We have seen how to apply the connectives '~', '&', and 'v' to atomic sentences such as 'A' and 'B' to get compound sentences such as '~A', 'A&B', and 'AvB'. But could we now do this over again? That is, could we apply the connectives not just to atomic sentences 'A', 'B', 'C', etc., but also to the compound sentences '~A', 'A&B', and 'AvB'? Yes, of course. For example, we can form the conjunction of '~A' with 'B', giving us '~A&B'. Using our current transcription guide, this transcribes into 'Adam does not love Eve and Adam is blond.'
As another example, we could start with the conjunction 'A&B' and take this sentence's negation. But now we have a problem. (This is the problem you encountered in trying to work exercise 1~2, e~i.) If we try to write the negation of 'A&B' by putting a '~' in front of 'A&B', we get the sentence we had before. But the two sentences should not be the same! This might be a little confusing at first. Here is the problem: We are considering two ways of building up a complex sentence from shorter parts, resulting in two different complex sentences. In the first way, we take the negation of 'A', that is, '~A', and conjoin this with 'B'. In the second way, we first conjoin 'A' and 'B' and then negate the whole. In English, the sentence 'It is not the case both that Adam loves Eve and Adam is blond.' is very different from the sentence 'Adam does not love Eve, and Adam is blond.' (Can you prove this by giving circumstances in which one of these compound sentences is true and the other one is false?)
In order to solve this problem, we need some device in logic which does the work that 'both' does in English. (If you are not sure you yet understand what the problem is, read the solution I am about to give and then reread the last paragraph.) What we need to do is to make clear the order in which the connectives are applied. It makes a difference whether we first make a negation and then form a conjunction, or whether we first form the conjunction and then make a negation. We will indicate the order of operations by using parentheses, much as one does in algebra. Whenever we form a compound sentence we will surround it by parentheses. Then you will know that the connective inside the parentheses applies before the one outside the parentheses. Thus, when we form the negation of 'A', we write the final result as '(~A)'. We now take '(~A)' and conjoin it with 'B', surrounding the final result with parentheses:
This says, take the sentence '(~A)' and conjoin it with 'B'. To indicate that the final result is a complete sentence (in case we will use it in some still larger compound), we surround the final result in parentheses also. Note how I have used a second style for the second pair of parentheses-square brackets-to make things easier to read.
with which means that one is to conjoin 'A' with 'B' and then take the negation of the whole.
In the same kind of way we can compound disjunctions with conjunctions and conjunctions with disjunctions. For example, consider
Sentence (6) says that we are first to form the conjunctions of 'A' with 'B' and then form the disjunction with 'C'. (7), on the other hand, says that we are first to form the disjunction of 'B' with 'C' and then conjoin the whole with 'A'. These are very different sentences. Transcribed into English, they are 'Adam both loves Eve and is blond, or Eve is clever.' and 'Adam loves Eve, and either Adam is blond or Eve is clever.'
We can show more clearly that (6) and (7) are different sentences by writing out truth tables for them. We now have three atomic sentences, 'A', 'B', and 'C'. Each can be true or false, whatever the others are, so that we now have eight possible cases. For each case we work out the truth value of a larger compound from the truth value of the parts, using the truth value of the intermediate compound when figuring the truth value of a compound of a compound:
Let's go over how we got this truth table. Columns a, b, and c simply give all possible truth value assignments to the three sentence letters 'A', 'B', and 'C'. As before, in principle, the order of the cases does not matter. But to make it easy to compare answers, you should always list the eight possible cases for three letters in the order I have just used. Then, for each case, we need to calculate the truth value of the compounds in columns d through h from the truth values given in columns a, b, and c.
Let us see how this works for case 5, for example. We first need to determine the truth value to put in column d, for '(A&B)' from the truth values given for this case. In case 5 'A' is false and 'B' is true. From the truth table definition of '&', we know that a conjunction (here, 'A&B') is false when the first conjunct (here, 'A') is false and the second conjunct (here, 'B') is true. So we write an 'f' for case 5 in column d. Column e is the disjunction of 'B' with 'C'. In case 5 'B' is true and 'C' is true. When we disjoin something true with something true, we get a true sentence. So we write the letter 't', standing for the truth value t, in column e for case 5.
Moving on to column g, we are looking at the disjunction of '(A&B)' with 'C'. We have already calculated the truth value of '(A&B)' for case 5~that was column d~and the truth value of 'C' for case 5 is given in column c. Reading off columns c and d, we see that '(A&B)' is false and 'C' is true in case 5. The sentence of column g, '[(A&B)vC]', is the disjunction of these two components and we know that the disjunction of something false with something true is, again, true. So we put a 't' in column g for case 5. Following the same procedure for column h, we see that for case 5 we have a conjunction of something false with something true, which gives the truth value f. So we write 'f for case 5 in column h.
Go through all eight cases and check that you understand how to determine the truth values for columns d through h on the basis of what you are given in columns a, b, and c.
