Why do we dislike it when bad things happen to us?

It is easy to give a theistic answer to the question in the title:

1. Bad things should be avoided, and so it is likely that God would make rational beings dislike them.

Presumably, the naturalistic story is going to be roughly something like this:

2. We tend to avoid things we dislike (this may even be analytic), and bad things tend to be detrimental to our fitness, so there is selection for dislike of bad things.

But there is still a puzzle: Why is it that bad things tend to be detrimental to our evolutionary fitness? Is it not a great coincidence on a naturalistic account that such highly varied qualities as ignorance, loss of limb and cowardice have both the property of badness and the property of being detrimental to fitness?

Of course some folks may say that there is no puzzle here, because our belief that these qualities are bad is caused by the fact that they are detrimental to fitness. However, that only answers why it is that there is a correlation between being believed to be bad and being detrimental to fitness, while the puzzle was about the correlation between being actually bad and being detrimental to fitness. Some of the folks I am imagining will go on to say that there is no such thing as badness, only beliefs about badness, and others will go relativistic and say that to be bad is to be believed to be bad. The problems with these options are obvious and well-known.

The sensible naturalist had better be a realist about the good and the bad. And then the correlation between badness and lack of fitness is, indeed, puzzling.

How surprising is evil?

According to the argument from evil:

1. The evils of this world are much more surprising given theism than given atheism.

But if (1) were true, then we would expect:

2. Theists tend to be much more surprised by evil than atheists.

However, I do not think (2) is in fact observed, and this provides evidence against (1).

Objection 1: Theists are irrational, and irrational people may not be surprised by the objectively surprising.

Response: This proposed explanation of the non-occurrence of (2) would itself lead to a further prediction:

3. The more rational a theist, the more likely she is to be surprised by evil.

But (3) is definitely not observed. In fact, the contrary is probably the case.

Objection 2: This is a version of the problem of old evidence. In old evidence cases, one is not surprised by the evidence as one already knew it.

Response: Still, if (1) is true, we would at least expect:

4. Theists, and if not in general then at least the more rational ones, are significantly more surprised than atheists to learn of new and particularly heinous evils.

But I do not think this is actually observed.

None of this is a conclusive refutation of (1). But it does decrease the likelihood of (1).

Some naive thoughts on syntax

I am neither a linguist nor a philosopher of language, so what I will say is naive and may be completely silly.

It seems to be common to divide up the task of analyzing language between syntax and semantics. Syntax determines how to classify linguistic strings into categories such as "sentence", "well-formed formula", "predicate", "name", etc. If the division is merely pragmatic, that's fine. But if something philosophical is supposed to ride on the division, we should be cautious. Concepts like "sentence" and "predicate" are ones that we need semantic vocabulary to explain—a sentence is the sort of thing that could be true or false, or maybe the sort of thing that is supposed to express a proposition. A predicate is the sort of thing that can be applied to one or more referring expressions.

If one wants syntax to be purely formal, we should see it as classifying permissible utterances into a bunch of formal categories. As pure syntactitians, we should not presuppose any particular set of categories into which the strings are to be classified. If we are not to suppose any specific semantic concepts, the basic category should be, I think, that of a "permissible conversation" (it may well be that the concept of a "conversation" is itself semantic—but it will be the most general semantic concept). Then, as pure syntactitians, we study permissible conversations, trying to classify their components. We can model a permissible conversation as a string of characters tagged by speaker (we could model the tagging as colors—we put what is spoken by different people in different colors). Then as pure syntactitians, we study the natural rules for generating permissible conversations.

It may well be that in the case of a human language, the natural generating rules for speakers will make use of concepts such as "sentence" and "well-formed formula", but this should not be presupposed at the outset.

Here is an interesting question: Do we have good reason to suppose that if we restricted syntax to something to be discovered by this methodology, the categories we would come up with would be at all the familiar linguistic categories? I think we are not in a position to know the answer to this. The categories that we in fact have were not discovered by this methodology. They were discovered by a mix of this methodology and semantic considerations. And that seems the better way to go to generate relevant syntactic categories than the road of pure syntax. But the road that we in fact took does not allow for a neat division of labor between syntax and semantics, since many of our syntactic categories are also natural semantic ones, and their semantic naturalness that goes into making them linguistic relevant.

