Modal Matters – Page 7 – Maverick Philosopher

A Closer Look at Material Composition and Modal Discernibility Arguments

(For David Brightly, whom I hope either to convince or argue to a standoff.)

Suppose God creates ex nihilo a bunch of TinkerToy pieces at time t suitable for assembly into various (toy) artifacts such as a house and a fort. A unique classical mereological sum — call it 'TTS' — comes into existence 'automatically' at the instant of the creation ex nihilo of the TT pieces. (God doesn't have to do anything in addition to creating the TT pieces to bring TTS into existence.) Suppose further that God at t assembles the TT pieces (adding nothing and subtracting nothing) into a house. Call this object 'TTH.' So far we have: the pieces, their sum, and the house. Now suppose that at t* (later than t) God annihilates all of the TT pieces. This of course annihilates TTS and TTH. During the interval from t to t* God maintains TTH in existence.

I set up the problem this way so as to exclude 'historical' and nonmodal considerations and thus to make the challenge tougher for my side. Note that TTH and TTS are spatially coincident, temporally coincident, and such that every nonmodal property of the one is also a nonmodal property of the other. Thus they have the same size, the same shape, the same weight, etc. Surely the pressure is on to say that TTH = TTS? Surely my opponents will come at me with their battle-cry, 'No difference without a difference-maker!' There is no constituent of TTH that is not also a constituent of TTS. So what could distinguish them?

Here is an argument that TTH and TTS are not identical:

1. NecId: If x = y, then necessarily, x = y.

2. If it is possible that ~(x = y), then ~(x = y). (From 1 by Contraposition)

3. If it is possible that TTS is not TTH, then TTS is not TTH. (From 2, by Universal Instantiation)

4. It is possible that TTS is not TTH. (God might have assembled the parts into a fort instead of a house or might have left them unassembled.)

5. TTS is not TTH. (From 3, 4 by Modus Ponens)

The gist of the argument is that if x = y, then they are identical in every possible world in which both of them exist. But there are possible worlds in which TTS and TTH both exist but are not identical. (E.g., a world in which the pieces are assembled into a fort instead of a house.) Therefore, TTS andf TTH are not identical.

If you are inclined to reject the argument, you must tell me which premise you reject. Will it be (1)? Or will it be (4)?

Your move, David.

Necessitas Consequentiae versus Necessitas Consequentiis

Take the sentence, 'If I will die tomorrow, then I will die tomorrow.' This has the form If p, then p, where 'p' is a placeholder for a proposition. Any sentence of this form is not just true, but logically true, i.e., true in virtue of its logical form. Now every sentence true in virtue of its logical form is necessarily true. (The converse, however, does not hold: there are necessary truths that are not logically true.) Thus we can write, 'Necessarily(if p, then p)' or

1. Nec (p –>p).

The parentheses show that the necessity attaches to the consequence, represented by the arrow, and not to the consequent, represented by the terminal 'p.' (When speaking of conditionals, logicians distinguish the antecedent from the consequent, or, trading Latin for Greek, the protasis from the apodosis.) Thus the above is an example of the necessitas consequentiae. This, however, must not be confused with the necessitas consequentiis, which is exemplified by

2. p–>Nec p.

In (2) the necessity attaches to the consequent. It should be obvious that (1) does not entail (2), equivalently, that (2) does not follow from (1). For example, although it is necessarily true that if I will die tomorrow, then I will die tomorrow, it does not follow, nor is it true, that if I will die tomorrow, then necessarily I will die tomorrow. Proving fatalism cannot be that easy. For even if I do die tomorrow, that will be at best a contingent occurrence, not something logically necessitated. (Think about it.)

To confuse (1) and (2) is to confuse the necessity of the consequence with the necessity of the consequent. This is an example of what logicians call a fallacy, i.e., a typical error in reasoning, and in particular a modal fallacy in that it deals with the (alethically) modal concepts of necessity and possibility and their cognates.

Class dismissed.

From the Mailbag: Faith and Modality

An astute reader e-mails,

First, sometime ago I recommended John Bishop's Believing by Faith: An Essay in the Epistemology

and Ethics of Religious Belief . If you have yet to read the book, I would recommend his new article

on Faith in the Stanford Encyclopedia of Philosophy. You may be particularly interested in

sections 7-10.

Second, I do not know a great deal about possible worlds semantics, and am taking a great risk of embarrassing myself in questioning your argument that "Necessarily, something exists" – but I think

that I will take a stab at it!

