 Hello, welcome to the e-lecture Predicate Logic 1. Predicate Logic is the most widely used meta language for the representation of the internal structure of simple propositions. The topic is too complex to be covered in one e-lecture. Thus I will split it into two parts, into Predicate Logic 1, this part, where I am going to explain the basic machinery of Predicate Logic and into a second part, Predicate Logic 2, where the principles of quantification will be discussed and where the combination of propositional and predicate logic will be exemplified. But why do we need predicate logic in the first place? Couldn't we say that truth conditional semantics is sufficient for the description of sentence meaning? Well, not really. Propositional logic treats simple propositions as atomic, that is, as if they had no internal structure. However, this assumption limits its usefulness as a tool for the description of meanings. Here are two examples. The first one is concerned with quantification. Propositional logic treats propositions as either identical or totally different. However, it may be the case that two propositions are only partially different, as in these two examples which differ in terms of quantification. P, all linguists are bald versus Q, some linguists are bald. Now, all versus some is the key issue here. Since there is no way of expressing quantification, all or some, in propositional logic, it is not possible to point out the differences between these two propositions and to simultaneously capture what they have in common. Note that I've already used some symbols used in predicate logic, for example the universal quantifier, the upside down A, the existential quantifier, and here some variables that are part of my examples. The second problem for propositional logic can be referred to as the problem of relations. Since propositional logic has no access to the internal structure of propositions, it fails to notice important relationships among them. For example, if we look at the following compound statement. John is taller than Mary, and Mary is taller than Paul, and Paul is taller than Linda. From such a compound statement, we can conclude that John is in fact taller than Linda. But this conclusion is based on our logic of the meaning of the relation X is taller than Y, holding inside each component proposition. It cannot be drawn from the corresponding expression in propositional logic. To handle such phenomena successfully, it is necessary to consider simple propositions not as unanalyzable units, but as having an internal structure that corresponds to the structure of the state of affairs in the external world. The machinery that describes the internal structure of propositions is referred to as predicate logic, also called predicate calculus or first order logic. It shifts the focus from the logical relations that hold between sentences to those that hold within sentences. The essential idea is that each proposition can be defined as a predication. Predications themselves consist of a predicate, and the predicate itself always occurs with an initial capital letter, so it is capitalized, and a set of arguments represented with small initial letters. Well, and you can read it as follows. So, here we have the predication Px, and it can be read as X is a P. So, let's look at an example to illustrate this in more detail. The proposition John loves Mary can be broken apart into the following components, into a predicate love, and into two arguments John and Mary. So, X would be John and Y is Mary. Whereas the corresponding predication would look like this, love John Mary, or with variables love X, Y. As we can see, predications consist of two basic elements, predicates and their arguments. Each predication may have only one predicate, but it may have one or more arguments. So, let's look at predicates in more detail. A simple predicate is one that assigns some property to its argument. So, here we have the predicate country with one argument Great Britain, and it can be read as Great Britain is a country. Other examples would be something like John is tall, Mary is pretty, and so on. More complex predicates represent relationships between the entities denoted by their arguments, if there is more than one. So, here we have a predicate, love, and two arguments, John and Mary, and the proposition that can be associated with this predication is John loves Mary. Or take this one, Jane sends Paul a letter, that's our proposition R, and the predication would be send with the predicate send, and three arguments, Jane, Paul and letter. As already mentioned by convention, the labels for predicates always start with a capital letter. And according to the number of arguments required by a predicate, often a distinction is made between one place predicates with one argument, two place predicates with two arguments, and three place predicates with three arguments. Well, and if there are more arguments, you could call them n place predicates with n, with any number of arguments. The argument structure itself specifies not only the number of its arguments, the number of the arguments of a predicate, but also their order. For example, the proposition the hunters shot someone can be represented by the predication that you can see here. The predicate is shoot and hunters and someone are the two arguments. But what about the proposition Q, someone shot the hunters? Well, this different state of affairs is represented in the order of the arguments of the predicate shoot. So in order to represent this adequately, we would have to swap the arguments. And this is our new predication, same predicate, same arguments, but different order of arguments. So the order of the arguments of a predicate is highly influential with regard to the meaning of a predication. Well, and what about the argument types? Well, the arguments of a predicate are entities minimally involved in the property or relation expressed by the predicate. If one is missing or unspecified, the predication is incomplete. The arguments of a predicate themselves may be any logical term. The simplest argument type is referred to as individual constant. An individual constant is a term that refers to a particular individual in the world. And the range of individuals includes the people, things and places of everyday life that can be recognized as distinct and identifiable. For example, cat, dog, Great Britain, John and so on and so forth. It should be noted that the task of giving a name to each individual is much easier in simplified artificial worlds than it is in the real world. And then there are so called individual variables. While individual constants stand for particular individuals, an individual variable stands for an arbitrary individual. In other words, variables range over all individuals. And variables are denoted with a lower case characters x, y, z and so on. With variables, we can form expressions such as teach x. So something like your going to teaching. Or we can create this predication, read x, y. So in this case, the children are reading a book. By themselves, these predications do not express any proposition because the variables x and y have no fixed reference. Well, finally, arguments can be more complex than simple constants and variables. In fact, they can be whole predications themselves, each with its own predicate argument structure. Let us construct such a predication. Here is a complex proposition. The teacher saw that the children were reading a book. And in fact, this complex proposition can be broken down into two propositions. P, the teacher saw something and Q, the children were reading a book. And both P and Q can be translated into individual predications. Here is the first C teacher. So the teacher saw something, C teacher. And here is the second. The children were reading a book, read children book. In putting them together, we now use the second predication as an argument of the first. So this is what we have to do. Here is our first predication C teacher. And then we need a second argument. Well, the second argument is the entire second predication Q. So this is now our complex predication with two arguments. The first argument is an individual constant. And the second argument is a predication. So predications can be arguments of predicates. Shall we practice a little bit? Okay, let's look at some examples. Here is the first. John is a linguist. The predicate is linguist. There is one argument, John. And this over here would be the predication linguist John. Another one, Mary snores. Again, a one-place predicate. Snore. One argument, Mary. And the present predication. Well, you should now understand the system. Let's take a two place predicate next. Admire is the predicate with two arguments, Bill and John. And the predication admire Bill John. And what if we reverse the situation John admires Bill? Well, then we virtually have the same. We still have the same predicate, but the order of the arguments has to be reversed. So now we have admire John Bill. Let's take a three-place predicate next. Linda gave Paul the book. Predicate give. Linda Paul book, three arguments. And here is the predication. And finally, here we have a complex predication with one predicate, C. And the predicate itself has two arguments where the first argument is John and the second argument is what John saw, namely that Mary left. So we have the first argument a constant and the second argument is a predication in its own right. So you should have understood the system I think. So let's stop here and let's summarize this first part. We first showed the limitations of propositional logic for the description of sentence meaning and introduced the idea of splitting up propositions into predications, which in turn consists of predicates and their corresponding arguments. Having understood this basic formalism of predicate logic, we can now integrate principles of quantification, as well as the set of logical connectives that I introduced in my e-lecture on propositions. How this can be done will constitute the focus of my second e-lecture on predicate logic. So see you there.