Saturday, July 29, 2006

We're Back

My wife and I just got back from Maui this morning. We had a really great time. Very relaxing.

Now it's back to the grind. I have three projects that I must finish by the end of August. And another two that I have to finish by the end of September. So I'm going to be real busy for the next two months. I will continue blogging during that time, but I will have to keep it fairly light - probably just a couple posts a week.

Nor can I respond to all comments. I have been pleased to see in my absence, however, that my turning off the comment moderation constraint has facilitated entended conversations among my readers. Thanks Tom, Ocham, HammsBear, and C Grace for your contributions.

Thursday, July 20, 2006

Going to Maui

Faithful readers,

My wife and I are leaving tomorrow morning for a week in Maui to celebrate our third anniversary. So no more blogging until I get back.


PS: I've decided to try turning off comment moderation. So your comments will now appear immediately after you post them.

Further Thoughts on Excluded Middle

In my previous post I raised some questions about the "Law of Excluded Middle" (LEM), which states that
(LEM) For any proposition p, either p is true or p is not-true.
The gist of my concern was that if LEM applies across the board then this implies that reality is discrete all the way down, and thus that genuine continuity is really an illusion.

But subsequently I've come to see that my argument trades on a possibly illicit shift from propositions to reality. As defined, LEM applies to propositions. Propositions, I gather, are usually taken to be abstract representations of conceivable states-of-affairs that are true when those states-of-affairs obtain and false otherwise. It's this abstractness that I failed fully to grapple with.

An abstraction always leaves something behind. For example, when I notice that my car needs a washing I temporarily ignore or abstract from other details of the car (its location, directional orientation, level of gas in the tank, etc.).

Now, the question of whether reality is fundamentally continuous is not a question about abstractions but about what's concrete, about reality-in-itself. Hence, if we think of propositions as invariably abstract, then continuity in reality can't pose any problems for LEM.

Of course, there is the issue of whether propositions are invariably abstract. I think they are so for us, because as finite beings situated in space and time we can only approach reality in bits and pieces from one perspective or another. So the only way we can get clear on anything is to focus on some things to the exclusion of other. In other words, human thought requires abstractions.

A transcendent, omniscient being, however, wouldn't know reality in bits and pieces or from any particular perspective but rather all-at-once. Such a being would have, I think, a non-abstract representation of the maximal state-of-affairs that is reality. If such a being is so much as possible, then propositions (understood simply as representations of conceivable states-of-affairs) are not necessarly abstract.

But all that means is that LEM may not apply for beings like God.

There's another way in which the abstractness of propositions (as they occur in human minds) that is relevant to LEM, though. Consider my car again. When I say to myself that it needs a wash, I make a mental judgment that it is a car and that it needs a wash. In those two respects I've mentally categorized it as this and not-that. Hence, LEM applies to my thought with respect to those categories. But I've made no judgment about other features of the car. As far as my conception goes at that moment, whether the car is moving or not is indeterminate, whether it needs gas or not is indeterminate, whether it is pointed east or west is indeterminate, and so forth. In other words, the determinateness of my conception of the car as needing a wash does not entail anything about how the car is to be determinately categorized in other respects, much less whether the car can in principle be determinately categorized in all respects. The proposition in my mind is neutral on all that.

Well, I've gotta run now. So to abruptly conclude: (1) Vagueness concerns what's not brought into focus in our conceptions. With regard to what is brought into focus, LEM applies. So insofar as there is a determinate abstract proposition at all, it obeys LEM. (2) LEM implies nothing, however, about how things are in reality. So one can't draw any grandiose metaphysical conclusions from LEM, like "apparently continuous space and time are really just sets of discrete points". LEM is merely a law of human thought insofar as those thoughts are determinate.

Sunday, July 16, 2006

Is the "Law of Excluded Middle" Really a Law?

The so-called "Law of Excluded Middle" (LEM) is often taken to be one of the most fundamental laws of logic. It may be expressed as follows:
(LEM) For any proposition p, either p is true or p is not-true.
LEM is to be distinguished from the Principle of Bivalence (BV), which states:
(BV) For any proposition p, either p is true or p is false.
BV entails LEM, but the converse does not hold because "not-true" may not be equivalent to "false" (That they are equivalent is what BV claims. LEM makes no commitment on the matter.)

Anyway, it occurs to me that LEM works whenever things admit of sharp classification into either this or not-this, and fails insofar as things do not admit of sharp classification. In other words, LEM works insofar as things are precise and fails insofar as they are vague. To say that a concept is vague is to say that it has fuzzy boundaries, that there is a penumbra or "grey area" in which it is simply not clear whether a given item should fall inside or outside the boundary, or whether it should simply stay on the boundary.

