Semantics Some More Notes On Propositions and Information

Some more notes on propositions and information We think of a proposition as being a specification of those possible worlds (or more accurately, world-time pairs) in which it is true. John left is true in all and only those possible worlds in which John left (at some time before now). Mary left is true in all and only those possible worlds in which Mary left (at some time before now). John left and Mary left is true in only some of those possible worlds in which John left—it’s the intersection of those worlds in which John left and those worlds in which Mary left, namely, the worlds in which both John and Mary left. If the facts are that John left and that Mary left, then if you know the facts, you know the proposition that John left and Mary left. More generally, an information state is a proposition that is a conjunction of the facts. It is the set of possible worlds in which all of the facts are true. These are the possible worlds that stand a chance of being the actual world. We aren’t always concerned with the actual world, however. We can also have opinions, false beliefs, dreams. These would be information states but relative to a different reference world (other than the actual world). If John thinks Mary left, we don’t know anything about the actual world, but we know something about what John believes the actual world to be like. If I dreamt Mary left I have a view of what my dream world is like. Verbs like think, believe, know, hope, want, dream are propositional attitude verbs, expressing a relation between the subject and a proposition. Certain propositional attitude verbs are also presuppositional as well. If John knows Mary left, then we know something about what John believes (that is, we know what worlds might be John’s “belief world”), but at the same time we’re making a statement about the actual world as well. X knows p presupposes p. You can’t very well say John knows Mary left, but he’s wrong. Individuals in the universe can be named (like John) or described (like the president). We assume that individuals who are named are constant across worlds, names are rigid designators, they somehow refer to the same individual (although with possibly different properties) in any possible world. It is possible, perhaps, that there is a possible world in which John was never born—in such worlds John does not exist would be true. But what we can’t allow is that John in some possible worlds is Mary. (Things can get even more complicated if we dig a little bit deeper—I dreamt I was walking with John, except John 1 wasn’t John, he was Donald Rumsfeld. For this semester, we’ll stick to the rigid designation idea). If individuals are described, like a student or the president, they may quite easily refer to a different individual in different worlds, however. There are fairly easily imagined possible worlds in which the president refers to the same individual that Al Gore refers to, for example. This gives rise to an ambiguity with propositional attitude verbs. Descriptions in a proposition can be interpreted with respect to reference world of the proposition (the “believe” world, the “dream” world), or with respect to one of the other reference worlds (e.g., the reference world the propositional attitude verb itself is evaluated with respect to, or the actual world). A description that is evaluated with respect to the proposition it is part of is evaluated de dicto. We might say “literally”—of what is said (that’s the same root as in dictum, dictate, etc.). A description that “transcends” the proposition, referring to the individual as described in a higher world (e.g., the actual world), is evaluated de re (of the thing, in “reality”, you might say). John believes the president likes pretzels. John stands in a relation to a proposition (the set of possible worlds he believes stand a chance of being the actual world). That proposition could be either of two things: Worlds in which the president in that world likes pretzels (de dicto), or worlds in which the president in the actual world likes pretzels (de re). The president in the actual world is an individual we could name (George W. Bush), so what John believes on the de re construal is that George W. Bush likes pretzels (true of any world in which George W. Bush likes pretzels). John may not know that George W. Bush is the president in the actual world. On the other hand, on the de dicto interpretation, John believes that whoever the president describes is someone who likes pretzels (true of any world in which the individual the president describes likes pretzels, including worlds in which Al Gore is president and likes pretzels). John may of course have a separate belief about who the president is, but that’s not part of the proposition The president likes pretzels. John dreamt that every student in LX502 was 7 feet tall. De dicto: John’s dream world is one in which x being is a student in LX502 implies that x is 7 feet tall. De re: x being a student in LX502 (in the actual world) implies that John’s dream world is one in which x is 7 feet tall. 2 Dynamics of discourse, continued… In conversation, apart from asserting propositions, one can also ask questions. A question basically serves to divide the set of possible worlds into propositions, requesting that the hearer pick one of the propositions as being the one containing the actual world. To put it another way, questions specify the propositions that serve as possible answers. For example, Who left? specifies that Pavarotti left, Bond left, Loren