Non-semantic definitions of truth

Here is a good reason to think that Tarski-style attempts at a definition of truth that do not make use of semantic concepts are going to fail. Such attempts are likely to make use of concepts like predicate and name. But these concepts are semantic concepts. A predicate is something can be applied to a name, and a name is something to which a predicate can be applied, and application is a semantic concept. Moreover, the definition of truth is going to have to presuppose an identification of the application function for the language (which takes a predicate and one or more names or free variables, and generates well formed formula, say by taking the predicate, appending a parenthesis, then appending a comma-delimited list of the names/variables, and then a parenthesis). But there is a multitude of functions from linguistic entities to linguistic entities, and to say which of them is application will be to make a semantic statement about the language.

Vices

The virtues support each other in two ways: (i) having one helps gain another; (ii) each helps to achieve the ends of the others. In regard to (ii), note that it is easier to achieve the goals of prudence if one is chaste, sober and eats in moderation, to achieve the goals of generosity if one is prudent and brave, to achieve the self-knowledge that humility aims at if one is wise and sober, and so on. This is partly distinct from (i).

The vices, on the other hand, support each other in sense (i), but hamper each other in sense (ii). Thus, laziness may lead to gluttony (having nothing better to do, one may just eat) and lust may lead to greed (in order to impress potential sexual partners): having a vice helps one gain another. But, in fact, the goals of the vices hamper one another. Lust is expensive, and hence hampers the goals of greed. Wrath makes it harder to make money and keep sexual partners. All the vices, including vanity itself, hamper the goals of vanity by making one appear ridiculous. Conversely, sloth and cowardice hamper the goals of all the other vices.

So, while type (i) support among the virtues is a delightful thing, because the virtues also help to achieve one another's goals, type (i) support among the vices is a baneful thing, because the vices hamper the achievement of one another's goals, but nonetheless the vices lead to one another.

This is a fine, and very broadly both Kantian and Aristotelian, answer to the question of why be virtuous.

S4

Here is an argument for S4. We want metaphysical necessity to be the strongest kind of necessity without arbitrary restrictions. If one responds that conceptual or strictly logical necessity are stronger, the answer is that they are, nonetheless, arbitrarily restricted, being dependent on a particular set of rules of inference and axioms. (The only non-arbitrary way to specify which which axioms are permitted is to say that it is all the fundamental metaphysically necessary propositions that are axioms, and then we presumably get metaphysical necessity.) Now, if L is a necessity operator, then LL is also a necessity operator. If LL is not equivalent to L, then LL is a stronger necessity operator. If LL counts as arbitrarily restricted, then we have reason to think that so does L, since L is even more restricted than LL, and it seems arbitrary to work with L instead of LL or LLL. And if LL doesn't count as arbitrarily restricted, then L is not the strongest non-arbitrarily restricted necessity operator. So if L is metaphysical necessity, L and LL are equivalent.

The dual of this argument is that metaphysical possibility is the most fundamental sort of possibility. But if M is metaphysical possibility, and MM is not equivalent to M, then MM will be a more fundamental possibility. So, if M is metaphysical possibility, M and MM are equivalent.

What's wrong with Tarski's definition of application?

Tarski's definition of truth depends on a portion which is, essentially, a disjunctive definition of application. As Field has noted in 1974, unless that definition of application is a naturalistically acceptable reduction, Tarski has failed in the project of reducing truth to something naturalistically acceptable. Field thinks the disjunctive definition of application is no good, but his argument that it is unacceptable is insufficient. I shall show why the definition is no good.

In the case of English (or, more precisely, the first order subset of English), the definition is basically this:

1. P applies to x₁, x₂, ... (in English) if and only if:
   • P = "loves" and x₁ loves x₂, or
   • P = "is tall" and x₁ is tall, or
   • P = "sits" and x₁ sits, or
   • ...