Now, back to the point that got us started on this example. I wanted to prove that the sentences '[(A&B)vC]' and '[A&(BvC)]' are importantly different. Note that in cases 5 and 7 they have different truth values. That is, there are two assignments of truth values to the components for which one of these sentences is true and the other is false. So we had better not confuse these two sentences. You see, we really do need the parentheses to distinguish between them.
Actually, we don't need all the parentheses I have been using. We can make two conventions which will eliminate the need for some of the parentheses without any danger of confusing different sentences. First, we can eliminate the outermost parentheses, as long as we put them back in if we decide to use a sentence as a component in a larger sentence. For example, we can write 'A&B' instead of '(A&B)' as long as we put the parentheses back around 'A&B' before taking the negation of the whole to form '~(A&B)'. Second, we can agree to understand '~' always to apply to the shortest full sentence which follows it. This eliminates the need to surround a negated sentence with parentheses before using it in a larger sentence. For example, we will write '~A&B' instead of '(~A)&B'. We know that '~A&B' means '(~A)&B' and not '~(A&B)' because the '~' in '~A&B' applies to the shortest full sentence which follows it, which is 'A' and not 'A&B'.
This section still needs to clarify one more aspect of dealing with compound sentences. Suppose that, before you saw the last truth table, I had handed you the sentence '(A&B)vC' and asked you to figure out its truth value in each line of a truth table. How would you know what parts to look at? Here's the way to think about this problem. For some line of a truth table (think of line 5, for example), you want to know the truth value of '(A&B)vC'. You could do this if you knew the truth values of 'A&B' and of 'C'. With their truth values you could apply the truth table definition of 'v' to get the truth value of '(A&B)vC'. This is because '(A&B)vC' just is the disjunction of 'A&B' with 'C'. Thus you know that '(A&B)vC' is true if at least one of its disjuncts, that is, either 'A&B' or 'C', is true; and '(A&B)vC' is false only if both its disjuncts, 'A&B' and 'C', are false.
And how are you supposed to know the truth values of 'A&B' and of 'C'? Since you are figuring out truth values of sentences in the line of a truth table, all you need do to figure out the truth value of 'C' on that line is to look it up under the 'C' column. Thus, if we are working line 5, we look under the 'C' column for line 5 and read that in this case 'C' has the truth value t. Figuring out the truth value for 'A&B' for this line is almost as easy. 'A&B' is, by the truth table definition of conjunction, true just in case both conjuncts (here, 'A' and 'B') are true. In line 5 'A' is false and 'B' is true. So for this line, 'A&B' is false.
Now that we finally have the truth values for the parts of '(A&B)vC', that is, for 'A&B' and for 'C', we can plug these truth values into the truth table definition for v and get the truth value t for '(A&B)vC'.
Now suppose that you have to do the same thing for a more complicated sentence, say
Don't panic. The principle is the same as for the last, simpler example. You can determine the truth value of the whole if you know the truth value of the parts. And you can determine the truth value of the parts if you can determine the truth value of their parts. You continue this way until you get down to atomic sentence letters. The truth value of the atomic sentence letters will be given to you by the line of the truth table. With them you can start working your way back up.
You can get a better grip on this process with the idea of the Main Connective of a sentence. Look at sentence (8) and ask yourself, 'What is the last step I must take in building this sentence up from its parts?" In the case of (8) the last step consists in taking the sentence '[Av~C]&[Bv(~A&C)]' and applying '~' to it. Thus (8) is a negation, '~' is the main connective of (a), and '[Av~C]&[Bv(~A&C)]' is the component used in forming (8).
What, in turn, is the main connective of '[Av~C]&[Bv(~A&C)]'? Again, what is the last step you must take in building this sentence up from its parts? In this case you must take 'Av~C' and conjoin it with 'Bv(~A&C)'. Thus this sentence is a conjunction, '&' is its main connective, and its components are the two conjuncts 'Av~C' and 'Bv(~A&C)'. In like manner, 'Bv(~A&C)' is a disjunction, with 'v' its main connective, and its components are the disjuncts 'B' and '~A&C'. To summarize,
The Main Connective in a compound sentence is the connective which was used last in building up the sentence from its component or components.
Now, when you need to evaluate the truth value of a complex sentence, given truth values for the atomic sentence letters, you know how to proceed. Analyze the complete sentence into its components by identifying main connectives. Write out the components, in order of increasing complexity, so that you can see plainly how the larger sentences are built up from the parts. In the case of (8), we would lay out the parts like this:
A, B, C, ~A, ~C, Av~C, ~A&C, Bv(~A&C), [Av~C]&[Bv(~A&C)], ~[Av~C]&[Bv(~A&C)]}
You will be given the truth values of the atomic sentence letters, either by me in the problem which I set for you or simply by the line of the truth table which you are working. Starting with the truth values of the atomic sentence letters, apply the truth table definitions of the connectives to evaluate the truth tables of the successively larger parts.
For each of the following sentences, state whether its main connective is '~', '&', or 'v' and list each sentence's components. Then do the same for the components you have listed until you get down to atomic sentence letters. So you can see how you should present your answers, I have done the first problem for you.
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.