MAT 101 How to Analyze a Symbolic Argument

Criterion for Validity. An argument is valid if the conclusion always follows from the premises. In other words, the conclusion of a valid argument is true whenever all the premises in the argument are true. If this is not the case, then the argument is invalid.

Standard Forms of Valid Arguments

MODUS PONENS MODUS TOLLENS (Law of Detachment ) (Law of Contraposition)

푝 → 푞 푝 → 푞 푝 ~푞 ______∴ 푞 ∴ ~푝

LAW OF SYLLOGISM DISJUNCTIVE SYLLOGISM (Law of Transitivity) 푝 ∨ 푞 푝 → 푞 ~푝 푞 → 푟 ______∴ 푞 ∴ 푝 → 푟

Standard Forms of Invalid Arguments

FALLACY OF THE CONVERSE FALLACY OF THE INVERSE

푝 → 푞 푝 → 푞 푞 ~푝 ______∴ 푝 ∴ ~푞

Consider the following argument:

“If the movie is directed by Martin Scorsese, then I want to see it. The movie must include at least two actors I like otherwise I do not want to see it. Joe Pesci is the only actor I like in this movie. Therefore, the movie is not directed by Martin Scorsese.”

This argument is, no doubt, a silly one. Regardless of that fact, we want to find out whether the conclusion follows logically from the premises. In other words, we want to know if this argument is valid. To answer this important question, we break down the argument to its symbolic form and proceed to analyze it using various method, as shown below.

1. Breakdown of Argument to its Symbolic Form

Let 푝 be the component “The movie is directed by Martin Scorsese.” Let 푞 be the component “I want to see the movie.” Let 푟 be the component “The movie must include at least two actors I like.”

The argument then has the following symbolic form:

푝 → 푞 (Premise 1) 푟 ∨ ~푞 (Premise 2) ~푟 (Premise 3) ______∴ ~푝 (Conclusion)

2. Analysis of Argument: Checking for Validity

Many different methods and approaches can be used to analyze a given argument. Below we use three methods to analyze this argument and check whether it is valid or invalid.

. Reducing to a Standard Form

This is probably the most popular method as it is relatively simple and does not require the construction of a truth table. However, the method does require the proper use of logical equivalences and a strategic symbolic manipulation of the premises.

Note that premise 2 is the disjunction 푟 ∨ ~푞, which can be rewritten as ~푟 → ~푞 using the disjunctive form of a conditional statement. This conditional premise is then equivalent to its contrapositive 푞 → 푟.

The argument is thus rewritten as follows:

푝 → 푞 (Premise 1) 푞 → 푟 (Premise 2) ~푟 (Premise 3) ______∴ ~푝 (Conclusion)

By the law of syllogism, premises 1 and 2 combine to give the new premise 푝 → 푟. The argument then reduces to

푝 → 푟 (Premises 1 & 2) ~푟 (Premise 3) ______∴ ~푝 (Conclusion)

Here we recognize the standard form of Modus Tollens. We therefore conclude that the argument is valid.

. Constructing a Truth Table

If you don’t recognize a standard form, the analysis can still be done through a truth table. This is the most straightforward method, but it typically requires the most amount of work.

Start by constructing a truth table where all three premises and the conclusion are placed as columns (ideally located sequentially in the table). Then identify the rows that have all three premises true. If the conclusion is false in any of these rows, you have uncovered a fallacy and the argument is invalid. If the conclusion is true in all of these rows, then the argument is valid.

As shown below, the only row we identify as having all true premises is row 8. Since the conclusion is true in this row, we confirm that the argument is indeed valid.

PREMISE 1 PREMISE 2 PREMISE 3 CONCLUSION

푝 푞 푟 ~푞 풑 → 풒 풓 ∨ ~풒 ~풓 ~풑

T T T F T T F F

T T F F T F T F

T F T T F T F F

T F F T F T T F

F T T F T T F T

F T F F T F T T

F F T T T T F T

F F F T T T T T

. Direct Approach

You could also show the validity of this argument using truth values directly. In that case, force the premises to be true and check that this necessarily leads to a true conclusion. This is an actual application of the criterion for validity.

This type of analysis leads to the elegant domino effect shown below.

. Premise 3 is only true if 풓 is false. . If 푟 is false, then 풒 must be false in order for premise 2 (the disjunction) to be true. . If 푞 is false, then 풑 must also be false in order for premise 1 (the conditional) to be true. . If 푝 is false, then the conclusion ~풑 must be true. The argument is, therefore, valid.