left are propositions one could give in response. We can partition possible worlds by which answer is true in that world. Who left? ———|—|————— Mary ———|—|————— left | | John left | | Just like there are certain verbs that take a proposition (the propositional attitude verbs), there are other verbs that take a question. John asked who left, John wonders who left. These basically mean John asked (someone, perhaps himself) the question. Although John may know quite a lot about what possible worlds are candidates for being the actual world, if they are divided up into propositions like x left, he’s not sure which proposition contains the actual world. Know is a kind of special verb, in that it seems to be able to take a question (as in John knows who left) but what it means is still that John knows a proposition—the proposition John knows is the (true and complete) answer to the question Who left?. If John knows who left and Mary left, John knows that Mary left. If John knows who left and Bill did not leave, John knows that Bill did not leave. Since John knows the true and complete answer to Who left? and Bill left is not among the propositions in that answer, it must be that Bill didn’t leave. (Things are more complicated, of course, but this is a start. For example, John knows where to buy the New York Times doesn’t seem to entail that if you can’t buy the NYT at the Natick oasis, John knows that you can’t buy the NYT at the Natick oasis. It’s also possible to say John knows for the most part who cheated on the exam. All interesting issues, but they’d take a whole semester to deal with.). In a cooperative discourse, an answer to a question must rule out some of the possible answers, but it need not always be a complete answer. For example, you can answer Where 3 did John go? by saying John either went to Texas or to California, and that seems fine (unlike answering I have feet, for example). The conditions on what make a discourse “appropriate” are complex, but certain things seem to affect that. One example is in question-answering, as above. Another is in the use of discourse markers, which can be viewed as expressing a relation between the utterance they mark and (some set, or perhaps just one, of) the preceding utterances. For example, John ate muffins. However, he did not touch the butter—here, however indicates that the proposition he did not touch the butter runs counter to the expectations set up by the preceding proposition John ate muffins. Scales and informativity Often in determining the meaning of an utterance, we make use of scales, a way to order things. One place in which scales are useful is in the computation of implication. Numbers are a good scale: If I have 3 children, then I have more children than if I have 2 children. Also, if I have 3 children, I also have 2 children (plus another one). I have 3 children implies that I have 2 children. I am 7 feet tall implies that I am 6 feet tall and that I am 2 feet tall. Because part of being cooperative is to be informative, a statement is generally assumed to be as informative as it can be without being false. Saying I have 3 children implies two other propositions (I have 2 children and I have 1 child). If I have 3 children, but say I have 2 children, this is true, but it is not as informative as it could have been. If I say I have 4 children, that’s more informative, but it’s also false. Thus, I should say I have 3 children. From this, the hearer, if the hearer assumes I am being cooperative, can assume that I do not have 4 children (since it would have been more informative to say so if it were actually true). This kind of implicature is called a scalar implicature. When we think of descriptions, we can also think of them as being in a kind of scalar relation. If something is a tomato, it is also a fruit (according to botanists). If something is a red tomato is it also red and a tomato. (Incidentally, another part of being cooperative is not being overly informative, but just to be as informative as is warranted. If someone says What’s that vegetable-looking thing are you eating? it would be unnecessarily informative to say I am eating a reddish-green organically grown tomato originally 3 inches in diameter—but now three-fifths consumed—that I bought at Bread & Circus last Thursday at 3:14pm for $1.99 using my Visa card). The point: A green bean is both green and a bean. Being a green bean implies being green, and also implies being a bean. 4 When we put a description of this sort in certain sentences, they imply other sentences. This is a red tomato implies This is red and This is a tomato. Or Everything I eat is a red tomato implies Everything I eat is red and Everything I eat is a tomato. These implications only go one way; if everything I eat is red, it is not necessarily true that everything I eat is a tomato. In these cases, replacing a description with a more general description (red tomato with red) preserves truth. In some environments, it goes the other way. If I did not a red thing, I did not eat a red tomato. By replacing a description with a more specific term (one which implies the more general term, like red tomato implies red) truth is preserved. An environment in which replacing the specific with the general preserves truth is an upward entailing environment. I ate a red tomato implies I ate a tomato. An environment in which replacing the general with the specific preserves truth is a downward entailing environment. I did not eat a tomato implies I did not eat a red tomato. Every tomato is delicious implies that Every red tomato is