In these notes, I will discuss the indirect truth table method in determining the validity of an argument in symbolic logic.
In my other notes (look for “Propositional Logic: Truth Table and Validity of Arguments” in Studypool search engine), I discussed the truth table method in determining the validity of an argument in symbolic logic. But the problem of the truth table method is that it can hardly be used in determining the validity of longer arguments.
Consider the example below.
- If the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth, then the captain either had good motives for controlling the world or his motives were evil. The airship Albatros had powerful weapon only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground. It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax. It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax. Furthermore, the captain’s motives for controlling the world were not evil. Therefore, his motives were good. (A, D, W, G, E, S, B, P)
As we can see, the argument contains 8 constants, namely, A, D, W, G, E, S, B, and P. If we employ the truth table method in determining the validity of this argument, then this means that we need to construct a truth table that contains 256 rows. Needless to say, that’s going to be a long and arduous process. It is for this obvious reason that logicians invented a shorter, more efficient method of determining the validity of arguments, namely, the indirect truth table method.
Let us determine the validity of the argument above using the indirect truth table method.
First, let us symbolize the argument above proposition by proposition or sentence by sentence to avoid confusion. In case one does not know yet how to symbolize arguments in logic, please refer to my previous post titled “Truth Table and Validity of Arguments”, http://philonotes.com/index.php/2018/03/26/truth-table-and-validity-of-arguments/. See also “Symbolizing Propositions in Symbolic Logic”, http://philonotes.com/index.php/2018/02/14/symbolizing-propositions-in-symbolic-logic/.
Proposition 1:
If the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth, then the captain either had good motives for controlling the world or his motives were evil.
[(A ⊃ D) • (D ⊃ W)] ⊃ (G v E)
Proposition 2:
The airship Albatros had powerful weapon only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground.
(A ⊃ S) • (S ⊃ D)
Proposition 3:
It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax.
(D ⊃ W) v (B ⊃ P)
Proposition 4:
It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax.
~ (B ⊃ P)
Proposition 5:
Furthermore, the captain’s motives for controlling the world were not evil.
~ E
Conclusion:
Therefore, his motives were good.
/∴ G
In the end, the argument above is symbolized as follows:
Now, in determining the validity of the argument above using the indirect truth table method, what we need to do is try to make the conclusion false and all the premises true. This is because if we recall our discussion on the rule in determining the validity of an argument in symbolic logic, we learned that an argument is invalid if the conclusion is false and all the premises are true. Thus, in using the indirect truth table method in determining the validity of an argument, we aim to make the argument invalid. If it is possible for us to make the argument invalid, then obviously the argument is invalid. If it is impossible for us to make the argument invalid, then obviously the argument is valid.
But how do we make the argument invalid?
First, let’s write the premises and the conclusion in a horizontal manner for convenience’s sake.
And second, assign truth-values to the conclusion and the premises that would result in the form “false conclusion and all true premises”. In doing so, start with the conclusion and assign the value “false”, and then go back to the premises and try to make all of them true. In making the premises true, always start with the first premise in order to avoid confusion and, of course, save time. Consider the example below.
As we can see, the argument above is valid because, although the conclusion is false, we cannot make all of the premises true. No matter what we do, Premise #5 cannot be true. Let me explain this further.
As said, let us always start with the conclusion and assign a false value to it. Please note that we should not assign a true value to the conclusion because if we do so, then we are defeating the purpose. This is obviously because if we assign the value true for the conclusion, then the argument will already appear valid. So, again, we should assign the value false for the conclusion. Look at the illustration below.
Please note that in this example, we can easily make the conclusion false by assigning the truth-value “false” to it because the conclusion above is a simple proposition. But even if it’s a compound proposition, we can still easily make it false if we have mastered the rules in compound propositions in symbolic logic. For example, if the conclusion is p ⊃ q, then we just need to assign the value true for p and false for q to make the proposition (conclusion) false. If we recall, the conditional proposition is false if the antecedent is true and the consequent false.
Now, since we have made the conclusion false, let’s go back to the premises and try to make all of them true. And let’s start with the first premise. Look at the illustration below.
As we can see, the first premise is a conditional proposition whose antecedent is [(A ⊃ D) • (D ⊃ W)] and the consequent is (G v E). Because this is a conditional proposition, and since our goal here is to make this premise true, then all we need to do is assign truth-values that would make the consequent (G v E) true. Please note that we need not assign any values to the antecedent [(A ⊃ D) • (D ⊃ W)] because whatever its truth-value, the premise is already true since the consequent is true. Again, the only instance wherein the conditional proposition becomes false is when the antecedent is true and the consequent false. Thus, whenever the antecedent is false or the consequent is true, the conditional proposition becomes automatically true.
