Know that we Jews have three kinds of books: the first is the Bible, and we all believe in this with perfect faith; the second is called the Talmud, and it is an explication of the commandments of Torah, for there are 613 commandments in the Torah, and every single one of them is explicated in the Talmud, and we believe in this explication of the commandments; and we have also a third book which is called Midrash, which means "Sermons". This is just as if a bishop were to stand up and make a sermon, and one of his hearers liked it so much that he wrote it down. And as for this book, the Midrash, if anyone wants to believe in it, well and good, but if someone does not believe in it, there is no harm....Moreover, we call the Midrash a book of "Aggadah", which means "razionamiento", that is to say, merely things that a man relates to his fellow.
This is from the Vikuah of Nahmanides, an account of a disputation in Barcelona between the great rabbi and Fray Paul before King James I of Aragon. Nahmanides, also known as Moses ben Nahman or Ramban (not to be confused with Rambam, who is Maimonides), was the last great figure in the waning years of what has been called the Spanish Golden Age for the Jews. In Spain, unlike elsewhere, the perpetual conflict between Christian and Muslim had forced both sides to make alliance with Jews. The result on both sides was a flowering of Jewish culture thatwas unrivalled until the Haskallah, or Jewish Enlightenment, and perhaps not even then. It was not a perfect time for Jews. While the Christian-Jewish alliance in Christian Spain (which, of course, was becoming a larger and larger part of Spain through much of the late Middle Ages) gave Jews more room to flourish on their own terms than ever before, it was nonetheless precarious; Spanish monarchs were perpetually under external and internal pressure to enforce the anti-Jewish laws that were still technically on the books. While Jews could and did do very well for themselves, there was always a sort of implicit qualification: they rose to power, wealth, and influence in the way foreigners with needed skills may rise to it.
As Maccoby notes in his discussion of the Barcelona Disputation (which took place in 1263), the fact that the disputation took place at all was a sign of just how far the Golden Age had waned. Nonetheless it is a remarkable tribute to Spain at the time that this Disputation, while not perfectly fair (it was, after all, Catholic Spain, and Nahmanides had to watch his step in a way the Catholics did not), was nonetheless very close to it, and we find in Nahmanides' Vikuah one of the greatest and most courageous Jewish minds of the Middle Ages meeting Christianity head on and not backing down. One of the features of the Disputation is that, while the playing field was Christian, the Jews had an advantage in the debate. At this period in time, the Dominicans were pioneering a new way of seeking Jewish conversion. Instead of preaching at them how stiffnecked they were (which at the time was a surprisingly common approach to persuading Jews to follow Christ), they made an attempt to speak to Jews in their own terms, appealing to the Talmud and to various books of Midrash to argue (abstracting entirely from the question of whether Jesus was the Messiah) that it was possible to establish a Jewish defense of the claims (1) that Messiah had come; (2) that Messiah had to die; and (3) that Messiah had to be divine. While politically the home court advantage always went to the Catholics, this new missionary tactic had the consequence that the terms of the debate favored the Jews, at least in principle. Jewish converts like the Fray Paul Christiani mentioned in the passage above used their pre-conversion study as a jumping-off point for disputing with the Jews.
Of course, the Jews themselves were not inclined to like this much more than any other sort of dispute. They wanted to be left alone; and, since they could not disobey the magistrates who summoned them to these debates, they generally sought as much as they could to evade any direct conflict head on. However, in this one case we have a Jew who felt he could speak more plainly arguing, in front of a monarch who, despite being definitely Catholic, had considerable sympathies with the Jews, many of who he depended on for running his kingdom, against Christian opponents who, while intelligent and learned in their own right, were no match for his genius. It's a fascinating result, particularly since we get Nahmanides's own perspective on the Disputation, and a very firm insistence on the claim that Judaism is a more rational religion than Christianity. We also get interesting cases in which Nahmanides struggles to convey to Christians some insight into aspects of Jewish life -- the primal importance of law, the secondary importance of doctrine, the concepts of midrash and haggadah, and so forth -- that were very foreign to Christian minds. To do so he has to simplify, of course (as he does in the above passage), but, being Nahmanides, he is still able to convey the basic point.
[Translation of passage from: Hyam Maccoby, ed & tr. Judaism on Trial: Jewish-Christian Disputations in the Middle Ages. Fairleigh Dickinson University Press (East Brunswick, NJ: 1982) 115-116.]