delicious. If I eat a tomato I get hives implies that if I eat a red tomato I get hives. No tomato is good enough implies that No red tomato is good enough. Few things in my refrigerator are tomatoes implies that Few things in my refrigerator are red tomatoes. A little factoid to store away, hypothesized by Ladusaw back in 1980, is that phrases like anything, any apple, an apple anyone gave me, etc. (sometimes called negative polarity items) occur only in downward entailing environments. *I ate any tomato. But: I did not eat any tomato. Every tomato anyone gives me is delicious. If I eat any tomato, I get hives. No tomato anyone gives me is good enough. Few things in my refrigerator are any good. We can understand this in terms of informativity: You can use any when it creates a stronger statement. If we take any to mean roughly “the most general” then because all of the statements that have something more specific in place of the any phrase are entailed, you’ve said something highly informative (I didn’t eat a tomato implies that I didn’t eat a red tomato). Not so in an upward entailing environment—if you replace “the most general” with something specific there, truth is not preserved (I ate a tomato does not imply I ate a red tomato). Kinds of individuals and measurement Nouns seem to come in two kinds mass and count. Count nouns can be counted (I have three chairs), mass nouns cannot (*I have three barley). The main property of mass nouns that distinguishes them from count nouns is that you use the same noun to describe 5 subparts or combinations: soup and soup is still soup, half of the barley I have is barley. Furniture and machinery are mass nouns, chair and machine are count nouns. It is possible to coerce a mass noun into a count noun by assuming a measure: I bought three coffees. Yet we feel the coercion, we can tell that we’ve done some semantic work, that we’ve assumed a measurement of a unit, like a cup. You can of course provide the measure explicitly: I bought two cartfuls of flour. Bare plurals like chairs or machines have the same properties as mass nouns, usually—because they are cumulative… chairs and chairs are chairs just like soup and soup is soup. We explored a specific phenomenon in connection with this, the use of two kinds of measurement terms. One kind is the pseudopartitive, which is a measurement that makes use of the word of, like One centimeter of wire. The other kind is the compound, which is a measurement that just puts the measurement and the noun together (no of) like one centimeter wire. Intuitively, we know where the pseudopartitive is appropriate and where the compound is appropriate. After investigating it in some detail, we discovered that it has to do with a specific property of how the noun is measured. If the metric we choose is one that will be less for a subpart of the noun, and if the noun has subparts that are named with the same noun (like soup, which has subparts that are also soup), then we must use the pseudopartitive. Otherwise, we must use the compound. So, two feet of snow is measured by something like depth or length (in a line), where half of the snow would measure less than all of the snow. And 80° water or a 20mph wind uses a compound because half of the water would not measure less (it would still be 80° or 20mph). Two pounds of chair requires viewing chair as a mass noun (as basically chair-stuff) since the pseudopartitive requires a monotonic measure (as described, where there are parts described with the same noun and the measurement of subparts is less). Considering plurals, they seem to be collections of individuals of some kind, like a set. However, plurals can often be interpreted as either a group or as individuals. So if Three students lifted a piano, they might have lifted it together (collective responsibility) or each individually lifted a piano (a distributive interpretation). We can tell from predicates like touch that a group is really something like the sum of the individuals: I touched the ceiling means that some edge of me is in contact with some spot on the (edge of) the ceiling. If the boys touched the ceiling, they could all be in a pyramid with only one boy actually touching the ceiling (the group reading), or of course they could each individually be touching the ceiling (the distributive reading). Or there could be two pyramids (but then both have to have a boy on top touching the ceiling). You can group collections in a lot of different ways: The boys and the girls like each other. This could 6 mean that the boys like the girls, or that the boys like the boys and the girls like the girls, or that there are like-pairings between individuals in the bunch of people constituted by the boys and the girls together. There are a few predicates that actually require groups (collections, but considered as a unit). The committee gathered. And others that make no sense with groups. The students resemble each other. Some nouns must be groups, like committee. (*The committee resemble each other). Events as individuals Language seems to also allow reference to events as if they were just regular individuals. John saw Mary (seeing a regular individual) and John saw Mary leave (seeing an event). We can even quantify over events: Every student is happy (quantifying over regular individuals—if every student is happy, we can conclude from John is a student that John is happy), Every burning consumes oxygen (quantifying over events—if every burning consumes oxygen, we can conclude from Ray burned a book that oxygen was consumed). Events come in different