essentially human’ becomes ‘N(x)(x=m –> Hx)’ (Necessarily, for any x, if x is Maitzen then x is human), not ‘NHm’ (Necessarily, Maitzen is human).
    So in your argument 1-4 above, 1 does assert the possible existence of a necessarily (and not just essentially) tri-omni being (and hence the possibility of a necessarily existing being).
    (2) I think there’s some sloppiness in the wording of the last two sentences of your comment. None of us is identical to his/her individual essence; we’re concrete, whereas essences aren’t. So it doesn’t follow from the existence of my essence in/at every world that I exist in/at every world. Further, it’s only our essences (or God’s essence) that are literally instantiated; neither we nor God are literally instantiated.

    June 10, 2010 — 8:25
  • Jeff: Many thanks for your helpful comment. You rightly point out that
    (1) K implies CBF.
    I’ve been pointing out that
    (2) (K + B) implies BF.
    So if I’m right that B is the culprit here, then BF must be implausible in a way in which CBF isn’t. I think that’s so. I’ve been arguing that theism generates counterexamples to BF because theism holds that God could have made an individual that God will in fact never make. (I think there are also counterexamples to BF that don’t depend on theism.) Now, are there counterexamples to CBF? The only ones that strike me as potential counterexamples require treating existence (or ‘exists’) as a predicate, such as
    N(x)(Ey)y=x –> (x)N(Ex)y=x,
    whose antecedent I accept (Necessarily, for any x there exists a y such that y=x) but whose consequent I reject (For any x, necessarily there exists a y such that y=x). In this case, what’s predicated of x is ‘x exists’, and maybe we should take to heart Kant’s warning that ‘exists’ isn’t a predicate. Similarly for this alleged counterexample to CBF:
    N(x)x=x –> (x)Nx=x,
    whose antecedent (but not consequent) I accept. The worry that ‘exists’ isn’t a predicate motivates the worry that ‘is self identical’ or ‘is identical to x’ aren’t predicates either. I think Kant’s criticism is right about at least some versions of the ontological argument. Do you think it’s fair to extend his criticism to alleged counterexamples to CBF? Are there counterexamples to CBF that don’t depend on treating ‘exists’ or ‘is self-identical’ as predicates?

    June 10, 2010 — 8:52
  • Mike Almeida

    Steve,
    1. Here is the reference to Williamsons, ‘Necessary Existents’ http://www.philosophy.ox.ac.uk/__data/assets/pdf_file/0012/1326/rip.pdf
    You might also want to check out Graeme Forbes S5 without B in his The Metaphysics of Modality p. 28 ff.
    2. I’m not sure why you balk at reading the narrow scope necessity operator (for instance, (Ex)NFx) as something is essentially F. It is a pretty standard way to read it. All the talk about degrees of essentialism that a logic commits us to are formulated in that way (see, for instance, T. Parsons, ‘Essentialism and Quantified Modal Logic’). There are two ways to read ‘necessity’. In one (narrow scope) I’m necessarily F just in case I am F in every world in which I exist. In another, (wide scope) necessarily I’m F just in case it is true in every world that I’m F. What S5 + LPC does is conflate these. They turn out to be equivalent in that logic, which is part of what makes it so interesting.
    3. I was not assuming that we are identical to our individual essences. For Plantinga, we are represented by individual essences in other worlds. He’s an actualist, so he doesn’t think there are literally objects in other worlds. But we do have representatives in other worlds. By comparison, Lewis uses counterparts as representatives of us in other worlds. So, when you are quantifying over, say, human beings in other worlds, the most those things could be are our individual essences. And all individual essences exist in every world. Our representatives are not literally human beings for Plantinga, though they are of course for Lewis.
    4. The ontological argument can be run in a weaker logic that includes B but not the derived rule 4. This way you get none of the commitments of S5 + LPC. Those who like the argument might want to run the argument that way.

    June 10, 2010 — 9:31
  • Mike: Thanks for the Williamson reference. It’s what I thought it would be: astoundingly weird claims about (e.g.) concreteness. I’ll give it another look, though. Thanks too for the other references. I’ll think about B minus DR4.
    Now I hope you’ll forgive me, but our discussion won’t be as productive as it could be if you won’t admit to having said something sloppy. In 3 above you say, “I was not assuming that we are identical to our individual essences.” But in the antepenultimate sentence of your previous message you said, “Since all individual essences exist at every world, it is true that all of us necessarily exist.” How is that latter inference valid if you’re not identifying us with our individual essences? It’s frustrating to see you deny having assumed (or implied) that identification and then just change the subject.
    Regarding 2 above: If you want to use ‘x is essentially F’ where I would use instead ‘x is necessarily F,’ that’s fine. I think your usage conflates a distinction that’s worth preserving, but never mind. The real point is not the label we choose but the logical form of premise 1 in your 1-4 ontological argument: it has the logical form ‘Possibly, there exists an x such that in every world [period] x is tri-omni’. There’s no restriction on the worlds in the logical form of 1. That’s the key point.

    June 10, 2010 — 10:14
  • Note that if BF indeed follows from S5 in the right logic, then Lewis’s modal realism is incoherent, since it verifies S5 but not BF.
    It seems to me that the derivation of BF from S5 is simply an artifact of the non-freeness of the logic. One needs to move to Kripke’s quantified modal logic, and the issue disappears.

    June 10, 2010 — 10:27
  • Alex (if I may; we met in Chicago): If non-freeness is the culprit, then non-free K should imply unacceptable consequences? Does it? It does imply CBF, but I’m not convinced that CBF is a problem (see my reply to Jeff). However, non-free (K + B) implies BF, which is a problematic consequence especially for theists. That suggests that B is the culprit.