The iteration here is finite and goes through all the predicates of English.

Before we handle this definition, let's observe that this is a case of a schematic definition. In a schematic definition, we do not give every term in the definition, but we give a rule (perhaps implicitly by giving a few portions and writing "...") by which the whole definition can be generated.

Now consider another disjunctive definition that is generally thought to be flawed:

2. x is in pain if and only if:
   • x is human and x's C-fibers are firing, or
   • x is Martian and x's subfrontal oscillator has a low frequency, or
   • x is a plasmon and x's central plasma spindle is spinning axially, or
   • ...

Why is this flawed? There is a simple answer. The rule to generate the additional disjuncts is this: iterate through all the natural kinds K of pain-conscious beings and write down the disjunct "x is a K and F_K(x)" where F_K(x) is what realizes pain in Ks. But this definition schema is viciously circular, even though the infinite definition it generates is not circular. If all the disjuncts were written out in (2), the result would be a naturalistically acceptable statement, with no circularity. However, the rule for generating the full statement—the rule defining the "..." in (2)—itself makes two uses of the concept of pain (once when restricting the Ks to pain-conscious beings and the other when talking of what realizes pain in Ks). Thus, giving the incomplete (2) does not give one understanding of pain, since to understand (2) one must already know what the nature of pain is. (The same diagnosis can be made in the case of Field's nice example of valences. To understand which disjuncts to write down in the definition in any given world with its chemistry, one must have the concept of a valence.)

Now, the Tarskian definition of application has the same flaw, albeit this flaw does not show up in the special cases of English and First Order Logic (FOL). The flaw is this: How are we to fill in the "..." in (1)? In the case of English we give this rule. We iterate through all the predicates of English. For each unary predicate Q, the disjunct is obtained by first writing down "P =", then writing down a quotation mark, then writing down Q, then writing down a quotation mark, then writing down a space followed by "and x₁" flanked by spaces, then writing down Q. Then we iterate through all the binary predicates expressible by transitive verbs, and write down ... (I won't bother giving the rule—the "love" line gives the example). We continue through all the other ways of expressing n-ary predicates in English, of which there is a myriad.

Fine, but this is specific to the rules of English grammar, such the subject-verb-object (SVO) order in the transitive verb case. If we are to have an understanding of what truth and application mean in general, we need a way of generating the disjuncts that is not specific to the particular grammatical constructions of English (or FOL). There are infinitely many ways that a language could express, say, binary predication. The general rule for binary predication will be something like this: Iterate through all the binary predicates Q of the language, and write down (or, more generally, express) the conjunction of two conjuncts. The first conjunct says that P is equal to the predicate Q, and the second conjunct applies Q to x₁ and x₂. We have to put this in such generality, because we do not in general know how the application of Q to x₁ and x₂ is to be expressed. But now we've hit a circularity: we need the concept of a sentence that "applies" a predicate to two names. This is a syntactic sense of "applies" but if we attempt to define this in a language independent way, all we'll be able to say is: a sentence that says that the predicate applies to the objects denoted by the names, and here we use the semantic "applies" that we are trying to define.

It's worth, to get clear on the problem, to imagine the whole range of ways that a predicate could be applied to terms in different languages, and the different ways that a predicated could be encapsulated in a quoted expression. This, for instance, of a language where a subject is indicated by the pattern with which one dances, a unary predicated applied to that subject is indicated by the speed with which one dances (the beings who do this can gauge speeds very finely) and a quote-marked form of the predicate is indicated by lifting the left anterior antenna at a speed proportion to the speed with which that predicate is danced. In general, we will have a predicate-quote functor from predicates to nominal phrases and an application functor from (n+1)-tuples consisting of a predicate plus n nominal phrases to sentences. Thus, the Tarskian definition will require us to distinguish the application functor for the language in order to form a definition of truth for that language. But surely one cannot understand what an application functor is unless one understands application, since the application functor is the one that produces sentences that say that a given predicate applies to the denotations of given nominal phrases.

