 The definition of meaning on the basis of words alone is problematic and the question arises as to whether the basic unit of meaning is not just the word but the sentence. This lecture covers the main approaches towards sentence meaning. First, it'll be shown how the meaning of a sentence can be defined in terms of simple or complex propositions and how propositions can be related by means of so-called entailment relationships. And finally, how the machinery of predicate logic can be used to describe the internal structure of propositions. First of all, however, let's define the central problems that arise with word meaning. Over and above the non-semantic problems of word definition, there are many difficulties in defining word meaning. For example, the problem of handling function words. How can we specify the meaning of items such as AND or however? Or we face cross-linguistic problems where across languages, the same concept may involve different degrees of transparency, such as thimble in English and finger hoot, literally meaning finger hat in German. Furthermore, there are numerous sentences whose meaning cannot be defined on the basis of the meaning of their words. In metaphors such as time is money, for example, the words involved do not reflect the overall meaning. The same applies to idiomatic expressions such as it is raining cats and dogs and often Paralinguistic features such as gestures, facial expressions, etc., accompany the uttering of linguistic constructions and influence their meaning. For this reason, semantics is not confined to word meaning but should incorporate the meaning of sentences as well. A central approach to what sentence meaning assigns truth values to sentences. This approach is referred to as propositional logic and it treats sentences as propositions. In logic, the meaning of sentences is defined in terms of statements or propositions and not just in terms of the sentence itself. One argument in favor of this treatment is that different sentences may be converted into identical propositions. Let's illustrate that. Here are John and Mary and obviously John is pointing a gun at Mary. So one sentence that could be derived from this situation is that John is killing Mary. Another one could be that Mary is being killed by John. A further sentence could be John is doing something to Mary and so on and so forth. However, in looking at these sentences in terms of their meaning, they essentially mean the same thing and this meaning is defined in terms of so-called propositions. Let us see how we can formalize sentence meaning in terms of propositions and how the machinery of propositional logic works. A proposition can be defined as the basic semantic content of a sentence or in more philosophical terms what a sentence says about the world. Propositions are labeled with small letters from P which stands for the initial letter of proposition upwards. So if you have one proposition only, it is always labeled P. If you have two propositions like in this sentence here, the sky is blue and it is raining, then we would have P and Q. P would be the sky is blue and it is raining is Q. Well, and if you have three propositions like the sky is blue, it is raining, it is cold, you would use P, Q and R and so on and so forth. Propositions have a truth value. So we could ask the question under what conditions is this proposition true where the capital T stands for true or false where the capital F stands for false? Well, this truth value by and large depends on our world knowledge, but also on a number of additional parameters. They include, for example, the general content. This proposition P is true if the content is in line with our basic understanding of the world. Another influential factor can be the speaker. This proposition P may be true, but what if a blind speaker utters this sentence? Can we say anything at all about its truth value? Well, depending on whether the blind speaker is right or not and then there may be the local or temporal context in within which the sentence is uttered. The proposition is only true. Our proposition here, the sky is blue, is only true if the sentence is uttered during daytime. So here it is false. Simple propositions can be turned into complex propositions by means of so-called logical constants or connectives. Depending on the type of connective, which is typically realized by an adverb or a conjunction in natural language, different expectations about the truth values of the individual propositions and the complex propositions can be inferred. Propositional logic has shown interest in five such connectives. So you see them here in the table. If you have something like P and Q, you would use one of these two symbols as the connector and the compound proposition is referred to as conjunction. A compound proposition is referred to as disjunction. If it is linked by any of these two symbols, the natural language equivalent would be or something like P or Q. If P then Q is referred to as the implication and here are the symbols. You see, you always have choices between several symbols, so logicians are a little bit lax on this. Negation simply means not P or not Q and you would use one of these symbols. And last but not least, if and only if P then Q, you would call this equivalence