    June 10, 2010 — 10:50
  • Mike Almeida

    “Since all individual essences exist at every world, it is true that all of us necessarily exist.” How is that latter inference valid if you’re not identifying us with our individual essences? It’s frustrating to see you deny having assumed (or implied) that identification and then just change the subject.
    I guess I’m not sure what the problem is. I’m sorry we’re not communicating on this point. I said this,
    There is a way Plantinga might manage this, and it echos a way that Williamson suggests (I’ll get the reference). Distinguish between existing at a world and being instantiated in a world. Since all individual essences exist at every world, it is true that all of us necessarily exist. But it is false that all of us is necessarily instantiated. But God is necessarily instantiated.
    You seem to think that it’s impossible that I (or anything) could necessarily exist if I am world-bound. But that’s false, right? We can define necessary existence in the way Lewis does in CTQML (see p. 31 ff. in PP I) where it come out true that everything actual necessarily exists, though everything is world-bound. I had in mind something analogous for Plantinga, as a way of managing the implications of S5 + LPC. Hope that’s less frustrating.
    The real point is not the label we choose but the logical form of premise 1 in your 1-4 ontological argument: it has the logical form ‘Possibly, there exists an x such that in every world [period] x is tri-omni’. There’s no restriction on the worlds in the logical form of 1. That’s the key point.
    No, I’m deliberately not reading M(Ex)NOOx as ‘possibly, something is OO in every world simpliciter’. My point is that there is no reason to read it ‘N’ as strong necessity (in the way you just suggested) since, in S5 + LPC, anything that exists in any world also exists in every world. Since that it true, the following two are equivalent.
    1. Possibly, something has OO in every world in which it exists.
    2. Possibly, something has OO in every possible world simpliciter.
    Something has OO in every world in which it exists just in case it has OO in every possible world simpliciter, since if it exists in any world it exists in all worlds. That’s how we get an ontological argument without the assumption that possibly, God necessarily exists. All we need is the claim that possibly, God is essentially OO.
    But I will retract the overstatement above that we get the reduction of de re and de dicto. We don’t quite get that in S5 + LPC.

    June 10, 2010 — 10:53
  • Mike Almeida

    It seems to me that the derivation of BF from S5 is simply an artifact of the non-freeness of the logic. One needs to move to Kripke’s quantified modal logic, and the issue disappears.
    That’s a nice point Alex. Surprized Menzel hasn’t shown up.

    June 10, 2010 — 11:03
  • You seem to think that it’s impossible that I (or anything) could necessarily exist if I am world-bound. But that’s false, right? We can define necessary existence in the way Lewis does…
    You seem to be appealing here to Lewis’s counterpart theory in order to defend your inference “Since all individual essences exist at every world, it is true that all of us necessarily exist” without having to identify us with our individual essences. For Lewis, each individual is world-bound, but a necessary being has a counterpart in every other world. So, if I’m reading you right, you’re suggesting that our individual essences are our Lewisian counterparts, and those counterparts exist in all other worlds. But it’s scarcely more plausible to identify our Lewisian counterparts and our individual essences than it is to identify us and our individual essences. Surely none of my Lewisian counterparts are abstract objects.
    …since, in S5 + LPC, anything that exists in any world also exists in every world.
    Whoa. Did I read that correctly? Are you saying that S5 implies that whatever could have existed exists necessarily? Where does that come from, and why would anyone be attracted to S5 if it did have that implication?

    June 10, 2010 — 11:14
  • Mike Almeida

    Steve,
    I suggested that these individual essences–which are not identical to me any more than counerparts are–might serve as representatives of me. The fact that they’re abstract objects does not make them any less candidates for representatives, does it? That’s all I had in mind.
    …since, in S5 + LPC, anything that exists in any world also exists in every world. .. Are you saying that S5 implies that whatever could have existed exists necessarily? Where does that come from, and why would anyone be attracted to S5 if it did have that implication?
    Maybe I have this wrong, so I’m happy to run through it. We have the proof that S5 + LPC entails BF, right? That’s what Hughes and Cresswell show on p. 247? And you agree with that conclusion, I’m pretty sure. But, unless I’ve got this wrong, the Barcan Formula is invalid if domains vary from world to world. Does that seems right? Here’s BF.
    BF. M(Ex)Fx –> (Ex)MFx
    Let F be the property of ‘being identical to x’. We have it true that if possibly, something is identical to x, then there is something that is possibly identical to x. But anything that is possibly identical to x is identical to x. So, it looks like any object in any world is also an actual object. But BF is true in every world, so it looks like every object that exists in any world exists in every world. Cresswell does not offer this proof, but he worries about this objection on pp. 274 (bottom)-275.

    June 10, 2010 — 11:34
  • Steve:
    The reason non-free K may not suffice to prove BF may simply be that K just doesn’t capture enough of our modal intuitions. 🙂
    Here’s a move, which may not work as I haven’t worked out the details. Let K@ be K enriched with a second modal operator: @ = “actually” or “at the actual world”, where this is understood rigidly (so, @p iff Nec @p). Add appropriate axioms and rules of inference to make it all come out reasonably. I don’t know what those are going to be. Basically, whichever ones my argument below needs. But it’s intuitively pretty clear that something like this had better be doable, since @ expresses a notion that we can certainly make sense of.
    Anyway, now define the modal operators “Nec+” and “Poss+”. Nec+ p holds iff both @p and Nec p, and Poss+ p holds iff either @p or Poss p, where Nec and Poss are the operators from K.
    If we set the axiom and rules of inference correctly, it’s pretty clear that Nec+ and Poss+ had better satisfy B. For suppose p. We’re supposed to show Nec+ Poss+ p. This means we must show @ Poss+ p and Nec Poss+ p. But Poss+ p is (@p or Poss p). So, we must show @(@p or Poss p) and Nec(@p or Poss p). But both of these hold since @p can be proved from p (using appropriate rules of inference, which I’ve waved my hand at), and hence @@p from @p, while Nec @p follows from @p by the unshiftiness of @.
    But BF is just as absurd for Nec+/Poss+ as for Nec/Poss.
    So, anybody who wants to deny BF, has to deny the possibility of defining a modal operator @ that lets you derive @p from p and that is such that Nec @p follows from @p. But intuitively, we have such an operator, and so much the worse for any logic that can’t handle it.

