In the next section we will use what we have learned about constructing statements to build arguments with logical statements. We will also use more Venn diagrams to evaluate whether an argument is logical, and introduce how to use a truth table to evaluate a logical statement.

Arguments

A logical argument is a claim that a set of premises support a conclusion. There are two general types of arguments: inductive and deductive arguments.

Many scientific theories, such as the big bang theory, can never be proven. Instead, they are inductive arguments supported by a wide variety of evidence. Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. The commonly known scientific theories, like Newton’s theory of gravity, have all stood up to years of testing and evidence, though sometimes they need to be adjusted based on new evidence. For gravity, this happened when Einstein proposed the theory of general relativity.

A deductive argument is more clearly valid or not, which makes them easier to evaluate.

It is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are only concerned with whether the premises are enough to prove the conclusion.

In addition to these categorical style premises of the form “all ___,” “some ____,” and “no ____,” it is also common to see premises that are implications.

Some arguments are better analyzed using truth tables.

The previous problem is an example of a syllogism.

Lewis Carroll, author of Alice in Wonderland, was a math and logic teacher, and wrote two books on logic. In them, he would propose premises as a puzzle, to be connected using syllogisms.

Logical Inference

Suppose we know that a statement of form [latex]P{\rightarrow}Q[/latex] is true. This tells us that whenever P is true, Q will also be true. By itself, [latex]P{\rightarrow}Q[/latex] being true does not tell us that either P or Q is true (they could both be false, or P could be false and Q true). However if in addition we happen to know that P is true then it must be that Q is true. This is called a logical inference: Given two true statements we can infer that a third statement is true. In this instance true statements [latex]P{\rightarrow}Q[/latex] and P are “added together” to get Q. This is described below with [latex]P{\rightarrow}Q[/latex] stacked one atop the other with a line separating them from Q. The intended meaning is that [latex]P{\rightarrow}Q[/latex] combined with P produces Q.

Two other logical inferences are listed above. In each case you should convince yourself (based on your knowledge of the relevant truth tables) that the truth of the statements above the line forces the statement below the line to be true.

Following are some additional useful logical inferences. The first expresses the obvious fact that if P and Q are both true then the statement [latex]P{\wedge}Q[/latex] will be true. On the other hand, [latex]P{\wedge}Q[/latex] being true forces P (also Q) to be true. Finally, if P is true, then [latex]P{\vee}Q[/latex] must be true, no matter what statement Q is.

An Important Note

It is important to be aware of the reasons that we study logic. There are three very significant reasons. First, the truth tables we studied tell us the exact meanings of the words such as “and,” “or,” “not,” and so on. For instance, whenever we use or read the “If…, then” construction in a mathematical context, logic tells us exactly what is meant. Second, the rules of inference provide a system in which we can produce new information (statements) from known information. Finally, logical rules such as DeMorgan’s laws help us correctly change certain statements into (potentially more useful) statements with the same meaning. Thus logic helps us understand the meanings of statements and it also produces new meaningful statements.

Logic is the glue that holds strings of statements together and pins down the exact meaning of certain key phrases such as the “If…, then” or “For all” constructions. Logic is the common language that all mathematicians use, so we must have a firm grip on it in order to write and understand mathematics.

Logical Fallacies in Common Language

In the previous discussion, we saw that logical arguments can be invalid when the premises are not true, when the premises are not sufficient to guarantee the conclusion, or when there are invalid chains in logic. There are a number of other ways in which arguments can be invalid, a sampling of which are given here.