 Hello, in the e-lecture sentence semantics we saw that the definition of meaning on the basis of words alone is problematic and that a reasonable alternative looks at sentence meaning. This additional e-lecture is a continuation of the discussion of sentence meaning with emphasis on propositional logic. We will look at the machinery of propositional logic. We will draw a distinction between simple and complex propositions and we'll finally talk about the limitations of propositional logic. In the introductory e-lecture on sentence meaning we looked at various word-based phenomena to justify the fact that sentence meaning is not a composition of word meaning. We identified problems of handling function words, we looked at cross-linguistic problems and we discussed numerous sentences whose meaning cannot be defined on the basis of the meaning of their words. For example, metaphors such as time is money and idioms such as it's raining cats and dogs. These observations led to an approach which is often termed formal or truth conditional semantics. This approach uses the machinery of propositional logic to assign formal meaning descriptions to sentences and to specify meaning relations between sentences in terms of entailment and presuppositions. Let us start with the central ideas of truth conditional semantics. Truth conditional semantics focuses on the referential properties of language that is how linguistic expressions in particular, how sentences relate to reality. Sentences or more precisely their semantic content here translated into a conceptual dependency representation are held to be expressions which can be judged true or false. That is they can be assigned a truth value. Knowing the meaning of a sentence involves knowing what the world has to be like for the sentence to be true. In other words, its truth conditions are known. To describe these truth conditions a meta language is used. Do you know what a meta language is? Well, a meta language is typically formal language used to make statements about a language. Let's say you want to make statements about the natural languages for example about Japanese. Then you need a general symbolic system with which you can make statements about Japanese and many other languages. That is a meta language. Here you see some of these symbols such as noun phrase pro, infill, verb, past and so on. Meta languages that have been proposed for the formalization of meaning include among others languages that deal with meaning components or the system of predicate calculus to name just two of them. In truth conditional semantics the meta language uses logical formula which can be subjected to rigorous truth tests. The basic elements in these formula are referred to as propositions. 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. One argument in favor of using such an abstraction is that different sentences may be converted into identical propositions. For example, all these sentences here, Brutus killed Caesar, Brutus cost Caesar to die and so on. All these sentences denote essentially the same thing. That is they have the same meaning. This meaning can be defined in terms of so-called propositions. Propositions are labeled with small letters starting with P, the initial letter of the word proposition, and then using letters upwards in the alphabet. So if you have one proposition only it is always labeled P, the sky is blue. If you have two propositions well then you take the next letter Q, then you have P and Q and if you have three the next letter in the alphabet would be R, the sky is blue and it is raining and it is called three propositions P, Q and R. The following properties are normally ascribed to propositions. Propositions, first of all, have a truth value. That is they can be true and then we would write a capital T for true or false which is denoted by means of a capital F. Propositions may be known, they may be believed, they may be doubted, they may be asserted, they may be denied or queried. So very often the truth value depends on a number of additional facts. We will come back to this in a second. And thirdly, propositions remain constant even when they are translated between languages. So the proposition P is identical, only their translation into German, French, English and Spanish is different. However, the truth value of propositions by and large depends on our world knowledge and on a number of additional parameters. They include, for example, the general content. So this proposition, the sky is blue, is true if the content is in line with our basic understanding of the world. So if the sky is really blue, another parameter which can influence the truth value is the speaker who utters the sentence. So this proposition P, the sky is blue, may be true or not depending on the speaker. So if a blind speaker like this one over here utters this sentence, we cannot tell what the speaker knows about the truth value of this utterance. And it may depend on the temporal and local context within which the sentence is uttered. So this proposition is only true if the sentence is uttered during daytime. Some propositions are true or false by virtue of the meaning relations between the words contained in them, regardless of what the situation in the world might be. Such propositions are called analytically true or depending on their truth value analytically false. Examples include so-called tautologies which are always true. A tautology is a statement that is always true and therefore gives no information. So either it is raining or it is not raining is certainly true but the information in it is virtually nil. Stupid people