Have you ever built a club house, tree house, or fort with your friends? If so, you and your friends likely started by gathering some tools and supplies to work with, such as hammers, saws, screwdrivers, wood, nails, and screws. Hopefully, at least one member of your group had some knowledge of how to use the tools correctly and helped direct the construction project. After all, if your house isn’t built on a strong foundation, it will be weak and could possibly fall apart during the next big storm. This same foundation is important in logic.

In this section, we will begin with the parts that make up all logical arguments. The building block of any logical argument is a logical statement, or simply a statement. A logical statement has the form of a complete sentence, and it must make a claim that can be identified as being true or false.

When making arguments, sometimes people make false claims. When evaluating the strength or validity of a logical argument, you must also consider the truth values, or the identification of true or false, of all the statements used to support the argument. While a false statement is still considered a logical statement, a strong logical argument starts with true statements.

[latex]\begin{array} {|c|c|} \hline \textbf{Logical statement}& \textbf{Negation} \\ \hline \text{Gordon Ramsey is a chef.} & \text{Gordon Ramsey is not a chef.} \\ \hline \text{Tony the Tiger does not have spots.} & \text{Tony the Tiger has spots.}\\ \hline \end{array}[/latex]

The way you phrase your argument can impact its success. If someone presents you with a false statement, the ability to rebut that statement with its negation will provide you with the tools necessary to emphasize the correctness of your position.

Negating Logical Statements Symbolically

The symbol for negation, or not, in logic is the tilde, [latex]\sim[/latex]. So, not [latex]p[/latex] is represented as [latex]\sim{p}[/latex]. To negate a statement symbolically, remove or add a tilde. The negation of not (not [latex]p[/latex]) is [latex]p.[/latex] Symbolically, this equation is [latex]\sim{(\sim{p})}=p[/latex].

Translating Negations between Words and Symbols

In order to analyze logical arguments, it is important to be able to translate between the symbolic and written forms of logical statements.

Expressing Statements with Quantifiers of All, Some, or None

A quantifier is a term that expresses a numerical relationship between two sets or categories. For example, all squares are also rectangles, but only some rectangles are squares, and no squares are circles. In this example, all, some, and none are quantifiers. In a logical argument, the logical statements made to support the argument are called premises, and the judgment made based on the premises is called the conclusion. Logical arguments that begin with specific premises and attempt to draw more general conclusions are called inductive arguments.

Consider, for example, a parent walking with their three-year-old child. The child sees a cardinal fly by and points it out. As they continue on their walk, the child notices a robin landing on top of a tree and a duck flying across to land on a pond. The child recognizes that cardinals, robins, and ducks are all birds, then excitedly declares, “All birds fly!” The child has just made an inductive argument. They noticed that three different specific types of birds all fly, then synthesized this information to draw the more general conclusion that all birds can fly. In this case, the child’s conclusion is false.

The specific premises of the child’s argument can be paraphrased by the following statements:

• Premise: Cardinals are birds that fly.
• Premise: Robins are birds that fly.
• Premise: Ducks are birds that fly.

The general conclusion is “All birds fly!”

All inductive arguments should include at least three specific premises to establish a pattern that supports the general conclusion. To counter the conclusion of an inductive argument, it is necessary to provide a counterexample. The parent can tell the child about penguins or emus to explain why that conclusion is false.

On the other hand, it is usually impossible to prove that an inductive argument is true. So inductive arguments are considered either strong or weak. Deciding whether an inductive argument is strong or weak is highly subjective and often determined by the background knowledge of the person making the judgment. Most hypotheses put forth by scientists using what is called the “scientific method” to conduct experiments are based on inductive reasoning.

In the following example, we will use quantifiers to express the conclusion of a few inductive arguments.

Negating Statements Containing Quantifiers

Recall that the negation of a statement will have the opposite truth value of the original statement. There are four basic forms that logical statements with quantifiers take on.

• All [latex]A[/latex] are [latex]B[/latex].
• Some [latex]A[/latex] are [latex]B[/latex].
• No [latex]A[/latex] are [latex]B[/latex].
• Some [latex]A[/latex] are not [latex]B[/latex].