I am assuming that "Necessarily something exists" = "In at least one possible world, at least one

thing exists." Is this correct?

No. The first sentence is equivalent to 'In every possible world, at least one thing exists.' In other words, there is no possible world that is empty: every world has at least one item in it. But this is consistent with there being no one item that exists in every possible world. Suppose every being is contingent, where a contingent being is one that exists in some but not all possible worlds. Then there would be no one being that exists in all worlds, and 'Necessarily something exists' would be made true by the fact that each world has something or other in it. But if there is a necessary being (defined as a being that exists in all worlds), then of course the sentence in question is also true.

1. Does not your argument depend upon the assumption of 'modal realism' – that all possible worlds

actually exist, a highly questionable assumption?

No, because I am using the 'possible worlds' language only as a façon de parler, a graphic and intuitive way of representing modal relationships which I find helpful. (Unfortunately, most of my readers are completely 'thrown' by it!) In other words, I could have stated the argument without mentioning possible worlds. Here is a partial schedule of intertranslation, where 'world' is short for 'broadly logically possible world':

X is a necessary being =df X exists in all worlds

X is a contingent being =df X exists in some but not all worlds

X is an impossible being =df X exists in no world

X is an actual being =df X exists in the actual world

(Note that if x is contingent, it doesn't follow that x is actual, nor conversely)

X is a possible being =df X exists in some world

X is essentially F =df X instantiates F-ness in every world in which X exists.

X is accidentally F =df X instantiates F-ness in some but not all worlds in which X exists

X is necessarily F =df X instantiates F-ness in every world in which X exists, and X exists in every world.

(Example: God is necessarily, not just essentially, omnipotent.)

Proposition p is necessarily true =df p is true in all worlds

P is contingently true =df p is true in some but not all worlds

And so on.

Now isn't that neat? The modal notions are explicated in terms of the familiar quantifiers of predicate logic. The schema facilitates modal reasoning. For example, if x is necessary, does it follow that x is possible? Yes, because if x exists in all worlds, then it exists in some worlds. If x is possible, does it follow that x is contingent? No, because if x exists in some worlds, that leaves it open that it exists in all worlds. If x is noncontingent, does it follow that x is necessary? No, because if it is not the case that x exists in some but not all worlds, it does not follow that x exists in all worlds: x might exist in no world.

You characterize modal realism as the doctrine that "all possible worlds actually exist." No philosopher maintains that every world is absolutely actual. There is only one possible world that is absolutely actual: all the rest are merely possible. Now there is a philosopher, David K. Lewis, who maintains that there is a plurality of worlds, all on an ontological par, and thus all equally real; but he denies that there is such a property as absolute actuality. For him each world is actual at itself, but no world is actual simpliciter or absolutely. I reject Lewis's view which could be called extreme modal realism. Almost everyone rejects it. Lewis's idea, which is both brilliant and crazy at the same time, is that modality can be reduced to purely extensional terms via definitions like the ones I gave above. But few follow him in that. The above definitions do not allow one to eliminate modality by quantifying over worlds, because the worlds in question are possible, and 'possible' is a modal term.

So, to answer what I take to be your question, my argument does not presuppose extreme modal realism. In fact, it does not require that we take any stand at all on what exactly possible worlds are. But I do presuppose realism to this extent: I asssume that modality is not merely epistemic. Thus the possibility that I be sleeping now instead of blogging is a 'real possibility' in that it is subsists independently of what I or anyone know or believe. It is not possible merely in the epistemic sense of 'possible for all I know,' but possible independently of what I know.

2. Does not your use of 'exists' in premises 4 and 5 treat it as, or assume that it can be used as, a

'real predicate' rather than merely a 'grammatical predicate' (B. Russell) – again a questionable if

not false assumption?

I discuss this is various articles and in my 2002 book A Paradigm Theory of Existence. I argue, among other things, that Russell's theory of existence, which is closely related to Frege's, is a complete nonstarter, wrong from the ground up. There is something on Russell's theory in Paul Edwards' Heidegger's Confusions: A Two-Fold Ripoff.

In 'possible worlds' lingo, we say things like this: There are possible worlds in which Socrates exists but is not the teacher of Plato. Now of course those worlds are all merely possible because we know (or reasonably believe) that in the actual world Socrates is the teacher of Plato. So what does it mean to say that Socrates exists in those worlds? Let W be a merely possible world. To say that x exists in W is to say that, had W been actual, x would have existed. A merely possible world in which Socrates is not the teacher of Plato is a world which is such that, had it been actual, then Socrates would have existed without being the teacher of Plato.

