Truth Accessibility Relations among Times

In my two previous posts, I have explained the notion of 'truth at a time' and argued that for a (contingent) proposition to be true at T it must supervene upon what exists at T. In short, truth at T requires a truthmaker at T.

A question that remains concerns accessibility relations among times. If a proposition is true at some earlier time, is it necessary that it be true at all later times that it was true at that earlier time? And if a proposition is true at some later time, is it necessary that it be true at all earlier times that it will be true at that later time? In common tense logical notation, these questions concern the theses:

1. p → FPp
   
2. p → PFp

(1) says that if p is true at some time, say T, then it will be true at all times subsequent to T that p was true at T. (2) says that if p is true at T, then it has been true at all times prior to T that p will be true at T.

I believe that (1) is true and (2) is false. In other words, I believe that earlier times are accessible to later times, but I don't think that later times are accessible to earlier times. This antisymmetry means that I take S4 and not S5 to be the right system of modal logic as far as times are concerned.

S4 ensures reflexivity and transitivity among times. Reflexivity means that each time is accessible to itself. Hence, if it is true at T that p, then it is true at T that it is true at T that p, etc. Transitivity means that if T1 is accessible to T2, and T2 is accessible to T3, then T1 is accessible to T3. Hence, if it is true at T1 that p, and if it is true at T2 that it is true at T1 that p, then it is true at T3 that it is true at T1 that p.

To reflexivity and transitivity, S5 adds symmetry. This means that if T1 is accessible to T2, then T2 is accessible to T1. Hence, if it is true at T1 that p, then it is true at T2 that it is true at T1 that p, and it is true at T1 that it is true at T2 that it is true at T1 that p, etc.

Now, if S5 is right, then both (1) and (2) have to be accepted - what's true at any time is mirrored at all times. If S4 is right, however, then we cannot accept both (1) and (2), for that would violate antisymmetry. The questions, then, are whether we should accept antisymmetry or not and, if so, whether we should go with (1) or (2).

My argument turns on the relation between time and causality and on the falsity of determinism. Earlier events can cause later events, but not vice-versa (or so most of us think). That motivates antisymmetry, but to establish it we have to appeal to the idea that what's true at a time depends upon what exists at that time. Suppose event E1 at T1 indeterministically causes event E2 at T2. It is true, therefore, at T2 that E2 is occurring. Given symmetric accessibility between T1 and T2, it is true at T1 that it is true at T2 that E2 is occurring. But this is false if the causal relation between E1 and E2 is indeterministic, for in that case we do not have a truthmaker at T1 for "E2 is occurring at T2".

If determinism is true, then a unique sequence of later times is implied by earlier times. It can be true at T1, therefore, that "E2 is occurring" will be true at T2. Hence, T2 is implicitly accessible from T1. But if determinism is false, then no unique sequence of later times is implied by earlier times. Let p represent the occurrence of some event that is causally contingent from the perspective of T1 (e.g., "E2 occurs at T2"). From that perspective, it may be that p is true at T2, and it may be that p is false at T2. Moreover, there is nothing at T1 sufficient for making it true either that p will be the case at T2 or that p will not be the case at T2. Since what's true at T1 depends on what exists at T1, it cannot be true at T1 either that p will be true at T2 or that p will not be true at T2. Given indeterminism, therefore, it follows that (2) is false. If, for example, there being a sea battle at T2 is causally contingent at T1, then even if a sea battle does occur at T2, it does not make "A sea battle will occur at T2" true at T1, contrary to (2).

But what about (1)? It would be very counterintuitive if (1) were false. It's falsity would allow, for example, that "Caesar crossed the Rubicon" might be true in, say, 1000 A.D., but not true in 2000 A.D. So I want to insist that (1) is true, but I emphasize that it's truth is not a mere matter of logic. If we accept that truth supervenes on being, then (1) commits us to a substantive metaphysical position. Since what's (contingently) true at T2 depends upon what exists at T2, for it to be true at T2 that some event E occurred at T1, there must be something at T2 sufficient for making that true. E's occurring at T1 is not enough. There must also be something that carries E's occurring at T1 forward in time and that constitutes at T2 E's having occurred at T1. As to what that "something" is, I refer the reader to this forthcoming paper of mine.

In summary, if determinism is false, as I believe, then (2) is false. (1), however, is true - it would be intuitively quite odd (to say the least) if it weren't - though it comes at what some will regard as an unacceptable metaphysical price. Those folks either have to deny that truth supervenes on being or reject (1). I think either is a much bigger price to pay.