A not unrelated problem also appears in the fact that a Tarskian definition of the language presupposes an identification of the functors corresponding to truth-functional operations like "and", "or" and "not". But it is not clear that one can explain what it is for a functor in a language to be, say, a negation functor without somewhere saying that the functor maps a sentence into one of opposite truth value. And if one must say that, then the definition of truth is circular. (This point is in part at least not original.)

The Tarskian definition of truth can be described in English for FOL and for English. But to understand how this is to be done for a general language requires that one already have the concept of application (and maybe denotation—that's slightly less obvious), and we cannot know how to fill out the disjuncts in the disjunctive definition, in general, without having that concept.

Perhaps Tarski, though, could define things in general by means of translation into FOL. Thus, a sentence s is true in language L if and only if Translation(s,L,L*) is true in L*, where L* is a dialect of FOL suitable for dealing with translations of sentences of L (thus, its predicates and names are the predicates and names take from L, but its grammar is not that of L but of FOL). However, I suspect that the concept of translation will make use of the concept of application. For instance, part of the concept of a translation will be that a sentence of L that applies a predicate P to x will have to be translated into the sentence P(x). (We might, alternately, try to define translation in terms of propositions: s* translates s iff they express the same proposition. But if we do that, then when we stipulate the dialect L* of FOL, we'll have to explain which strings express which propositions, and in particular we'll have to say that P(x) expresses the proposition that P applies to x, or something like that.) The bump in the carpet moves but does not disappear.

None of this negates the value of Tarski's definition of truth as a reduction of truth to such concepts as application, denotation, negation (considered as a functor from sentences to sentences), conjunction (likewise), disjunction, universal quantification and existential quantification.

Parthood

I've run this argument before. But let's do it again, maybe more clearly. If some things can have non-mereological parts, the following scenario is possible: an entity has m parts to begin with, and then it loses k and is left with n=m−k parts. It would be really weird if this couldn't happen in the case where n=1, but could happen in the case where n=2, say. So, plausibly, this can happen in the case where n=1. Suppose Fred, thus, loses all but one of his parts. The remaining part is not identical with Fred—if it were identical with Fred, then prior to the loss of the other parts, Fred would have been identical with a proper part of himself. So at the end, Fred has one part. But the following two claims seem plausible, too:

1. x is a proper part of y if x is a part of y and x is not identical with y
2. if x is a proper part of y, then y has at least one other proper part than x.

And the case contradicts the conjunction of (1) and (2).

I take it that the advocate of non-mereological parts will have to deny (2). This introduces a new class of quasi-simples. A quasi-simple is an entity that has at most one proper part. Like a simple, it is not possible to subdivide a quasi-simple any further. But unlike a simple, a quasi-simple is allowed to have a proper part. This is weird indeed.

It is a puzzling question when two or more simples compose a whole. But once one allows quasi-simples, we get the further puzzling question when a simple composes a quasi-simple.

Is anything worth doing?

"Is anything worth doing?" (in a broad sense of "doing") is a question which, if it is worth thinking about, needs to be answered in the positive. But it is clear that the question is worth thinking about. Hence, the answer to it is positive.

I am confident that it can be established that if something is worth doing (or even if the words "is worth doing" express a concept), then naturalism is false. Thus, naturalism is not worth believing in. For if it is false, it is not worth believing in, and if it is true, nothing is worth doing and hence in particular nothing is worth believing in. (I love these sorts of arguments!)

How to establish that if something is worth doing, then naturalism is false? One approach is this. If something is worth doing, then "is worth doing" expresses a property. The only plausible fully naturalistic accounts of the expressiveness of our language are going to have a heavy dollop of causation (or at least explanation) in them. But if naturalism is true, then something's being worth doing (perhaps as opposed to that something is believed to be worth doing) does not enter into causal relations, and is explanatorily inert (unlike perhaps the truthmakers of mathematical truths, which do not enter into causal relations on naturalistic views, but may be explanatorily potent). Of course, this argument sketch has giant holes. But my intuition is that the holes can be filled in.