My correspondent continues with several more questions/objections which I don't understand. In any case the above gives us plenty to discuss.

A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions

A commenter in the 'Nothing' thread spoke of possible worlds as sets. What follows is a reposting from 1 March 2009 which opposes that notion.

…………….

In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions. If this is right, then the actual world, which is of course one of the possible worlds, is the maximally consistent set of true propositions. But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, Cantor's theorem implies that possible worlds cannot be maximally consistent sets of propositions.

1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. The power set of a set S is the set whose elements (members) are all of S's subsets. Recall the difference between a member and a subset. The set {Socrates, Plato} has exactly two elements, neither of which is a set. Since neither is a set, neither is a subset of this or any set. {Socrates, Plato} has four subsets: the set itself, the null set, {Socrates}, {Plato}. Note that none of the four sets just listed are elements of {Socrates, Plato}. The power set of {Socrates, Plato}, then, is {{Socrates, Plato}, { }, {Socrates}, {Plato}}.

Could There Have Been Just Nothing At All?

No doubt, things exist. At least I exist, and that suffices to show that something exists. But could it have been the case that nothing ever existed? Actually, there is something; but is it possible that there have been nothing? Or is it rather the case that necessarily there is something? Is it not only actually the case that there is something, but also necessarily the case that there is something? I will argue that there could not have been nothing and that therefore necessarily there is something. (Image credit.)

My thesis, then, is that necessarily, something (at least one thing) exists. I am using 'thing' as broadly as possible, to cover anything at all, of whatever category. If I am right, then it is impossible that there have been nothing at all. The type of modality in question is what is called 'broadly logical' or 'metaphysical.'

Note that Necessarily something exists does not entail Something necessarily exists. I am not asserting the second proposition, but only the first. The second says more than the first. In the patois of possible worlds, the second says that there is some one thing that exists in every possible world, whereas the first says only that every possible world is such that there is something or other in it. The first proposition is consistent with the proposition that every being is contingent, while the second is not. So the first and second propositions are logically distinct and the first does not entail the second. I am asserting only the first.

What I will be arguing, then, is not that there is a necessary being, some one being that exists in all possible worlds, but that every world has something or other in it: every possible circumstance or
situation is one in which something or other exists. That is, there is no possible world in which there is nothing at all.

You can think of merely possible worlds as maximal or total ways things might have been, and you can think of the actual world as the total way things are. My thesis is that there is no way things might have been such that nothing at all exists. But if you are uncomfortable with the jargon of possible worlds, I can translate out of it and say, simply, that it is impossible that there have been, or be, nothing at all. As a matter of metaphysical necessity, there must be something or other!

The content of my thesis now having been made clear, I proceed to give a reductio ad absurdum argument for thinking it true.

1. Let S = Something exists and N = Nothing exists, and assume for reductio that N is possibly true.
2. If N is possibly true, then S, which is true, and known to be true, is only contingently true.
Therefore
3. There are possible worlds in which S is false and possible worlds in which S is true. ( From 2, by definition of 'contingently true')
4. In the worlds in which S is true, something exists. (Because if 'Something exists' is true, then something exists.)
5. In the worlds in which S is false, it is also the case that something exists, namely, S. (For an item cannot have a property unless it exists, and so S cannot have the property of being false unless S exists)
6. Every proposition is either true, or if not true, then false. (Bivalence)

Therefore
7. Every world has something in it, hence there is no world in which nothing exists.
Therefore
8. N is not possibly true, and necessarily something exists.

If you disagree with my conclusion, then you must either show that one or more premises are either false or not reasonably maintained, or that one or more inferences are invalid, of that the argument rests on one or more dubious presuppositions.

Modality and Existence

Steven Nemes, who may prove to be my nemesis, e-mails:

I'm enjoying your book so far. I'm starting the constructive half of it now, and am going to reread the chapter "The Ground of the Contingent Existent" after a quick skim over it recently. I don't want to sound arrogant or anything, but upon hearing some of the theories of existence you cover in the book, the thought in my head is "Man, this obviously can't be right. How could anyone think this?" But the philosophers in question are much smarter than me, so maybe my surprise at their theories is improper.

I have a question now regarding possible worlds, what is true-in-W, etc.

You make the point in your book that it is the fact that my existence is contingent that makes it true in some worlds that I exist and false in others. And it is the necessary existence of the Paradigm that would make it true in every possible world that he exists, rather than vice versa. This all seems very correct to me, but I am wondering about its consequences.