The negation of logical statements that use the quantifiers all, some, or none is a little more complicated than just adding or removing the word not.

For example, consider the logical statement “All oranges are citrus fruits.” This statement expresses a subset relationship. The set of oranges is a subset of the set of citrus fruit. This means that there are no oranges that are outside the set of citrus fruit. The negation of this statement would have to break the subset relationship. To do this, you could say, “At least one orange is not a citrus fruit.” Or, more concisely, “Some oranges are not citrus fruits.” It is tempting to say “No oranges are citrus fruits,” but that would be incorrect. Such a statement would go beyond breaking the subset relationship, to stating that the two sets have nothing in common. The negation of “[latex]A[/latex] is a subset of [latex]B[/latex]” would be to state that “[latex]A[/latex] is not a subset of [latex]B[/latex],” as depicted by the Venn diagrams below.

The statement “All oranges are citrus fruits” is true, so its negation, “Some oranges are not citrus fruits,” is false.

Now consider the statement “No apples are oranges.” This statement indicates that the set of apples is disjointed from the set of oranges. The negation must state that the two are not disjoint sets, or that the two sets have at least one member in common. Their intersection is not empty. The negation of the statement, “[latex]A[/latex] intersection [latex]B[/latex], is the empty set” is the statement that “[latex]A[/latex] intersection [latex]B[/latex] is not empty,” as depicted in the Venn diagram below.

The negation of the true statement “No apples are oranges” is the false statement “Some apples are oranges.”

The following table summarizes the four different forms of logical statements involving quantifiers and the forms of their associated negations, as well as the meanings of the relationships between the two categories or sets [latex]A[/latex] and [latex]B[/latex].

[latex]\begin{array} {|l|l|} \hline \textbf{Logical statements with quantifiers}& \textbf{Negation of logical statements with quantifiers} \\ \hline \text{Form: All } A \text{ are } B. & \text{Form: Some } A \text{ are not } B. \\ \text{Means: } A \text{ is a subset of } B\text{, } A \subset B. & \text{Means: } A \text{ is not a subset of } B\text{, } A \not\subset B. \\ \text{All zebras have stripes. (True)} & \text{Some zebras do not have stripes. (False)} \\ \hline \text{Form: Some } A \text{ are } B. & \text{Form: No } A \text{ are } B. \\ \text{Means: } A \text{ intersection } B\text{ is not empty, } A \cap B \neq \emptyset . & \text{Means: } A \text{ intersection } B \text{ is empty, } A \cap B = \emptyset. \\ \text{Some fish are sharks. (True)} & \text{No fish are sharks. (False)} \\ \hline \text{Form: No } A \text{ are } B. & \text{Form: Some } A \text{ are } B. \\ \text{Means: } A \text{ intersection } B\text{ is empty, } A \cap B = \emptyset . & \text{Means: } A \text{ intersection } B \text{ is not empty, } A \cap B \neq \emptyset. \\ \text{No trees are evergreens. (False)} & \text{Some trees are evergreens. (True)} \\ \hline \text{Form: Some } A \text{ are not } B. & \text{Form: All } A \text{ are } B. \\ \text{Means: } A \text{ is not a subset of } B \text{, } A \not\subset B. & \text{Means: } A \text{ is a subset of } B\text{, } A \subset B. \\ \text{Some horses are not mustangs. (True)} & \text{All horses are mustangs. (False)} \\ \hline \end{array}[/latex]

As a reminder from the Sets chapter (Chapter 5), “[latex]A \text{ is a subset of } B[/latex]” means that every member of set [latex]A[/latex] is also a member of set [latex]B[/latex]. The intersection of two sets [latex]A[/latex] and [latex]B[/latex] is the set of all elements they share in common. If [latex]A[/latex] intersection [latex]B[/latex] is the empty set, then sets [latex]A[/latex] and [latex]B[/latex] do not have any elements in common. The two sets do not overlap. They are disjoint.

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