    June 10, 2010 — 12:09
  • Mike: As you know, I’m no friend of S5, B, or BF, and I’ve been arguing that no theist should befriend them either. So I don’t object in principle to your argument that S5 has the weird consequence that everything exists necessarily. But your argument assumes that ‘being identical to x’ is a property (or perhaps ‘is identical to x’ is a predicate). I worry slightly about that assumption, since it might give CBF (which I accept) consequences that I reject (compare my reply to Jeff). If it doesn’t, I’m OK with it.
    I surmise that defenders of S5 would reply to your argument in the way suggested by Alex: use a free logic! But, as I said to Alex, I don’t regard non-freeness as defective, since non-free K has (I think) no unacceptable consequences whereas non-free (K + B) does.

    June 10, 2010 — 12:28
  • Mike Almeida

    I surmise that defenders of S5 would reply to your argument in the way suggested by Alex: use a free logic! But, as I said to Alex, I don’t regard non-freeness as defective, since non-free K has (I think) no unacceptable consequences whereas non-free (K + B) does.
    I think most S5 people don’t realize the implication, and will reconsider when they do. There is the move to S5 without BF that Forbes describes, and there is Kripke’s S5 without BF. Either of these is a better option.

    June 10, 2010 — 13:11
  • Chris Menzel

    Thanks to Mike for letting me know about this thread; I hadn’t checked the RSS feed over the past few days. A few comments, in more or less random order. Let SQML (“Simplest Quantified Modal Logic”) be S5+LPC with identity.
    • SQML does indeed entail that everything necessarily exists; more exactly, the following is a theorem of SQML:
    NE: ∀x⎕∃y(y=x)
    Here’s a proof:

    1. ∀y(y≠x) ⊃ x≠x     Instance of UI
    2. x=x ⊃ ∃y(y=x)     1, contraposition, def of ∃
    3. ⎕(x=x ⊃ ∃y(y=x))  2, Necessitation
    4. ⎕x=x ⊃ ⎕∃y(y=x)   3, K schema, and MP
    5. x=x               Theorem of SQML
    6. ⎕x=x              5, Necessitation
    7. ⎕∃y(y=x))         4,6 by K
    8. ∀x⎕∃y(y=x)        7, UG
    

    • As Alexander Pruss suggests, modifications to SQML can be applied that undermine the proof of BF (as well as NE). He points to Kripke’s system KQML as an example of such a modified system. KQML only has closed theorems — no free variables allowed. Hence, the above proof of NE is blocked both at line 1 and at line 5, neither of which is a theorem of KQML. Unfortunately, KQML is rather badly crippled, as (among other things) the language of the logic contains no names and hence does not enable you to express simple de re modal assertions like “Socrates could have been a bricklayer.” (The reason names are disallowed is that one would otherwise be able to reconstruct the proofs of BF and NE in the system.)
    • Tom Jager developed a Kripke-style logic/semantics (call it “JQML”) that embodies Plantinga’s haecceitist metaphysics in “An Actualist Semantics for Quantified Modal Logic” (Notre Dame Journal of Formal Logic, 23(3) (1982) 335–49). In the “intended” model for this semantics, instead of the apparent possibilia of Kripke’s semantics one has haecceities. In this semantics, ‘Fx’ is true at a world w iff (roughly) the haecceity h assigned to ‘x’ is coexemplified with the property expressed by ‘F’ at w. In the semantics of JQML, therefore, BF (evaluated at the actual world) expresses (roughly) that, if there is a world where some haecceity is coexemplified with F, then some actually exemplfied haecceity is coexemplified with F at some world. Al has expressed is approval of JQML in a couple of places. Unlike the “intended” model of KQML, the intended model of JQML is actualist-friendly because haecceities are properties and, hence (???), are necessary — hence, actually existing — beings. JQML avoids BF by following Kripke in restricting theorems to closed formulas only. The system is arguably preferable to Kripke’s in a number of ways, particularly in its treatment of de re modalities. Unfortunately, it is still crippled in ways similar to Kripke’s — notably, again, the language does not contain individual constants.
    • In the present context, I suppose the central question is whether JQML is enough to reconstruct the modal ontological argument. I’m quite certain that it is. However, one then also has to ask whether a commitment to a crippled logic like KQML or JQML has untoward consequences in other contexts where one needs a QML. Given their restrictions, it seems to me obviously the case that KQML and JQML are not the logics you want for general philosophical purposes.
    • Pruss suggests that “the derivation of BF from LPC+S5 is simply an artifact of the non-freeness of the logic” and suggests moving instead to KQML. He seems to be implying thereby that Kripke’s quantification theory is free; it is not.
    • It is quite true, however, that BF and NE can both be avoided by moving to a free quantification theory. Fine, notably, presents such a logic in “Model Theory for Modal Logic, Part I — The DE RE / DE DICTO Distinction” (JPL 7 (1978), 125-156). In a free quantification theory the argument for NE above never gets off the ground as, instead of line 1, the free UI schema only yields ‘∀y(y≠x) ⊃ (E!x ⊃ x≠x)’, where ‘E!’ is the existence predicate.
    • Jettisoning classical quantification theory is, in my opinion, still draconian. Fortunately, it is also not necessary; one can make other modifications instead.
    • Shameless self-promotion 1: In my paper “The True Modal Logic” (JPL 20 (1991) 331-374), I develop an S5-based QML with a classical quantification theory in which both BF and NE are invalid. There are two secrets to the system’s success. The first is that the quantification theory, while classical (i.e., it includes all of LPC with identity), is free in modal contexts. The second is that Necessitation is restricted to theorems that are not derived using axioms of the form ‘α=α’, for terms α. The latter is what clobbers the above proof for NE, as the inference from 5 to 6 is invalid. The justification for this restriction on Necessitation is simply that identity is a relation. If, like me, you are a serious actualist, you do not think that contingent beings have any properties or stand in any relations to anything in worlds where they don’t exist. Hence, in particular, I wouldn’t have stood in the identity relation to myself if I hadn’t existed. Hence, it is not necessary that I am identical to myself. What is necessary is the weaker conditional proposition that if I exist, then I am identical to myself, i.e., more generally, we have ‘⎕(∃y(y=x) ⊃ x=x)’ as a theorem of the system.
    • Shameless self-promotion 2: Many of the issues raised in this thread are discussed in excruciating formal and philosophical detail in my SEP article Actualism, to which Alexander kindly linked in an earlier post.

