5.1.4 Laws of Logic

Logic, like the sciences, has laws. But while the laws of science are meant to accurately describe observed regularities in the natural world, laws of logic can be thought of as rules of thought. Logical laws are rules that underlie thinking itself. Some might even argue that it is only by virtue of these laws that we can have reliable thoughts. To that extent, laws of logic are construed to be laws of reality itself. To see what is meant by this, let’s consider the law of noncontradiction.

Noncontradiction

To understand the law of noncontradiction, we must first define a few terms. First, a statement is a sentence with truth value, meaning that the statement must be true or false. Statements are declarative sentences like “Hawaii is the 50th state to have entered the United States” and “You are reading an online philosophy book.” Sometimes philosophers use the term “proposition” instead of “statement,” and the latter term has a slightly different meaning. But for our purposes, we will use these terms as synonyms. Second, a negation of a statement is the denial of that statement. The easiest way to turn a statement into its negation is to add the qualifier “not.” For example, the negation of “My dog is on her bed” is “My dog is not on her bed.” Third, a contradiction is the conjunction of any statement and its negation. We may also say that any statement and its opposite are contradictory. For example, “My dog is on her bed” and “My dog is not on her bed” are contradictory because the second is the negation of the first. And when you combine a statement and its opposite, you get a contradiction: “My dog is on her bed and my dog is not on her bed.”

The law of noncontradiction is a law about truth, stating that contradictory propositions cannot be true in the same sense, at the same time. While my dog may have been on her bed earlier and now she’s off barking at squirrels, it cannot be true right now that my dog is both on her bed and not on her bed. However, some of you may be thinking about dogs who lie half on their beds and half on the floor (Josie, the dog belonging to the author of this chapter, is one of them). Can it not be true that such a dog is both on and not on their bed? In this instance, we must return to the phrase in the same sense. If we decide that “lying on the bed” means “at least 50% of your body is on the bed,” then we must maintain that definition when looking at propositions to determine whether they are contradictory. Thus, if Josie is half out of the bed with her head on the floor, we can still say “Josie is on the bed.” But notice that “Josie is not on the bed” remains false since we have qualified the meaning of “on the bed.”

For Aristotle, the law of noncontradiction is so fundamental that he claims that without it, knowledge would not be possible—the law is foundational for the sciences, reasoning, and language (Gottlieb 2019). Aristotle thought that the law of noncontradiction was “the most certain of all principles” because it is impossible for someone to believe that the same thing both is and is not (1989, 1005b).

The Excluded Middle

The law of the excluded middle is related to the law of noncontradiction. The law of the excluded middle states that for any statement, either that statement is true, or its negation is true. If you accept that all statements must be either true or false and you also accept the law of noncontradiction, then you must accept the law of the excluded middle. If the only available options for truth-bearing statements are that they are true or false, and if a statement and its negation cannot both be true at the same time, then one of the statements must be true while the other must be false. Either my dog is on her bed or off her bed right now.