Truth at a Time and Truth Simpliciter

As a follow-up to my previous post, I'd like to say a bit about how truth at a time and truth at a world relate to truth simpliciter.

Unlike 'truth at a world' and 'truth at a time', truth simpliciter is not evaluated from the perspective of any given time or world, but absolutely. In other words, it is evaluated from the proverbial "God's eye" or absolute perspective, that is, in relation to what exists simpliciter. Consequently, what is true simpliciter depends on the correct metaphysics.

Actualism is the thesis that the only possible world that exists simpliciter is the actual world, whereas possibilism is the thesis that all possible worlds exist simpliciter. Presentism is the thesis that only what exists now exists simpliciter, whereas eternalism is the thesis that all past, present, and future world states exist simpliciter. If actualism and presentism are correct, then what exists simpliciter is coextensive with what exists now. Moreover, truth simpliciter coincides with what is true now. Hence, "Obama is now President" is true simpliciter.

In contrast, if actualism and eternalism are correct, then all instantaneous world states (however foliated into past, present, and future) exist simpliciter. As a result, truth simpliciter and truth at a time can come apart. Omnitemporal truths, which are true at all times, hold distributively across the entire time series and are true simpliciter. So are transtemporal truths, which hold collectively of the entire time series. These can, but arguably need not, be true at all times in the series. For example, "At some time or other, Obama is President" is true simpliciter. It is also true now. But it may not have been true at all earlier times because the fact that makes it true may not be accessible from all earlier times. One way in which this could happen (defended by Peirce, Prior, Hartshorne, and Ryle) is through reference failure. Arguably, before Obama was conceived, there was no such individual for propositions to refer to. Finally, there may be (though this is controversial) atemporal truths, truths about matters that stand in no relation to time whatsoever. These truths, if they exist, are true simpliciter, but fail to be true at any time. (Some people think mathematical truths fall into this category. I doubt this. I say that '2+2=4' is omnitemporally true, not atemporally true. It sounds rather odd to say that it isn't true now.)

Finally, if possibilism and eternalism are correct, then all worlds and all instantaneous world states in all worlds exist simpliciter. As before, truths that hold distibutively over all world-time pairs (i.e., at all times in all worlds) will be true simpliciter. Similarly, anything that is true at all worlds (i.e., necessary truths) will be true simpliciter. Further, truths that hold collectively of the entire world/history ensemble (e.g., "At some time in some world Obama is President") will be true simpliciter. This latter truth is true now in alpha (the actual world), but it need not be true true at all times in all worlds for it may not be accessible from all times in all worlds.

Truth-at-T Depends on What Exists-at-T

In my previous post I made the following claim without argument: "for a proposition to be true now, what it represents as being the case must correspond to ... what is the case now." I'm now going to give that argument.

Every argument for any categorical conclusion requires some categorical assumptions which are not defended within the scope of that argument. Here are some of my assumptions:

1. Truth supervenes on being (TSB). This means that what is true depends on what exists for its being true. Moreover, every difference in truth corresponds to a difference in being, such that if anything that is true had not been true, then there would have been a corresponding difference in reality.
2. Correspondence theory of truth (CT). A proposition is true if and only if the state of affairs (STOA) that it represents obtains. In other words, a proposition p is true if and only if the STOA that would be posited were p asserted obtains.
   
3. It makes sense to speak of a proposition's being true at a time. This notion is compatible with, but does not entail, the idea that truth is temporally invariant.

I won't defend these claims here, though if pressed I might be willing to restrict (1) and (2) to contingent truths.

Now, the key to establishing my claim that what is true at a time depends on what is real or what exists at that time is to understand the notion of 'truth at a time'. I want to suggest that we should think about this notion in a manner analogous to the idea of 'truth at a world'.

In metaphysics, possible worlds are typically distinguished in terms of what is true 'at' them. Thus, worlds W and W* are distinct if and only if something is true at W that is not true at W*, or vice-versa. To consider what is true at a world is essentially to ask what would be true if that world were actual. Alternatively, since each world is actual from its own perspective, we can ask how things stand from the perspective of that world. Finally, what is true at one world may be reflected in what is true at another world. Whether this is so depends upon the accessibility relations that hold among the worlds in question. Thus, if p is true at W, and if W is accessible to W* (that is, if one can "see" W from the perspective of W*), then it is true at W* that p is true at W.