    June 10, 2010 — 15:29
  • Chris Menzel

    My first remark in my previous post (sorry for the length) was that it is indeed a theorem of SQML that everything exists necessarily. I neglected to add that its necessitation, i.e.,
    NNE: ⎕∀x⎕∃y(y=x)
    follows immediately by, of course, the rule of Necessitation. Hence, it is also a theorem of SQML that (as Steve put it) “whatever could have existed exists necessarily.”

    June 10, 2010 — 15:49
  • Chris: Many thanks for your detailed comments, and thanks to Mike for alerting you to this thread. I look forward to digesting your comments and the references and links you provided and seeing what’s left of my theistic challenge to the MOA. I promise to return and comment once I’ve done that.
    Mike: Looking back over this very helpful thread, I confess that one thing still puzzles me. In comment #8 above, you say, “I’m not denying the equivalence you note.” The equivalence at issue was the equivalence between your (universally quantified) version of BF and my (existentially quantified) version of BF. Yet in comment #3, you had replied, “That’s mistaken, as far as I can tell” to my assertion in comment #2 that “The version of BF you gave above straightforwardly implies” my version of BF. And so I provided a derivation of the latter from the former, both by email and in comment #6. The referent of “That” in your “That’s mistaken” is pretty clear (you even quoted it): my claim that your BF implies my BF. How is it, then, that you never denied their equivalence? In comment #3, you denied that the first even implies the second.

    June 10, 2010 — 16:21
  • Mike Almeida

    Steve, this should be easy to settle. If you take BF to entail that if Jim is necessarily F then necessarily Jim is F, then you’re reading the notation incorrectly.
    Steve, I took you to be claiming (something like) BF entails that ‘if Jim is necessarily F then necessarily Jim is F’. If you were claiming that, I said, then it misreads BF. It turns out that you were not claiming that at all. I just misread what you said, which was entirely correct. This is what comes from reading posts too quickly. Apologies.

    June 10, 2010 — 16:39
  • Chris (if you’re still checking this): Thanks again for all that material. While I make my way through it, let me ask you a different question. Your modal logic is evidently, as you say, S5-based. I wonder, then, how you’d respond to the sorites-style arguments against propositional S5 (and S4, and even B) due to Hugh Chandler (1976) and Nathan Salmon (1989). Those arguments seem to need only two very plausible assumptions: (1) some version of the essentiality of origins; (2) enough modal tolerance to deny mereological essentialism. I myself find the arguments pretty compelling. They strike me as good independent grounds for rejecting at least S4 and S5 (if not also B), regardless of any problems arising from BF or NE in particular. (Granted, sorites paradoxes crop up everywhere, but that’s no reason not to avoid them when we can.)

    June 11, 2010 — 7:58
  • Dr. Menzel:
    Here I have to confess to ignorance. I am not a logician, and know very little about particular formal developments of logic. (I am a mathematician. I find working with sets much more intuitive than formal logic. I was surprised when I taught logic to learn that philosophy grad students found propositional logic easier to understand than set theory.) So the stuff below is no doubt half-baked and ignorant, and I am waiting to be corrected (hopefully gently).
    It doesn’t seem to me to be at all a cost that KQML only has closed theorems. Theorems in a correct logic are true. Only propositions, or things that express propositions, can be true. But open formulae don’t express propositions–it is sentences that do that. Open formulae are functions from tuples of objects to sentences, or something like that. (I kind of like the thought of developing logic in a non-standard completely Platonic way. Then open formulae are n-ary relations for n>0, and sentences are 0-ary relations, which we call “propositions”. Unary operators are functions from n-ary relations to n-ary relations; quantifiers are functions from n-ary relations to (n-1)-ary relations; etc. The point I am making still goes through: it is only 0-ary relations that can be true.)
    Until yesterday, I didn’t even know that there were ways of formulating FOL in such a way that one got to assert free sentences. When I read the proof of BF in your SEP entry, the first step in the proof threw me for a big loop: I just didn’t understand what it was asserting, since it wasn’t a sentence. It took some more reading to realize that there were logical systems where one got to do that.
    The one time I taught logic, I used Barwise and Etchemendy, and one doesn’t get to assert free sentences there–instead, one does subproofs using fake names (there is no doubt some technical term for these, but as I said, I’m not a logician), and the stuff in the subproof does not count as asserted. I think that whether the proofs of BF and NE go through in a FOL like that is going to depend on what constraints, if any, there are on using necessitation in a subproof. It does not seem to me to be a cost to restrict necessitation in a subproof. The intuition behind necessitation is that propositions that are provable hold necessarily. But the text in a subproof that uses fake names does not in general express propositions, and hence in particular does not express necessary truths. So the intuition behind necessitation does not apply. Of course, there is still going to be the issue that anything that has a name will have necessary existence if we allow unrestricted necessitation in the main proof.
    I also have to say that it doesn’t seem to me that it would be a cost to leave classical quantificational logic for a free theory, for a couple of reasons:
    1. The classical theory has the counterintuitive consequence that it’s a logical truth that if everything is an F, then there is an F. (I grant that the claim is necessarily true, but only because God is a necessary being, which doesn’t seem to be a matter of logical truth.)
    2. The classical theory has the counterintuitive consequence that anything for which there is a name in the theory has existence as a logical truth. But it’s not a logical truth that Obama exists.
    3. It seems to me that historically the classical theory was developed for non-modal extensional purposes. It is not surprising that once one adds modal stuff, it doesn’t all work well.
    In regard to names, I have the following half-baked idea. When a FOL language has a bunch of non-fake names, we should regard the “theorem that p” in that FOL as expressing the proposition that if such-and-such objects exist, then p. Thus, although “(Ex)(x=Obama)” is a theorem, it doesn’t express the proposition that Obama exists. Rather, it expresses the proposition that if Obama and Bush and … exist, then (Ex)(x=Obama). But this means that it is perfectly intuitive to restrict necessitation even in the outer proof. For the intuition behind necessitation is that any proposition that is provable holds necessarily, and the provable proposition is not that (Ex)(x=Obama), but that if Obama and Bush and … exist, then (Ex)(x=Obama).