Frequentism

According to frequentism, the probability of an event E happening is equal to the limit of N_E(n)/n as n goes to infinity, where N(n) is the number of times that E-type outcome occurs in the first n independent trials. (If there are only finitely many trials in the history of the universe, we've got a serious problem, since then we get the conclusion—surely inconsistent with current physics—that all probabilities are rational numbers. I am guessing that in that case we need to make a counterfactual move—if we were to go to infinity, what limit would we get?)

But now here is a puzzle for the frequentist: Why is it that N(n)/n in fact has a limit at all? The non-frequentist has an answer—the Law of Large Numbers implies that, with probability one, N(n)/n converges to the probability of E, if E has a probability. But it would be circular for the frequentist to offer this explanation.

From the Grim Reaper paradox to the Kalaam argument

A Grim Reaper (GR) timed to go off at t₀ is an entity which does the following at exactly t₀. If Fred is not alive at t₀, the GR does nothing at t₀. If Fred is alive at t₀, the GR instantaneously annihilates Fred. (If instantaneous action is not logically possible, one can complicate the situation by allowing shorter and shorter time intervals for these actions.) The GR Paradox then is this scenario. Fred is alive at 11:00 am today, and that he does not die today unless killed by a GR and he does not get resurrected today. There are infinitely many GRs, timed to go off in a staggered way at the respectively times t₁,t₂,... where t_n is equal to 11:00 am + 1/n minutes. Well, by 11:02 am, Fred is certainly dead, since it is impossible that he survive a time at which a GR is timed to go off. But when was he killed? He wasn't killed by the 11:00 am + 1 minute GR, because if he were alive just before 11:01 am, then he would have been alive at 11:00 am + 1/2 minute, when another GR went off, and he can't survive a GR going off. It seems that none of the GRs could have killed him, because before each, there was another. So we have a contradiction: he both was and was not killed. Somebody has suggested that Fred is killed by the mereological sum of all the GRs, but that's mistaken in the present setting because the GRs check if Fred is already dead before they do anything, so in the present setting, none of them actually do anything—and if they don't do anything, how can they kill Fred?

The Kalaam argument needs the premise that there couldn't be a backwards infinite sequence of events. Here is an argument for this:

1. If there could be a backwards infinite sequence of events, Hilbert's Hotel would be possible.
2. If Hilbert's Hotel were possible, the GR Paradox could happen.
3. The GR Paradox cannot happen.
4. Therefore, there cannot be a backwards infinite sequence of events.

Actually, one could make steps 1 and 2 into a single step, but this is more fun, and, if it works, establishes the interesting corollary that Hilbert's Hotel couldn't exist.

Argument for (1): If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed. An infinite number of hotel rooms would then be the result.

Argument for (2): If Hilbert's Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes. And that would result in the GR Paradox.

The argument for (3) was already given at the beginning of the paper.

For about two years, I've smelled this argument coming, but I think my vanity has kept me from seeing it. I still have to confess that I have a really hard time accepting the corollary that Hilbert's Hotel couldn't exist—that corollary seems extremely counterintuitive to me. I wish I had some good way out.

On the other hand, establishing a major premise of an argument for the existence of God is a very happy outcome.

The surprising effectiveness of non-rigorous mathematics in physics

It is well-known how surprisingly effective mathematics is in science. But it is perhaps even more surprising, I think, how effective non-rigorous mathematics is. Physicists by and large do not do mathematics with the rigor with which mathematicians do it (not that mathematicians are that rigorous—basically, I think of the "proofs" published by mathematicians as informal arguments for the existence of a proof in the logician's sense). But, amazingly enough, it works. Neither Newton's nor Leibniz's calculus was rigorous. Yet physics based on calculus did just fine before the 19th century when calculus was made rigorous. Physicists often make approximations—for instance, taking the first term or two in some expansion—without proving any bounds on the approximation, but tend to get it right. Likewise, it is, I suspect, not uncommon for a scientist to write down a set of partial differential equations governing some system, and then say things like "Solutions must be like this..." without ever proving that the equations in fact have a solution. (It won't do, logically speaking, to say: "It must have a solution since it describes a physical system." For in practice none of the equations describe physical systems—they describe approximations to physical systems.)

