(p & q) t t t t f f f f t f f fSo, no matter what the sentence is, as long as it is a conjunction, it will be true in only one situation or possible world, or under one interpretation, that is, when both conjuncts are true. Below is an example of a longer table. Take the sentence: [L & (-S & W)]. Here is the proper way to render the truth table:
[L & (-S & W)] 1 t f ft f t 2 t f ft f f 3 t t tf t t 4 t f tf f f 5 f f ft f t 6 f f ft f f 7 f f tf t t 8 f f tf f fI've numbered the "rows" in the above table for explanation purposes only, as we don't normally do this. You can see that since we have 3 different components ('L', 'S', and 'W'), we need 8 rows. OK, first, note that the conjuncts here are 'L' and '(-S & W)'. So, one of the conjuncts is itself a conjunction.
Second, see that I started with the letter closest to the letter 'Z' in the alphabet (which is 'W') and given it 't' and 'f' 8 times alternating by ones. Then I went to 'S' and gave it 't' and 'f' alternating by twos; then to 'L' and gave it 't' and 'f' alternating by fours. This way, we've got every possible combination of truth values for the sentence itself. That is, there is a row where all the letters are true; a row where they're all false; a row where L is true, S is false and W is true; and so on, for every possible combination.
How then, does one come up with the truth values under the logical symbols?
Take special notice of '-S'. If the truth value of S is true, then the truth value of -S is going to be false. Say someone says, "Saturn is the seventh planet from the sun", and let's represent that sentence with 'S'. If Saturn is the seventh planet from the sun, then S is true. Now, if someone comes along and says that S is false (in effect, saying '-S'), then what that person said is itself false, since Saturn is the seventh planet from the sun. So, to come up with the truth values under the bar ('-') in front of the S, one looks at the value under the S. So in row #3, for example, since S is false, the value we place under the bar must be true. One crucial point here is that once you have put the values under all the letters themselves, then you go to the logical symbols and start with the logical symbol which has the smallest scope. In this case it would be the bar before the S.
The logical symbol with next smallest scope is the ampersand ('&') between -S and W. One puts the truth values under this symbol next by referring to the truth table for the conjunction. For example, where -S is true and where W is false, then, since the table for conjunction says that if either conjunct is false the entire conjunction is false, one puts an 'f' under the ampersand between -S and W (this example corresponds to row #8). In effect, one looks at the values under the bar and under the W to determine the value for the ampersand. The table for the conjunction is the key here; it is like a how-to manual and it covers every single possible situation for truth values of conjunctions.
The ampersand directly after L is the major connective, because it has the widest scope and ranges over both L and the conjunction of -S and W. As such, it receives its truth values last, by looking at the values under L and the ampersand between -S and W. And, when one is trying to determine the truth value of the entire sentence itself, one looks under the major connective.
So, then what is the actual truth value of the sentence? To answer this question definitively, we need to know the truth of at least some of the components of the sentence. Let: L = Richard Lionheart was King of England; S = the Saxons attacked Spain in 1043; W = Westminster Abbey was built in 1045. We know that L is true and that S and W are false. These truth values correspond to row #4. Looking under the first ampersand in row #4, I see 'f'. I know, then, that this sentence is actually false.
Let's move now to disjunctions. The table for all disjunctions is this:
(p v q) t t t t t f f t t f f fWe see that disjunctions are true whenever at least one of the disjuncts is true, and also true when both disjuncts are true. The only situation where a disjunction is false is where both disjuncts are false. Now, say we have the following sentence:
(D v L) t t t t t f f t t f f fRow #2 is the row corresponding to the actual world, where D is true and L is false.
However, if I know the truth values of the various non-logical components in a sentence, then it is something of waste of time to give the full truth table for the sentence. It's good practice for an exam, or it may be fun just to construct tables (the longer the better), but if one's goal is to discover the value of a sentence, then one really has no need to do the full table. Consider the following sentence:
[N v (E & W)] t t f f fWe see, then, that the sentence itself is true, since it basically says that at least one of the disjuncts is true, and in fact one is true, that is, N is true.
Here is the full truth table for the sentence above:
[N v (E & W)]
t t t t t
t t t f f
f t t t t
f f t f f
t t f f t
t t f f f
f f f f t
f f f f f
Notice how I started with the letter closest to 'Z', which is 'W', and gave it 't' and 'f' alternating by ones. Then I moved to the next letter closest to 'Z', the 'N', and gave it 't' and 'f' alternating by twos. Then I went to 'E', giving it 't' and 'f' alternating by fours. All the possible combinations of truth values are given by the eight rows, and there are eight rows because there are 3 different letters. Notice that the 6th row is the "actual world row". Hence, this sentence is true (you look under the major connective, which is the wedge ("v") to find the truth value of the sentence). Let's work up a sufficiently difficult sentence, translate it, and get at it's truth value via a table. Sentence:
Some of us don't know who invented truth tables. In that case we'd have to do a full table for the translation. The full table would look like this:
[(R v W) & -(R & W)] t t t f f t t t t t f t t t f f f t t t t f f t f f f f t f f fQuestion: Why are there 4 rows in the table? Answer: Because there are 2 different letters in the sentence (R and W). For those who know who invented truth tables, we can get away with only 1 row in the table. Wittgenstein invented them and Russell didn't. So, row #3 is the row corresponding to the actual world here. The short table would look like this:
[(R v W) & -(R & W)] f t t t t f f tWe see, then, by looking at the truth value under the major connective (which is in this case the first ampersand), that this sentence is true.
End of lecture.