Syntactic self-reference without diagonal lemma or Gödel numbers

For the proof of Goedel's incompleteness theorem and in work on the Liar Paradox it is usual to use the Diagonal Lemma to secure self-reference. The challenge of self-reference is this. Given a predicate Q, find a syntactically definable predicate P such that

1. (s)(P(s) → R(s))

is provably the one and only sequence of symbols satisfying P. Then (1) says that Q holds of itself. (To get the (Strengthened) Liar Paradox, just make R(s) say that s is not true.) But the proof of the diagonal lemma is hard to understand.

I find the following way of securing self-reference easier to understand. Start with a language that has nestable quotation marks, which I'll represent with ‘...’, and some string manipulation tools. I'll use straight double quotation marks for meta-language quotation. Add to the language a new symbol "@" which is ungrammatical (i.e., no well-formed formula may contain it). For any sequence of symbols s, we define two new sequences of symbols N(s) and Q(s) by the following rules. If s contains no quoted expressions or contains imbalanced opening and closing quotation marks, N(s) and Q(s) are just "@". If s contains a quoted expression, Q(s) is the first quoted expression, without its outermost quotation marks (but with any nested quotations being included), and N(s) is the result of taking s and replacing that first quoted occurrence of Q(s), as well as its surrounding single quotation marks, with "@". Thus:

2. Q("abc‘def‘ghi’’+jkl")="def‘ghi’"
3. N("abc‘def‘ghi’’+jkl")="abc@+jkl".

It is easy to see that Q and N are syntactically defined. Now, let M(s) be equal to N(s) if N(s)=Q(s) and let M(s) be an empty sequence "" otherwise. Again, M(s) is syntactic. Now consider this sentence:

4. (s)(‘(s)(@=M(s) → R(s))’=M(s) → R(s)).

It is easy to prove (given a bit of string manipulation resources) that the only sequence s that satisfies the antecedent of the conditional is (4) itself. So we have constructed the syntactic predicate P(s). It is: ‘(s)(@=M(s) → R(s))’=M(s).

One can also adapt this to work with Goedel numbers and hence presumably for use in proving incompleteness.

[Removed a nasty typo.]

Desiderata for a theory of atonement

My previous post on atonement implicitly identified one constraint ("must") and one desideratum ("should") for a theory of atonement:

1. The theory must be able to apply in cases where the person saved lacks personal sin.
2. The theory should not require explicit beliefs on the part of the person saved.

There is another desideratum that I think is important but somewhat vague:

3. At least one of the facts that Jesus Christ actually lived among us, died on the cross and rose again should in every case be central to the mechanism of salvation.

This condition rules out theories on which the mechanism of atonement is that we are transformed by the example of Jesus Christ (this will be a subset of what my previous post calls "epistemic theories"). For in those theories, the central part of the mechanism of atonement is not that Jesus Christ actually lived, died and rose again, but that we believe that Jesus Christ actually lived, died and rose again. The reason Jesus Christ had to actually live, die and rise again is not for the mechanism of salvation to work, but only because God is not a deceiver and so God could not teach us that Jesus Christ lived, died and rose again unless this was actually true. But the soteriologically important thing on such theories is the belief that this happened, not that this happened. And hence such theories are unsatisfactory.

Leviathan

I was once rather amused by an undergraduate in my Philosophy of Love and Sex class who complained that the sexual ethics material we were reading only applied to humans. On that topic, I rather enjoyed this story.

