For he is the best student who does not read his thoughts into the book, but lets it reveal its own; who draws from it its sense, and does not import his own into it, nor force upon its words a meaning which he had determined was the right one before he opened its pages.
St. Hilary of Poitiers, On the Trinity, 1.18
Saturday, March 21, 2009
Friday, March 20, 2009
Ideas Without Comment, on Truth Values and Modal Operators
I've been intending to write a post or two on truth values and modal operators, but I am currently rather more busy than I expected to be, with grading, revising, and other things. So instead here are some ideas about modal operators and truth values, without much comment, that I've been toying with recently.
1. There is no fundamental difference between modal operators and truth values. You can have a logic with no truth values, using True and False as modal operators, and you can have a modal logic with no modal operators, only an expanded range of truth values. That is, you can have your truth values as purely operational, or you modal operators as part of your truth table. There is no significant underlying difference. You can go back and forth, if you want; this, in a sense, is what we do, since we treat T and F as truth values and other modalities as operators.
2. Negation can be treated as a modal operator not distinct from False.
3. Connectives are not constant between different modal systems; the only part of their truth tables that is invariant across such systems is the purely alethic part, i.e., the part using True and False.
4. Truth values admit of being multiplied (conjoined) or summed (disjoined). Thus, you can have not only T and F as truth values, even in a purely alethic table, but also TT, TF, FT, FF, TTTT, etc. In classical logic, any string of T's only is equivalent to T; any string of F's or F's and T's where there is an even number of F's is equivalent to T; any string of F's or F's and T's where there is an odd number of F's is equivalent to F. T+T is equivalent to T, F+F is equivalent to F, F+T trivially applies to everything. There is no particular need to have a classical truth table, though. You can deny that FF is equivalent to T, for instance, and that is like rejecting double negation; you can deny that F+T trivially applies to everything, and that is like rejecting excluded middle. You can deny commutativity (i.e., that TF is equivalent to FT) and that gets you something else again. There are modal logics where we in fact do things analogous to any and all of these. You could also reject associativity; I'm not sure if I've ever come across a modal logic that does this. You can do stranger things with truth tables than anyone seems to have done before.
5. Truth tables need not be deterministic. That is, one could have a deterministic truth table in which, for instance, (Tp & Tq) always has the truth value T (by standard conjunction rules). But you could also have a truth table in which the truth value of (Tp & Tq) is T or ◊. This 'or' is naturally interpreted as additive conjunction: that is, you can have T, or you can have ◊, as you deem fit. In fact, most of the modal systems we use make use of nondeterministic truth tables in this way. It is possible that we can think of truth values as resources, and describe them by means of linear logic; for instance, you can have a logic where Necessary linearly implies True which linearly implies Possible. If that's genuinely possible, that would, I think, be a more interesting example of the value of linear logic than the usual vending machine examples.
6. You could focus on truth values almost entirely, treating propositions as simply ways of indexing truth values.
7. Of course, I've put it all in terms of truth tables, but, again, with regard to (1), you could instead do it all in terms of modal operators and the rules governing them. For some things that we might like to do in either case, you would need to have variables into which we could plug modal operators. For instance, for a given proposition p you might not know its truth value; is it necessary, true, possible, false, etc.? So we can use a variable. Likewise, we can, with the right information, solve for such variables in exactly the way you might solve for any variable.
8. If you can do any of this, there seems no good reason to deny that you can quantify over truth values / modal operators. And why wouldn't you want to play that game! But it would seem to be tricky business.
1. There is no fundamental difference between modal operators and truth values. You can have a logic with no truth values, using True and False as modal operators, and you can have a modal logic with no modal operators, only an expanded range of truth values. That is, you can have your truth values as purely operational, or you modal operators as part of your truth table. There is no significant underlying difference. You can go back and forth, if you want; this, in a sense, is what we do, since we treat T and F as truth values and other modalities as operators.