Analogously, 'truth at a time' means truth as judged from the perspective of a given time. By a 'time' I mean a complete time-slice of reality, an instantaneous world state. To determine what is true at a time we ask what would be true if that time were present. Alternatively, since each time is present from its own perspective, we can ask how things stand from the perspective of that time. If T is such that Caesar is ruling in Rome, then "Caesar is ruling in Rome" is true at T. If T* is such that Nero (and not Caesar) is ruling in Rome, then "Caesar is ruling in Rome" is false at T*. As with worlds, what is true at one time may be reflected in what is true at another time. Whether this is so depends upon the accessibility relations that hold among the times in question. Thus, if p is true at T, and if T is accessible to T* (that is, if one can "see" T from the perspective of T*), then it is true at T* that p is true at T. If, in addition, T* is earlier than, simultaneous with, or later than T, then accessibility may be expressed in terms of tense. Thus, if p is true at T, and if T is accessible to T*, and if T is earlier than T*, then it is true at T* that it was the case that p is true at T.

The thesis I set out to defend follows from what I have said above. Just as what is true at a world depends upon what obtains at that world, so also what is true at a time depends upon what obtains at that time. This neither precludes nor entails there being accessibility relations among times such that what is true at one time either has been or will be true at another time. Accordingly, it cannot be true now, for example, that there will be a sea battle tomorrow, unless this truth supervenes upon present reality. Depending on accessibility relations, however, it may be true now that it will (probably) be true tomorrow that there is a sea battle.

A Recent Objection to the Alethic Openness of the Future

The future is 'alethically' open at time T with respect to possible state of affairs X and future time T* > T if and only if neither "X will obtain at T*" nor "X will not obtain at T*" is true at T.

I believe that the future is alethically open with respect to future contingent states of affairs. Thus, if the present state of the world is indeterministic with respect to whether a sea battle occurs tomorrow (i.e., if that's a future contingent), then on my view neither "A sea battle will occur tomorrow" nor "A sea battle will not occur tomorrow" is now true. Indeed, I submit that both propositions are now false.

There's an interesting challenge to my view that has recently been proposed by Alex Pruss of Baylor Univ. His argument is encumbered with the technical trappings of probability theory, but the basic objection can be stated quite simply as follows:

1. On the open future view that I espouse, there are no “will” or “will not” truths about future contingents.
2. But we have excellent (inductive) reasons for believing that there are “will” or “will not” truths about future contingents (e.g., “S will not win the lottery” - assume, e.g., a standard lotto and that S has only one lotto ticket).
3. Therefore, we have excellent reasons for believing that the open future view is false.

This is a potentially devastating argument against my position. The logic is clearly valid, I'm already committed to (1), and premise (2) looks very plausible. Clearly, I have to deny (2). The challenge is to make it clear why that's a reasonable thing to do.

In the first place, it is essential to get as clear as we can on exactly what proposition is expressed by the sentence "S will not win the lottery". This is important because linguistic expressions do not wear their meaning on the sleeve, so to speak. We have to interpret them as best we can in light of whatever contextual clues are available. Due to factors like vagueness, ambiguity, lack of information about the context, and so forth, interpretation is often difficult and frequently inconclusive.

In the second place, there is an important distinction that we have to make between modally inflected uses of "will" and "will not" and amodal uses. Interpreted in the former sense, "S will not win the lottery" may be read as saying something like "S will probably not win the lottery". Here the word "probably" attributes an objective probability to the event S's not winning the lottery. Different objective probabilities can be captured with different qualifiers (e.g., "will definitely", "will possibly", or, more generally, "will, with probability p"). Interpreted in the amodal sense, "S will not win the lottery" should be read as saying something like "Subsequently, S does not win the lottery". This says nothing about what it's objective probability is, other than implying that it is non-zero. Probabilities applied to this proposition have be read as epistemic probabilities, which qualify the attitude that a subject has toward a proposition without entering in the meaning of the proposition. The contrast, then, is between a modal, objective probability reading "S will probably not win the lottery" and an amodal reading "Subsequently, S does not win the lottery".

In the third place, there is a potential ambiguity in the scope of the word "not" in "S will not win the lottery". Does it apply just to the predicate alone or to the rest of the proposition in its entirety? On the first reading, "S will not win the lottery" means "S will not-(win the lottery)". On the second reading, "S will not win the lottery" means "Not-(S will win the lottery)". These readings are distinct since the former entails the latter, but not vice-versa. It is possible, after all, for "Not-(S will win the lottery)" to be true without "S will not-(win the lottery)" being true - for example, when S does not exist.