Saturday, September 09, 2006
Sigh....
Why is it that people will actually criticize others for being reasonable? That is precisely what Jason Rosenhouse did in a recent post on people who are religious and 'moderate', i.e., don't close their eyes to scientific progress. That's a broad, vague category; it would be utterly silly to think that there is any one criticism that can be leveled against the category as a whole, but Rosenhouse comes very close to trying to do this. He argues:
Of course, if we take enough trouble to do our homework, we can find pretty easily that 'the world was created by an act of God's will for the amusement of humans' has been regarded as an absurd oversimplification for hundreds of years. Indeed, it has been commonly denied; and it takes no great literary competence to see that reading this into Genesis requires very selective reading.
And that is what precisely this sort of thing is: eisegesis rather than exegesis. Rosenhouse and others like him make the mistake of assuming that fundamentalists are reading more rationally than those who take explicit pains to read more carefully. On the basis of what do they make this surprising assumption? In and of itself it should give them pause enough to reflect on whether they might be on the verge of sophism; but, since they don't pause at all, but rush headlong into the conclusion without any questioning of their assumptions, the sophism moves right along as if it were a good piece of reasoning.
Further, it is clear that Rosenhouse is confusing this issue with other issues. For instance, he argues:
But there is no 'explaining it after the fact' here. Rather, what's going on is exactly what one would expect of a reasonable person: if you want to know how the world works, you go to the world and look at how it works. And there's nothing about believing that the world is created by God that would change this. Rosenhouse says that God "could simply have snappend his fingers and brought his world in being". Perhaps so. But if you want to know what world was brought into being, snapped fingers or no, you will, if you are a reasonable person, look to see what world was brought into being. And the world that we find if we look into this matter is a world consisting of a vast number of galaxies, each with a vast number of stars, around one of which we find ourselves on a planet. Rosenhouse summarizes the history of the planet as "a mechanism for creation that got hung up at the bacteria stage for three billion years, and then needed an assist from several mass extinctions after clearing that hump." How he or anyone else would know that it got 'hung up' at one stage and 'needed an assist' at another he doesn't say; and he apparently forgets that if God can "snap his fingers" and make the whole universe exist, there's no reason to assume he couldn't have three billion years of bacteria alone and a bunch of mass extinctions on one tiny planet in a vast array of stars. There's no room here for 'getting hung up' or 'needing an assist'; it's either rhetoric backed up by no reasoning, or a rather serious category mistake.
And, again, if you consider yourself to have reason to believe that God created the universe, and you want to know about this universe that God created, as many of us do, you look and see. If the heavens are telling the glory of God, you don't close yourself up in a room and speculate groundlessly about what sort of universe God would make; you inquire into the universe He did make, you raise your head to the heavens and see.
Rosenhouse then goes on to say:
Yet if we were actually to look at the real historical evidence, we would find that what was more often thought a big deal was the claim that the sun was the immovable center of the universe; in the Galileo controversy, for instance, this is explicitly treated as the more serious problem. People did, of course, believe that the earth was at the center of the universe, and did think it a big deal when other people began to hold that it was not. But on what evidence shall we say that they believed this because they thought that "the Earth is the point of it all"? One wonders; because in medieval cosmology it is usually clear that the Earth is not the point of it all. If we are going to talk about importance, the reason they thought the earth was at the center of the universe was that it was the least important body in the heavens. All the other bodies in the heavens moved in circles, and moving in circles was, in Aristotelian and Platonic eyes alike, the most perfect sort of motion. A body that did not move in circles (like the stationary Earth) was thereby shown to be an imperfect sort of body. Further, the Earth did not even have the distinction of bearing the most important people in the universe. Human beings, far from being considered the most intelligent creatures in the guide, were considered the least intelligent: of all the intelligences in all the universe, we were the least impressive, because we were on the edge -- we were just intelligent enough to be considered genuine intelligences (we could think in universal terms and act morally) but were still rather dimwitted in comparison with other intelligences the medievals recognized. Far more impressive were the intelligences associated with the heavenly bodies; far more impressive than they were the intelligences of the empyrean. On such a view of the universe, what sense could be made of the claim that "The Earth is the point of it all"? So if Rosenhouse has an argument here, we need more information -- the evidence on which he is basing this claim -- in order to know what it is.