For example, under normal conditions a given lightbulb must be either ON or OFF, with no in-between state. Consequently, in this case we have an instance where LEM applies. If p is "This lightbulb is on" then p is either true (the lightbulb is ON) or it is not true (the lightbulb is not-ON, i.e., OFF). Contrast that example with a man who has lost most, but not nearly all, of the hair on his head. Is he bald or not-bald? Well, it may not be so clear. Depending on how his remaining hair is distributed across his scalp we might lean more toward one answer or the other, but this case just isn't as sharply defined as with the lightbulb. Perhaps we should say that he's partly bald and partly non-bald or bald in some places and non-bald in others.

Anyway, I think my point is clear that vague cases create difficulties for LEM. The crucial question is whether all such cases merely pose difficulties in the application of LEM, or whether some of them reflect a genuine limitation on LEM itself.

To say that LEM ought to hold always and everywhere is tantamount to saying that reality is perfectly sharp and determinate all the way down and that vagueness is always merely epistemic, a function of the inexactness of our language and of our concepts. This is not obviously correct. In fact, there is evidence from physics that when we go down far enough things really do get "fuzzy" - think of wave-particle duality and such. Now this fuzziness may in the final analysis turn out to be merely epistemic, but it's going quite a ways beyond the available evidence at this point to suppose that this must be the case.

Continuity is another issue where LEM naturally comes into question. Our natural conception of space, time, and motion is that they are smoothly continuous and that points, lines, events, and such are interruptions of sorts in an already existing continuum. The arithmetization program in mathematics, however, proposes to analyze such continua into sets of discrete points. If we adopt this latter course as being the metaphysical truth of the matter, then we can apply LEM to such continua all the way down. But this is a large metaphysical assumption to make. It essentially amounts to saying that continua are not metaphysically basic and can, therefore, be reductively analyzed in terms of discrete non-continua (i.e., points). Conversely, one who takes our ordinary conception of continuity to be closer to the metaphysical truth of the matter - in other words, one who holds that continuity is metaphysically more basic than discontinuity - thereby commits himself to a denial of LEM at the continuum level.

One can't have it both ways. If LEM holds across the board, then continua cannot be metaphysically fundamental. If continua are metaphysically fundamental, then LEM can't hold across the board. Since I am partial to the latter view, I say that LEM is not really a law.

Sunday, July 09, 2006

Presentism, Truthmakers, and God's Memories: Reply to an Objection

I've argued here that the truthmaker objection to presentism is best met by appealing to God's memories of the past as the truthmakers of true propositions about the past. (Note: Readers of the linked paper should be aware that I am still revising it. Constructive feedback is welcome.) Instead, I will briefly explain how my proposal is supposed to work and then I'll present an objection sent to me by Patrick Todd, a grad student at the University of Missouri who independently arrived at the same idea, but found that it was not well received by his colleagues.

First, the basic idea is this: Truths about the past need truthmakers, some parcel of reality the existence of which grounds and, thereby, makes those truths true. This poses a prima facie problem for presentism, which is the view that whatever exists, exists now, in the present. Somehow the presentist needs to supply presently existing facts to make true every truth about the past, but it's not at all obvious how this could be done. What present fact, for example, could make it true that Columbus discovered American in 1492? Presumably it would have to be some sort of causal trace of that past event, but the available physical evidence is not sufficient to guarantee the truth of that event. The same goes for every other contingent truth about the past.

Enter God's memories. A presentist who is prepared to accept theism has, I think, a relatively straightforward answer to the truthmaker objection. First of all, the combination of theism and presentism requires that God exist now, not atemporally. Hence, God was present in 1492 and directly experienced Columbus's voyage. As omniscient, God retains a perfectly detailed memory of that event as a consequence of his experience of it, even though the event itself no longer exists. God's memories, therefore, contain presently existing traces of that past event and every other past event. Those memories represent past events in full detail in the exact sequence in which they occurred. And since those memories would not have been what they are if past events had not occurred as they did, those memories suffice to guarantee the truth of every true proposition about the past.

Now for the objection. I'm quoting the email I received from Patrick here:
They said that the position leads to a severe Euthyphro problem:

Is p true because God remembers that p, or does God remember that p because 'p happened' is true? Or, said differently and more simply, Is it true because God remembers it or does God remember it because it is true?

