Euler Circles and Truth Tables can be used to determine the validity of arguments and statements. People of all walks of life; hear, read, and see arguments and persuasive statements every day. Truth Tables and Euler Circles are effective tools to evaluate such messages. Using Euler Circles is a great way to get a visual representation of what the premises and conclusion are stating, when most of the information is available. Conversely, Truth Tables will be more effective when considering several unknowns; since they provide every possible outcome for each variation of all available variables.
Before constructing a test, one must understand the premises by which the argument is concluded. The conclusion, “therefore, some B is not C” (Bluman, 2005) does not make sense unless the premises, “No A is B, and some C is A,” (Bluman, 2005) are understood. To make it more clear, elements can be given to each group. Let A symbolize all even numbers less than 20, B symbolize all odd numbers less than 20, and C symbolize all prime numbers less than 20. Thus, the statement reads, “No even number is an odd number, and some prime numbers are even; therefore, some odd numbers are not prime numbers.”
Plotting an Euler circle is not difficult if the rules are known and understood. No A is B tells the tester that these circles do not touch or connect. This is true with the elements, because no even number is also an odd number. The second premise places circle C connecting with Circle A, because “2” is both a prime number and an even number. Since there are also some odd numbers that are also prime numbers, the circles B and C also connect. The Euler Circle for this statement looks like the one in figure 1-a. Giving elements numerical properties can be helpful in understanding how to plot Euler Circles, but it is not necessary. If the elements are unknown, and only the statements are available, then there are two possible Euler Circle outcomes for this first argument (fig. 1-b). Both of which prove the statement’s validity.
Another option is to use a Truth Table to test the validity of an argument. Before plugging variables and their truth values into a table, however, one must understand the basic symbols used. When all of one group is equivalent to another, an arrow is placed between the two groups. This reads, “if this…than that…” If none of the first group is equivalent to the second, a negation sign is placed before the second group and after the arrow. This reads, “If this…then the opposite of that…” If some of one group is equivalent to the other, a carrot symbol is placed between them. This is read, “Some of this, and that” If some of one group is not equivalent to the other, a negation sign is placed before the second group and after the carrot. This would read, “Some of this…but not that…” (Bluman, 2005).
Using the same argument, “No A is B, and some C is A; therefore, some C is not B,” a Truth Table can be constructed. Since the standard symbols used to represent groups or sets in Truth Tables are p, q, and r; the statement would look like this: [(p → ~q) ^ (r ^ p)] => (r ^ ~q). The first step in plotting truth values is to plot the variable; a T is placed for every chance that the variable is true, an F is placed for every chance it is false. When applying three variables, there are eight possibilities total. These combinations are placed in the far left three columns of the table. The next step is to work out the statement or argument. This is done in the right columns.
Based on the truth values of each variable, plot the truth values of each piece of the statement or argument. This is not a simple arithmetic problem worked left to right. Instead it is a compound statement, which follows a basic order of operations (i.e. Work parenthesis first, then negations, the junctions, and last the conditionals).
Once the table is plotted, it can validate or nullify the argument. The only way for the argument to be considered valid is if the table reveals a tautology. That is, the statement is true for every variation possible (Bluman, 2005). If there is even one false result, the statement is invalid. In the above case, the argument is valid because the truth table reveals a tautology (fig. 1-c).
Even though Euler Circle and Truth Table tests show the same results, sometimes one does work better than the other. When all of the elements are known, an Euler circle is the easiest way to test the conclusion. However, if there are several unknowns the Euler Circle test can be inconclusive at first glance. On the other hand, the Truth Table test considers every possibility, and tests the conclusion based on all the combinations of every possibility. Therefore, Truth Tables can be a more accurate way to evaluate an argument if there are several unknowns.
Consider the following statement, “All B is A, and all C is A; therefore, all C is B” (Bluman, 2005). There are three possible Euler Circles one could draw up to show this argument. One diagram will validate the statement; whereas, the other two will nullify it (figure 2-a). If the person testing the statement doesn’t think about the other two possibilities, they may come to the conclusion that it is a valid argument. However, if the information is plugged into a Truth Table, all of the variations will be presented, and every outcome considered. The final conclusion will be that the statement is invalid, because there are occasions in which the end result is a negative (figure 2-b). Based on the understanding of truth tables, and tautologies, this contradicts the argument because a tautology is not revealed.
Bluman, A.G. (2005) Mathematics in our World. McGraw-Hill Companies: New York.