So, how do we make the consequent (G v E) true?
Since the consequent is an exclusive disjunctive, then we need to see to it that (G v E) should not have the same truth-value in order for it to become true. If we recall our discussion in exclusive disjunction, an exclusive disjunction is true if one disjunct is false and other is true, and vice versa. And since we already have the value false for G in the conclusion, then we cannot make it true in the premise. Please note that in indirect truth table method, once the variable or constant has a fixed truth-value, then we cannot change it. Thus, if we change the truth-value of one variable or constant, then we need to change the truth-value of the same variable or constant in the entire indirect truth table. Since G is false, then we are forced to assign the truth-value “true” for E to make (G v E) true. So, since G is false and E is true, then the exclusive disjunction (G v E) is now true. And since the proposition is a conditional one, and because the consequent (G v E) is true, then Premise #1 is now true.
Let’s proceed to the second premise and try to make it true. Look at the illustration below.
Premise #2 is a conjunctive statement whose conjuncts are both conditional propositions. If we recall our discussion on conjunctive statements, we learned that a conjunctive statement is true if both conjuncts are true. Hence, if one conjunct or both are false, then the conjunctive statement is false. Since our goal here is to make Premise #2 true, then we have to see to it that both conjuncts must be true. In other words, (A ⊃ S) and (S ⊃ D) must be true.
How do we make (A ⊃ S) and (S ⊃ D) true?
Let’s start with (A ⊃ S). Since we don’t have a value for A and S yet, then we are free to assign whatever value that will make (A ⊃ S) true. So, if we assign a true value for both A and S, then (A ⊃ S) is true. Hence, as you can see in the diagram above, A is true and S is true.
The second conjunct is (S ⊃ D). Please note that we already have a value for S, which is true. Hence, we cannot assign a false value to D because it will make the proposition false. So, we are forced to assign the value true for D. Since S and D are now true, then the proposition (S ⊃ D) is true.
And since the conjuncts (A ⊃ S) and (S ⊃ D) are now true, then Premise #2 is now true (see diagram above).
Let’s proceed to Premise #3. Look at the illustration below.
Premise #3 is an inclusive disjunction whose disjuncts are both conditional propositions. If we recall our discussion on inclusive disjunction, we learned that an inclusive disjunction is true if at least one of the disjuncts is true. So, in Premise #3, we just need to make either of the disjuncts true in order for it to become true. And in the illustration above, we just made the first disjunct (D ⊃ W) true.
How do we make the first disjunct (D ⊃ W) true?
Since we already have the value true for D (see Premise #2), and since the proposition is conditional, that is, (D ⊃ W), then we cannot assign a value false for W; otherwise, we are making the proposition false. Hence, we are forced to assign the value true for W. Now, Since D is true and W is true, then the first disjunct (D ⊃ W) is true. If we look at the illustration above, we do not assign a value to the second disjunct (B ⊃ P). Of course, we are free to assign a value for (B ⊃ P), but that is not necessary because whatever value we have for (B ⊃ P), the premise (D ⊃ W) v (B ⊃ P) is already true because the first disjunct is true.
Let’s proceed to Premise #4. Look at the illustration below.
Premise #4 is a conditional proposition, but it is completely negated. In this case, it is relatively easy for us to make this premise true. All we need to do is make B ⊃ P false; so that if B ⊃ P is false, then ~ (B ⊃ P) is true.
How do we make B ⊃ P false?
Because B ⊃ P is a conditional proposition, there is only one way to make it false, that is, assign a true value to the antecedent B and false to consequent P. If we recall our discussion on conditional propositions, we learned that a conditional is false if the antecedent is true and the consequent false. Hence, if B is true and P is false (see illustration above), then B ⊃ P false. Again, since B ⊃ P false, then ~ (B ⊃ P) is true (see illustration above).
Lastly, let us make Premise #5 true. Look at the illustration below.
Premise #5, as we can see, is just a simple proposition. So, it is very easy for us to make this premise true. Since the premise is ~ E, all we need to do is assign the value false for E. This is because if E is false, then ~ E is true.
However, if we go back to Premise #1, we notice that we have assigned the value true for E. And since in indirect truth table method we are not allowed to change the value of a variable or constant, then we are forced to use the value true for E in Premise #5. So, if E is true, then ~ E is false (see illustration above).
At the end of it all, it’s impossible for us to make the argument above invalid. Therefore, the argument is absolutely valid.
Just a final note. There are, of course, several ways of making a conclusion false and all the premises true in indirect truth table method. For example, we can make Premise #1 in the argument above by making the antecedent [(A ⊃ D) • (D ⊃ W)] false. But I have exhausted all the ways in making the argument above invalid but to no avail. The argument, therefore, is valid.