[...] On the one hand, if we say that X is true because God remembers X, then it looks like God could just remember any old X and it would be true that X. ... And on the other, if we say that God remembers X because "X happened" is true, then it looks like there is some past state of affairs to which God's memory presently corresponds. And in that case, we haven't solved the grounding problem.
Here's my reply. Let P be the present-tense proposition "Columbus discovers America". We can then represent the past-tense "Columbus discovered America" (or, alternatively, "It was the case that Columbus discovers America") as WAS(P). Let Q=WAS(P). And let T be the present state of affairs that serves as the truthmaker for Q.

The proposal on the table is that T is one of God's memories. The important question is which memory? Here's where the Euthyphro-type objection as formulated above goes wrong. The truthmaker for Q cannot be God's memory that Q. Why not? Well, to remember something is to represent it as being past, which is like tacking on a past-tense operator, WAS(). So, since Q = WAS(P), if God remembers that Q, then he remembers that WAS(P), which means the representational content of God's thought becomes WAS(WAS(P)), a past-perfect construction. But God's memory of P has the representational content WAS(P), a simple past construction. Since WAS(P) = Q, that's what we want.

So T = God's memory that P = God's belief that WAS(P) = God's belief that Q.

(I realize that memories cannot in general be equated with past-tense beliefs, but for God this seems unproblematic. He has directly experienced and thus remembers every past event, and he has past-tense beliefs about every past event. Hence, God's memories and past-tense beliefs are coextensive.)

A possible rejoinder at this point is a refomulation of the Euthyphro-type dilemma: Is (a) it true that Q because God believes that Q, or (b) does God believe that Q because it is true that Q?

On the view I'm defending, (b) gets things explanatorily backwards. It is God's memories, or if you prefer his past-tense beliefs, that make true propositions about the past true. So I must reject (b). The objector might wonder, though, if it is not I who have things explanatorily backwards. Doesn't God believe what he believes because it's true? To that I say No. God doesn't believe anything because it is true. Rather, God believes what he believes because he is immediately acquainted with the reality that makes it true. (For my defense of this claim and some discussion go here.)

What about (a)? The worry is that this could render God's beliefs about the past arbitrary, such that God's could just concoct some story about the past, convince himself of it, and thereby make "true" propositions about a past that never was. But it should be obvious that this worry is misplaced. God's beliefs about the past cannot be arbitrary because, in the first place, they are themselves causal traces of the past--God's belief that WAS(P) is a causal trace of the very event that made P true when P was true, namely, the event of Columbus's discovering America. And, in the second place, because God is essentially omniscient, he could not have believed WAS(P) otherwise. So this version of the Euthyphro-type objection fails as well.

Propositions and States of Affairs - IV

Awhile back, I did a series of three posts (I, II, and III) in which I was trying to work out the relations between propositions ("props" for short) and states of affairs ("sofas" for short). I've since been rethinking things a bit. So here's my new and (hopefully) improved theory.

Earlier I had written that propositions are assertoric units of meaning, but I now see that this is ambiguous between their being assertions and their being assertible. The second reading is correct, and so "assertoric" should be replaced with "assertible" for clarity's sake.

One problem with the first reading is that, strictly speaking, it is only persons that make assertions, so defining propositions as assertions makes it analytic that there can be no propositions without persons. But it is doubtful that that should be admitted as an analytic truth. There is no clear contradiction is supposing that some proposition or other being true or false.

A second problem with the first reading is that it makes it difficult to construe the common element in "Is p true?" and "p is true", where the latter expresses an assertion and the former does not. More generally, if we treat propositions as assertions, then to handle non-assertoric "propositional attitudes" we have to treat them as second-order qualifications of assertions. Thus, doubting that p becomes doubting whether to assert that X; withholding that p becomes withholding whether to assert that X; and so forth. X here, stands for an abstract entity that is just like a proposition except for being attitudinally neutral. Philosopher Roderick Chisholm employs abstract states of affairs to fill that role, and until recently I basically followed him in that. On this view, a proposition p posits an abstract state of affairs S and is true just in case that abstract state of affairs corresponds to a concrete state of affairs that "obtains".

But all this is awkward and, I now think, needlessly complex. If we simply say that propositions are assertible units of meaning, then we can treat all propositional attitudes in the same way and we don't need to bloat our ontology by admitting abstract states of affairs in addition to propositions.