I've tried not to be completely harsh in this post. The sort of argument Rosenhouse is making is one of my pet peeves, because it seems to me to be a very bad strategy all around: fighting pseudoscience is not a sufficient justification for propagating pseudohistory; arguing against irrationality is not a good reason for failing to provide good reasoning oneself. It is simply unwise to attack fundamentalists for not making proper and reasonable distinctions and then go on in the next breath to avoid making proper and reasonable distinctions oneself. It is unreasonable to criticize reasonable people for being reasonable about the world. And a lot of this argument is just argument of the "I know better than you do, from a superficial reading, what the text you've studied all your life says" kind of argument. And as I've pointed out before in a very different context, this is usually absurd through and through. Let's move beyond this to a reasonable approach.
The problem is that theological moderation is even harder to defend than fundamentalism.
A lot of people say a lot of things in the dispute between science and religion. There are arguments and counter arguments ad nauseum on both sides. Many of these arguments are fascinating and reward careful study, which, indeed, is why I spend so much time on them at this blog.
But when you strip away all the logic chopping and the careful parsing of Genesis you're left with a simple truth that no theologian has yet been able to make go away. It is this: Evolution by natural selection just isn't what you expect from a world created by an act of God's will for the amusement of humans.
Of course, if we take enough trouble to do our homework, we can find pretty easily that 'the world was created by an act of God's will for the amusement of humans' has been regarded as an absurd oversimplification for hundreds of years. Indeed, it has been commonly denied; and it takes no great literary competence to see that reading this into Genesis requires very selective reading.
And that is what precisely this sort of thing is: eisegesis rather than exegesis. Rosenhouse and others like him make the mistake of assuming that fundamentalists are reading more rationally than those who take explicit pains to read more carefully. On the basis of what do they make this surprising assumption? In and of itself it should give them pause enough to reflect on whether they might be on the verge of sophism; but, since they don't pause at all, but rush headlong into the conclusion without any questioning of their assumptions, the sophism moves right along as if it were a good piece of reasoning.
Further, it is clear that Rosenhouse is confusing this issue with other issues. For instance, he argues:
You can explain it after the fact, of course. You can say that evolution is God's means of creation, or you can transform Genesis from an unambiguous sequence of historical events into a parable meant to teach theological truths, or you can gush that God put in place a system of natural laws so wonderful that it was sufficient to bring about his creative ends, or you can argue that somehow humanity or something like it was the inevitable result of evolution. The fact remains that God chose a mechanism for creation that got hung up at the bacteria stage for three billlion years, and then needed an assist from several mass extinctions after clearing that hump. This, when he could simply have snapped his fingers and brought his world into being.
But there is no 'explaining it after the fact' here. Rather, what's going on is exactly what one would expect of a reasonable person: if you want to know how the world works, you go to the world and look at how it works. And there's nothing about believing that the world is created by God that would change this. Rosenhouse says that God "could simply have snappend his fingers and brought his world in being". Perhaps so. But if you want to know what world was brought into being, snapped fingers or no, you will, if you are a reasonable person, look to see what world was brought into being. And the world that we find if we look into this matter is a world consisting of a vast number of galaxies, each with a vast number of stars, around one of which we find ourselves on a planet. Rosenhouse summarizes the history of the planet as "a mechanism for creation that got hung up at the bacteria stage for three billion years, and then needed an assist from several mass extinctions after clearing that hump." How he or anyone else would know that it got 'hung up' at one stage and 'needed an assist' at another he doesn't say; and he apparently forgets that if God can "snap his fingers" and make the whole universe exist, there's no reason to assume he couldn't have three billion years of bacteria alone and a bunch of mass extinctions on one tiny planet in a vast array of stars. There's no room here for 'getting hung up' or 'needing an assist'; it's either rhetoric backed up by no reasoning, or a rather serious category mistake.
And, again, if you consider yourself to have reason to believe that God created the universe, and you want to know about this universe that God created, as many of us do, you look and see. If the heavens are telling the glory of God, you don't close yourself up in a room and speculate groundlessly about what sort of universe God would make; you inquire into the universe He did make, you raise your head to the heavens and see.