Mints, cats, double effect and proportionality

It's noon. You and two other innocents, A and B, are imprisoned by a dictator in separate blast-proof cells. All the innocents are strangers, and you know of no morally relevant differences between them (whether absolutely or relative to you). A's and B's cells both contain bomb and timer apparatuses that A and B cannot do anything about. B's bomb timer is turned off. A's timer is set to blow her up at 1:00 pm. In your cell, there is a yummy mint on a weight-sensitive switch connected to the apparatus in B's cell. If the mint is removed, B's timer will be set to go off at 1:00 pm. The dictator will check up on the situation shortly before 1:00 pm, and will turn off A's timer if you've done something that caused B's timer to turn on. Anybody who survives past 1:00 pm will then be released.[note 1]

So you reason to yourself. "I like mints. If I eat the mint, I will cause B's death, but A will be saved. My causing of B's death will be non-intentional, and on balance the consequences to human life are neutral. But I get a mint out of it. So the Principle of Double Effect should permit me to eat the mint."

If this reasoning is good, the Principle of Double Effect is close to useless. Strict deontologists think it's wrong to kill one innocent to save millions. Most think it's wrong to kill one innocent to save two. But just about every deontologist will say that it's wrong to kill one innocent to save one innocent and one cat. Now, consider this case. The dictator hands you a gun, and tells you that if you don't kill innocent B, she'll kill innocent A and a cat. You clearly shouldn't. But if you thought it was acceptable to take the mint, then you could reason thus: "It would be interesting to see what a bullet hole in a shirt pocket looks like (and the shirt doesn't belong to B—it is prison attire, belonging to the dictator). If I aim the gun at B's shirt pocket, and press the trigger, the bullet will make a hole in the shirt pocket. And as an non-intended side-effect, it will subsequently cause B's death. But that's fine, because on balance the consequences to human life are neutral, as then B will be saved—plus a cat!" And since you can always think up some minor good that is served by pulling a trigger (finger exercise, practice aiming, etc.), you will get results any deontologist should reject.

So something is wrong with the reasoning—or Double Effect is wrong. I do not think, however, that Double Effect is wrong—I think it's indispensible. So what I will say is this. Double Effect requires that the evil effect not be intended and that there be a proportionality between the side-effect and the intended effect. What the above cases show is that, as a number of authors have noted, proportionality is not a matter of utilitarian calculation. Not only should we have on-balance positive consequences, but the intended effect should be a good proportionate to the foreseen evil. And the foreseen evil is not "that one person fewer will be alive than otherwise", but the foreseen evil is that a particular person should die. The deaths of different people are incommensurable evils even when we know no morally significant differences between the people.

In some cases the virtuous agent may count the numbers of people. But not in these cases. It is callous and unloving to get a mint or produce a bullet hole at the cost of B's death. It trivializes the value of B's life. There is a dilemma here. Either one is acting in the way that causes B's death for the sake of saving A, or not. If one is not, then B literally died so that one might have a mint or be intellectually gratified by the sight of a bullet hole. And so one trivializes B's life. If one is acting to save A, then one is not trivializing B's life. But in that case one is intending B's death, and deontology forbids that.

Here is a variant analysis that comes to the same thing, perhaps. There are cases where one can only do something in one of two ways: by intending a basic evil or by having a morally vicious set of intentions. The cases I gave are like that: one can only take the mint or produce the bullet hole by intending B's death or by having a set of intentions that trivialize B's life. In either case, one is unloving to B. It's hard to say which is the worse.

(This is related to the looping trolley case. There, I think one is either intending the absorption of kinetic energy by the one person, which is problematic, or one is intending a slight increase in length of life or slightly increase in probability of survival on the part of the five, which trivializes the death of the one.)

Epistemic theories of the atonement

Every orthodox Christian agrees that:

1. Salvation occurs at least in part because of Christ's death on the cross.

The "at least in part" is because Christ's earlier life and subsequent resurrection no doubt play a role. It is also uncontroversial that this has something to do with atonement and sin, but there are many theories here. Epistemic theories say:

2. The explanatory connection between Christ's death on the cross and the salvation of an individual always involves the individual's epistemic encounter with Christ's crucifixion.