are stupid, similar, again the informational value of this true proposition is useless. And then we have contradictions which are analytically false. Contradictions are statements which exhibit an internal consistency either in itself as in today is not today or with another proposition as in Brutus killed Caesar but Caesar is not dead. There are other propositions whose truth or falsity does not automatically follow from the meanings of their elements but it depends on whether they are accurate descriptions of reality or not. The truth of such propositions is synthetic rather than analytic in nature. They cannot be verified without checking the state of affairs in the world. So here we have two examples. My father is a rocket scientist. To verify the truth value of this you need some background knowledge and the other one all linguists are unhappy. Well the same thing. We have to take into account what we know about the world of linguists. Are they really all unhappy? Well I'd rather doubt it. Okay thus far we have mainly been talking about simple propositions. Let's now see how complex propositions can be analyzed. The central means to build complex propositions by combining simpler ones are the so-called connectives or logical constants. There are five such connectives which in natural language are typically realized by an adverb or a conjunction. Well here they are. We have the conjunction and we have a disjunction. Natural language equivalent is all. In the middle you see the symbols that are used in propositional logic. Then we have an implication, the if-then relationship. Then we have a relationship called negation. Not is the natural language equivalent. And finally a relationship of equivalence which or whose natural language equivalent is if and only if, often abbreviated to if with two f's. Note that in each case you have alternative options to select one of these symbols. So the truth value of a complex proposition is now fully determined by the truth value of its component propositions and the specified effect of each logical connective involved. The possible combinations of truth values and the effects of the connectives are shown in so-called truth tables. So let us go through all these connectives in detail. Let's start with the conjunction. The conjunction closely resembles and in natural language. Here is an example. The bank was robbed, P, and the police are on their way, Q. And is used in logic to construct compound propositions which are true if all simple propositions, the so-called conjuncts, are true. So let's write that down in a truth table. So we have P, the bank was robbed, Q, the police are on the way. So if P is true and Q is true, well then the whole conjunction is true. The bank was robbed is true and the police are on the way. If P is true, the bank was robbed, but the police are not on their way. This is false. Well then, what happens to the conjunction? It's of course false. And the other way around, if the bank was not robbed but the police are on their way, well then the conjunction, the bank was robbed and the police are on their way is of course false too. Well and if both conjuncts are false, well then the overall conjunction is false too. So in other words, we can summarize and say if any simple conjunct is false, the whole proposition is false too. Let's continue with the disjunction. The disjunction closely resembles or in natural language. Given our present knowledge, the following statement should be true. Mars is a satellite of the sun or a planet of the solar system. So let us create the corresponding truth table again. If both disjuncts, now here the two propositions are not called conjuncts but disjuncts, if both disjuncts are true, well Mars is a satellite of the sun and Mars is a planet of the solar system, well then the disjunction is true. If one of these disjuncts is true and the other one is false, well then quite interestingly, Mars is a satellite of the sun, true, or a planet of the solar system false, the whole disjunction is still true. And the same the other way around. So if one of the disjuncts is true, the disjunction is true. Well, if both are false, well then of course the disjunction is false too. But what about this sentence here, these statements. Mars is either a star, well there the A is missing, a star. Mars is either a star or a planet. Well in such a case, we would have to construct an extra truth table. This is a stricter type of disjunction often referred to as exclusive disjunction. It is true only if one of the disjuncts is true. So in this particular case, if both are true, which is really the case, the disjunction becomes false. So in other words, this is the new version of, or this line replaces the top line in the truth table of the inclusive or. The disjunction is used in logic to create a compound proposition which is true if one disjunct is true. That summarizes both cases of the disjunction. Let's look at the implication next. The implication, if then, is less similar to its corresponding expression in ordinary language than any other connective. In logic, it is usually called the material implication and it is only false if its antecedent p, the if clause, is true and its consequence, the then clause, is false. Here is an example. If it rains antecedent, then I'll do my homework, the consequence. So let us now construct the truth table using this example. Well, and here we can postulate the following. This implication can only be false if it rains and the person producing the sentence doesn't do his homework. Okay? So in other words, we have this particular situation, p is true and q is