As I recall, my thought was along the following lines. The biconditionals

N. x is a necessary being iff x exists in all metaphysically possible worlds

C. x is a contingent being iff x exists in some but not all possible worlds

are neutral with respect to reductions of the RHS to the LHS or vice versa. So we can legitimately ask: Is a necessary being necessary because it exists in all worlds, or does it exist in all worlds because it is necessary? And: Is a contingent being contingent because it exists in only some worlds, or does it exist in only some worlds because it is contingent? My answer was that existence in all/some worlds is grounded in, and explained by, the different ways of existing of the Paradigm and what depends on it.

It seems the principle, then, is that what is possible depends upon what is actual, depends upon the potentialities that exist in what is actual, etc. Would you agree to this?

That's the next step, but my principle was merely that possible worlds talk is a very useful façon de parler, a graphic manner of speaking that allows us to picture modal relations in extensional terms using the machinery of quantification, but that necessity and contingency of existence cannot consist in, or be constituted by, existence in all/some worlds.

But I do take the next step, though I didn't work it out in the book. The Paradigm is the numero uno necessary existent and as such the ground of all actualities other than itself, but also the ground of all possibilities. Mere possibilities, after all, are not nothing, and so have some ontological status shy of actuality. So I had the not entirely original thought that mere possibilities could be identified with powers of the Paradigm.

Are there bad consequences of this, however? It seems like there is nothing actual sufficient to ground the truth of a typical counterfactual of creaturely freedom about nonexistent agents, like "If Bill the Bald Bostonian were offered the chance, he'd freely agree to murder the Yankees star pitcher". Does that mean it isn't true in any possible world? Can there be any truths about nonexistent agents and their free actions at all, assuming the only kind of free action is libertarian-free action? Can their be any truths in other possible worlds about what existent agents would freely do?

Underlying your question is whether there could be nonexistent but possible individuals. The conclusion I came to in the book was that all mere possibilities are general in nature, hence not involving specific individuals. Before Socrates came into existence there was no merely possible Socrates, though there was the possibility of there existing a snub-nosed sage, married to a shrewish wife, who was given to moments of abstraction when he communed with his daimon, etc. To get a feel for the issue here, imagine someone prophesying the coming of Socrates, master dialectician, fearless questioner of powerful men, who ran afoul of them, got sentenced to death, etc. Imagine the prophet being asked, after Socrates is on the scene, whether the Socrates in existence is the one he prophesied, or a numerically different one. My claim is that this question makes no sense. Before Socrates came into existence, there was no individual Socrates.

I was pushed into this view by my arguments against haecceity properties and also by my vew that existence is not a property added to a pre-formed fully individuated essence, but the unity of an individual's constitutents. Accordingly, existence individuates so that there is no individuation apart from existence, hence no merely possible individuals.

Broadly Logical Modality

David Brightly has difficulty with the notion of broadly logical modality. Let me see if I can clarify this notion sufficiently to satisfy him. It might be best to begin with the notion of narrowly logical impossibility. I'll number my paragraphs so that David can tell me exactly where he disagrees or finds obscurity.

1. There are objects and states of affairs and propositions that can be known a priori to be impossible because they violate the Law of Non-Contradiction (LNC). Thus a plane figure that is both round and not round at the same time, in the same respect, and in the same sense of 'round,' is impossible, absolutely impossible, simply in virtue of its violation of LNC. I will say that such an object is narrowly logically (NL) impossible. Hereafter, to save keystrokes, I will not mention the 'same time, same respect, same sense' qualification which will be understood to be in force.

2. But what about a plane figure that is both round and square? Is it NL-impossible? No. For by logic alone one cannot know it to be impossible. One needs a supplementary premise, the necessary truth grounded in the meanings of 'round' and 'square' that nothing that is round is square. We say, therefore, that the round square is broadly logically (BL) impossible. It is not excluded from the realm of the possible by logic alone, which is purely formal, but by logic plus a 'material' truth, namely the necessary truth just mentioned.

3. If there are BL-impossible states of affairs such as There being a round square, then there are BL-necessary states of affairs such as There being no round square. Impossibility and necessity are interdefinable: a state of affairs is necessary iff its negation is impossible. It doesn't matter whether the modality is NL, BL, or nomological (physical). It is clear, then, that there are BL-impossible and BL-necessary states of affairs.

