In these notes, I will discuss the topic truth table and validity of arguments, that is, I will discuss how to determine the validity of an argument in propositional logic using the truth table method.
However, it must be noted that there are two basic methods in determining the validity of an argument in symbolic logic, namely, truth table and partial truth table method. Again, in this post, I will only discuss the truth table method, thus the topic “truth table and validity of arguments”. I will discuss the partial truth table method in my next post.
Validity and Invalidity of Arguments
How do we know whether an argument is valid or invalid?
On the one hand, a truth-functionally valid argument form is an argument that is composed of propositions that have truth-functional forms such that it is impossible for its premises to be all true and its conclusion false. In other words, an argument is valid if it does not contain the form “all true premises and false conclusion”.
On the other hand, a truth-functionally invalid argument form is an argument that is composed of propositions that have truth-functional forms such that it is possible for its premises to be all true and its conclusion false. In other words, an argument is invalid if all of its premises are true and its conclusion false.
Let’s consider the example below.
1. If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights. The squatters are going to settle here. Therefore, there will be a fight for water rights. (S, C, F)
So, how do we determine the validity of the argument above?
Before we can apply the truth table method in determining the validity of the argument above, we need to symbolize the argument first. After symbolizing the argument, we will construct a truth table for the argument, and then apply the rule in determining the validity of arguments in symbolic logic.
But how do we symbolize the argument above?
In case one does not know how to symbolize arguments in symbolic logic, please refer to my previous post titled “Symbolizing Propositions in Symbolic Logic”, http://philonotes.com/index.php/2018/02/14/symbolizing-propositions-in-symbolic-logic/.
In symbolizing arguments in symbolic logic, we just need to apply the techniques that we employed in symbolizing propositions. Hence, we symbolize arguments in symbolic logic proposition by proposition or sentence by sentence.
Now, if we look at the argument above, the first proposition is “If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights.” And then we see the constants “S, C, and F” at the end of the argument.
If we recall my discussion on symbolizing propositions, we learned that the variables or constants provided after the proposition (argument in this case) represent the propositions in the entire proposition (argument in this case) respectively. Hence, the constant S stands for “The squatters settle here”, C for “The cattlemen will be angry”, and F for “There will be a fight for water rights”. Thus, the first proposition “If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights” is symbolized as follows:
S ⊃ (C • F)
The next proposition in the argument above says “The squatters are going to settle here”. As we notice, this proposition is just a repeat of the proposition in the previous statement, and this proposition is symbolized by the constant S. Hence, the second proposition “The squatters are going to settle here” is symbolized as follows:
S
The third and last proposition is obviously the conclusion because of the signifier “therefore”. This proposition is also a repeat of the proposition in the first sentence, which is symbolized by the constant F. Hence, the conclusion “Therefore, there will be a fight for water rights” is symbolized as follows:
F
At the end of it all, the argument “If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights. The squatters are going to settle here. Therefore, there will be a fight for water rights. (S, C, F)” is symbolized as follows:
S ⊃ (C • F)
S /∴ F
Please note that in the symbolized form of the argument above, S ⊃ (C • F) is the first premise, S is the second premise, and F is the conclusion.
Now, how do we construct a truth table for this argument?
First, we need to construct a truth table that contains columns for the variables or constants and columns for the premises and the conclusion. In order to do this, we will use the formula 2 raised to the power n (2n), where 2 is constant and n is a variable.
The n in the formula 2n represents the number of variables or constants used in the argument. Since the argument above contains 3 constants, namely, S, C, and F, then the formula now reads:
23
So, 23 = 8. This means that we need to construct a truth table that contains 8 rows. But first we have to draw columns for the constants and the premises and the conclusion, which will look like this:
After drawing the columns for the constants and the premises and conclusion, we will now draw 8 rows. With this, the truth table will now look like this:
Now that we have constructed the truth table that contains the columns for the constants and the premises and conclusion, let us provide the truth values of the variables S, C, and F. We need to do this because the truth values of the premises and the conclusion will be based on the truth values of the variables or constants.
But how do we do this?
First, it must be noted that the product 8 above (23 = 8) also represents 4 true values and 4 false values for the variable or constant (S in the case of the example above). And the rule here is that we write the true values first and then the false values. So, the truth table will now look like this:
For the next column (that is, the column for C), we need to divide the 4 true and false values by 2. Thus, we will have 2 true values and 2 false values. The rule is we will write 2 true values first and then 2 false values. For the remaining rows, we will write 2 true values and 2 false values alternately. So, the truth table will now look like this:
For the next column (that is, the column for F), we need to divide the 2 true and false values by 2. Thus, we will have 1 true and 1 false value. The rule is we will write 1 true value first and then 1 false value. So, the truth table will now look like this:
For the remaining rows, we will write 1 true value and 1 false value alternately. So, the truth table will now look like this:
Since we already have the truth values of the constants S, C, F, we can now determine the truth values of the premises and the conclusion. Please note that we need to provide the truth values of all the premises and the conclusion before we can apply the rule in determining the validity of an argument. So, let’s provide the truth values of the premise.