and here are the two symbols that can be used. Let us look at conjunction to take one of the our examples here. The conjunction closely resembles and in natural language. Here is an example. The bank was robbed and the police are on their way and is used in logic to construct compound propositions which are true. If all simple propositions, that is all the conjuncts are true. If any simple conjunct is false, the whole proposition would be false. This can be manifested by means of so-called truth tables. So the bank was robbed, let us assume this is true. The police are on their way, this is true, so P and Q is also true. If the bank was robbed is false, it was dropped at all, but the police are on their way, then the compound proposition is false too. And the other way round, if the bank was robbed, but the police are not on their way, then we have another false compound proposition. Well, and if both conjuncts are false, the whole compound proposition is also false. Such truth tables can be constructed for all logical connectives. The important point is that we now have a mechanism of defining sentence meaning in terms of the truth values of its propositions. And from the propositions which are true, we can draw inferences and these inferences are referred to as entailment. Now, formally entailment is defined as any true inference from a true proposition. It is symbolized by this symbol over here and we can say if P is true and Q is true, then P entails Q. Or the other way round, P entails Q if and only if P is true and Q is also true. Here are two examples. John regrets studying maths. Now P is John regrets studying maths and Q is, for example, John studies maths and clearly P entails Q. Or take this one. Brutus killed Caesar. P is Brutus killed Caesar. Q, well let's assume Q is Caesar died and certainly P entails Q. So the fact that Brutus killed Caesar entails that Caesar is dead. But how can we test this? Well there is an interesting entailment test. Let us illustrate this test which works on four individual steps. Here is our first example. Brutus killed Caesar. Now clearly P is Brutus killed Caesar. Now the first step is assume any proposition Q that is entailed or that could be entailed by P. So let's assume here Caesar died. So that is the inference, the entailment. Step two, make Q negative. Okay let's do it. Caesar did not die. Step three, conjoin P and not Q using and or bat or any such connective. P and not Q. So here we get Brutus killed Caesar and Caesar didn't die. Now think about the logic behind this. If the result is nonsense then P entails Q. If the result makes sense P doesn't entail Q. So that's the rule for entailment testing. So does this make sense Brutus killed Caesar and Caesar didn't die? Well not at all really. So it is not logical. In other words P entails Q. Let's try this test again. Again we take the same sentence and the same proposition P is Brutus killed Caesar. Now let's assume Q, Caesar was a slave. That is step one. We assume any proposition Q to be entailed by P. Step two, make Q negative. Here we go. Caesar was not a slave. Conjoin P and not Q. Here we go. Brutus killed Caesar and Caesar was not a slave. Step four, does it make sense or not? Well this could make sense because Caesar could have been anyone or we don't know whether he was a slave or not. In other words this is logical so P does not entail Q. With entailment we can now define further semantic relationships between propositions. For example we can define the relationship of paraphrase. Paraphrase can be defined as mutual entailment relation between two propositions. So if P entails Q then Q also entails P. The most typical natural language example of a paraphrase relation is the active passive relationship as over here. So P, John killed Mary, entails Q, Mary is being killed by John and the other way around. I'm sorry about these examples but they nicely illustrate my point and they can be reused in other propositional relationships. The relationship of contradiction holds if one proposition asserts the opposite of another. In terms of entailment P entails not Q. In other words we have some sort of sentence autonomy. Cases of contradiction in natural language typically involve negation as in John killed Mary versus John didn't kill Mary as indicated here by Q1. Or examples of the opposition of semantic features John killed Mary contradicts Mary is alive. Well and then we have inclusion. Inclusion can be defined as a unidirectional entailment between two propositions. If P entails Q then Q does not entail P. Natural language examples of inclusion typically involve like seems that are in a hyponymy relationship. So for example John flew to London includes John went to London but not vice versa. Or here in our case John killed Mary entails that John did something to Mary but John did something to Mary doesn't entail John killed Mary. Propositional logic as introduced here treats simple propositions as atomic as if they have no internal structure. However with this limitation we cannot describe relations or quantification. To handle such phenomena successfully it is necessary to consider simple propositions as having an internal structure that corresponds to the structure of the state of affairs in the external world. The meta language