Rosenhouse then goes on to say:
And it's not just evolution. The Catholic Church used to think it was a very big deal to claim that the Earth was not the center of the universe. They were right to think it a big deal. If the Earth is the point of it all then it is rather hard to explain why God also created billions of other galaxies with stars orbited by lifeless worlds. Since theological reasoning is constrained only by the imagination of the reasoner, it is possible to conjure up explanations for this fact. And who knows? Maybe you're one of those people who can actually talk yourself into believing those explanations.
Yet if we were actually to look at the real historical evidence, we would find that what was more often thought a big deal was the claim that the sun was the immovable center of the universe; in the Galileo controversy, for instance, this is explicitly treated as the more serious problem. People did, of course, believe that the earth was at the center of the universe, and did think it a big deal when other people began to hold that it was not. But on what evidence shall we say that they believed this because they thought that "the Earth is the point of it all"? One wonders; because in medieval cosmology it is usually clear that the Earth is not the point of it all. If we are going to talk about importance, the reason they thought the earth was at the center of the universe was that it was the least important body in the heavens. All the other bodies in the heavens moved in circles, and moving in circles was, in Aristotelian and Platonic eyes alike, the most perfect sort of motion. A body that did not move in circles (like the stationary Earth) was thereby shown to be an imperfect sort of body. Further, the Earth did not even have the distinction of bearing the most important people in the universe. Human beings, far from being considered the most intelligent creatures in the guide, were considered the least intelligent: of all the intelligences in all the universe, we were the least impressive, because we were on the edge -- we were just intelligent enough to be considered genuine intelligences (we could think in universal terms and act morally) but were still rather dimwitted in comparison with other intelligences the medievals recognized. Far more impressive were the intelligences associated with the heavenly bodies; far more impressive than they were the intelligences of the empyrean. On such a view of the universe, what sense could be made of the claim that "The Earth is the point of it all"? So if Rosenhouse has an argument here, we need more information -- the evidence on which he is basing this claim -- in order to know what it is.
I've tried not to be completely harsh in this post. The sort of argument Rosenhouse is making is one of my pet peeves, because it seems to me to be a very bad strategy all around: fighting pseudoscience is not a sufficient justification for propagating pseudohistory; arguing against irrationality is not a good reason for failing to provide good reasoning oneself. It is simply unwise to attack fundamentalists for not making proper and reasonable distinctions and then go on in the next breath to avoid making proper and reasonable distinctions oneself. It is unreasonable to criticize reasonable people for being reasonable about the world. And a lot of this argument is just argument of the "I know better than you do, from a superficial reading, what the text you've studied all your life says" kind of argument. And as I've pointed out before in a very different context, this is usually absurd through and through. Let's move beyond this to a reasonable approach.
Thursday, September 07, 2006
The Plans of Burnaby
While doing a bit of reading on the subject of eighteenth century theories of taste, I came upon an amusing incident in the history of scholarship that I thought I would pass on. For some time it had been known that Frances Reynolds, the sister of Sir Joshua Reynolds and a correspondent of the great Samuel Johnson, had printed as a pamphlet in 250 copies, an essay on taste. Very little was known of it, however. It seemed to be quoted by James Northcote in his early nineteenth-century Life of Sir Joshua Reynolds, but none of the quotations ever seemed very helpful; and people began to suspect that either Northcote was mistaken about the source (i.e., mistakenly thought the text he was quoting was by Reynolds) or he was quoting an early draft that may have been very different from the one actually printed. (It turned out they were right.)
In the meantime, scholars were trying to hunt down another, even more mysterious work of Reynolds's, which she had sent to Johnson for comments. Johnson had responded (8 April 1782) kindly but critically. The letter everyone had access to was that printed by Croker and later by Hill:
High and low scholars looked for this otherwise unknown novel or short story about the plans of Burnaby. It never came to light.
Which is not surprising, as there was no Burnaby. Johnson's actual letter never said anything about the plans of Burnaby; it was a faulty transcription. The actual letter, instead of 'The plans of Burnaby' had 'The Ideas of Beauty'. The Burnaby story was really the essay on taste!
In any case, having more information in hand it was possible to identify the work. You can read it, and, in the introduction, James Clifford's account of the history of the work, at Project Gutenberg. It strikes me as a nice little parable for all scholars everywhere. Every discipline has its own versions of the plans of Burnaby, whether on a small scale or a grand one -- that stunning little piece of evidence that sends you on a wild goose chase because it is, unbeknownst to you, all wrong. Beware the plans of Burnaby!