    June 11, 2010 — 10:49
  • Dr. Maitzen:
    Salmon’s argument uses one more assumption: that the origins of an object could have been slightly different. Call the doctrine that the origins of an object couldn’t be at all different “radical essentiality of origins”. So, Salmon has shown that essentiality of origins plus S4 or B implies radical essentiality of origins. We can take this as an argument against S4 and against B. Or we can take this is an argument against the essentiality of origins. Or we can take this is an argument for the radical essentiality of origins. I do the last of these.
    I think there is independent reason to believe radical essentiality of origins: it is a neat alternative to haecceitism. We individuate objects by their complete origins (both material and causal).
    The cost is the Identity of Indiscernibles (II) in a strong form: one cannot have two distinct objects that prior to some time were indiscernible. Distinct objects must always have been discernible (at least historically). II is widely denied, but I think it has something intrinsically going for it. For instance, as Leibniz noted, if II is false, then we have a whole slew of additional facts to be explained or to be taken as brute. If II is false, there is presumably some world where I have an exact duplicate. But then what explains why I exist actually and not that duplicate? Of course, theistic haecceitists have a story here–God just chooses which haecceity to actualize–but we still are multiplying the number of things that God has to “just choose”.

    June 11, 2010 — 11:09
  • Mike Almeida

    Chris,
    I wonder how close we could come to proving NE: ∀x⎕∃y(y=x)(or something equivalent) if we use only closed formulas. Here’s a first pass at it. Let ‘F’ be the property ‘is an element or part of W’, where ‘W’ is a name of our world. Let ‘M’ be the possibility operator.
    1. M(∃x)Fx ⊃ (∃x)MFx … BF
    2. M(∃x)(∀y)(Fy & (x ≠ y)) … Assume for reductio
    3. (∃x)M(∀y)(Fy & (x ≠ y)) … 1,2,
    4. (∀x)(∃y)(Fy ⊃ (x = y)) … theorem, 3,4 contradiction
    5. (∀x)(∃y)(Fy ⊃ (x = y)) … 2, RAA
    From (5) we get that every possible object is identical to something in our world.

    June 11, 2010 — 11:41
  • Alex: Thanks for your comment. (By the way, I hope you’ll call me “Steve”; even people who haven’t had dinner with me do that.) You’re certainly right that Salmon and Chandler assume non-radical essentiality of origins. That’s what I tried, perhaps unsuccessfully, to refer to by “enough modal tolerance to deny mereological essentialism.” I suppose we can distinguish mereological essentialism from radical essentiality of origins: you could hold that objects depend for their identity on all of their parts at their origin but not thereafter.
    In my opinion, the main cost isn’t sacrificing II. It’s sacrificing the intuition that my bicycle could have been manufactured with a better paint-job (it would’ve been a better bicycle had that been true of it!). Also, individuation by complete causal origins threatens to imply that I wouldn’t exist had the prior history of the universe been different in the slightest way. I’d find that hard to accept too. I realize that I invite a sorites argument by trying to steer a middle course, which is why I favor abandoning any modal logic that lets the argument go through.

    June 11, 2010 — 11:54
  • Mike:
    “From (5) we get that every possible object is identical to something in our world.”
    I don’t see this. (∀x)(∃y)(Fy ⊃ (x = y)) is trivially true in any world in which there is a y such that ~Fy.

    June 11, 2010 — 12:16
  • Chris Menzel


    Chris (if you’re still checking this): Thanks again for all that material. While I make my way through it, let me ask you a different question. Your modal logic is evidently, as you say, S5-based. I wonder, then, how you’d respond to the sorites-style arguments against propositional S5 (and S4, and even B) due to Hugh Chandler (1976) and Nathan Salmon (1989). Those arguments seem to need only two very plausible assumptions: (1) some version of the essentiality of origins; (2) enough modal tolerance to deny mereological essentialism. I myself find the arguments pretty compelling. They strike me as good independent grounds for rejecting at least S4 and S5 (if not also B), regardless of any problems arising from BF or NE in particular. (Granted, sorites paradoxes crop up everywhere, but that’s no reason not to avoid them when we can.)

    Steve,
    I have to confess that I haven’t put a great deal of thought into those arguments but I guess I don’t find them terribly compelling because they depend so heavily on strong intuitions about identity conditions for artifacts that I don’t really share, at least not to a comparable degree. Since S5 seems the right logic for so many other modal matters, my inclination is therefore to think the problem is with the Chandler/Salmon arguments rather than with S4/S5. If I were to pursue this more responsibly, I suspect my first inclination would be to defend mereological essentialism, as I in fact find it to be rather appealing. But the other thing is that if I start thinking hard about these arguments I might have to change my mind about a lot of things and that would be upsetting. 🙂
    -chris

    June 11, 2010 — 12:31
  • Mike Almeida

    I don’t see this. (∀x)(∃y)(Fy ⊃ (x = y)) is trivially true in any world in which there is a y such that ~Fy.
    Well, it get’s us the right result, Alex. But it gets us an unintuitive result. There is some actual object that is a member of every world, since there are necessaruly existing objects. So, there is no world in which Fy is unsatisfied. But then everything in every world will be identical to some actual object. But in some worlds that object will be God or some abstract objects such as properties. Not quite the intuitive result we wanted. But it’s just a first pass.

    June 11, 2010 — 12:53
  • Mike Almeida

    It gets us to the conclusion that everything in every world is identical to some actual object. But we want it to be true that every object is such that necessarily there is some object to which it is identical. There is another way to do this, if we let F be the predicate ‘exists in every world’. In that case, (∀x)(∃y)(Fy ⊃ (x = y)) says that necessarily, everything is such that there is something y that exists in every world only if it is identical to y. This looks like it makes everything identical to numbers or God or propositions or some other necessarily existing thing. But not so. It entails that everything in the actual world necessarily exists. Unfortunately, the proof then begs the question at (4).

