To recap, we have gone through truth tables for sentences, both full tables (that is, tables for sentences whose components have unknown truth values) and tables for sentences where we know one or more of the actual truth values of the components. We have also gone through tables for arguments (both full tables, as well as the "short-cut" method). Some examples:
Here is a full truth table for a sentence.
[(L -> N) & -G] t t t f ft t f f f ft f t t f ft f t f f ft t t t t tf t f f f tf f t t t tf f t f t tf
Since we don't know any of the truth values of 'L', 'N', and 'G', we have to give the full truth table. The major connective in this sentence is the ampersand (&) and you will notice that under the ampersand there are 5 f's and 3 t's. This tells me that in some possible worlds this sentence is true (in 3 of them) and in some possible worlds this sentence is false. Since I don't know which possible world is the actual world, I see that logic alone cannot tell me what the actual truth value of the sentence is. Say, however, that I know that 'G' is false, that 'N' is true and that 'L' is false. That set of values corresponds to the 7th row (possible world). So, the 7th row represents the actual world. I now know that this sentence is true (by looking under the ampersand in the 7th row).
Now, there is no need to do a full truth table for a sentence when you know one or more of the truth values of the components of the sentence. So, for the sentence above, if we know that 'G' and 'N' are false and that 'L' is true, we can do an abbreviated table, like so:
[(L -> N) & -G] t f f f tf
In this case, again, we know the truth value of the sentence is false. And again, if we know that 'G' and 'N' are true but we don't know the value of 'L', then we need two rows, like so:
[(L -> N) & -G] t t t f ft f t t f ft
One row is for 'G', 'N' and 'L' all being true and one row is for 'G' and 'N' being true and 'L' being false (since we don't know the value of 'L', we must give both values for 'L' and see whether that makes a difference in the truth value of the entire sentence). This sentence is false no matter whether 'L' is true or false (when both 'G' and 'N' are true).
Three final things on tables for sentences: If any sentence has all true's under the major connective when you do a full truth table, then that sentence is called a logical truth, or a tautology, or simply L-true. If any sentence has all false's under the major connective when you do a full truth table, then that sentence is called a contradiction, or logical falsehood, or simply L-false. If any sentence has at least one true under the major connective and at least one false under the major connective, when you do a full truth table, then that sentence is called a contingent sentence, or logically indeterminate, or simply L-indeterminate. Examples:
1. (L -> L) 2. (-L & L) 3. (P v -R) t t t ft f t t t ft f t f tf f f t t tf f f ft f t tf
Tables for arguments.
Arguments are comprised of sentences. If you can do a truth table for sentences, then you can do one for an argument. The tables for an argument consist in a table for each of the sentences in the argument; one table for each premise and one table for the conclusion. There will be no table for the conclusion indicator ('//') and there should appear no truth values under this symbol. Here is an example of a full set of truth tables for an argument:
Premise Premise Conclusion Indicator Conclusion (S -> H) (-H & E) // -S t t t ft f t ft f t t ft f t tf t f f tf t t ft f t f tf t t tf t t t ft f f ft f t t ft f f tf t f f tf f f ft f t f tf f f tf
We have 3 different components (letters), hence we need 8 rows (possible worlds). Note how we must be consistent in the assignment of truth values. That is, the same letter gets the same truth value in the same row. For example, in the second row, 'H' is true and we must give 'H' true in the second row whereever it occurs (in this case 'H' appears in both the first and second premises). Same with 'S' in the first premise and in the conclusion.
Now the crucial part: The object here is to discover whether the argument is valid or invalid. To do that, go through each row and ask, "Are each of the premises true and the conclusion false?" If the answer is "yes" for any row, then the argument is invalid. If the answer is "no" for every row, then the argument is valid. This argument is valid; we know this because there is no row in which each of the premises is true and the conclusion false.
Try a different argument:
(J -> O) (-O v W) // -W t t t ft t t ft t t t ft f f tf t f f tf t t ft t t f tf t f tf f t t ft t t ft f t t ft f f tf f t f tf t t ft f t f tf t f tf
Here we see that this argument is invalid, because each of the premises is true in the first row while the conclusion is false in that row. Now try the same premises but with a different conclusion. Here's the table:
(J -> O) (-O v W) // W t t t ft t t t t t t ft f f f t f f tf t t t t t f tf t f f f t t ft t t t f t t ft f f f f t f tf t t t f t f tf t f f
This argument is also invalid. The fourth row has all true premises and a false conclusion. So, from these premises, neither does '-W' follow, nor does 'W' follow.
The Short-Cut Method of Tables for Arguments.
To do the short-cut method,
Example:
(A -> R) -A // -R ft
Note there is only one way to make the conclusion false here; assign 't' to 'R'. Since 'R' has been given 't' in the conclusion, we'll go into the set of premises and give it 't' wherever it occurs, like so:
(A -> R) -A // -R
t ftThe next step is to go into the premises and try to make each true. I'll start with the easy one, i.e., '-A'. There is only one way to make the second premise true; assign 'f' to 'A', making '-A' true, like so:
(A -> R) -A // -R
t tf ftNow, since we gave 'A' 'f' in the second premise, we have to give it 'f' in the first premise as well, like so:
(A -> R) -A // -R f t tf ft
And, when we do that, we see that the first premise will be true, because when the antecedent is false and the consequent true in a conditional, the conditional itself is true. Here is how this would look.
(A -> R) -A // -R f t t tf ft
We have one row here instead of 4. Hence, "the short-cut" method.
Some preliminary talk about truth trees.
The method of truth trees uses what is known as the concept of reductio ad absurdum, where one assumes the conclusion of the argument to be false and then tries, using that assumption, to show that contradictions are generated in each possible world. The idea behind this idea is this: If some set of premises leads validly to a certain conclusion, then if we assume this conclusion to be false, it should be possible, using the premises and the "denied conclusion" to get contradictions of the form "p & -p". For any argument, if you deny the conclusion and generate a contradiction on each branch of the tree, then you know the argument is valid, that is, that the original argument, with the original conclusion, is valid.
Set up the argument in a different way from the way for tables. Set up the argument for trees vertically, first premise on top, second premise under the first, and so on. Draw a line under the last premise. Write the conclusion directly under this line. Then, deny the conclusion by putting a negation sign in the appropriate location. Next, to the extreme right of the conclusion, write the letters "DC" (which indicates that the conclusion has been denied). Here's how to set it up.
(F -> Q) (Q -> -C) -(F -> -C) DC
Now begin to apply the rules, beginning anywhere you like. I'll begin with the conclusion and work up. (This lecture, and the tree below, presumes you have some familiarity with the truth tree rules themselves. In the book, I repsresent them in two ways: 1. In English, and 2. I their schematic, symbolic forms.)
(F -> Q) *
(Q -> -C)*
-(F -> -C) * DC
F
--C *
C
/\
/ \
/ \
-Q -C
| X
/\
/ \
/ \
-F Q
X X
In the above tree, note that there are 'X's" under '-C', '-F' and 'Q'. This means, for '-F', that there is a contradiction on the branch on which it appears. '-C' contradicts 'C', '-F' contradicts 'F', and so on. The asterisks indicate that a rule has been applied to that sentence (we use a check mark usually). You need apply a rule to a sentence only once. Since a contradiction appears on each branch of the tree, we know the oroginal argument is valid.