For instance, it may be that Christ's death expresses to the sinner the weight of the sinner's sin and seeing the free acceptance of the penalty transforms the sinner. Epistemic theories as I defined them need not hold that the epistemic encounter is the whole story. Someone could, for instance, hold that there are two essential components to atonement, one of them an epistemic component and the other a penal substitution component. Such a theorist would count as an epistemic theorist.

But there is a plausible argument against this:

3. Nobody is saved except because of Christ's death on the cross.
4. Some are saved who have no epistemic encounter with Christ's crucifixion.
5. Hence, the explanatory connection between Christ's death on the cross and salvation does not always involve an epistemic encounter with Christ's crucifixion.

And so, it seems, epistemic theories of atonement are false.

I think (3) is a central part of Christian orthodoxy, assuming that by "nobody" we mean no human beings other than Christ (contextually restricted quantifiers!). One way to see this is to consider the debate over Mary's Immaculate Conception. The doctrine says that Mary was conceived without original sin. Probably the deepest theological objection to the doctrine has centered on arguments that the doctrine is incompatible with (3). If rejecting (3) were an option for a Christian, the defenders of the doctrine would have had ample motivation to reject (3). But they didn't—instead, they offered theories that attempted to reconcile (3) with the Immaculate Conception. It is not my point to evaluate the arguments for or against the Immaculate Conception (though of course I do accept the Immaculate Conception) but simply to note that both sides admitted that (3) is non-negotiable.

Now, it may seem that (4) directly contradicts the epistemic view (2), and hence begs the question. That's not quite right. Claim (2) is that whenever there is an explanatory connection between Christ's sacrifice and salvation, that connection is at least in part epistemically mediated. As far as that goes, this is compatible with the possibility, denied by (3), that some are saved without any such explanatory connection.

Why accept (4)? Because of the following three classes of persons:

• Jews and gentiles who were saved prior to the time of Christ.
• Those who are saved without ever hearing about Christ's death.
• Those (e.g., at least baptized infants) who are saved despite dying prior to having developed an ability to have an epistemic encounter with Christ's crucifixion.

In each of these types of cases, it certainly seems that we have (4).

I want to consider now one kind of reply. We could modify (2) by restricting the quantifiers. For instance, we could apply (2) only to those who have achieved the age of reason and positing that all who die prior to the age of reason are saved, thereby ruling out the third class of cases as offering an argument for (4). This would be an unacceptable variant of Pelagianism. The person who died in infancy would be saved not by Christ, but by natural causes—namely, the causes of death. If some who die in infancy are saved—and certainly at least those baptized people who die in infancy are saved—even they had better be saved only by Christ.

Or we could, if we were willing to bite the bullet on the case of infants in some way, restrict the quantifiers in (2) not to apply to those who died prior to Christ's death, thereby ruling out the first class of examples as offering an argument for (4). I think this, too, is a kind of Pelagianism. Moreover, consider the weirdness of supposing that an Inuit who died at 2:59 pm on Good Friday could be saved not by the cross, while an Inuit who died two minutes later needed to be saved by the cross.

Another move one might make would be to deny that, at least since the time of Christ's death, anyone is saved without ever hearing the Gospel. This is a hard-line response to my argument. For sociological reasons, I suspect this response to my argument is not going to be that popular. I suspect that most of the people who take a hard-line on those who die without hearing about Christ's death take some substitutionary sacrifice theory of the atonement. This is not because there is a good logical connection between these two views—indeed, substitutionary sacrifice theories of the atonement appear to me to be our best bet for explaining how one can be saved without expressly hearing the Gospel—but simply because the kind of tough-mindedness that inclines one to a hard-line on salvation outside the apparent boundaries of the Church is apt to incline one to a substitutionary sacrifice theory.

A different response is that a transformative epistemic encounter with the crucifixion occurs after death for those who are saved despite having died without hearing about Christ's death. Such a view would not only be committed to post-death purgation—i.e., to purgatory. That is not a problem. But it would, further, require the thesis that baptized infants who die prior to hearing about Christ's sacrifice go to purgatory, if only for an instant, and that view simply seems wrong. For one, it downplays the effects of baptism.

