I have said that arguments are composed of declarative sentences. Some logicians prefer to say that arguments are composed of the things we say with sentences, that is, statements or propositions. Sentences can be problematic in logic because sentences are often ambiguous. Consider this sentence:
(3) I took my brother's picture yesterday.
I could use this sentence to mean that yesterday I made a photograph of my brother. Or I could use the sentence to mean that I stole a picture that belonged to my brother. Actually, this sentence can be used to say a rather amazingly large number of different things.
Ambiguous sentences can make a problem for logic because they can be true in one way of understanding them and false in another. Because logic has to do with the truth and falsity of premises and conclusions in arguments, if it is not clear whether the component sentences are true or false, we can get into some awful messes. This is why some logicians prefer to talk about statements or propositions which can mean only one thing. In a beginning course, I prefer to talk about sentences just because they are more familiar than statements and propositions. (What are statements and propositions supposed to be, anyway?) We can deal with the problem of ambiguity of sentences by insisting that we use only unambiguous sentences, or that we specify the meaning which a possibly ambiguous sentence will have in an argument and then stick to that meaning.
Actually, in most of our work we will be concerned with certain facts about the logical form of sentences and we won't need to know exactly what the sentences mean in detail. All we will need in order to avoid problems about ambiguity is that a given sentence be either true or false (although we usually won't know which it is) and that the sentence should not change from true to false or from false to true in the middle of a discussion. As you will see very soon, the way we will write sentences will make it extremely easy to stick to these requirements.
In fact, by restricting our attention to sentences which are either true or false, we have further clarified and extended our restriction to declarative sentences. Questions ('Is Adam happy?'), commands ('Cheer up, Adam!'), and exclamations ('Boy, is Adam happy!') are not true or false. Neither, perhaps, are some declarative sentences. Many people don't think "The woman who landed on the moon in 1969 was blond." is either true or false because no woman landed on the moon in 1969. In any case, we shall study only sentences which are definitely one or the other, true or false.
We will initially study a very simple kind of logic called Sentence Logic. (Logicians who work with propositions instead of sentences call it Propositional Logic.) The first fact on which you need to focus is that we won't be concerned with all the details of the structure of a sentence. Consider, for example, the sentence 'Adam loves Eve.' In sentence logic we won't be concerned with the fact that this sentence has a subject and a predicate, that it uses two proper names, and so on.
Indeed, the only fact about this sentence which is relevant to sentence logic is whether it happens to be true or false. So let's ignore all the structure of the sentence and symbolize it in the simplest way possible, say, by using the letter 'A'. (I put quotes around letters and sentences when I talk about them as opposed to using them. If this use of quotes seems strange, don't worry about it-you will easily get used to it.) In other words, for the moment, we will let the letter 'A' stand for the sentence 'Adam loves Eve.' When we do another example we will be free to use 'A' to stand for a different English sentence. But as long as we are dealing with the same example, we will use 'A' to stand for the same sentence.
Similarly, we can let other capital letters stand for other sentences. Here is a transcription guide that we might use:
A: Adam loves Eve.
B: Adam is blond.
C: Eve is clever.
'A' is standing for 'Adam loves Eve.', 'B' is standing for 'Adam is blond.', and 'C' is standing for 'Eve is clever.' In general, we will use capital letters to stand for any sentences we want to consider where we have no interest in the internal structure of the sentence. We call capital letters used in this way Atomic Sentences, or Sentence Letters. The word 'atomic' is supposed to remind you that, from the point of view of sentence logic, these are the smallest pieces we need to consider. We will always take a sentence letter (and in general any of our sentences) to be true or false (but not both true and false!) and not to change from true to false or from false to true in the middle of a discussion.
Starting with atomic sentences, sentence logic builds up more complicated sentences, or Compound Sentences. For example, we might want to say that Adam does not love Eve. We say this with the Negation of 'A', also called the Denial of 'A'. We could write this as 'not A'. Instead of 'not', though, we will just use the negation sign, '~'. That is, the negation of 'A' will be written as '~A', and will mean 'not A', that is, that 'A' is not true. The negation sign is an example of a Connective, that is, a symbol we use to build longer sentences from shorter parts.
We can also use the atomic sentences in our transcription guide to build up a compound sentence which says that Adam loves Eve and Adam is blond. We say this with the Conjunction of the sentence 'A' and the sentence 'B', which we write as 'A&B'. 'A' and 'B' are called Conjuncts or Components of 'A&B', and the connective '&' is called the Sign of Conjunction.
Finally, we can build a compound sentence from the sentence 'A' and the sentence 'B' which means that either Adam loves Eve or Adam is blond. We say this with the Disjunction of the sentence 'A' and the sentence 'B', which we write as 'AvB'. 'A' and 'B' are called Disjuncts or Components of 'AvB', and the connective 'v' is called the Sign of Disjunction.
You might wonder why logicians use a 'v' to mean 'or'. There is an interesting historical reason for this which is connected with saying more exactly what 'v' is supposed to mean. When I say, 'Adam loves Eve or Adam is blond.', I might actually mean two quite different things. I might mean that Adam loves Eve, or Adam is blond, but not both. Or I might mean that Adam loves Eve, or Adam is blond, or possibly both.
If you don't believe that English sentences with 'or' in them can be understood in these two very different ways, consider the following examples. If a parent says to a greedy child, 'You can have some candy or you can have some cookies,' the parent clearly means some of one, some of the other, but not both. When the same parent says to an adult dinner guest, 'We have plenty, would you like some more meat or some more potatoes?' clearly he or she means to be offering some of either or both.
Again, we have a problem with ambiguity. We had better make up our minds how we are going to understand 'or', or we will get into trouble. In principle, we could make either choice, but traditionally logicians have always opted for the second, in which 'or' is understood to mean that the first sentence is true, or the second sentence is true, or possibly both sentences are true. This is called the Inclusive Sense of 'or'. Latin, unlike English, was not ambiguous in this respect. In Latin, the word 'vel' very specifically meant the first or the second or possibly both. This is why logicians symbolize 'or' with 'v'. It is short for the Latin 'vel,' which means inclusive or. So when we write the disjunction 'AvB', we understand this to mean that 'A' is true, 'B' is true, or both are true.
1-2 Transcribe the following sentences into sentence logic, using 'G' to transcribe 'Pudding is good.' and 'F' to transcribe 'Pudding is fattening.'
- Pudding is good and pudding is fattening.
- Pudding is both good and fattening.
- Pudding is either good or not fattening.
- Pudding is not good and not fattening.
You may well have a problem with the following transcriptions, because to do some of them right you need to know something I haven't told you yet. But please take a try before continuing. Trying for a few minutes will help you to understand the discussion of the problem and its solution in the next section. And perhaps you will figure out a way of solving the problem yourself!
- Pudding is not both good and fattening.
- Pudding is both not good and not fattening.
- Pudding is not either good or fattening.
- Pudding is either not good or not fattening.
- Pudding is neither good nor fattening
To summarize this section:
Sentence logic symbolizes its shortest unambiguous sentences with Atomic Sentences, also called Sentence Letters, which are written with capital letters: 'A', 'B', 'C' and so on. We can use Connectives to build Compound Sentences out of shorter sentences. In this section we have met the connectives '~' (the Negation Sign), '&' (the Sign of Conjunction), and 'v' (the Sign of Disjunction).
Paul Teller (UC Davis). The Primer was published in 1989 by Prentice Hall, since acquired by Pearson Education. Pearson Education has allowed the Primer to go out of print and returned the copyright to Professor Teller who is happy to make it available without charge for instructional and educational use.