Crossing the above distinctions gives us four basic readings of "S will not win the lottery":

1. S will probably not-(win the lottery).
2. Not-(S will probably win the lottery).
3. Subsequently, S does not-(win the lottery).
4. Not-(subsequently, S does win the lottery).
   

Now, presumably, we have good (inductive) reasons to believe all of these to be true. (a) and (b), however, pose no difficulty for my position, since I am not committed to saying that "will probably" propositions are false when it comes to future contingents. It comes down, then, to the amodal propositions (c) and (d). To refute me the objector has to show that we have good reason for believing that either (c) or (d) concerns a future contingent and is true now. It is here that I make my stand. I deny that these conditions are satisfied by (c) or (d).

For a proposition to be true, what it represents as being the case must correspond to reality, to what is the case. Likewise, for a proposition to be true now, what it represents as being the case must correspond to present reality, to what is the case now. Now, (c) clearly represents what is, from our present perspective, a future contingent state of affairs. It represents S's not winning the lottery subsequent to a contextually specified index date. I deny, however, that (c) is true now or that we have good reason for believing it to be true now on the grounds that there is no present reality that it corresponds to. We have good reason for believing that it will probably come to be true, but not for believing that it is true. Thus, after the drawing of the lottery, either S will have won subsequently to the index date or not. At that point, assuming S does not win, (c) will be true. But not until then.

With respect to (d), I grant that it is true now and that we have good reason for believing it to be true now, but I deny that it directly concerns the future. It concerns, rather, a present fact, for it says it that it is not (now) the case that subsequently S does not win the lottery. This is provably correct. Future events do not obtain now.

In summary, I reject premise (2) of the objector's argument on the grounds that (i) the sorts of "will not" propositions about future contingents that we have good reasons for believing to be true now are not the sorts of propositions to which the doctrine of alethic openness applies, and (ii) the sorts of "will not" propositions about future contingents to which alethic openness applies are not the sorts of propositions that we have good reason for believing to be true now.

Could God Infallibly Know that He Is Omniscient?

I've been reflecting a bit about the following analogy proposed by Enigman:

Given someone who knows that she can move freely anywhere within an infinitely dimensional space, does she know that she has complete freedom of motion? The problem is that such a space is isomorphic to a hyperspace containing such a space as a mere slice.

This analogy is supposed to call into question whether God can be certain that there is no reality beyond his ken. Couldn't the entire space of God's knowledge simply be a mere slice of a larger hyperspace containing things of which not even God is aware?

Traditional theism answers with an emphatic NO. But can that answer be solidly justified, or is it simply a dogmatic prejudice?

One strategy for justifying the traditional answer is to go Thomistic. On Thomas's account, God is defined as ipsum esse subsistens. His essence is identical to his existence. That trait is necessarily unique to God. Hence, in everthing else there must be a real distinction between existence and essence. As such, beings other than God cannot account for their own existence. Hence, they must all owe their existence to God as their creator. And since God has perfect self-knowledge, he knows his own creative activity, and so knows of the existence of whatever else other than himself that there may be.

My problem with that response is that I think Thomistic metaphysics creates more problems than it solves. For one thing, it's hard to see how Thomas can account for real contingency. Identity is a necessary relation (if two things are identical then they are necessarily identical). It follows that God's essence = his existence, across all possible worlds. Now, either there are multiple, distinct possible worlds, or there are not. If not, then everything is necessary, which is counterintuitive in excelsis. If, however, there are multiple possible worlds, then they differ in some respects, which means that there are contingencies. Hence, what God has to know in order to know everything must vary from world to world, which means that the content of God's knowledge must to some extent be contingent. But if God's existence is really identical to God's essence, then everything about God must be essential and thus invariant across possible worlds in which God exists, including the content of God's knowledge, which is inconsistent with the assumptions that there are multiple possible worlds and that God's knowledge is exhaustive. In short, the following three assumptions are mutually inconsistent:

1. God has exhaustive knowledge of reality.
2. God's essence is identical with his existence.
3. Some facts are contingent.
   

Since 3 is undoubtedly true, either 1 or 2 is false. My preference is to deny 2.