One might think that a mathematical proof that is not logically valid is like tracing your ancestry to Charlemagne with only two gaps. But it's not like that at all in the sorts of mathematical arguments physicists use. They do tend to get it right, despite not doing things rigorously.

Yet another argument against naturalism

The following argument is valid:

1. (Premise) Every reasonable desire can be fulfilled.
2. (Premise) The desire for moral perfection is reasonable.
3. (Premise) Moral perfection requires being such that one is morally responsible and yet cannot do wrong.
4. (Premise) If naturalism is true, a state that entails moral responsibility and an inability to do wrong is not attainable.
5. Therefore, naturalism is false.

The argument being valid, the question is whether it is sound. I think (1) is plausible if we take "reasonable" in a strong enough sense. It is easy to argue for (4), since our best theories involve such a degree of indeterminism that, if they are complete descriptions of human beings, the possibility of doing wrong will always be there. That leaves (2) and (3). There is an argument from authority for (2): Kant thought so (and made an argument somewhat similar to this one). It does seem that a part of the moral life is the pursuit of moral perfection, and the moral life is reasonable in a strong sense.

That leaves (3). Let's consider two alternate views of moral perfection.

"Moral perfection only requires that one be morally responsible and never any longer actually do wrong." This is too weak, surely. It would mean that anybody whose existence ends with a morally responsible choice to do something right achieves moral perfection just prior to that choice.

"Moral perfection requires having all the virtues to a complete degree. Having the virtues to a complete degree is incompatible with self-initiated wrongdoing, but is not incompatible with losing the virtues or being forced through neurological manipulation into wrongdoing." This view is plausible, but I think the argument can still be run on this view, albeit with some complications. The challenge is whether an analogue of (4) is still true. I think it is. The morally perfect person is not blind to temptation—i.e., she is aware of the goods that temptation offers. (Courage is not achieved by not noticing danger.) But if she is aware of these goods and naturalism holds, then it is surely possible for her to choose these goods, where the choice is constituted by an indeterministic event in the brain, even if she has brain structures that are virtuously pointed the right way. And such a choice would be, surely, a morally responsible one, being a choice of a (lesser) good that comes from one's appreciation of that good. (If it be said that only deterministic choices are morally responsible, then moral responsibility is not available given naturalism in our indeterministic universe, and, again, moral perfection is unattainable.)

Could materialism be true in heaven?

This post is inspired by Mike Almeida's argument against materialism.

1. In heaven, it is impossible to sin.
2. Necessarily, if our minds supervene on physical brains like ours and these brains are not potentially interfered with, it is possible for us to sin, because of non-deterministic processes in physical brains like ours.
3. Therefore, in heaven, our minds either do not supervene on physical brains like ours or else these brains are potentially interfered with.

So the Christian materialist has to say that something changes in heaven. Either we get different kinds of brains from the ones we now have (either through matter being moved about or through the laws of the functioning of that matter being changed—which I think also counts as a change of the kind of brains), or else materialism ceases to be false, or else our indestructable righteousness in heaven depends on potential interference with our brain's functioning. This isn't a knock-down argument that materialism can't be true in heaven, but it should give the materialist pause.

Strangeness

Sometimes I am struck with how "strange" the Christian faith is—it just seems a bit incredible. But this reflection, I think, helps: we have very good reason to think that the correct physics and cosmology is going to be very strange, too. (Even if, and maybe especially if, it turns out to be quite simple and elegant.) What is prior in the order of knowledge is posterior in the order of being, the Aristotelians tell us, and so we would expect the ultimate explanations of reality to be removed from ordinary experience. I always find amusing the story of how St John Chrysostom had to preach against Arian heretics who used arguments like "If God is a Trinity, then God's essence is incomprehensible; but God's essence is comprehensible; hence God is not a Trinity." St John was preaching against the second premise.