The first premise is S ⊃ (C • F). As we can see, the first premise is a conditional proposition whose consequent is a conjunction. We also need to remember that in determining the truth value of this premise, we need to apply the rules in conditional and conjunctive propositions. If one does not know yet the rule in conditional propositions, one may visit our previous post titled “Conditional Propositions”, http://philonotes.com/index.php/2018/02/11/conditional-propositions/. And for the rule in conjunctive statements, please see “Conjunctive Statement”, http://philonotes.com/index.php/2018/02/03/conjunctive-statements/.
Now, before we can apply the rule in conditional proposition in the premise S ⊃ (C • F), we need to simplify first the consequent (C • F). This can be done my determining its truth value using the rule in conjunction. Since the rule in conjunction says “A conjunction is true if both conjuncts are true”, then the truth table will now partially look like this:
Please note that in the truth table above, I temporarily removed the column for S to avoid confusion, that is, in order to show that we are just using the values for the columns C and F.
Since we have already simplified (C • F), then we can now proceed to determining the final truth values of the first premise S ⊃ (C • F). So, the truth table will now look like this:
Please note that the final truth values of the first premise S ⊃ (C • F) are the ones in bold red. Let me illustrate how I arrived at those values. But let me just illustrate the first two rows. In the first row, S is true, C is true and F is true. So,
In the second row, S is true, C is true, and F is false. So,
For the truth values of the second premise which is S, we just need to copy the truth values of S in the first column. This is obviously because the second premise is a simple proposition. So, the truth table will now look like this:
For the truth values of the conclusion which is F, we just need to copy the truth values of F in the third column. This is obviously because the conclusion is a simple proposition. So, the truth table will now look like this:
As we can see, the truth table is now complete. So, we may now apply the rules in determining the validity of arguments in symbolic logic. But before we proceed to that, let us remove the columns for the variables in order to avoid confusion. It must be remembered that the rule talks about the premises and the conclusion only, and so we may now drop them. Of course, as we can see, the columns for the variables/constants are needed only in determining the truth values of the premises and conclusion. So, the truth table of the argument above will finally look like this:
If we recall, the rule in determining the validity of an argument in symbolic logic says that an argument is valid if it does not contain the form “all true premises and false conclusion” and an argument is invalid if “all of its premises are true and its conclusion false”. Please note that in applying the rule, we need to consider all rows in the truth table.
Now, the easiest and most convenient way to do it is to look for an invalid form in each row, that is, a row that contains all true premises and a false conclusion. Thus, if we cannot find one, then the argument is obviously valid.
If we look at the final truth table of the argument above, we cannot find at least one row that contains the form all true premises and a false conclusion. Therefore, at the end of the day, the argument “If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights. The squatters are going to settle here. Therefore, there will be a fight for water rights” is absolutely valid.
Finally, let me give an example of an invalid argument so we can fully understand why the argument above is valid. Consider the example below.
2. If Marco had been a poor businessman, then he would have had to undertake extensive lecture hours. He did undertake extensive lecture hours. Hence, he must be a poor businessman. (p, q)
I need not explain again here why we have come up with the truth table above. The discussion above is enough for us to know how to construct a truth table.
Now, although not necessary, let’s remove the columns for the variables in the truth table above to avoid confusion. So, the truth table will now look like this:
If we recall, the rule in determining the validity of an argument in symbolic logic says that an argument is valid if it does not contain the form “all true premises and false conclusion” and an argument is invalid if “all of its premises are true and its conclusion false”. Please note that in applying the rule, we need to consider all rows in the truth table.
The easiest and most convenient way to do it is to look for an invalid form in each row, that is, a row that contains all true premises and a false conclusion. Thus, if we cannot find one, then the argument is obviously valid.
Now, if we look at the final truth table of the argument above, we can indeed find one row (row 3) that contains the form all true premises and a false conclusion. Thus, at the end of the day, the argument “If Marco had been a poor businessman, then he would have had to undertake extensive lecture hours. He did undertake extensive lecture hours. Hence, he must be a poor businessman. (p, q)” is absolutely invalid.