2. Negation can be treated as a modal operator not distinct from False.
3. Connectives are not constant between different modal systems; the only part of their truth tables that is invariant across such systems is the purely alethic part, i.e., the part using True and False.
4. Truth values admit of being multiplied (conjoined) or summed (disjoined). Thus, you can have not only T and F as truth values, even in a purely alethic table, but also TT, TF, FT, FF, TTTT, etc. In classical logic, any string of T's only is equivalent to T; any string of F's or F's and T's where there is an even number of F's is equivalent to T; any string of F's or F's and T's where there is an odd number of F's is equivalent to F. T+T is equivalent to T, F+F is equivalent to F, F+T trivially applies to everything. There is no particular need to have a classical truth table, though. You can deny that FF is equivalent to T, for instance, and that is like rejecting double negation; you can deny that F+T trivially applies to everything, and that is like rejecting excluded middle. You can deny commutativity (i.e., that TF is equivalent to FT) and that gets you something else again. There are modal logics where we in fact do things analogous to any and all of these. You could also reject associativity; I'm not sure if I've ever come across a modal logic that does this. You can do stranger things with truth tables than anyone seems to have done before.
5. Truth tables need not be deterministic. That is, one could have a deterministic truth table in which, for instance, (Tp & Tq) always has the truth value T (by standard conjunction rules). But you could also have a truth table in which the truth value of (Tp & Tq) is T or ◊. This 'or' is naturally interpreted as additive conjunction: that is, you can have T, or you can have ◊, as you deem fit. In fact, most of the modal systems we use make use of nondeterministic truth tables in this way. It is possible that we can think of truth values as resources, and describe them by means of linear logic; for instance, you can have a logic where Necessary linearly implies True which linearly implies Possible. If that's genuinely possible, that would, I think, be a more interesting example of the value of linear logic than the usual vending machine examples.
6. You could focus on truth values almost entirely, treating propositions as simply ways of indexing truth values.
7. Of course, I've put it all in terms of truth tables, but, again, with regard to (1), you could instead do it all in terms of modal operators and the rules governing them. For some things that we might like to do in either case, you would need to have variables into which we could plug modal operators. For instance, for a given proposition p you might not know its truth value; is it necessary, true, possible, false, etc.? So we can use a variable. Likewise, we can, with the right information, solve for such variables in exactly the way you might solve for any variable.
8. If you can do any of this, there seems no good reason to deny that you can quantify over truth values / modal operators. And why wouldn't you want to play that game! But it would seem to be tricky business.
Threefold Truth
...[N]othing is known or understood except the true. Truth, however, is triple or is triply, according as each thing has being triply, namely, in its own genus, in the created understanding, and in the eternal exemplar, as Augustine says in book V of the Literal Commentary on the Genesis and Anselm in the book on Truth in chapter X. Therefore we can speak of truth in three ways: by comparison to the matter which it informs, since (in that every created nature represents according to its grade the art by which it was made) the form itself or nature or essence, through which it imitates the very art or exemplar, is its truth; and it is true only in so far as it expresses that exemplar. In the second place we can consider truth by comparison to the understanding which it excites; and this is the created understanding an dmost of all the human understanding, since we are speaking of that mode. For there is no nature which does not manifest and declare itself to the understanding as it is able. In the third place we can consider truth by compraison to the exemplar from which it emanates; this is the divine light and art, by which all things have been made.
Matthew of Aquasparta, Ten Disputed Questions on Knowledge, Question II, in Selections from Medieval Philosophers, Vol. II,, Richard McKeon, ed. Scribners (New York: 1958) 281-282.