In the meantime, scholars were trying to hunt down another, even more mysterious work of Reynolds's, which she had sent to Johnson for comments. Johnson had responded (8 April 1782) kindly but critically. The letter everyone had access to was that printed by Croker and later by Hill:
Your work is full of very penetrating meditation, and very forcible sentiment. I read it with a full perception of the sublime, with wonder and terrour, but I cannot think of any profit from it; it seems not born to be popular.
Your system of the mental fabric is exceedingly obscure, and without more attention than will be willingly bestowed, is unintelligible. The plans of Burnaby will be more easily understood, and are often charming. I was delighted with the different beauty of different ages.
I would make it produce something if I could but I have indeed no hope. If a Bookseller would buy it at all, as it must be published without a name, he would give nothing for it worth your acceptance.
High and low scholars looked for this otherwise unknown novel or short story about the plans of Burnaby. It never came to light.
Which is not surprising, as there was no Burnaby. Johnson's actual letter never said anything about the plans of Burnaby; it was a faulty transcription. The actual letter, instead of 'The plans of Burnaby' had 'The Ideas of Beauty'. The Burnaby story was really the essay on taste!
In any case, having more information in hand it was possible to identify the work. You can read it, and, in the introduction, James Clifford's account of the history of the work, at Project Gutenberg. It strikes me as a nice little parable for all scholars everywhere. Every discipline has its own versions of the plans of Burnaby, whether on a small scale or a grand one -- that stunning little piece of evidence that sends you on a wild goose chase because it is, unbeknownst to you, all wrong. Beware the plans of Burnaby!
Wednesday, September 06, 2006
On/Off
I've run across arguments like this quite often, but I've never understood the reasoning:
[Christopher Hughes, On a Complex Theory of a Simple God. Cornell (Ithaca, 1989) p. 27]
This particular version of the argument is directed at Aquinas; I'm setting aside the question of whether 'existence' as we usually understand it is an adequate translation of Aquinas's esse (I don't think it is, outside of certain narrow contexts; the notions overlap, of course, but there are a number of reasons to regard 'existence' as a term that fails to coincide very well with esse in meaning). Instead I want to point out that arguments like this seem to make an unwarranted leap. Suppose that X is an 'on/off' property; you either have it or you don't. It doesn't follow from this that you can't have degrees or gradations of X when it's 'on'. Why would it? The mere fact of there being no tertium quid between 'on' and 'off' tells us nothing about the nature of 'on'.
What people appear to be doing in these arguments is confusing 'degrees of X' with 'degrees between X and non-X'. The two are not the same, however similar the answers to questions about them may be in particular contexts. The first is a scale of X; the second is a scale between X and non-X, and these are two different scales. To have the former, you just need X to admit of intension and remission -- X's need to have some feature or characteristic that can be described by 'more' or 'less' in comparison with other X's. To have the latter, you need to have a characteristic or feature that X and non-X share in different intensities and or strengths. For the former, X can be a very precise sort of thing -- sharp boundaries between X and non-X -- without causing problems for the scale. For the latter, the line between X and non-X has to be fuzzy, vague, blurred. However similar they may seem, they really are extremely different.
So, even if existence cannot admit of degrees of 'fullness of existence' or 'perfection of existence qua existence' or what have you, the mere fact that things either exist or don't is not a good reason for thinking so. So it is with other things. To have degrees of truth, for instance, you don't have to admit that there are degrees between truth and falsity (likewise with degrees of falsity). Being committed to the claim that there are degrees of beauty is not enough to commit you to the claim that there are degrees between the beautiful and the non-beautiful. Commitment to the claim that there are degrees of goodness isn't a commitment to the claim that there are degrees between being good and not being good.
I think I can understand the difference between having whiteness according to its full power and having whiteness according to something less than its full power, because, I take it, it is the difference between being less than perfectly white and being perfectly white. By contrast, there does not seem to be a difference between being perfectly existent and being less than perfectly existent. Existence is an on/off property: either you're there or you're not. Because existence is on/off, it would seem, either you have it according to its full power or you don't have it at all.