    June 11, 2010 — 13:40
  • Mike:
    I don’t see how it gets that result. Suppose in world w2, there are objects { a, b, c, N }, where N is a necessary being, and in the actual world there are { d, e, f, N }. Then it’s false that everything in w2 is identical with something in the actual world. But (∀x)(∃y)(Fy ⊃ (x = y)) is still true in w2, because Fa ⊃ (x = a) is always true.
    Steve (sorry!):
    “In my opinion, the main cost isn’t sacrificing II. It’s sacrificing the intuition that my bicycle could have been manufactured with a better paint-job (it would’ve been a better bicycle had that been true of it!). Also, individuation by complete causal origins threatens to imply that I wouldn’t exist had the prior history of the universe been different in the slightest way. I’d find that hard to accept too. I realize that I invite a sorites argument by trying to steer a middle course, which is why I favor abandoning any modal logic that lets the argument go through.”
    But unless we bring in vagueness (which maybe you don’t mind–but I do mind, since I think there cannot be vagueness when there is anything fundamental, and identity and possibility are fundamental) we will have the implausible result that it is possible that I had originated in a world where causal history H1 took place but not in a world where causal history H2 took place, even though H2 differs only in respect of one elementary particle. This seems implausible for pretty much the same reasons that radical origins essentialism (ROE) is. Maybe it’s a slightly smaller bullet to swallow, but if one swallows the smaller one, one might as well swallow the bigger one, and get the advantages of S5! 🙂 Moreover, I think this view has a puzzle that the radical thesis does not: the puzzle of explaining why the line is drawn where it’s drawn rather than somewhere else.
    Let me try to expand on the latter puzzle. Let M be the possibility operator we’ve hitherto been talking about, and L the corresponding necesssity. Then consider the modal operator MM. Now, observe that although it’s M-impossible that I had history H2 but M-possible that I had history H2, it is MM-possible that I had history H2. Of course, for MM, there will be (unless ROE holds for MM) a transition elsewhere, maybe between H3 and H4. So, why is the transition for the “ordinary metaphysical possibility” between H1 and H2 rather than between H3 and H4? What’s special there, between H1 and H2? And why is it M that is the ordinary metaphysical possibility operator, rather than, say, MM or MMM or M*, where M*p iff Mp or MMp or MMMp or …?
    Observe that if M satisfies Distribution and T, so does M*, respectively. Moreover, if we have infinite L-distribution (L(p1 & p2 &amp …) iff L(p1) & L(p2) & …), then M* satisfies S4. Infinite L-distribution seems hard to deny. So, if we grant infinite L-distribution, we get S4, if not for L/M, then for some derived modal operator. (I don’t remember what is the minimum axiom system that yields L-distribution. If it’s S4, then I haven’t shown anything interesting! But if it’s S4, then that’s a nice argument for S4.) But now Salmon’s problem arises for M*: either I could* have had completely different origins or I couldn’t* have had the least different origins.
    Besides, maybe M* is the right account of metaphysical possibility? I look at S4 as telling us that the metaphysical possibility operator is in an important sense “ultimate” or “fundamental”. If S4 is false, there is something more fundamental than M, namely MM or M* or the like.
    I don’t have as neat an argument for B as I do for S4. But I have a high level reason to believe B: my official account of the nature of possibility entails S5 (actually, it’s kind of weird: it entails that S5 is true at every world other than the actual one; I then make the technical assumption that S5 is either true at all worlds or false at some non-actual world), even though I didn’t build it in.
    Here’s another approach. Suppose we’re realists about worlds (whether of Lewis variety or some other variety). Then we can define an S5-compatible modal operator by taking the initial non-S5 accessibility relation and replacing it with its symmetric and transitive closure. And why not say that that operator is the right one?

    June 11, 2010 — 14:01

  • Dr. Menzel:

    “Chris”, if you please. 🙂

    Here I have to confess to ignorance. I am not a logician, and know very little about particular formal developments of logic. (I am a mathematician. I find working with sets much more intuitive than formal logic. I was surprised when I taught logic to learn that philosophy grad students found propositional logic easier to understand than set theory.) So the stuff below is no doubt half-baked and ignorant, and I am waiting to be corrected (hopefully gently).

    Well, having struggled through a number of your very interesting and often formally challenging posts, I don’t expect much ignorance or half-baked-ness. But I certainly always try be respectful in any case.

    It doesn’t seem to me to be at all a cost that KQML only has closed theorems.

    I don’t think I ever said that a restriction to closed theorems a serious cost; in fact, I don’t think it is. It simply reflects one of two ways of thinking about free variables. One way is that they are always implicitly universally quantified. This is a pretty natural way to think about them that is in fact often reflected in actual practice; e.g., the commutativity of addition is often expressed just by saying that m+n=n+m, leaving the universal quantifiers implicit. But free variables can also be thought of more along the lines of names. The difference here has some concrete (although fairly trivial) consequences. On the former approach (exemplified in Mendelson’s well-known text), {Px} ⊢ ∀xPx and on the latter (exemplified in Enderton’s text) {Px} ⊬ ∀xPx. I myself like to think of free vars more along the lines of constants, so my preference is for a logic that allows them to occur in theorems.
    The much bigger cost in KQML is the loss of individual constants.

    Theorems in a correct logic are true. Only propositions, or things that express propositions, can be true. But open formulae don’t express propositions–it is sentences that do that. Open formulae are functions from tuples of objects to sentences, or something like that.

    Again, whether or not open formulae express propositions, or have definite truth values, depends on the treatment of free vars. It is in my view a mere historical artifact (due mostly to Tarksi) that we treat free vars the way we do, as different from constants and requiring a separate assignment function over and above an interpretation. There is no reason they can’t simply be treated as “quantifiable constants” that receive a definite denotation up front. (This is in fact how I treat them in the formal development of the system in “The True Modal Logic”.) In fact, there isn’t any formal reason to distinguish names and variables at all. (Not that the world is all a-quiver about it, but I’m in fact working on a paper on this (in a somewhat broader context) at the moment.) Note that I am not saying that it was a mistake to treat the semantics of variables the way we have, only that it was not inevitable.