Wednesday, March 18, 2009
St. Cyril of Jerusalem on the Body
Tell me not that the body is a cause of sin. For if the body is a cause of sin, why does not a dead body sin? Put a sword in the right hand of one just dead, and no murder takes place. Let beauties of every kind pass before a youth just dead, and no impure desire arises. Why? Because the body sins not of itself, but the soul through the body. The body is an instrument, and, as it were, a garment and robe of the soul: and if by this latter it be given over to fornication, it becomes defiled: but if it dwell with a holy soul, it becomes a temple of the Holy Ghost. It is not I that say this, but the Apostle Paul hath said, Know ye not, that your bodies are the temple of the Holy Ghost which is in you? Be tender, therefore, of thy body as being a temple of the Holy Ghost. Pollute not thy flesh in fornication: defile not this thy fairest robe: and if ever thou hast defiled it, now cleanse it by repentance: get thyself washed, while time permits.
Catechetical Lectures
Tuesday, March 17, 2009
On Suits on Games
There has been some discussion recently about the Suits analysis of games. In The Grasshopper: Games of Life and Utopia, Bernard Suits famously gave an account of what it is to be a game, in which he holds that games have the following three elements:
(1) They are aimed at goals that can be described independently of the games themselves. For instance, in golf you aim at getting your ball in the hole; but, of course, you don't need to recognize the rules of the game to recognize that a ball goes in a hole in as few strokes as possible.
(2) They have rules that place impediments in the way of doing things in the most efficient ways. For instance, soccer players cannot pick up the ball with their hands and run down the field.
(3) In playing the game you voluntarily accept these rules because they make the game possible.
As Suits put it, a game is a voluntary attempt to overcome unnecessary obstacles. What is noteworthy is that these criteria, if not read in a very particular way, make following traffic laws a game. If I am trying to drive to a conference, I have a goal that can be described independently of driving and the rules governing it. The rules that govern driving, namely, traffic laws, impede doing things in the most efficient ways; for instance, even if there would be, in my circumstances, no harm in violating the rules, I have to stay on the road, and stay in my lane, and if a sign says, "No Right Turn," I can't turn right even if the conference is right there, to the right. And I voluntarily accept traffic laws because they make my driving possible. We just have no good name for the game, and a lot of people appear not to have any fun playing it. Indeed, most rule-following activities where the rules are not necessary but are reasonable turn out to be games on this account. When evangelical Christians have a passover seder -- which happens on occasion at many evangelical churches in the United States -- they are involved in a voluntary attempt to overcome unnecessary obstacles. Rabbinical law puts up a lot of obstacles; they are being accepted by the evangelicals because they are the rules for a passover meal. And the particular things you do in a passover can be described in a way independent of the passover itself. So is it a game? Perhaps, but we seem to be stretching the term beyond recognition, with no definite benefit in doing so.
There is a further issue that the account seems better suited for describing games that are sports than all games. If you play games with children to any great extent, it's hard to deny that there are games that have no unnecessary obstacles; to take an extroardinarily simple card game, in 52 Pick-Up you throw a pack of cards up into the air, and then the game begins: everyone picks them up as quickly as they can. And that's the whole game (there is no way to win, you just play), a game for little children, but a game nonetheless. Unnecessary obstacles make games challenging; they don't make them games.
But there is something useful about Suits's analysis. The basic elements are themselves attempt to specify a more general analysis in which games have three components: 'prelusory goals', 'lusory means', and a 'lusory attitude'. And the idea, a right one, is that playing a game is to use means to an end, in accordance with rules, with a particular sort of attitude. The problem with (3) as it stands is that it pretty clearly falls short of capturing the sort of attitude we have with regard to a game we are playing (an attitude notably lacking on the highway, for instance). One problem with Suits's argument is that he is unable to capture the attitude element in a noncircular way: the lusory attitude can't be just any sort of attitude a game-player might have (so that it doesn't matter whether, for instance, someone is playing a game for fun or for money), but, as the Grasshopper describes it, the attitude without which it is not possible to play the game. The problem with (2) is that it pretty clearly goes beyond what is required for something to be a lusory means; in fact, the lusory means merely have to be appropriate to the prelusory goal, and the rules that determine what counts as appropriate may be of just about any sort, as long as they are consistent with the lusory attitude and the prelusory goal.