[Christopher Hughes, On a Complex Theory of a Simple God. Cornell (Ithaca, 1989) p. 27]
This particular version of the argument is directed at Aquinas; I'm setting aside the question of whether 'existence' as we usually understand it is an adequate translation of Aquinas's esse (I don't think it is, outside of certain narrow contexts; the notions overlap, of course, but there are a number of reasons to regard 'existence' as a term that fails to coincide very well with esse in meaning). Instead I want to point out that arguments like this seem to make an unwarranted leap. Suppose that X is an 'on/off' property; you either have it or you don't. It doesn't follow from this that you can't have degrees or gradations of X when it's 'on'. Why would it? The mere fact of there being no tertium quid between 'on' and 'off' tells us nothing about the nature of 'on'.
What people appear to be doing in these arguments is confusing 'degrees of X' with 'degrees between X and non-X'. The two are not the same, however similar the answers to questions about them may be in particular contexts. The first is a scale of X; the second is a scale between X and non-X, and these are two different scales. To have the former, you just need X to admit of intension and remission -- X's need to have some feature or characteristic that can be described by 'more' or 'less' in comparison with other X's. To have the latter, you need to have a characteristic or feature that X and non-X share in different intensities and or strengths. For the former, X can be a very precise sort of thing -- sharp boundaries between X and non-X -- without causing problems for the scale. For the latter, the line between X and non-X has to be fuzzy, vague, blurred. However similar they may seem, they really are extremely different.
So, even if existence cannot admit of degrees of 'fullness of existence' or 'perfection of existence qua existence' or what have you, the mere fact that things either exist or don't is not a good reason for thinking so. So it is with other things. To have degrees of truth, for instance, you don't have to admit that there are degrees between truth and falsity (likewise with degrees of falsity). Being committed to the claim that there are degrees of beauty is not enough to commit you to the claim that there are degrees between the beautiful and the non-beautiful. Commitment to the claim that there are degrees of goodness isn't a commitment to the claim that there are degrees between being good and not being good.
A Better Resurrection
I have no wit, no words, no tears;
My heart within me like a stone
Is numbed too much for hopes or fears.
Look right, look left, I dwell alone;
I lift mine eyes, but dimmed with grief
No everlasting hills I see;
My life is in the falling leaf:
O Jesus, quicken me.
My life is like a faded leaf,
My harvest dwindled to a husk:
Truly my life is void and brief
And tedious in the barren dusk;
My life is like a frozen thing,
No bud nor greenness can I see:
Yet rise it shall--the sap of spring;
O Jesus, rise in me.
My life is like a broken bowl,
A broken bowl that cannot hold
One drop of water for my soul
Or cordial in the searching cold;
Cast in the fire the perished thing;
Melt and remould it, till it be
A royal cup for Him, my King:
O Jesus, drink of me.
[Christina Rossetti (1830-1894)]
Sunday, September 03, 2006
Sommers-Englebretsen Term Logic, Part VI
In previous posts in this series I've been attempting to provide a rough characterization of basic SETL. In Part I, I noted the basics of SETL, in a rough way. In Part II and Part III, I discussed briefly some special cases and how SETL handles them. In Part IV, I discussed some basics of argument using SETL. Part V looked at some simple arguments for which SETL gives us a better sense of what's going on than ordinary predicate logic does. (There will be a post at the end of the series giving references and supplementary readings.)
In this part I want to look a bit at how one might go about incorporating modality into SETL. The work that has been done to this end is still incomplete, but turns out to be immensely promising.
Propositional vs. Predicate Modality
When we have a modality, like necessity, should we take it as modifying the predicate or as modifying the whole proposition? In language we sometimes seem to do one or the other. For instance, we might say,
It is possible that the new animal is a mammal.
And that seems to be propositional modality. However, we might also say,
The new animal is possibly a mammal.
This makes it look like a predicate modification. One could, perhaps, hold that they are equivalent; but this seems very clearly not to be true. A more abstract example might help. Compare:
Necessarily, no B is nonB.
No B is necessarily nonB.
'Necessarily' is clearly doing different work here. In the one, "No B is nonB" is taken as a necessary truth. The second, however, does not imply this. It doesn't rule out the possibility that some B is nonB; the only possibility ruled out is that some B is necessarily nonB. It could be true, for all that we can tell from this sentence alone, that some B is non-necessarily nonB. Very different sorts of modality.