4. We can now introduce the term 'BL-noncontingent' to cover the BL-impossible and the BL-necessary.

5. What is not noncontingent is contingent. (Surprise!) The contingent is that which is possible but not necessary. Thus a contingent proposition is one that is possibly true but not necessarily true, and a contingent state of affairs is one that possibly obtains but does not necessarily obtain. We can also say that a contingent proposition is one that is possibly true and such that its negation is possibly true. The BL-contingent is therefore that which is BL-possible and such that its negation is BL-possible.

6. Whatever is NL or BL or nomologically impossible, is impossible period. If an object, state of affairs, or proposition is exckluded from the realm of possible being, possible obtaining, or possible truth by logic alone, logic plus necessary semantic truths, or the (BL-contingent) laws of nature, then that object, state of affairs or proposition is impossible, period or impossible simpliciter.

7. Now comes something interesting and important. The NL or BL or nomologically possible may or may not be possible, period. For example, it is NL-possible that there be a round square, but not possible, period. It is BL-possible that some man run a 2-minute mile but not possible, period. And it is nomologically possible that I run a 4-minute mile, but not possible period. (I.e., the (BL-contingent) laws of anatomy and physiology do not bar me from running a 4-minute mile; it is peculiarities not referred to by these laws that bar me. Alas, alack, there is no law of nature that names BV.)

8. What #7 implies is that NL, BL, and nomological possibility are not species or kinds of possibility. If they were kinds of possibility then every item that came under one of these heads would be possible simpliciter, which we have just seen is not the case. A linguistic way of putting the point is by saying that 'NL,' 'BL,' and 'nomological' are alienans as opposed to specifying adjectives: they shift or 'alienate' ('other') the sense of the noun they modify. From the fact that x is NL or BL or nomologically possible, it does not follow that x is possible. This contrasts with impossibility. From the fact that x is NL or BL or nomologically impossible, it does follow that x is impossible. Accordingly, 'NL,' 'BL,' and 'nomological' do not shift or alienate the sense of 'impossible.'

9. To appreciate the foregoing, you must not confuse senses and kinds. 'Sense' is a semantic term; 'kind' is ontological. From the fact that 'possible' has several senses, it does not follow that there are several species or kinds of possibility. For x to be possible it must satisfy NL, BL, and nomological constraints; but this is not to say that these terms refer to species or kinds of possibility.

God: Necessary or Noncontingent?

Many theists in the tradition of Anselm and Aquinas define God as a necessary being. But if God is a necessary being, then he cannot not exist: he exists in all broadly-logically possible worlds. The actual world is of course one of these worlds. So it would seem to follow from the very definition of God favored by Anselmians that God exists. But surely the existence of God cannot be fallout from a mere definition!

I have hammered the Objectivists (Randians) for their terminological mischief as when they rig up 'existence' in such a way that the nonexistence of the supernatural is achieved by terminological fiat. So doesn't fairness demand that I hammer the Anselmians equally? (This is one way of attaching sense to Nietzsche's notion of philosophizing with a hammer, although it is not what he had in mind.)

The trouble with defining God as a necessary being is that 'necessary being' conflates modal status and existence. For any item we ought to distinguish its modal status (whether necessary, impossible, or contingent) from its existence or nonexistence.

The concept of God as "that than which no greater can be conceived" is the concept of a being that exists in every possible world if it exists in any world. But from this one cannot validly infer that God exists. For it might be (it is epistemically possible that) God exists in no world, in which case he would be impossible. God is either necessary or impossible: that was Anselm's great insight. He cannot be a contingent being.

If we want one word to express this disjunctive property of being either necessary or impossible, that word is 'noncontingent.' So we should not say that God is a necessary being. We should say that he is a noncontingent being.

Companion post: Necessary, Contingent, Impossible: A Note on Nicolai Hartmann

A Modal Aporetic Tetrad

Here is a four-limbed aporetic polyad:

1. The merely possible is not actual.

2. To be actual is to exist.

3. To exist is to be.

4. The merely possible is not nothing.

Each limb is plausible, but they cannot all be true. The first three limbs, taken together, entail the negation of the fourth. Indeed, any three, taken together, entail the negation of the remaining limb, as you may verify for yourself.

Now which limb ought we reject in order to avoid logical inconsistency? (1) is non-negotiable because purely definitional. Everything actual is possible, but not everything possible is actual. 'Merely possible,' by definition, refers to that which is possible but not actual. This leaves us three options.