And so, in the end, what we find out from Suits is that a game is an attempt to reach a goal appropriate to a game by means appropriate to that goal, according to rules that are accepted because they are the way you play the game. That does indeed clarify what a game is. It even provides necessary and sufficient conditions, since every game will meet these conditions, and meeting these conditions is playing a game.* And because of its circularity that makes it resistant to counterexamples. I don't think there is anything wrong with this. But if we use it as an explanation of what makes a game a game, it is, we should be quite clear, a Father Noel account of games.**
----
* The finding of necessary and sufficient conditions is often seen as an important goal in analytic philosophy, or, at least, it has been seen as such. But it's misleading to leave it at that. No matter what you're talking about, you always already have in hand a necessary and sufficient condition for the thing you're talking about: itself, since everything is both a necessary condition for itself and a sufficient condition for itself. But this sort of necessary or sufficient condition is not counted, of course. And it's possible to have necessary and sufficient conditions that are different from the thing being explained. For instance, in a list of biological species it may be that being a renate is necessary and sufficient for being a chordate. But this sort of necessary or sufficient condition doesn't illuminate anything, at least on its own, and therefore this doesn't seem to be what is primarily in view. What people are really looking for is a definition (in a broad sense of the term), and the 'necessary and sufficient conditions' part is just two of several requirements being used to sort out good attempts at definition from bad attempts at definition -- in particular, the adequately proportioned to each other, or equivalent. And when analytic philosophers look for either necessary or sufficient conditions, they don't look for just any necessary or sufficient conditions; they look for those that will contribute to a definition that will clarify and illuminate. In other words, they are looking for elements of definitions that fit certain values; explicative definitions, they are called. And this goal, being value-dependent, allows all sorts of judgment calls about what counts as the right sort of necessary or sufficient condition.
** Father Noel was one of the objects of Pascal's criticism; according to Pascal, he defined light as the luminary motion of a luminous body, which meant one could not understand the definition unless you already knew what was supposed to be defined. Such 'definitions' are not, contrary to what Pascal wanted to imply, useless; for instance, this one makes a claim relating light to motion and to body, which could be the beginnings of an advance in one's understanding of light. But regardless of whether it is true, and regardless of whether it can be the beginning of a research project, it will not help you to understand what light is. It is not what you are really looking for; at best it is a first baby-step toward it.
(1) They are aimed at goals that can be described independently of the games themselves. For instance, in golf you aim at getting your ball in the hole; but, of course, you don't need to recognize the rules of the game to recognize that a ball goes in a hole in as few strokes as possible.
(2) They have rules that place impediments in the way of doing things in the most efficient ways. For instance, soccer players cannot pick up the ball with their hands and run down the field.
(3) In playing the game you voluntarily accept these rules because they make the game possible.
As Suits put it, a game is a voluntary attempt to overcome unnecessary obstacles. What is noteworthy is that these criteria, if not read in a very particular way, make following traffic laws a game. If I am trying to drive to a conference, I have a goal that can be described independently of driving and the rules governing it. The rules that govern driving, namely, traffic laws, impede doing things in the most efficient ways; for instance, even if there would be, in my circumstances, no harm in violating the rules, I have to stay on the road, and stay in my lane, and if a sign says, "No Right Turn," I can't turn right even if the conference is right there, to the right. And I voluntarily accept traffic laws because they make my driving possible. We just have no good name for the game, and a lot of people appear not to have any fun playing it. Indeed, most rule-following activities where the rules are not necessary but are reasonable turn out to be games on this account. When evangelical Christians have a passover seder -- which happens on occasion at many evangelical churches in the United States -- they are involved in a voluntary attempt to overcome unnecessary obstacles. Rabbinical law puts up a lot of obstacles; they are being accepted by the evangelicals because they are the rules for a passover meal. And the particular things you do in a passover can be described in a way independent of the passover itself. So is it a game? Perhaps, but we seem to be stretching the term beyond recognition, with no definite benefit in doing so.