The distinction noted here is often called the de re/de dicto distinction for modality; I'll instead continue to use the terms I've already introduced and call it the distinction between predicate and propositional modality.
Predicate and Propositional Modality in SETL
How would we go about incorporating predicate modality into SETL? It turns out that this is very easy. A proposition with predicate modality is simply a proposition with a modalized predicate. Thus, we can simply add modal functors to our predicates. Then we can simply use our ordinary SETL rules, just adding a few basic modal laws (e.g., if +S+P, we can conclude +S+◊P). Let's put up two axioms:
Axiom 1. If x, then x.
Axiom 2. If x, then x◊.
For instance, +S+P implies +s+◊P; +S+P implies +S+P. With these we can handle all the important logical relations that hold for predicate modality.
Propositional modality is a more complicated affair, and to see how we could handle it, we should perhaps take a moment to look at domains again. Every proposition is put forward relative to a domain of discourse; to contradict each other "There are dragons" and "There are no dragons" have to presuppose the same domain (if the domain of one is fictional beast mentioned in Piers Anthony novels and the domain of the other is real animals, for instance, they simply don't conflict). Any logical relation between two propositions, in fact, requires some connection of domains. We could state this by saying that every statement denotes its domain; and what a statement signifies is what it says of that domain. "Dragons are blue" says of the domain of the sentence that it includes blue dragons. When we say a statement is true, we mean that the domain it denotes includes the feature it signifies.
Putting it this way, however, simplifies matters considerably, because some statements, while having a domain, don't have a determinate domain -- or, if you prefer, their domain is a meta-domain, a domain of domains -- and are therefore harder to pin down. As it happens, all propositionally modalized propositions are of this sort. "Possibly at least one S is P" can't be interpreted as saying that at least one P-characterized S is in a particular domain; rather, it has to be interpreted as saying that there is at least one domain that includes at least one P-characterized S. Likewise, "Necessarily at least one S is P" doesn't mean that there is at least one P-characterized S in a particular domain; it means that in every domain there is at least one P-characterized S.
When we recognize this, it turns out to be fairly easy to see that the contradictory of (Every S is P) is not, as one might think, (Some S is not P), but instead, ◊(Some S is not P); and the contradictory of (No S is P) is ◊(Some S is P); which is actually rather similar to what one finds in predicate modality (the universal + necessity is contradicted by the particular + possibility). If we take SETL and add the following axioms, we can handle all propositionally modal propositions using necessity and possibility:
Axiom 3. If x, then x
Axiom 4. If x, then ◊x
And, it turns out, we can combine our two types of modality if we recognize that the following two statements are true:
If x, then x
If x◊, then ◊x.
This gives us a truly lovely square of opposition or, perhaps more accurately, square of implication, for modal statements, which I borrow from Englebretsen:
Each arrow indicates an implication or possible inference, and gives us an excellent sense of the strongest and weakest modal claims, and, for stronger claims, what weaker claims they entail. Claims that are both necessary and universal are at the top; claims that are both possible and particular are at the bottom; everything else is somewhere in-between.
Inferences with Modality in SETL
With our square of opposition we have already said something about how one would handle simple inferences in a modal version of SETL. But we need a way to handle mediate inference. Since predicate modalities are just modalized predicates, DDO, or the rule of mediate inference, is sufficient to handle them. But propositional modalities are more slippery here as elsewhere. DDO is not sufficient. We need to add another rule:
If x and y, then (x and y)
All this only gives a quick glimpse of a vast and important extension of basic SETL. In any case, any system that cannot distinguish predicate modality from propositional modality is hampered in its handling of real-life modal inferences; and one of the nice things about modalized SETL is that it can handle this distinction very well.
In the next post I'll give another quick glimpse of yet another interesting extension, precise quantification.
In this part I want to look a bit at how one might go about incorporating modality into SETL. The work that has been done to this end is still incomplete, but turns out to be immensely promising.
Propositional vs. Predicate Modality
When we have a modality, like necessity, should we take it as modifying the predicate or as modifying the whole proposition? In language we sometimes seem to do one or the other. For instance, we might say,
It is possible that the new animal is a mammal.
And that seems to be propositional modality. However, we might also say,
The new animal is possibly a mammal.