One might, however, suppose a miraculous epistemic encounter prior to death. God can miraculously make it possible for an infant, or even embryo, to understand the central doctrines of Christianity, whether explicitly or more vaguely. That this view posits a miracle is no objection. Salvation always involves a miracle. I do not know how plausible this way out will be for particular epistemic theorists. But I think in the end this is the only satisfactory account available to them.

So, unless one wants to posit a miraculous raising of intellectual abilities—and I do not reject this option—epistemic theories of atonement should be rejected.

But I don't think the substitutionary sacrifice theorist is off the hook either. For the above argument gives us a necessary condition for a theory of atonement: it must explain the connection between Christ's sacrifice and the salvation of an infant. If the theory is that Christ is paying the penalty for the individual's sin, then that theory will not be sufficient to account for the salvation of infants who have never committed any sins.

There are two separate issues here, I think. One is the issue of overcoming personal sin. That issue does not come up for the infant, as far as we know (I am inclined to some epistemic caution on this point). The other is the issue of attaining salvation. Many Catholic theologians have said that lack of personal sin is insufficient for salvation. A supernatural love is necessary and sufficient for salvation, a love that can only come from grace. Atonement is not only atonement for sin. It is, as its corny but apparently genuine "at-one-ment" etymology indicates, a matter of uniting us with God. While sin keeps us from union with God, union with God is not constituted by the absence of sin. It requires something more than absence of sin. And for fallen humanity, even in the case of non-sinful members such as infants, this "something" more must be held to come from the Cross. A puzzle or maybe even mystery, then, is how it is that the "something else", the supernatural agapê, comes from Christ's sacrifice. I am inclined to think that a crucial component here is that by our membership in the Body of Christ, Christ's sacrifice is our sacrifice, and the agapê of his sacrifice is our agapê.

Omnirationality

An agent is omnirational provided that

1. whenever he makes a decision, he is impressed by all the unexcluded reasons that there in fact are for him for all the relevant options, being impressed by a reason exactly to the extent to which he has reason to be impressed by it in virtue of the reason's force and the force of relevant higher order reasons
2. when he decides to do A, he does A for all the unexcluded reasons that he in fact has for doing A.

In the case of an omnirational being who has multiple potential unexcluded reasons for an action, there is no difficulty to the question which reasons he actually acted on—in fact, he acted on them all.

If God is simple, he is omnirational. And God is simple.

Non-natural facts explain

Consider this thesis:

1. Only natural facts explain contingent truths.

(So if there are non-natural facts, they are explanatorily epiphenomenal.) I will argue that (1) is false.

For:

2. Only facts knowable by the scientific method are natural facts.
3. The non-existence of immaterial beings that do not interact with the physical world is not knowable by the scientific method.

Now, consider the coherent but false theory that there are two generations of highly intelligent supernatural mathematicians, the first of which are called the "Great Ones" and the second of which are called the "Daughters of the Great Ones", and that they do not interact with the physical world. Then:

4. It is a contingent fact that the Daughters of the Great Ones don't exist.
5. That the Daughters of the Great Ones don't exist is explained by the fact that the Great Ones don't exist.
6. That the Great Ones don't exist is not knowable by the scientific method.

And so:

7. That the Great Ones don't exist is a non-natural fact that explains the contingent fact that the Daughters of the Great Ones don't exist.
8. Therefore, at least one non-natural fact explains a contingent truth.

And this contradicts (1). So, (1) is false. (And if negative states of affairs can be causes, then maybe we can even say that the non-existence of the Great Ones causes the non-existence of their Daughters.)