There is a further issue that the account seems better suited for describing games that are sports than all games. If you play games with children to any great extent, it's hard to deny that there are games that have no unnecessary obstacles; to take an extroardinarily simple card game, in 52 Pick-Up you throw a pack of cards up into the air, and then the game begins: everyone picks them up as quickly as they can. And that's the whole game (there is no way to win, you just play), a game for little children, but a game nonetheless. Unnecessary obstacles make games challenging; they don't make them games.
But there is something useful about Suits's analysis. The basic elements are themselves attempt to specify a more general analysis in which games have three components: 'prelusory goals', 'lusory means', and a 'lusory attitude'. And the idea, a right one, is that playing a game is to use means to an end, in accordance with rules, with a particular sort of attitude. The problem with (3) as it stands is that it pretty clearly falls short of capturing the sort of attitude we have with regard to a game we are playing (an attitude notably lacking on the highway, for instance). One problem with Suits's argument is that he is unable to capture the attitude element in a noncircular way: the lusory attitude can't be just any sort of attitude a game-player might have (so that it doesn't matter whether, for instance, someone is playing a game for fun or for money), but, as the Grasshopper describes it, the attitude without which it is not possible to play the game. The problem with (2) is that it pretty clearly goes beyond what is required for something to be a lusory means; in fact, the lusory means merely have to be appropriate to the prelusory goal, and the rules that determine what counts as appropriate may be of just about any sort, as long as they are consistent with the lusory attitude and the prelusory goal.
And so, in the end, what we find out from Suits is that a game is an attempt to reach a goal appropriate to a game by means appropriate to that goal, according to rules that are accepted because they are the way you play the game. That does indeed clarify what a game is. It even provides necessary and sufficient conditions, since every game will meet these conditions, and meeting these conditions is playing a game.* And because of its circularity that makes it resistant to counterexamples. I don't think there is anything wrong with this. But if we use it as an explanation of what makes a game a game, it is, we should be quite clear, a Father Noel account of games.**
----
* The finding of necessary and sufficient conditions is often seen as an important goal in analytic philosophy, or, at least, it has been seen as such. But it's misleading to leave it at that. No matter what you're talking about, you always already have in hand a necessary and sufficient condition for the thing you're talking about: itself, since everything is both a necessary condition for itself and a sufficient condition for itself. But this sort of necessary or sufficient condition is not counted, of course. And it's possible to have necessary and sufficient conditions that are different from the thing being explained. For instance, in a list of biological species it may be that being a renate is necessary and sufficient for being a chordate. But this sort of necessary or sufficient condition doesn't illuminate anything, at least on its own, and therefore this doesn't seem to be what is primarily in view. What people are really looking for is a definition (in a broad sense of the term), and the 'necessary and sufficient conditions' part is just two of several requirements being used to sort out good attempts at definition from bad attempts at definition -- in particular, the adequately proportioned to each other, or equivalent. And when analytic philosophers look for either necessary or sufficient conditions, they don't look for just any necessary or sufficient conditions; they look for those that will contribute to a definition that will clarify and illuminate. In other words, they are looking for elements of definitions that fit certain values; explicative definitions, they are called. And this goal, being value-dependent, allows all sorts of judgment calls about what counts as the right sort of necessary or sufficient condition.
** Father Noel was one of the objects of Pascal's criticism; according to Pascal, he defined light as the luminary motion of a luminous body, which meant one could not understand the definition unless you already knew what was supposed to be defined. Such 'definitions' are not, contrary to what Pascal wanted to imply, useless; for instance, this one makes a claim relating light to motion and to body, which could be the beginnings of an advance in one's understanding of light. But regardless of whether it is true, and regardless of whether it can be the beginning of a research project, it will not help you to understand what light is. It is not what you are really looking for; at best it is a first baby-step toward it.