Let me turn now to what I think is a better strategy. The key idea is that actuality is the sole delimiter of real possibility. What this means is that real possibility is always relative to some actuality. For example, what's physically possible depends upon what physical laws are actually in place. Now, suppose we stipulate that God is the sole delimiter of real possibility. If that be accepted, then there are no real possibilities that are not grounded somehow in God. For example, there could be no metaphysically possible world in which God did not exist. Now, if both God and actuality are delimiters of real possibility, then they must be related in some systematic way. Assuming that pantheism is false, that relation can't be identity. A theist, however, could subordinate the one to the other and say that God is the delimiter of possibility in virtue of being the delimiter of actuality. Thus, what's possible is grounded in what's actual, and what's actual is grounded either in God's nature or God's will. If that's right, then in knowing himself, God knows all actualities, and there is no possibility for there to be any actuality that is unrelated to God.

Does that solve the problem? Well, it's a good start at least. One might wonder how God can be sure that he is the sole delimiter of real possibility. If he is, then his perfect self-knowledge guarantees that he knows that he is, and the problem is solved. But if God is not sure, then he can conclude that either he isn't the sole delimiter of possibility or he doesn't have perfect self-knowledge.

Does Moral Realism and Infinite Spacetime Entail Atheism?

Vlastimil Vohánka send me an email recently asking for my thoughts on the following reconstruction of an argument recently proposed by Quentin Smith.

1. Necessarily, global moral realism is true (everything, or rather, every physical entity, has a positive amount of value). Premise
2. Necessarily, aggregative value theory is true (each physical location has a finite positive amount of value; a location can be a person, any other animal, a plant, a particular of matter or energy, a point of space or time, or some larger complex of particulars of these kinds, for example, a forest, an orchestra or an hour of time; values add up). Premise
2*. Necessarily, the performance of an action is morally indifferent iff the performance of that action neither increases nor decreases the amount of value in the universe. Premise, or from (2)
3. Contingently (and according to contemporary physics), spacetime is infinite, both temporally -- there are infinitely many non-overlapping future hours --, and, more controversially, spatially -- at each time there are infinitely many non-overlapping, equal sized cubes of space. Premise
3*. Necessarily, every human action has only finite effect on the amount of value in the universe. Premise
3**. Necessarily, neither any finite addition to nor any finite detraction from an infinite amount of value neither increases nor decreases the amount of value. Premise
4. Moral nihilism is contingently true (it does not morally matter what humans do, it does not matter what actions humans perform). From (1)-(3**)
5. Humans have no intrinsic dignity (from 1-3**); humans have no rights (from 1-3** or from 4); human life has no meaning (from 1-3** or from 4); every human's life is less valuable than the entire state of his being dead (from 1-3**).
5*. God (at least as traditionally conceived) does not exist. From (4) or (5)

The gist of the argument is that if something exists that is infinitely valuable, if some such thing would exist regardless of what we do, and if our contributions are at best finite, then nothing we do makes any real difference to the aggregate value of what exists (since ∞ + 1 = ∞). Hence, nothing we do really matters.

Now, I think that premise (3) of this argument is particularly questionable. I don't think there are any empirical results from physics that establish or that even could establish the existence of an infinite spacetime. But the argument could be reconstructed without that premise. For example, instead of infinite spacetime, we could start from the assumption that God necessarily exists and has infinite value. The result, if the rest of the argument is sound, would be a reductio ad absurdum of theism.

If there's a fundamental flaw in the argument, it has to do with its assumptions about the nature of value. There are several that I find particularly questionable.

I'm skeptical about (1). If all that ever existed was a diffuse cloud of hydrogen, would it have positive value? I'm inclined to think not. I suggest, rather, that there may be no value that is not value to someone or, more broadly, to some sentient being. This is reflected in the notion of the good as that which is desirable, a notion that has no application apart from beings capable of desiring.

I'm also skeptical about (2). Can values simply be added up? Not if there are incommensurable kind of value. It may well be that there is no common currency in which all values can be "cashed out" and then added up.

Finally, I'm skeptical about (3*). Why think that human actions have only a finite effect on the amount of value in the universe? Suppose, for example, that there is an afterlife and that one of the possibilities is heaven (infinite positive utility). Well, if I by my own choices and the grace of God can enter the kingdom of heaven, then haven't I done something that contributes infinite value? Or if inspire others to lives of virtue and godliness so that they enter the kingdom of heaven, then haven't I done something that contributes infinite value?