Deviant logic

In the chapter on deviant logic in his philosophy of logic book, Quine makes the claim that:

1. The deviant logician changes the subject rather than disagreeing with classical logic.

Thus, the logician who denies excluded middle is not using the words "or" and "not" to indicate disjunction and negation. She is, perhaps with good reason (though Quine is sceptical of that), using some other connectives. Thus, when she denies "not not p entails p", she is not disagreeing with us when we assert "not not p entails p". The basic thought running behind this is that:

2. The rules of classical logic are grounded in the meanings of the logical connectives (using "connectives" very widely to include negation, quantifiers, etc.)

and so any departure from the rules is a change of subject.

There is a powerful kind of argument against deviant logic here. Claims (1) and (2) seem to tell us that it is not really possible to disagree with classical logic without self-contradiction. I am either using my words in the sense that they have in classical logic, in which case I had better not disagree with classical logic on pain of contradiction, or else I am using the words in a different sense and hence not disagreeing.

I now want to describe a class of apparently non-classical logics that do not change the subject. Thus, either a deviant logician doesn't always change the subject, or else these logics are not actually deviant. The idea is this. We have rules like:

• You can infer p from p.
• If r is a conjunction of p with q, then you can infer r from p and q, p from r and q from r (conjunction introduction and elimination).
• If r is a negation of a negation of p, then you can infer p from r.
• If you can infer p and a negation of p from r, and s is a negation of r, then you can infer s from r.

And so on. The interesting thing is the second rule does not tell us that p and q have a conjunction. And indeed that is how I am imagining the system deviating from classical logic. We simply disallow certain conjunctions, negations, etc.—there will be sentences that perhaps have no negation, and pairs of sentences that perhaps have no conjunction. If we represent the language along the lines of First Order Logic, there may be cases where "A" is a sentence and "B" is a sentence but "A and B" counts as malformed. The rules of disallowing combinations may take all sorts of forms. For instance, we might simply prohibit any sentences that contain a double negation. This would result in a severe intuitionist-type limitation on what can be proved.

The logic, thus, has standard classical rules in an important sense. The rules are correct whenever they can be applied—whenever there are output sentences that work. The subject is not changed—"or" means or, "and" means and, and "not" means not—but it can be a substantive claim whether for a pair of sentences A and B, there is a sentence that we might wish to denote "A or B".

This restriction does not count as a change of subject. Indeed, Quine himself notes that there can be languages which are incapable of translating all the English truthfunctionally and quantificationally connected sentences, and he seems to think that these languages do have connectives that mean the same thing as English ones. In fact, English itself has restrictions on the formation of sentences. Past several levels of embedding, there just is no way to make distinctions. You probably can't express "(A or (A and not (B or (B and (C or D) and E) or F) and not A))" in English. Yet English does not have a deviant logic. It's just that English's logic is likely incomplete.

There are two ways of looking at this. One way is to say that what I have offered is a family of genuinely deviant logics that don't change the subject, and hence that Quine's argument against deviant logics fails. The other way—and it is what I prefer—is to say that what I have given is in an important sense a family of non-deviant, and even classical, logics, but one that differs from First Order Logic.

I think it could be a good thing to define the connectives in terms of valid inference (perhaps understood in terms of entailment). For instance, one might say that:

3. A partially-defined functor C that takes a pair of sentences p and q into a new sentence C(p,q) is a conjunction if and only if you can validly infer p as well as q from C(p,q) and C(p,q) from the pair of premises p and q whenever C(p,q) is defined.

(We also need an extension to wffs.) If we do this, then excluded middle is true by definition in the following sense:

4. Whenever p is a disjunction of q with a negation of q, then p is true.

But no claim is made that every sentence has a negation or that every pair of sentences has a disjunction. That would be a substantive claim. But whenever a sentence has a negation and can be disjoined with that negation, the result of the latter disjunction is true. That is a claim that is true by definition of "negation" and "disjunction".