My current view may be summed up as follows:
  • A state of affairs is a parcel, any parcel, of reality, where the real consists in whatever is and is as it is independently of what any (non-archtypal) intellect thinks about it. States of affairs can overlap and include other states of affairs. The actual world is the totality of reality, the one all-inclusive state of affairs.
  • A proposition is an assertible unit of meaning. It is an abstract representation of a state of affairs and, as such, is true if and only if the state of affairs represented obtains (= exists, is actual, is real). Propositions in themselves don't posit states of affairs; they merely represent them.
  • The meaning of a proposition consists in the sum total of its entailments. Two propositions are distinct if and only if they have different entailments. We make explicit the meaning of a proposition by considering what we would be commiting ourselves to were we to assert it.
  • We use statements (declarative sentences) to express propositions. A statement is true if and only if the proposition it expresses it true.
  • Any state of affairs that suffices to make a given proposition true and that includes no states of affairs distinct from itself that suffices to make that proposition true is a minimal truthmaker for that proposition. There can be multiple minimal truthmakers for a given proposition (e.g., "Some dogs exist" is made true by Fido's existing; it is also made true by Lassie's existing; it is also made true by Fido's and Lassie's existing, but the latter is not a minimal truthmaker.). Any state of affairs that includes a minimal truthmaker for a proposition is also a truthmaker for that proposition. Thus, the actual world, the all-inclusive state of affairs, is a truthmaker for all true propositions.

Friday, July 07, 2006

Three Types of Explanations - Law, Chance, and Design

As William Dembski has pointed out, there are three basic types of explanations we can give for any phenomenon, E:
  1. Law: We can posit some nomological regularity L which allows us to predict E as a (probable) consequence of antecedent conditions.
  2. Chance: We can say that E had no systematic cause but was simply a coincidence or luck.
  3. Design: We can say that E was intentionally brought about by some agent A.
In many cases, one of these types of explanations will strike us as the overwhelmingly most plausible type of explanation.

For example, suppose I pick up a rock, hold it suspended in midair, and let go. The rock falls and lands with a thud. Why? However we answer that, I'm confident that almost everyone would hazard a lawlike explanation as their best guess. Today we'd propose the law of gravity. Earlier ages would have proposed that rocks have a natural tendency to move toward the center of the universe. We probably wouldn't even consider appealing to chance or design in this case--although there's no strictly logical reason why we couldn't. There's no logical contradiction, after all, in supposing that the rock's falling was simply coincidental and that it could just as easily have gone upwards, sideways, or hung suspended in midair. Nor is there any contradiction in supposing that the rock fell because some invisible agent or spirit made it fall. But few would take such proposals seriously, and rightly so.

Contrast this with Paley's example of stumbling across a watch and noticing that it contains an intricate assembly of parts apparently working together toward a purpose, namely, measuring hours, minutues, and seconds. As Paley points out, a design type of explanation jumps right out at us, whereas law and chance type explanations strike us as woefully implausible. Again, strict logic does not force a design explanation on us. We could without contradiction suppose there to be some, perhaps very complex, natural law of watch formation or chalk the emergence of a watch up to the chance interplay of natural forces. But few would take such proposals seriously, and rightly so.

So when it comes to determining the best explanation for E, we always have a choice between three different types of answers--law, chance, or design--a choice that logic alone cannot settle for us. Other explanatory considerations must be brought in to assess plausibility, considerations like simplicity and coherence with existing background knowledge. Hence, determining whether to appeal to chance, law, or design is generally a complex matter that needs to be judged on a case-by-case basis. And there's no a priori guarantee that all rational and informed persons will arrive at the same determination.

There's another complicating issue as well. How are these three types of explanation related to each other? Considerations of parsimony encourage us to try and reduce one or more of these types of explanation to the others, and there are several proposals on how this might be done.

For example, metaphysical materialism holds that design can be reductively explained in terms of either law or chance or some combination of law and chance. There is some divergence of opinion here. On the one hand, it used to be commonly thought that everything could be reduced to law. Laplace, for example, famously claimed that given a complete understanding of the laws of nature and the state of the world at a given time, he could calculate with complete precision the state of the world at any other time. Chance, on the this view, is merely a cloak for our ignorance of the real causes of things. On the other hand, Charles Peirce at one point argued that design and law could be reduced to chance, with natural laws having evolved over countless eons from a primeval chaos.

Theistic and idealistic worldviews, on the other hand, usually try to reduce law and chance to design. Thus, why do we have the natural laws that we do? God designed the world that way. What is chance? Simply our ignorance of the real causes of things, whether law or design. Not every theist or idealist views chance as completely epistemological, however. And there are different views as to whether everything can be reduced to one designer or not. Theistic determinists like Calvin and Jonathan Edwards, hold that there is really only one designer, namely God, who has meticulously prescripted everything that comes to pass. Most theists, however, believe that God has granted a measure of free will to his creatures, such that many events cannot be fully explained without appealing to two or more designers.