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Subsection 1.3 Implications (If-Then Statements)

An implication is a statement that has a premise and a conclusion such as, "If I take birth control, then I can’t get pregnant". In this implication, the premise is the statement "I take birth control" and the conclusion is the statement "I can’t get pregnant". With an implication, the truth value of the premise and conclusion determine the truth value of the actual implication. Given two statements \(p\) and \(q\text{,}\) the statement: if \(p\text{,}\) then \(q\) can be represented by \(p\rightarrow q\) and read as "if \(p\text{,}\) then \(q\)", "\(p\) implies \(q\)", "\(p\) only if \(q\)", and "\(q\text{,}\) if \(p\)". Below is the truth table for the implication "if \(p\text{,}\) then \(q\)":

The truth value of implications can be counterintuitive at times. This occurs primarily when the premise is false because the implication will always be true. For example the implications below are true since the implications in all are false:

  1. If abortions are federally protected, then people will use abortions to terminate unwanted pregnancies.

  2. If abortions are federally protected, then people will not use abortions to terminate unwanted pregnancies.

  3. If abortions are a federal crime, then more women will be jailed.

  4. If abortions are a federal crime, then more women will be jailed.

In these statements, it seems as if only one of the first two statements should be true and one of the last two statements should be true as the conclusions are negations of one another. However, all four statements are true.

It turns out that the only way for an implication to be false is when the premise is true and the conclusion is false. An example would be, "If abortions are not a federally protected right, then no-one will get an abortion".

Another counterintuitive concept that arises with implications is their negation. The negation, \(\sim (p \rightarrow q)\) is equivalent to the statement \(p\wedge \sim q\text{.}\) This is verified by examining their truth tables.

Verify that the following statements are not equivalent to \(\sim (p \rightarrow q)\) using the truth table generator below:

  1. \(\displaystyle \sim p\rightarrow q\)

  2. \(\displaystyle \sim p\rightarrow \sim q\)

  3. \(\displaystyle p\rightarrow \sim q\)

  4. \(\displaystyle q\rightarrow p\)

Solution.

  1. Not equivalent

  2. Not equivalent

  3. Equivalent

  4. not equivalent

It is important to notice that the negation of an implication is not an implication. For example, the negation of "If abortions are federally protected, then people will not use abortions to terminate unwanted pregnancies" is "Abortions are federally protected and people use abortions to terminate unwanted pregnancies" which is a false statement.