 Hello again. In the e-lecture Propositions, I introduced the central machinery of propositional logic, that is, the main principles behind truth-conditional semantics, including the contribution of the logical connectives to the truth value of complex propositions. This e-lecture takes up the ideas of propositional logic in order to capture meaning relationships between sentences or more precisely propositions. In other words, we will talk about the relationship of entailment in general. We will see how entailments can be tested and we will make use of entailment in order to define semantic relationships between propositions. As you may recall, the essential idea behind propositional logic is that we now have a mechanism of defining sentence meaning in terms of the truth value of its propositions. For example, the complex proposition P and Q, the sky is blue and it is raining, must be false, according to our understanding of the world. The conjunction P and Q requires that P and Q must both be true, but this cannot be the case because it cannot at the same time rain and the sky is simultaneously blue. So one of the two conjuncts is false according to our understanding of the world, so the conjunction P and Q must be false. The interesting thing is that we can now draw logical inferences from the single propositions which are true. Assuming that the sky is blue, one such inference is that the sun is shining. And from it is raining, from Q we can infer that there is at least one cloud in the sky since according to our knowledge of the world one prerequisite for rain is that there are clouds in the sky. These logical inferences are referred to as entailment. Formally entailment is defined as any true inference from a true proposition. So if P is true then it may entail and this here is the logical symbol, the propositional symbol for entailment. If P is true it may entail any other proposition Q, R, S and so on and so forth. However, for a proposition to entail any other proposition a set of conditions has to be satisfied and also these entailments can be tested. Let us illustrate that. Here is the first example. Let's take the following proposition P Brutus killed Caesar. And this proposition entails that Caesar died. So here is our representation P entails Q. Let us now see under what conditions we have such an entailment relationship. Now let's first of all consider the relationship P entails Q. If P is true, that is Brutus actually killed Caesar, then Q, Caesar is dead, Caesar died is true. Two, so our entailment relationship holds it is true. If P is false, if Brutus did not kill Caesar, well then Caesar may be dead for any other reason or Caesar may still be alive so P does not entail Q. And let's now consider the other way around. Q versus P, so Q entails P. So if Q is true, that means if Caesar is really dead, well then we cannot really tell anything about P. Brutus may have killed Caesar, but it could also have been any other person who killed Caesar. So this is clearly no entailment relationship. Well and if both are false, so if Caesar is not dead, well then no one can have killed him. In other words, the entailment relationship is also not valid here. In other words, entailment is a logical relationship between two propositions such that the truth of the first proposition P guarantees the truth of the second proposition Q. And also the falsity of Q guarantees the falsity of P. Here is another example. All dogs are purple, so P is all dogs are purple. Well and this entails my dog is purple. And again we know that the truth of P entails the truth of Q and that if Q is false then P must also be false. So entailment is a semantic relationship between two propositions such that if and only if, spelled with two Fs, if and only if P is true then Q is true too. Well then this opens the door for an entailment test. So let's see how entailments can be tested. Let us illustrate the entailment test which is also sometimes referred to as negation test on the basis of a number of steps. This is how it works theoretically. Take any proposition P, that's step one. Step two, assume any proposition Q to be entailed by P. Then make Q the proposition which you assume to be entailed by P, make that proposition negative. Step four, conjoin P and not Q using logical and or you can also use any natural language equivalent such as but this sometimes makes the contrast more obvious. Well and then look at the result. If the result is a contradiction or even nonsense, P does entail Q. So then the entailment relationship holds. If the result is fully plausible and no contradiction at all then P does not entail Q. Let us apply this test to some real examples. Here is the first. Step one, take any proposition P, our proposition is all dogs are purple. Step two, assume any proposition Q that is to be entailed by P. My dog is purple is our entailment, our Q. Make Q negative. My dog is not purple. Now conjoin P and Q using logical and or any other natural language equivalent. The result is all dogs are purple and my dog is not purple. All dogs are purple but my dog is not purple. Well let's now evaluate the result. Well here we clearly have a contradiction. If all dogs are purple my dog cannot be not purple. In other words Q is correctly entailed by P. Here is another example. Again the same proposition all dogs are purple. Now step two we assume any proposition Q to be entailed by P. This time we assume that my dog likes cats is entailed by P. Step three we have to make Q negative. My dog does not like cats and now we conjoin P and not Q. All dogs are purple and my dog does not like cats or all dogs are purple but my dog does not like cats. And what about the result? Well here we do not have a contradiction at all. The fact that my dog doesn't like cats does not contradict the fact that all dogs are purple. So Q is not entailed by P. Having developed a suitable test for entailment relations we can now use entailment to define further semantic relationships between propositions. In fact there are three relationships that may be defined between propositions on the basis of entailment. We have the relationship of paraphrase or mutual entailment. We have the relationship of contradiction and the relationship of inclusion. Let us look at them in detail and start with paraphrase. Paraphrase can be defined as a mutual entailment relation between two propositions which can logically be expressed by a conjunction. P entails Q and Q entails P. The most typical natural language example of a paraphrase relationship is the active passive relation. Mary loves John, entails John is loved by Mary and John is loved by Mary, entails Mary loves John. So mutual entailment. Another paraphrase relation holds if Q contradicts the opposite of P. That is a sort of double negation. All dogs are purple, entails there is no dog that is not purple and there is no dog that is not purple entails all dogs are purple. The third type represented here rephrases P by taking up an essential semantic feature of it. Ignoring specific problems with gender and sex and age we can say John is a man, entails John is male. John is male and adult and John is male and adult entails that John is a man. Whereas paraphrase can be seen as some sort of sentence synonymy, contradiction our next relation realises some sort of sentence autonomy. The relationship of contradiction holds if one proposition asserts the opposite of another. In terms of entailment P entails not Q. Cases of contradiction in natural language typically involve negation as in Mary loves John which contradicts Mary does not love John. Or it may involve general cases of incompatibility where all dogs are purple contradicts my dog is gray. Here we have incompatibility between purple and gray as two colors of the same semantic field. Or we may have cases of semantic feature opposition. John is a man contradicts John is female again according to a default approach towards gender, sex and age. Finally we can define a relationship of inclusion if you wish some sort of sentence hyponymy. Inclusion can be defined as a standard entailment or unidirectional entailment a relationship between two propositions. If P entails Q then Q does not entail P. Natural language examples of inclusion typically involve leg seems that are in a hyponymy relationship. Mary loves John includes Mary is fond of John but not vice versa. Or take propositions such as Mary flew to London which includes Mary went to London. Other hyponymy relationships may involve simple hierarchies such as the color hierarchy. All dogs are purple includes my dog is colored but not the other way round. Or we can take semantic features that are captured by means of redundancy rules such as if something is male then it has to be human. So John is a man includes John is human. Well that's it. This lecture should have outlined the essential ideas of sentence semantics where sentence meaning is defined in terms of propositions which in turn can be true or false. Furthermore it should have demonstrated how we can draw inferences from true propositions the so called entailments and how entailments can be used to define propositional relationships. In fact we defined three propositional relationships the relationship of paraphrase if you wish some sort of propositional synonymy. The relationship of contradiction if you wish some sort of propositional antonomy. And the relationship of inclusion if you wish some sort of propositional hyponymy. I know that many students of language and linguistics are not particularly fond of logic. I'm not sure whether I could increase your interest in this field but I hope that I succeeded at least to some extent in making you aware that propositional logic is an important contribution to the definition of sentence semantics. So thank you very much.