This also lets one stipulate into place new connectives like tonk. Tonk is a connective such that one can infer q from "p tonk q" and "p tonk q" from p. The problem with tonk is that once one has the connective, it seems one derive anything (e.g., 1+1=2, so 1+1=2 tonk q, so q, for any q). But not quite. One can only derive everything with tonk if one adds the additional thesis that sufficiently many pairs of sentences have tonks. For instance, if we grammatically restrict tonking so that one is only allowed to tonk a sentence with itself, we can continue to have a sound logic.

Why care about such logics? Well, they might be helpful with the Liar Paradox. They might provide a way of doing the sort of thing that Field does to resolve the Liar by invoking a deviant logic but within a logic that has all the classical rules of inference.

I think Sorensen's "The Metaphysics of Words" [PDF] is very relevant to the above.

An argument for the material conditional account of indicatives

The material conditional account of indicatives is that "If s, then u" is true if and only if s is false or u is true or both.

1. (Premise) If the indicative conditional has the same truth values as the material conditional in the standard cases which are alleged to be counterexamples to the material conditional account, then the material conditional account is correct.
2. (Premise) The indicative conditional has mind-independent truth value.
3. (Premise) If the indicative conditional has mind-independent truth value, then it has the same truth values as the material conditional in the standard cases which are alleged to be counterexamples to the material conditional account.
4. Therefore, the material conditional account is correct.

In this argument, I am convinced of premises (1) and (3), but not sure of premise (2). Consequently, what the argument convinces me of is that either (2) is false or (4) is true. Premise (1) is not that controversial, I think. The material conditional account is simple and elegant, verifies modus ponens and contraposition, is well-defined and mind-independent. The only problem is that it appears to give the wrong answers for certain standard cases. If this appearance were undercut, the material conditional account would be the winner.

The hard work is going to be justify (3). Let us start by giving three representative alleged counterexamples, classified by the truth values of the antecedent and consequent:

5. "If I will have dinner with the queen tonight, I will eat dinner tonight in my pajamas." (Antecedent and consequent are both false.)
6. "If I will have dinner with the queen tonight, everyone that I will have dinner with tonight will be a family member." (Antecedent is false and consequent is true.)
7. "If it is snowing in the United States, it is snowing in Central Texas." (Suppose this was uttered a couple of days ago when it was snowing in Central Texas. Antecedent and consequent were both true.)

The material conditional account says that all three conditionals are true. But all three conditionals sound wrong (assuming I am not a member of the royal family and that I wouldn't wish to insult the queen).

I will argue that:

8. If the indicative conditional has mind-independent truth value, then (5)-(7) are all true.

The method of argument generalizes to all the standard counterexamples, and thus yields (3).

Here's the way I will argue for (8). Let "a" be the antecedent in the alleged counterexample. Let "c" be the consequent. Suppose I have the belief, justified or not, that at least one of "not-a" and "c" is true, and I have no further, more specific beliefs about the matters in a and in c. Since I believe that at least one of "not-a" and "c" is true, I should be able to sincerely say to someone:

9. I may not know much about the queen, dinners, pajamas, snow, etc., but I do believe that at least one of "not-a" and "c" is true. Hence, if a, then c.

This seems very reasonable.

Suppose now that I learn all the relevant facts about the queen, dinners, pajamas, snow, etc. In particular, I learn such facts as that people tend not to wear pajamas for dinner with the queen, that central Texas is one of the somewhat less likely places in the US to have snow, etc. I also learn the truth values of "a" and "c". None of the things I learn gives me reason to retract the claim that at least one of "not-a" and "c" is true. And neither have I any reason to retract the conclusion I drew, that if a, then c.

Therefore, when I said (9), I said something true. If it wasn't true, I would have reason to withdraw it. But the difference between the circumstances in my story in which I said the conditional in (9) and standard circumstances was in my beliefs—when I said (9), I lacked various beliefs that normal people in our culture have. Thus, if the indicative conditional has mind-independent true value, I have to conclude that actually the conditional "if a, then c" is also true. And so we have an argument for (8).