false. If it rains, it's true and then I don't do my homework, then of course the implication is false. If it doesn't rain, that is, if p is false. So that's these two cases. If it doesn't rain, the implication cannot be invalidated by whatever the person does. So if q is true or if q is false, it doesn't matter. So in both cases, the implication, if p then q, must be true. Well, in the first case, if both are true, well, that's quite simple. Then of course the implication is true too. So once more, the implication can only be false and that's this line here in the truth table. The implication can only be false if it rains, that is, p is true and the person producing the sentence doesn't do his homework despite his or her announcement. If it doesn't rain, the implication cannot be invalidated at all. So whatever the person does, it doesn't really matter. Okay? So that's a little bit more complicated than the disjunction and the conjunction. The next connective we have to deal with is negation. Now negation closely resembles not in natural language. It is used in logic to construct compound propositions with the opposite truth value of that of a simple proposition on which it operates. Assuming a standard knowledge about the world, the following proposition is true. Mary is not a woman. Well, and the truth table is now quite simple. If p is true, Mary is a woman, then not p must definitely be false. Well, and if Mary is a woman is false, so if Mary is a man, then Mary is not a woman would be true. Quite simple, isn't it? The final connective, equivalence, roughly corresponds to if and only if. And it is often abbreviated to if, spelled with two f's in ordinary language. Here is an example. Mary is a woman if she is female, if and only if she is female. The equivalence is used in logic to construct compound propositions which are true if all simple propositions have the same truth value. Thus the following compound propositions are true. It is quite simple in a truth table. So if p is true, if Mary is a woman and she is female, then Mary is a woman if she is female, if and only if she is female. If Mary is a woman but she is not female, well, then the equivalence is false. If Mary is not a woman but she is female, we have the same situation. And if Mary is not a woman and Mary is not female, well, then the equivalence relationship becomes true. So much for the discussion, for the detailed discussion of the connectives. Now propositional logic, as you may guess, does not capture all aspects of the meaning of a sentence. It just deals with the abstract propositional core. Let us look at some examples to illustrate these limitations. The first set of examples concerns the use of the connectives in logic and in natural language. The logical connectives can only capture the meaning of the propositional core. In many cases, however, there seem to be additional links between the component propositions. In the statements, Linda arrived late at the airport and missed her plane, versus Linda missed her plane and arrived late at the airport, the departure airport. We have the same truth values. We have a conjunction consisting of two propositions and both are true, so the conjunction must be true. However, in natural language, only the first statement should be acceptable due to the necessity of ordering the events. You cannot miss a plane before arriving late. So, in other words, this here is at least questionable. Take another example. My parents were poor and honest. My parents were poor but honest. Again, we have the same truth values. We have a conjunction consisting of two conjuncts in each case and both are true. But despite the identity of truth values, the sentences do not mean the same thing. Even though we can treat and and but in logic, like logical and, but add something to the two propositions. In this case, a sort of adversive meaning, a meaning of contrast. Or look at adverbial relations. Linda studied a lot and passed her exams. Linda didn't study at all and passed her exams. Again, we have two conjunctions, but the propositional content excludes adverbial relations such as causal, studying hard as a reason for passing the exams, or concessive relations, the sort of contrast between the fact that she didn't study and nevertheless passed. Furthermore, the force of words such as yet, still and already is ignored in propositional logic. As a result, these sentences here, Linda is outside, Linda is still outside, Linda is already outside, which denote one proposition in each case, which can be assumed to be true. They have all the same propositional content, but they are obviously neither identical in meaning nor appropriate in the same situation. And last but not least, we have other dimensions of meaning. You remember the dimensions in word semantics that were introduced, for example, by Jeffrey Leitch, such as stylistic meaning, effective meaning, attitudes indicated by different degrees of politeness. All these additional dimensions are not part of propositional logic. So, having discussed the basic machinery of propositional logic, the question arises, what can we do with it? To answer this question, we should proceed as follows. On the one hand, we should now find out how to draw true inferences from true propositions and how to relate propositions to one another. On the other hand, we should formalize the internal semantic structure of propositions. Both aspects are important within sentence semantics. The first is treated under the heading of entailment. The second is referred to as predicate logic. Both deserve special treatment in separate e-lectures. So, see you there.