Monday, March 16, 2009
Two Teresas on Entering into Oneself
Teresa of Avila, The Interior Castle, First Mansions, Section Seven:
Teresa Benedicta (Edith Stein), Finite and Eternal Being, Reinhardt, tr. ICS Publications (Washington, D.C.: 2002) p. 373:
Now let us return to our beautiful and charming castle and discover how to enter it. This appears incongruous: if this castle is the soul, clearly no one can have to enter it, for it is the person himself: one might as well tell someone to go into a room he is already in! There are, however, very different ways of being in this castle; many souls live in the courtyard of the building where the sentinels stand, neither caring to enter farther, nor to know who dwells in that most delightful place, what is in it and what rooms it contains.
Teresa Benedicta (Edith Stein), Finite and Eternal Being, Reinhardt, tr. ICS Publications (Washington, D.C.: 2002) p. 373:
The soul as the interior castle--as it was pictured by our holy mother Teresa--is not point-like as is the pure ego, but "spatial." It is a space, a "castle" with its many mansions in which the I is able to move freely, now going outward beyond itself, now withdrawing into its own inwardness.
Sunday, March 15, 2009
JTB and Locke
Analytic philosophers notoriously have very short traditions; a good example of this is the fact that the claim that knowledge is justified true belief is often called the 'traditional' account of knowledge, even though it is extremely difficult to find people who held it before the 20th century, or, indeed, anyone who has ever held it besides analytic philosophers or those who were taught epistemology by them. If you poke and prod the term 'justification' enough you might be able to attribute it to Plato, when he considers the idea that knowledge is what you get when you give true belief a rational account; but that poking and prodding requires a great many assumptions, both about justification and about what Plato is getting at, that need not be made. And JTB is even less plausibly attributed to others. Consider Locke.
Locke is very clear that knowledge consists in something radically different from belief: it is perception of the agreement or the disagreement between two ideas. On the basis of this knowledge we assent to various propositions that are proposed to the mind (and, indeed, cannot help but assent, any more than I could deny to myself that I am seeing the color white when it is taking up my field of vision; but Locke seems clearly enough to think that the knowledge itself is a perception, not a belief. (I think there is a good argument that this is actually the most common view of knowledge in the early modern period, on both the rationalist and empiricist sides; Locke is not being idiosyncratic here.) From Locke, Essay IV.1.2:
Indeed, Locke thinks that the sort of knowledge that most properly receives that name, the one that has "the utmost light and greatest certainty," is had simply by perceiving the agreement or disagreement between two ideas, and nothing else. This intuitive knowledge leaves no room for the intervention of a third idea, or anything else besides the perception itself of the two ideas, and is that by virtue of which everything else is known or understood.
Locke is very clear that knowledge consists in something radically different from belief: it is perception of the agreement or the disagreement between two ideas. On the basis of this knowledge we assent to various propositions that are proposed to the mind (and, indeed, cannot help but assent, any more than I could deny to myself that I am seeing the color white when it is taking up my field of vision; but Locke seems clearly enough to think that the knowledge itself is a perception, not a belief. (I think there is a good argument that this is actually the most common view of knowledge in the early modern period, on both the rationalist and empiricist sides; Locke is not being idiosyncratic here.) From Locke, Essay IV.1.2:
Knowledge then seems to me to be nothing but the perception of the connexion of and agreement, or disagreement and repugnancy of any of our ideas. In this alone it consists. Where this perception is, there is knowledge, and where it is not, there, though we may fancy, guess, or believe, yet we always come short of knowledge. For when we know that white is not black, what do we else but perceive, that these two ideas do not agree? When we possess ourselves with the utmost security of the demonstration, that the three angles of a triangle are equal to two right ones, what do we more but perceive, that equality to two right ones does necessarily agree to, and is inseparable from, the three angles of a triangle?
Indeed, Locke thinks that the sort of knowledge that most properly receives that name, the one that has "the utmost light and greatest certainty," is had simply by perceiving the agreement or disagreement between two ideas, and nothing else. This intuitive knowledge leaves no room for the intervention of a third idea, or anything else besides the perception itself of the two ideas, and is that by virtue of which everything else is known or understood.