3.5. ARGUMENTS Arguments Have Always Been Central to the Study Of

Arguments have always been central to the study of logic. Aristotle was the first to formally study and analyze the structure of syllogistic arguments, such as the classic “All men are mortal. Socrates is a man. So Socrates is mortal.” His systematic treatment of these standard forms of arguments remained influential for over 2,000 years. Today, arguments continue to play a fundamental role in logic. One needn’t go far to see the importance of arguments. All mathematical results, or theorems, are in some form or another arguments. Every case pleaded by an attorney before a judge in a court of law is an oral argument. Much of politics and science is grounded by the use of arguments. Critical theories, historical analyses, and social commentaries are all, in essence, rather elaborate arguments. In fact, one would be hard-pressed to find an intellectual endeavor that was established without the use of logical arguments.

Definition An argument consists of a finite set of statements called the premises that, in conjunction, imply a final statement called the conclusion. Essentially, an argument is an extension of a conditional statement. It can be represented symbolically by the conditional form

(푷ퟏ ∧ 푷ퟐ ∧ … ∧ 푷풏) → 푪,

where 푃1, 푃2, … , 푃푛 are the 푛 premises of the argument and 푪 is the conclusion of the argument.

Typically, an argument is written in the following vertical format:

푷ퟏ 푷ퟐ ⋮

푷풏 ____ ∴ 푪

Example Consider the following argument: “The bond market will fall in London if the stock market rises in Frankfurt. The bond market did not fall in London. Hence the stock market did not rise in Frankfurt.”

Q: How can we verify whether an argument such as this one is valid? A: By applying the following criterion.

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.

Checking for Validity: Analyzing an Argument

In practice, this criterion can be implemented in several ways. Here is a list of methods that can be used to check whether an argument is valid or invalid. Note that one or more of these approaches might be applicable.

Analyzing an Argument with Truth Tables To check for validity using truth tables, construct a truth table that shows all the possible truth values for each of the premises in the argument as well as all truth values for the conclusion. Make sure the columns for the premises and the conclusion are all displayed in sequential order at the end of the table. Applying the criterion for validity, you must then check that for every row in the table where all premises are true, the conclusion is also true. If this is not the case, then the argument is invalid. Any row where the premises are all true but the conclusion is false is called a fallacy.

Analyzing an Argument with Circuits The criterion for validity also implies that an argument is valid if and only if the compound statement (푃1 ∧ 푃2 ∧ … ∧ 푃푛) → 퐶 is a tautology. This is the symbolic form used when checking for the validity of an argument using a circuit. In that case, you must show that the light bulb on the right end of the circuit is always lit (“ON”) in all cases.

Analyzing an Argument with Standard Forms Perhaps the simplest way to check whether an argument is valid or not is by using standard forms (when applicable). Below are tables with some of the most commonly used standard forms for arguments.

Analyzing an Argument with Euler Diagrams Euler diagrams are set diagrams that are only applicable to syllogistic argument involving quantifiers such as “All”, “Every”, “Some”, “At least”, “No(ne)”, etc. In such cases, draw the Euler diagram based on the premises of the argument and check that the conclusion must follow from the picture. If you can draw a diagram that makes the conclusion of the argument false, then the argument is invalid. The Euler diagram of a valid argument cannot be drawn to make the conclusion false. 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

푝 → 푞 푝 ∨ 푞 푞 → 푟 ~푝 ______∴ 푝 → 푟 ∴ 푞

Standard Forms of Invalid Arguments

FALLACY OF THE CONVERSE FALLACY OF THE INVERSE

푝 → 푞 푝 → 푞 푞 ~푝 ______∴ 푝 ∴ ~푞

Note that there are many other forms of arguments that are either valid or invalid. The six classic forms shown above, however, are some of the most common ones used in propositional logic.