    (I kind of like the thought of developing logic in a non-standard completely Platonic way. Then open formulae are n-ary relations for n>0, and sentences are 0-ary relations, which we call “propositions”. Unary operators are functions from n-ary relations to n-ary relations; quantifiers are functions from n-ary relations to (n-1)-ary relations; etc. The point I am making still goes through: it is only 0-ary relations that can be true.)

    I’m certainly with you on treating propositions as 0-place relations. But to say that open n-ary formulae are identical with n-place relations seems too dangerously close to a use/mention confusion to me. (I am quite certain you are well aware of this.)

    Until yesterday, I didn’t even know that there were ways of formulating FOL in such a way that one got to assert free sentences. When I read the proof of BF in your SEP entry, the first step in the proof threw me for a big loop: I just didn’t understand what it was asserting, since it wasn’t a sentence. It took some more reading to realize that there were logical systems where one got to do that.

    Sorry for the shock. 🙂 Note that the proof in the SEP article is essentially Prior’s. Kripke replicates it in “Semantical Considerations on Modal Logic”. But note that the proof can be essentially replicated in a system that requires closed theorems but whose language includes individual constants, as one can (in a standard system) universally generalize on constants introduced by applications of UI. So there really isn’t anything radical going on here.

    The one time I taught logic, I used Barwise and Etchemendy, and one doesn’t get to assert free sentences there…

    That’s pretty common in intro texts.

    …instead, one does subproofs using fake names (there is no doubt some technical term for these, but as I said, I’m not a logician), and the stuff in the subproof does not count as asserted.

    I usually call them “arbitrary names” (in the context of an EI subproof) to indicate that they don’t have a fixed reference. To my mind, however, these arbitrary names function much more like free variables — we know that there are things that satisfy φ; suppose this name refers to any of those things.

    I think that whether the proofs of BF and NE go through in a FOL like that is going to depend on what constraints, if any, there are on using necessitation in a subproof. It does not seem to me to be a cost to restrict necessitation in a subproof.

    I agree there is warrant to do so in a subproof.

    The intuition behind necessitation is that propositions that are provable hold necessarily.

    Mistaken, I think. 🙂 What is logically provable should be analytic and a priori, but we know those two can come apart from each other and each of them from necessity.

    But the text in a subproof that uses fake names does not in general express propositions,…

    Seems to me that another way to put it is that we are reasoning about an arbitrary member of a collection of similar propositions. Of course, the idea of reasoning about arbitrary objects needs some spelling out here (a la Fine or some other way).

    …and hence in particular does not express necessary truths. So the intuition behind necessitation does not apply. Of course, there is still going to be the issue that anything that has a name will have necessary existence if we allow unrestricted necessitation in the main proof.

    This certainly seems like one way to go here.

    I also have to say that it doesn’t seem to me that it would be a cost to leave classical quantificational logic for a free theory, for a couple of reasons:
    1. The classical theory has the counterintuitive consequence that it’s a logical truth that if everything is an F, then there is an F. (I grant that the claim is necessarily true, but only because God is a necessary being, which doesn’t seem to be a matter of logical truth.)

    Granted, that one can prove that something exists might be considered an untoward consequence of classical logic. But one way to look at it is that it is a logic for the actual world where we know that that nearly uninformative proposition is true. This seems to me all the more plausible in the case of QML, which we can think of as a logic for modality from the perspective of the actual world. I’m not sure I share the intuition (which perhaps you have?) that logic ought to be so utterly scrubbed of ontological implications that the theorem in question is a philosophical embarrassment.

    2. The classical theory has the counterintuitive consequence that anything for which there is a name in the theory has existence as a logical truth. But it’s not a logical truth that Obama exists.

    And surely classical logic does not entail that it is. It would be only if Obama existed in every model of classical logic. All that classical logic requires is that, if you’re going to use a language that contains a name — ‘Obama’, say — it has to denote something in any interpretation of the language. So yeah, for such languages, we do have ⊢∃x(x=Obama), but it doesn’t follow that it is a logical truth that Barack Obama exists. Granted, this fact about classical logic does not make it the ideal logic for fiction and the like (though Zalta and others would disagree). Seems to me the conclusion is simply that classical logic is not the right logic for every philosophical context.

    3. It seems to me that historically the classical theory was developed for non-modal extensional purposes. It is not surprising that once one adds modal stuff, it doesn’t all work well.

    But it works just fine. 😉

    In regard to names, I have the following half-baked idea. When a FOL language has a bunch of non-fake names, we should regard the “theorem that p” in that FOL as expressing the proposition that if such-and-such objects exist, then p. Thus, although “(∃x)(x=Obama)” is a theorem, it doesn’t express the proposition that Obama exists. Rather, it expresses the proposition that if Obama and Bush and … exist, then (∃x)(x=Obama). But this means that it is perfectly intuitive to restrict necessitation even in the outer proof. For the intuition behind necessitation is that any proposition that is provable holds necessarily, and the provable proposition is not that (∃x)(x=Obama), but that if Obama and Bush and … exist, then (∃x)(x=Obama).

    I’m a little concerned here that ‘Obama’ in your informal English metalanguage, which has a fixed, definite reference (viz., the current US President), is being conflated with ‘Obama’ in your object language, which has no fixed definite reference outside the context of a model — wherein it might refer to anything at all. But I actually think your intuitions are better captured by placing more general restrictions on necessitation (such as, for example, the ones found in the system in “The True Modal Logic” 😉 that simply prevent the application of Necessitation to ‘∃x(x=Obama)’ despite its provability.
    -chris

    June 11, 2010 — 14:02
  • Mike Almeida

    I don’t see how it gets that result. Suppose in world w2, there are objects { a, b, c, N }, where N is a necessary being, and in the actual world there are { d, e, f, N }. Then it’s false that everything in w2 is identical with something in the actual world. But (∀x)(∃y)(Fy ⊃ (x = y)) is still true in w2, because Fa ⊃ (x = a) is always true.
    Alex, change the example. Suppose there is some dog that exists in every world and that is the only thing that exists in every world. Let P = (∀x)(∃y)(Fy ⊃ (x = y)) say that for every boy x there is some dog y such that, if y is in the world, then x loves y. Now in w2 we have this dog d. So, P is true in w2 only if every boy in w2 loves d. Suppose there is some boy b1 who is in w2 but does not love d who is in w2. Then P is falsified. It is true in w2 that (∃x)~(∃y)(Fy ⊃ (x = y)), there is some boy b1 such that for no dog y that exists in w2 does b1 loves y.