types. There are states (basically inactive, like know French), activities (activities, with some duration in time but without defined boundaries, like swim), achievements (instantaneous, like pop a balloon, spot a bald eagle, win the race, eat), and accomplishments (like an activity plus an achievement, like build a house, eat an apple). There are three dimensions along which we can differentiate these types. Events can be static vs. dynamic (states are static, the rest are dynamic). They can take time or be instantaneous (achievements are instantaneous, accomplishments and activities take time). They can have a natural/logical endpoint or not (accomplishments have an endpoint, the others don’t). EVENT qp STATIC DYNAMIC state qp INSTANTANEOUS TAKES TIME achievement qp HAS ENDPOINT DOESN’T HAVE ENDPOINT accomplishment activity Very often the endpoint is provided by one of the verb’s arguments. For example, John ate is an activity, but John at an apple is an accomplishment. This is generally because the 7 progress through the event can be measured in terms of progress through the argument. The apple-eating is half over when the apple is half gone, and can’t continue once the apple is fully eaten. This interacts with the kind of argument used—if you use a mass noun or a bare plural as the measuring argument, this is insufficient to provide the kind of measure of progress we need to be an accomplishment. So, John ate apples or John ate soup both describe activities (not accomplishments). A common way to diagnose activities is to test the event with a durative modifier like for an hour. You can swim for an hour, but you can’t as easily eat two apples for an hour (to the extent that I ate two apples for an hour is comprehensible, it is interpreted as an activity). To diagnose accomplishments, you can test the event with a frame adverbial like in an hour. You can eat an apple in an hour, but you can’t really eat apples in an hour. To diagnose statives or achievements (as opposed to accomplishments and activities), you can test them in the imperative. You can tell someone: Walk! or Build a house!, but you can’t tell someone: Know French! or Find a quarter! (to the extent you can, you have coerced them into activities). Given that events seem to be basically like individuals, we can take the activity/accomplishment (or atelic/telic) distinction in events to be basically the same as the mass/count distinction. A part of a swim event is also a swim event, two (consecutive) swim events is also a swim event, and you can swim for an hour. A part of an eat an apple event is not an eat an apple event (it’s probably an eat apple event, just like part of a chair is not a chair, it’s [something made of] chair-stuff), nor are two (consecutive) eat an apple events an eat an apple event (it’s an eat two apples event), and you can eat an apple in an hour. Lexical decomposition Even apart from the knowledge we have about how sentences relate to one another and to events and to possible worlds, we also have some pretty specific knowledge of (intricate!) details of the meanings of individual words. Words in the lexicon can be divided into classes in terms of their behavior. We looked at the differences between the verbs cut, break, touch, and hit. There are quite a number of verbs that pattern with each, but they have distinctive behaviors. Hit and cut type verbs can appear in the conative (he hit at the door, he cut at the meat, *he touched at the wall, *he broke at the bottle); cut and break type verbs can appear in the middle (this meat cuts easily, this vase breaks easily, *this wall touches easily, *this punching bag hits easily); all 8 but break allow body-part ascension (Pat hit Robin on the arm, Pat touched Robin on the arm, Pat cut Robin on the arm, *Pat broke Robin on the arm). We take this to mean that things that act the same way have some feature in common: All but break involve CONTACT—so body-part ascension is allowed with verbs that involve CONTACT. Cut and break involve CHANGE to the object—so middles are allowed where the object is affected. Hit and cut involve significant MOTION—so conatives are allowed where MOTION is involved. By identifying features of this sort we can simplify our descriptions of which verbs participate in which constructions. We also know that kill and die are related, in that if George killed the dragon, then The dragon died. Many verbs seem to have this kind of relationship (lift/raise, give/receive, persuade/believe). We can make sense of this if the meaning of kill x is actually cause x to die. That is, CAUSE is a component of the meaning of lift, give, kill, persuade. We can extend this point of view to try to find universal semantic categories—such that verbs in the lexicon are composed of combinations of these. We can decompose the meanings of sentences like Bill went into the house into much more basic components: This is a motion event moving Bill along a Path that ends at a place that is in a thing, the house. The goal of this kind of research is to find the basic elements of meaning, probably shared by all languages. 9 A couple of others, primarily from Squad Helps Dog Bite Victim, Columbia Journalism Review Random House 1980 and its followup Red Tape Holds up New Bridge (Perigee 1987, Gloria Cooper ed.). Grandmother of Eight Makes Hole in One Milk Drinkers are Turning to Powder Farmer Bill Dies in House Dealers Will Hear Car Talk at Noon British Left Waffles on Falkland Islands Juvenile Court to Try Shooting Defendant Stolen Painting Found by Tree A couple found about (I think from Leno): 10