(2)-Rejection. One might reject the equivalence of the actual and the existent analogously as one might reject the equivalence of the temporally present and the existent. Just as one might maintain that past events exist just as robustly as present events despite their pastness, one might maintain that merely possible items exist just as robustly as actual items. David Lewis' extreme ('mad dog') modal realism is an example of (2)-rejection. On his modally egalitarian scheme there is a plurality of possible worlds all on an ontological par. Each is a maximal mereological sum of concreta. Each of these worlds is actual at itself, but no one of these worlds is actual simpliciter. For each world w, w is actual-at-w, but no world is actual, period. Thus there is no such property as absolute actuality. It is not the case that one of the worlds is privileged over all the others in point of being actual simpliciter. What is true of a world is true of its occupants: I enjoy no ontological privilege over that counterpart of me who is bald now and living in Boston. Actuality is world-relative and 'actual' is accordingly an indexical term like 'now.' When I utter a token of 'now' I refer to the time of my utterance; likewise, on Lewis' theory, when I utter a token of 'actual,' I refer to the world I am in.

Having rejected (2), a Lewis-type philosopher could gloss the other limbs of the tetrad as follows. To say that the merely possible is not actual is to say that merely possible objects (e.g. bald Bill the Bostonian) are denizens of worlds other than this one. To say that to exist is to be is to say that there is no distinction between the existence of an object and its being in some world or other. To say that the merely possible is not nothing is to say that objects which are not denizens of this world are denizens of some other world or worlds.

I am tempted to say that this solution, via rejection of (2), is worse than the problem. For one has to swallow an infinity of equally real possible worlds. Further, my possibly being bald is not some counterpart of mine's being bald in another possible world. (This critique of course needs to be spelled out in detail.)

(3)-Rejection. A second theoretical option is to reject the equivalence of being and existence, of that which is and that which exists. Accordingly, there are things that are but do not exist. They have Being but not Existence. Everything is, but only some things exist. The early Russell, in the Principles of Mathematics from 1903, toyed with this view although he rejected it later in his career. If existents are a proper subset of beings, then one could locate merely possible items in among the beings that do not exist. The merely possible would then have Being but not Existence or Actuality.

This solution leads to an ontological population explosion much as the Lewis theory does.

(4)-Rejection. A third option is to deny (4) by affirming that the merely possible is nothing in reality, that it has no ontological status. One might construe the merely possible as merely epistemic, as being merely parasitic on our ignorance, or as having no status outside our thought. A view along these lines can be found in Spinoza.

Intuitively, though, it seems mistaken to say that there are no genuine, mind-independent possibilities. My writing desk, for example, is one inch from the wall, but it could have been two inches from the wall. It is not just that I can imagine or conceive it being two inches from the wall; it really could be two inches from the wall even though this possible state of affairs was never actual and never will be actual. (Moreover, what I CAN imagine or conceive refers to real but unactual possibilities of imagination and conception; or will you say that these possibilities are themselves derivative from acts of imagining or conceiving? If you do, then a vicious infinite regress is in the offing.))

Now suppose I had provided more rigorous and more convicing rejections of each of the three theoretical options. Suppose that a strong case can be made that all four propositions must be accepted. Then we would have four propositions each of which has a very strong claim on our acceptance, but which are collectively inconsistent. (Assume that the inconsistency is demonstrable.) What might one conclude from that? (A) One possibility is that we ought to abandon the Law of Non-Contradiction. (B) A second is that one of the solutions must be right even though we have good reason to think that every solution is mistaken. (C) A third is that the aporetic tetrad is an insoluble problem, a genuine intellectual knot that cannot be untied.

Note that (A), (B), and (C) form a meta-aporia. Each of them has a claim on our acceptance, but they cannot all be true.

Suppose there are genuine but absolutely insoluble philosophical problems. What would that show, if anything?

Contingent, Necessary, Impossible: A Note on Nicolai Hartmann

Nicolai Hartmann, Moeglichkeit und Wirklichkeit, p. 29: . . . denn das Zufaellige ist immerhim wirklich, und nur die Notwendigkeit negiert. Hartmann is saying in effect that everything contingent is actual, and that the contingent and the necessary are polar opposites: what is contingent is not necessary, and what is not necessary is contingent.

I beg to differ. First of all, not everything contingent is actual. My being asleep now and my being awake (= not asleep) now are both possible states of affairs. The second is actual, the first is not. But both are contingent. So not everything contingent is actual. The imagery of possible worlds ought to make this graphic for the modally challenged. A contingent state of affairs is one that obtains in some but not all possible worlds. Now my being asleep now obtains in some but not all possible worlds. Therefore, my being asleep now is contingent though not actual. So not everything contingent is actual.