    June 11, 2010 — 14:42
  • :
    I need to digest a lot of this. But two things I can comment on.
    As someone who likes radical essentiality of origins, the loss of names doesn’t bother me much: we can replace every name with a complete definite description of the object’s origins, which is necessary and sufficient for the object’s identity.
    “I’m certainly with you on treating propositions as 0-place relations. But to say that open n-ary formulae are identical with n-place relations seems too dangerously close to a use/mention confusion to me. (I am quite certain you are well aware of this.)”
    Just about anything–or maybe anything at all–could serve as a linguistic token. In particular, we can have a language where relations are identical with token formulae (this is in the spirit of Lagadonian languages) denoting them, and each type necessarily has only one token. I have this fantasy of a Platonic logic book that starts with relations, and then says that there are certain distinguished functions (or, if one prefers, functors). Thus, there is a function & from pairs of relations to relations (with arity equal to the sum of the arities). Then in the book (which is supposed to be a physical book; it’s Platonic as to its approach, not its own mode of existence) we can say things like: if p and q are propositions, p&q is a proposition. And we can say this without any implicit or explicit corner-quotes or use-mention confusions, because “p&q” denotes that 0-ary relation which is obtained by applying & to the ordered pair whose first member is p and whose second member is q.

    June 11, 2010 — 14:49
  • Chris:
    Somehow the “Chris” before the colon at the beginning of my last comment disappeared.
    Mike:
    “Let P = (∀x)(∃y)(Fy ⊃ (x = y)) say that for every boy x there is some dog y such that, if y is in the world, then x loves y. Now in w2 we have this dog d. So, P is true in w2 only if every boy in w2 loves d.”
    I don’t understand. What if in w2 there is some other dog e that does not exist in the actual world? Then P is still true even if every boy in w2 hates every dog. All you need for (∃y)(Fy ⊃ (x = y)) to hold for every x is that there be a y such that ~Fy. So as long as the world contains something that doesn’t exist in the actual world, (∃y)(Fy ⊃ (x = y)) holds for every x. What am I missing?

    June 11, 2010 — 14:56
  • Mike Almeida

    All you need for (∃y)(Fy ⊃ (x = y)) to hold for every x is that there be a y such that ~Fy. So as long as the world contains something that doesn’t exist in the actual world, (∃y)(Fy ⊃ (x = y)) holds for every x. What am I missing?
    I guess maybe the question is what am I missing. Suppose I say to you: for every boy in this world there is some dog such that, that dog speaks French only if the boy loves that dog. Now suppose I point to a bulldog in the corner speaking French to Sartre and I say truly, that’s the only dog in the world that speaks French.
    Now it is true that there is a dog that speaks French. And it is true of every boy that he is such that there is something or other meeting the description of being a French speaking dog ONLY IF he loves that dog. Well, there is something or other meeting that description in this world. And, well, these boys are such that they love something or other that meets that description. So, they havd better love that dog.

    June 11, 2010 — 16:32
  • Alexander wrote:
    Mike:
    “From (5) we get that every possible object is identical to something in our world.”

    I don’t see this. (∀x)(∃y)(Fy ⊃ (x = y)) is trivially true in any world in which there is a y such that ~Fy.

    Assuming the context here is classical logic with identity, it’s true in any world at all since it’s a logical truth. (It follows straightaway from ∀x∃y(x = y).)
    -chris

    June 11, 2010 — 18:36
  • Chris:
    I missed that! Of course.

    June 12, 2010 — 18:59
  • Mike Almeida

    Alex/Chris, Ok, a second (and final) pass. Tell me where it goes wrong. You know, you could help with this!
    1. M(∃x)Fx ⊃ (∃x)MFx … BF
    2. M(∃x)(∀y)(Fy ⊃ (x ≠ y)) … Assume for reductio
    3. (∃x)M(∀y)(Fy ⊃ (x ≠ y)) … 1,2,
    4. (∀x)(∃y)(Fy & (x = y)) … 3,4 contradiction
    5. (∀x)(∃y)(Fy & (x = y)) … 2, RAA
    All you have to do is grant me (at 4) that there is some world in which every object is necessary. Let that world be W. Run this argument at W (or conjoin the premises and conclusion and put a wide scope M). It follows that every object at every world is necessary.

    June 13, 2010 — 12:41
  • Alex, you wrote:
    I have this fantasy of a Platonic logic book that starts with relations, and then says that there are certain distinguished functions (or, if one prefers, functors). Thus, there is a function & from pairs of relations to relations (with arity equal to the sum of the arities). Then in the book (which is supposed to be a physical book; it’s Platonic as to its approach, not its own mode of existence) we can say things like: if p and q are propositions, p&q is a proposition. And we can say this without any implicit or explicit corner-quotes or use-mention confusions, because “p&q” denotes that 0-ary relation which is obtained by applying & to the ordered pair whose first member is p and whose second member is q.
    Seems to me your fantasy is already a reality. What you describe is a sort of reified version of Quine’s predicate functor calculus in “Variables Explained Away”. George Bealer in fact developed this idea and worked out a corresponding algebraic semantics of Platonic properties, relations and propositions in Quality and Concept (Oxford UP, 1980). The operators of the algebra function in just the manner you describe. Ed Zalta and Alan McMichael developed a similar semantics in “An Alternative Theory of Non-existent Objects” (JPL 9 (1980) 297-313. Zalta uses this framework extensively in his two books. I draw upon this work with theistic ends in mind in “Theism, Platonism, and the Metaphysics of Mathematics” (Faith and Philosophy 4 (1987) 365-382).

    June 17, 2010 — 14:01
  • Chris:
    Oh, good. So now I have references for if I want to use it, and I don’t have to do any of the work of writing it. Thanks. I knew Quine did a predicate-type version of this, but I am glad to hear that the Platonic version has been done, too.

    June 18, 2010 — 14:49