for the description of these internal structures is referred to as predicate logic. So let's look at predicate logic next. The simple proposition P. John loves Mary describes a state of affairs in the world where two individuals John and Mary are in a relationship with each other in which the former loves the latter. If we label this relationship like this X loves Y and replace X with John and Y with Mary we arrive at a representation of the internal structure of this proposition which looks like this. The essential idea behind such a presentation is that each proposition can be defined as a so called predication. Predications in turn consist of a predicate which is always capitalized that is the initial letter is by convention a capital letter. And a set of arguments theoretically unlimited which are always represented in small letters. Such a predication can be read as follows. Let's say the predicate is P and the argument is X then we could say X is a P. Let's look at some examples. Now here's the first. Great Britain is a country is the proposition. Now obviously we have a predicate country with one argument Great Britain. This is a so called one place predicate. A two place predicate has two arguments John loves Mary the predicate is love and the two arguments are John and Mary. Well and here is a three place predicate. Jane sends Paul her love send the predicate Jane Paul and love the arguments. Thus each predicate semantically defines its arguments. For example verbal predicates define the thematic role of their arguments. This is a well known procedure which crosses the borderline between syntax and semantics the so called thematic relations agent goal and theme. Additionally predications can involve quantifiers. While simple predications represent relations between entities the addition of quantifiers allows the formalization of relationships between sets of entities. For example here we have two sets the set of bald people and the set of linguists. Let's assume we want to express that all linguists are bald. Now all what does all denote all denotes the subset relation between two sets that is one set is totally included within the other. In the example the set of bald people includes the set of linguists so we have something like this. In other words there is no linguist who is not also bald. Well and here is the predicate logic representation. We have two predicates linguist with one argument and bald with one argument and the arguments are identical because we're talking about the same person. Well on the upside down A which stands for the universal quantifier simply means and this is how to understand this formula for all elements X it holds if X is a linguist then X is also bald. Here is another possibility let's assume the following relationship. Now here we do not have subsets but we have the relationship where there is at least one element that both sets have in common that is the two sets intersect. The predicate logic representation is this one and now the upside down A denotes the existential quantifier. It denotes the intersection relation between two sets. In other words we can interpret it like this there is at least one element X such that if X is a linguist then X is bald. And here is the third relationship the not relationship. Now how do we express this one well here the two sets do neither intersect nor are they included within one another. We simply express the relationship that we have a disjunction relation between two sets there is no intersection in this example. The set of linguists has no common element with a set of bald people in other words there are no linguists who are also bald. Well and the use of the symbol is quite simple there is no element X such that if X is a linguist then X is also bald. Let's look at further examples of predicate logic to deepen our understanding. Here is the first one all men love Mary. Well we already know this formula we need the universal quantifier which says for all X it holds if X is a man then X loves Mary. Number two a man sneezed well we need the existential quantifier there is at least one element X such that X is a man and X sneezed. Well and here's the final one nobody likes Mary and here for a change I use two different representation which essentially mean the same thing. For no X it holds if X is a person then X likes Mary or we could say and use the existential quantifier. It is not the case that there is at least one X for which it holds that X is a person and X loves Mary. So you see with predicate logic propositions can be analyzed internally allowing a detailed semantic formalization. Well this e-lecture could not unveil all details about logic this will be done in separate e-lectures about propositional logic and predicate logic. However it should have outlined the essential ideas of sentence semantics where sentence meaning is first defined in terms of propositions whose internal structure can be defined by means of predicate logic. Even though there are limitations for example the exclusion of non declarative aspects such as the function of sentences in particular contexts propositional and predicate logic are suitable tools for the formalization of sentence meaning. I hope that I've shown this at least to some extent so thank you and see you again.