Second, it is not the case that x is contingent if and only if x is not necessary. For there are states of affairs that are not necessary but also not contingent. My being both awake and not awake now is an impossible state of affairs. It is neither necessary nor contingent. Not necessary, because it does not obtain in every possible world. Not contingent, because it it does not obtain in some (but not all) possible worlds.

The polar opposite of the contingent is not the necessary but the the noncontingent. The noncontingent embraces both the the necessary and the impossible, that which exists/obtains in all worlds, and that which exists/obtains in no world. Reality, then, is modally tripartite:

The necessary: that which exists/obtains in all possible worlds. The contingent: that which exists/obtains in some but not all possible worlds. The impossible: that which exists/obtains in no possible world.

You say you are uncomfortable with the patois of possible worlds? The distinctions can be sliced without this jargon. The necessary is that which cannot not be. The contingent is that which is possible to be and possible not to be. The impossible is that which cannot be.

And that's all she wrote, modally speaking.

Another Example of a Necessary Being Depending for its Existence on a Necessary Being

The Father and the Son are both necessary beings. And yet the Father 'begets' the Son. Part, though not the whole, of the notion of begetting here must be this: if x begets y, then y depends for its existence on x. If that were not part of the meaning of 'begets'' in this context, I would have no idea what it means. But how can a necessary being depend for its existence on a necessary being? I gave a non-Trinitarian example yesterday, but it was still a theological example. Now I present a non-theological example.

I assume that there are mathematical (as opposed to commonsense) sets. And I assume that numbers are necessary beings. (There are powerful arguments for both assumptions.) Now consider the set S = {1, 3, 5} or any set, finite or infinite, the members of which are all of them necessary beings, whether numbers, propositions, whatever. Both S and its membership are necessary beings. If you are worried about the difference between members and membership, we can avoid that wrinkle by considering the singleton set T = {1}.

T and its sole member are both necessary beings. And yet it seems obvious to me that one depends on the other for its existence: the set is existentially dependent on the member, and not vice versa. The set exists because — though this is not an empirically-causal use of 'because' — the members exist, and not the other way around. Existential dependence is an asymmetrical relation. I suppose you either share this intuition or you don't. In a more general form, the intuition is that collections depend for their existence on the things collected, and not vice versa. This is particularly obvious if the items collected can also exist uncollected. Think of Maynard's stamp collection. The stamps in the collection will continue to exist if Maynard sells them, but then they will no longer form Maynard's collection. The point is less obvious if we consider the set of stamps in Maynard's collection. That set cannot fail to exist as long as all the stamps exist. Still, it seems to me that the set exists because the members exist and not vice versa.

And similarly in the case of T. {1} depends existentially on 1 despite the fact that there is no possible world in which the one exists without the other. If, per impossibile, 1 were not to exist, then {1} would not exist either. But it strikes me as false to say: If, per impossibile, {1} were not to exist, then 1 would not exist either. These counterfactuals could be taken to unpack the sense in which the set depends on the member, but not vice versa.

It therefore is reasonable to hold that two necessary beings can be such that one depends for its existence on the other. And so one cannot object to the notion of the Father 'begetting' the Son by saying that no necessary being can be existentially dependent upon a necessary being. Of course, this is not to say that other objections cannot be raised.

Can A Necessary Being Depend for its Existence on a Necessary Being?

According to the Athansian Creed, the Persons of the Trinity, though each of them uncreated and eternal and necessary are related as follows. The Father is unbegotten. The Son is begotten by the Father, but not made by the Father. The Holy Spirit proceeds from the Father and the Son. Let us focus on the relation of the Father to the Son. When I tried to explain this to Peter Lupu, he balked at the idea of one necessary being begetting another, claiming that the idea makes no sense. One of his arguments was as follows. If x begets y, then x causes y to begin to exist. But no necessary being begins to exist. So, no necessary being is begotten. A second argument went like this. Begetting is a causal notion. But causes are temporally precedent to their effects. No two necessary beings are related as before to after. Therefore, no necessary being begets another.

I first pointed out in response to Peter that the begetting in question is not the begetting of one animal by another, but a begetting in a different sense, and that whatever else this idea involves, it involves the idea of one necessary being depending for its existence on another. Peter balked at this idea as well. "How can a necessary being depend for its existence on a necessary being?" To soften him up, I looked for a non-Trinitarian case in which a necessary being stands in the asymmetrical relation of existential dependency to a necessary being. Note that I did not dismiss his problem the way a dogmatist might; I admitted that it is a genuine difficulty, one that needs to be solved.