This makes it look like a predicate modification. One could, perhaps, hold that they are equivalent; but this seems very clearly not to be true. A more abstract example might help. Compare:
Necessarily, no B is nonB.
No B is necessarily nonB.
'Necessarily' is clearly doing different work here. In the one, "No B is nonB" is taken as a necessary truth. The second, however, does not imply this. It doesn't rule out the possibility that some B is nonB; the only possibility ruled out is that some B is necessarily nonB. It could be true, for all that we can tell from this sentence alone, that some B is non-necessarily nonB. Very different sorts of modality.
The distinction noted here is often called the de re/de dicto distinction for modality; I'll instead continue to use the terms I've already introduced and call it the distinction between predicate and propositional modality.
Predicate and Propositional Modality in SETL
How would we go about incorporating predicate modality into SETL? It turns out that this is very easy. A proposition with predicate modality is simply a proposition with a modalized predicate. Thus, we can simply add modal functors to our predicates. Then we can simply use our ordinary SETL rules, just adding a few basic modal laws (e.g., if +S+P, we can conclude +S+◊P). Let's put up two axioms:
Axiom 1. If x, then x.
Axiom 2. If x, then x◊.
For instance, +S+P implies +s+◊P; +S+P implies +S+P. With these we can handle all the important logical relations that hold for predicate modality.
Propositional modality is a more complicated affair, and to see how we could handle it, we should perhaps take a moment to look at domains again. Every proposition is put forward relative to a domain of discourse; to contradict each other "There are dragons" and "There are no dragons" have to presuppose the same domain (if the domain of one is fictional beast mentioned in Piers Anthony novels and the domain of the other is real animals, for instance, they simply don't conflict). Any logical relation between two propositions, in fact, requires some connection of domains. We could state this by saying that every statement denotes its domain; and what a statement signifies is what it says of that domain. "Dragons are blue" says of the domain of the sentence that it includes blue dragons. When we say a statement is true, we mean that the domain it denotes includes the feature it signifies.
Putting it this way, however, simplifies matters considerably, because some statements, while having a domain, don't have a determinate domain -- or, if you prefer, their domain is a meta-domain, a domain of domains -- and are therefore harder to pin down. As it happens, all propositionally modalized propositions are of this sort. "Possibly at least one S is P" can't be interpreted as saying that at least one P-characterized S is in a particular domain; rather, it has to be interpreted as saying that there is at least one domain that includes at least one P-characterized S. Likewise, "Necessarily at least one S is P" doesn't mean that there is at least one P-characterized S in a particular domain; it means that in every domain there is at least one P-characterized S.
When we recognize this, it turns out to be fairly easy to see that the contradictory of (Every S is P) is not, as one might think, (Some S is not P), but instead, ◊(Some S is not P); and the contradictory of (No S is P) is ◊(Some S is P); which is actually rather similar to what one finds in predicate modality (the universal + necessity is contradicted by the particular + possibility). If we take SETL and add the following axioms, we can handle all propositionally modal propositions using necessity and possibility:
Axiom 3. If x, then x
Axiom 4. If x, then ◊x
And, it turns out, we can combine our two types of modality if we recognize that the following two statements are true:
If x, then x
If x◊, then ◊x.
This gives us a truly lovely square of opposition or, perhaps more accurately, square of implication, for modal statements, which I borrow from Englebretsen:
Each arrow indicates an implication or possible inference, and gives us an excellent sense of the strongest and weakest modal claims, and, for stronger claims, what weaker claims they entail. Claims that are both necessary and universal are at the top; claims that are both possible and particular are at the bottom; everything else is somewhere in-between.
Inferences with Modality in SETL
With our square of opposition we have already said something about how one would handle simple inferences in a modal version of SETL. But we need a way to handle mediate inference. Since predicate modalities are just modalized predicates, DDO, or the rule of mediate inference, is sufficient to handle them. But propositional modalities are more slippery here as elsewhere. DDO is not sufficient. We need to add another rule:
If x and y, then (x and y)
All this only gives a quick glimpse of a vast and important extension of basic SETL. In any case, any system that cannot distinguish predicate modality from propositional modality is hampered in its handling of real-life modal inferences; and one of the nice things about modalized SETL is that it can handle this distinction very well.
In the next post I'll give another quick glimpse of yet another interesting extension, precise quantification.
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