Frankfurt, flickers and voting

A standard example in the literature of Frankfurt cases is where a guy is deciding whom to vote for. If he isn't going to freely vote for the candidate Dr. Black wants him to vote for, then Dr. Black will force him to vote for that candidate. But he is going to freely vote for the candidate, so Dr. Black doesn't intervene.

Here's a funny thing about this case. Dr. Black can't force his victim to vote for a particular candidate. At most Dr. Black can force his victim to check a box, press a lever, or the like. But checking a box or pressing a lever is not voting, because the validity of a vote requires that one not have been compelled. For the same reason, you can't run a Frankfurt case where the action is a making of a promise, an entry into a contract, a marriage, etc. (I don't know if you can force someone to make an assertion.) Many of our actions are of a sort that logically cannot be compelled.

A lesson from Frankfurt cases

Here is one lesson one might take away from Frankfurt cases: Causal necessitation is not the same thing as logical necessitation by past conditions conjoined with laws. If the libertarian's intuitions are driven by the idea that free choices can't be causally necessitated, then Frankfurt cases have no effect, because the genius of the cases is precisely to construct cases where there is logical necessitation by past conditions conjoined with laws but no causal necessitation.

Metaphysically Aristotelian quantification

There is a sense in Aristotelian metaphysics that "there are only substances". They are all there is a focal sense. Yet if we can talk about and quantify over accidents or modes, surely there are accidents or modes.

Here, then, is a simple quantified logic that preserves the Aristotelian intuition. This logic is developed only in the case of modes (or tropes) that are non-relational—that subsist in a single substance. The logic has the standard resources of first order sentential logic, together with the standard universal quantifier symbols ∀x and ∃x which quantify over substances x. But additionally there are two new quantifier symbols: ∀_ax and ∃_ax which quantify over a's modes x. Thus, "Some table has an accident" becomes:

1. ∃x(Table(x) and ∃_xy(Accident(y,x))).

Then we can say that only the substances exist simpliciter—only they are quantified over by the standard quantifiers Ax and Ex. Modes "exist" only relative to the substance of which they are modes—they are grounded in that substance, as is indicated in the language by the subscripted quantifiers.

We can say that the mode-quantifier ∃_ax yields existential quantification in an analogical sense. And we can spell out the analogy at least to some degree by giving rules of inference that are structurally analogous to those for the focal-sense quantifier ∃x.

Here's another application of the notion of relative existence. We might, for instance, hesitate to say that characters in novels really exist, but we might think (I am hesitant about that, too) that novels really exist. We might then think that for any novel N, there is a pair of quantifiers ∃_Nx and ∃_Nx over the entities-in-N. If S is some Star Trek novel, then when we say that ∃_Sx(Klingon(x)), we are not really saying that there really are Klingons. We are saying that virtually, in-the-novel, relative-to-the-novel there are Klingons. This is not a fact about Klingons but about the novel, and our primary ontological commitment is to the novel. Of course then our logic then needs to be suitably designed so that we cannot infer from ∃_Sx(Klingon(x)) that ∃x(Klingon(x)). This can all be done, and what I shall do below for modes can be done for characters in novels. Again, quantification over characters is quantification in an analogical sense.

The rest of this post is almost entirely technical and can be skipped.

We leave the truth-functional rules unchanged. We modify the quantificational rules as follows:

Universal elimination: From ∀xF(x) and Substance(a), you get to infer F(a). From ∀_axF(x) and Mode(d,a) you get to infer F(d).

Universal introduction: If you have a subproof assuming Substance(c) and concluding with F(c), and the subproof cites nothing involving c from outside of itself, then you get to infer ∀xF(x). If you have a subproof assuming Mode(c,a) and concluding with F(c), and the subproof cites nothing involving c from outside of itself, then you get to infer ∀_axF(x).

