in the truth table for an invalid argument
To determine whether an argument is valid or invalid, you need to construct a truth table to assess all possible combinations of truth values for the premises and the conclusion. If there is at least one row in the truth table where the premises are true and the conclusion is false, then the argument is invalid. However, if there is no such row and the conclusion is always true when the premises are, then the argument is valid.
Here is an example of a truth table for an invalid argument:
Let's consider the following argument:
Premise 1: If it is raining, then the ground is wet.
Premise 2: The ground is wet.
Conclusion: Therefore, it is raining.
To construct the truth table, we need to determine all possible combinations of truth values for the premises and the conclusion. In this case, there are two premises and one conclusion, so we need to consider four rows:
Row 1: It is raining (P1=true), the ground is wet (P2=true) and it is raining (C=true).
Row 2: It is raining (P1=true), the ground is wet (P2=true), but it is not raining (C=false).
Row 3: It is not raining (P1=false), the ground is wet (P2=true), but it is not raining (C=false).
Row 4: It is not raining (P1=false), the ground is wet (P2=true), but it is raining (C=true).
In the truth table, you would input the truth values for each row based on the premises and conclusion. Once you have filled in all the rows with their respective truth values, you can examine the table to determine if there is any row where the premises are true and the conclusion is false.
In this case, in Row 2, the premises (P1=true and P2=true) are true, but the conclusion (C=false) is false. Therefore, this argument is invalid.
By constructing and analyzing a truth table for an argument, you can determine its validity or invalidity based on the truth values of the premises and the conclusion.