So I said to Peter: Look, you accept the existence of Fregean propositions, items which Frege viewed as the senses of sentences in the indicative mood from which indexical elements (including the tenses of verbs) have been removed and have been replaced with non-indexical elements. You also accept that at least some of these Fregean propositions, if not all, are necessary beings. For example, you accept that the proposition expressed by '7 + 5 = 12' is necessarily true, and you see that this requires that the proposition be necessarily existent. Peter agreed to that.

You also, I said to him, have no objection to the idea of the God of classical theism who exists necessarily if he is so much as possible. He admitted that despite his being an atheist, he has no problem with the idea of a necessarily existent God.

So I said to Peter: Think of the necessarily existent Fregean propositions as divine thoughts. (I note en passant that Frege referred to his propositions as Gedanken, thoughts.) More precisely, think of them as the accusatives or objects of divine acts of thinking, as the noemata of the divine noesis. That is, think of the propositions as existing precisely as the accusatives of divine thinking. Thus, their esse is their concipi by God. They don't exist a se the way God does; they exist in a mind-dependent manner without prejudice to their existing in all possible worlds. To cop a phrase from the doctor angelicus, they have their necessity from another, unlike God, who has his necessity from himself.

So I said to Peter: Well, is it not now clear that we have a non-Trinitarian example in which a necessary being, the proposition expressed by '7 + 5 = 12,' depends for its existence on a necessary being, namely God, and not vice versa? Is this not an example of a relation that is neither merely logical (like entailment) nor empirically-causal? Does this not get you at least part of the way towards an understand of how the Father can be said to beget the Son?

To these three questions, Peter gave a resounding 'No!' looked at his watch and announced that he had to leave right away in order to be able to teach his 5:40 class at the other end of the Valle del Sol.

Does Inconceivability Entail Impossibility?

In an earlier thread James Anderson makes some observations that cast doubt on the standard entailment from inconceivability to impossibility. (I had objected that his theological mysterianism seems to break the inferential link connecting inconceivability and impossibility.) He writes,

But even though we have no direct epistemic access to any other inconceivability than our own, and despite the formidable historical pedigree of the idea, it still strikes me as implausible to maintain that inconceivability to us entails impossibility. [. . .] For the principle in question is logically equivalent to the principle that possibility entails conceivability. But is it plausible to think that absolutely whatsoever happens to be possible in this mysterious universe and beyond must be conceivable to the human mind, at least in principle? Can this really be right?

I want to emphasize that I'm not advocating some form of modal skepticism, i.e., the view that our intuitions as to what is possible or impossible are generally unreliable. On the contrary, I think they're reliable. I just deny that they're infallible.

This does indeed give me pause. Anderson is certainly right that if inconceivability entails impossibility, then, by contraposition, possibility entails conceivability. These entailments stand or fall together. But is it plausible to maintain that whatever is possible is conceivable? Why couldn't there be possible states of affairs that are inconceivable to us?

But there may be an ambiguity here. I grant that there are, or rather could be, possible states of affairs that we cannot bring before our minds. These would be states of affairs that we cannot entertain due to our cognitive limitations. But that is not to say that a state of affairs that I can bring before my mind and in which I find a logical contradiction is a possible state of affairs. Thus we should distinguish two senses of inconceivable, where S is a state of affairs and A is any well-functioning finite cognitive agent:

S is inconceivable₁ to A =df A entertains S and finds a contradiction in S.

S is inconceivable₂ to A =df A is unable to entertain (bring before his mind) S.

Now it seems clear that inconceivability₂ does not entail impossibility. But I should think that inconceivability₁ does entail impossibility. For if S is contradictory, then that very state of affairs as the precise accusative of my thought that it is, cannot obtain. Its possibility in reality is ruled out by the fact that it cannot be entertained without contradiction.

Now does possibility entail conceivability? No, in that the possible need not be thinkable by us: there could be possibilities that lie beyond our mental horizon. But possibility does entail conceivability if what we mean is that possible states of affairs that we can bring before our minds must be free of contradiction.

So, in apparent contradiction to what Anderson is claiming, I urge that we can be infallibly sure that a state of affairs in which we detect a logical contradiction cannot obtain in reality. There is more to reality, including the reality of the merely possible, than what we can think of; but what we can think of must be free of contradiction if it is to be possible.

Conceivability without contradiction is no infallible guide to possibility. But inconceivability₁ is an infallible guide to impossibility. Where Anderson apparently sees symmetry, I uphold the traditional asymmetry.