Existential elimination: If you have ∃xF(x) and a subproof from (F(c) and Substance(c)) to S, where the subproof cites nothing involving c from outside of itself and c does not appear in S, then you get to infer S. If you have ∃_axF(x) and a subproof from (F(c) and Mode(c,a)) to S, where the subproof cites nothing involving c from outside of itself and c does not appear in S, then you get to infer S.

Existential introduction: From Substance(a) and F(a), you get to infer ∃xF(x), and from Mode(c,a) and F(c), you get to infer ∃_axF(x).

And we add an additional equality introduction rule: If you have Mode(c,a) and Mode(c,b), then you get to infer a=b.

Models contain a substantial domain S and a function m that assigns to each member of S a set of objects, with the property m(x) and m(y) have no elements in common if x and y are distinct. We can define interpretations and satisfaction in a straightforward way, restricting the interpretations of the Substance and Mode predicates in such a way that I(Substance) is always equal to S and I(Mode) is the set of all pairs (x,y) such that x is in S and y is a member of m(x). (We don't put this rule in in the case of existence-in-a-novel.)

I haven't checked it, but I expect that we have soundness and completeness.

If, like Spinoza and unlike Aristotle, we want to allow for nested modes, this can be done, too.

"Negand"

Sometimes one wants a word like "un-negation" or "de-negation"—a word for the sentence "s" as it relates to "~s". For instance, when teaching logic, one wants to say that if one is asked to prove a negated sentence, one's best bet is often to prove a contradiction from that sentence's un-negation. I just found out that there is a handy word for this. It's "negand". Shiny! I never knew that. So one can say things like:

1. Believing a negative proposition is the same as disbelieving its negand.

(I actually don't know if that's true, but one can say it.  But I am inclined to think it is true.)  There are also times when one wants to refer to a proposition and "a negation or negand of it".  I just love that word.

Last time I needed a word for this in class, I talked of "s" as the "de-negation" of "not-s", but "negand" is much better.

Book indexing script

I had to index my modality book, so I wrote a little perl script to help me (it also needs the Roman module from CPAN) and it generated this index.  The idea is that one inserts special plain-text codes in my Microsoft Word file for the book which mark the ranges to index for each term and that mark where the page breaks in the galleys are (actually, Logan Gage, my TA, marked the page breaks), and then one runs the perl script which generates an html file with the index (which one can then import into Word if one so sees fit).

The main special codes are these:

• {{entry name:}} This is put at the beginning of a passage that will be indexed under "entry name"
• {{:entry name}} This is put at the end of the passage
• {{nickname>official name}}  This specifies that any entries flagged with the nickname get re-indexed under the official name.  For instance, to save myself typing, in the body of the text I would use codes like {{EMR:}}...{{:EMR}}, and then I'd put an entry that says {{EMR>Extreme Modal Realism}}
• {{synonym~entry name}}  This generates a "see entry name" entry in the index, under synonym.
• @@n@@  This marks the beginning of page n.

There are no special facilities for generating an "n" after a page number for a footnote--one just surrounds the footnote superscript marker with {{entry name:}}...{{:entry name}} and gets a reference to the page it's on.  This won't be good for endnotes that need to be indexed.  There is no facility for "see also".

I also used a Word macro so that I could highlight some text, and it would surround it with the {{entry name:}}...{{:entry name}} codes (getting the name of the entry from the clipboard).

If you want to use the script for something and need help, email me.

Double Effect conference online tomorrow

The Anscombe Centre is running what looks to be a really good conference on Double Effect [PDF] on Saturday, February 19th, 9:30-17:30 GMT. They'll be accepting questions by email (and a selection of the email questions will be asked by the chair). They ask that you send an email to s.barrie@bioethics.org.uk if you're planning on attending electronically. No registration fee for online attendance, but they do accept donations.
Here is a fuller schedule.
I'll be there virtually, albeit sleepily, starting from around the second talk (the first starts at 3:30 am my time). If anybody wants to chat with me during or between sessions, go here, and make up a nickname (ideally one such that I'll know who you are). (That's not an official conference venue.)