Having expressed validity through logical implication, we now replace long truth-table verification with standard valid argument forms. Direct rules of inference are reusable patterns: once a rule is known to be valid, it may be applied whenever the same logical form appears. A common mistake is to remember only the name of a rule but not its exact pattern. In this lesson, direct rules of inference begin with Modus Ponens, the Law of the Syllogism, and Modus Tollens. :::definition[Rule of Inference] A rule of inference is a valid argument form that allows a conclusion to be deduced from given premises. If $$\begin{equation*} (p_1\land p_2\land\cdots\land p_n)\to q \end{equation*}$$ is a tautology, then the corresponding argument $$\begin{equation*} p_1,\ p_2,\ldots,p_n\ \therefore\ q \end{equation*}$$ is a valid rule of inference. ::: :::definition[Tabular Form of a Rule] Let $p_1,p_2,\ldots,p_n$ be premises and let $q$ be a conclusion. The form $$\begin{equation*} \begin{array}{c} p_1,\ p_2,\ldots,p_n\\ \hline q \end{array} \end{equation*}$$ means that from the premises $p_1,p_2,\ldots,p_n$, the conclusion $q$ may be inferred. ::: :::theorem[Rule of Detachment] Let $p$ and $q$ be statements. If $p$ is true and $p\to q$ is true, then $q$ is true. This rule is also called Modus Ponens: $$\begin{equation*} \begin{array}{c} p,\ p\to q\\ \hline q \end{array} \end{equation*}$$ ::: :::proof Given that $p$ is true and $p\to q$ is true. To prove that $q$ is true. The conditional $p\to q$ is false only when $p$ is true and $q$ is false. Since $p\to q$ is true while $p$ is true, $q$ cannot be false. Therefore $q$ is true. Hence the Rule of Detachment is valid. $\square$ ::: :::definition[Logical Form of Modus Ponens] The logical implication corresponding to Modus Ponens is $$\begin{equation*} [p\land(p\to q)]\to q. \end{equation*}$$ ::: :::example[Lottery Argument] Let $p$ mean “Lydia wins a ten-million-dollar lottery,” and let $q$ mean “Kay will quit her job.” The argument $$\begin{equation*} p,\ p\to q\ \therefore\ q \end{equation*}$$ has the form of Modus Ponens. Therefore, if Lydia wins the lottery and Lydia’s winning implies that Kay will quit her job, then Kay will quit her job. ::: :::example[Scholarship Argument] Let $p$ mean “Allison vacations in Paris,” and let $q$ mean “Allison won a scholarship.” The argument $$\begin{equation*} p\to q,\ p\ \therefore\ q \end{equation*}$$ is Modus Ponens. Therefore, Allison won a scholarship. ::: :::theorem[Law of the Syllogism] Let $p$, $q$, and $r$ be statements. If $p\to q$ is true and $q\to r$ is true, then $p\to r$ is true: $$\begin{equation*} \begin{array}{c} p\to q,\ q\to r\\ \hline p\to r \end{array} \end{equation*}$$ ::: :::proof Given that $p\to q$ and $q\to r$ are true. To prove that $p\to r$ is true. Assume that $p$ is true. Since $p\to q$ is true, $q$ is true. Since $q\to r$ is true, $r$ is true. Therefore, whenever $p$ is true, $r$ is true. Hence $p\to r$ is true. $\square$ ::: :::definition[Logical Form of the Law of the Syllogism] The logical implication corresponding to the Law of the Syllogism is $$\begin{equation*} [(p\to q)\land(q\to r)]\to(p\to r). \end{equation*}$$ ::: :::example[Divisibility Argument] Let $p$ mean “$35244$ is divisible by $396$,” let $q$ mean “$35244$ is divisible by $66$,” and let $r$ mean “$35244$ is divisible by $3$.” If $p\to q$ and $q\to r$ are true, then by the Law of the Syllogism, $$\begin{equation*} p\to r. \end{equation*}$$ Hence, if $35244$ is divisible by $396$, then $35244$ is divisible by $3$. ::: :::theorem[Modus Tollens] Let $p$ and $q$ be statements. If $p\to q$ is true and $\neg q$ is true, then $\neg p$ is true: $$\begin{equation*} \begin{array}{c} p\to q,\ \neg q\\ \hline \neg p \end{array} \end{equation*}$$ ::: :::proof Given that $p\to q$ is true and $\neg q$ is true. To prove that $\neg p$ is true. Since $\neg q$ is true, $q$ is false. If possible let $p$ be true. Then, because $p\to q$ is true, $q$ must be true. This contradicts that $q$ is false. Therefore $p$ is false. Hence $\neg p$ is true. $\square$ ::: :::definition[Logical Form of Modus Tollens] The logical implication corresponding to Modus Tollens is $$\begin{equation*} [(p\to q)\land\neg q]\to\neg p. \end{equation*}$$ ::: Direct rules of inference work because the premises force a specific conclusion. Choose a rule and set the truth values for the statements involved in that rule. The simulator checks whether the premises are all true and then reports whether the conclusion is forced. This helps separate the name of a rule from the exact logical pattern that makes the rule valid. :::scientific-preview[Direct Rule Simulator] ::: :::example[Sorority Election Argument] Let $p$ mean “Connie is elected president of Phi Delta sorority,” and let $q$ mean “Helen will pledge Phi Delta sorority.” The premises are $$\begin{equation*} p\to q,\ \neg q. \end{equation*}$$ By Modus Tollens, $\neg p$. Hence, Connie was not elected president of Phi Delta sorority. ::: :::solved-problem Establish the validity of the argument $p,\ p\to\neg q,\ \neg q\to\neg r\ \therefore\ \neg r$. ::: :::solution Let $p$, $p\to\neg q$, and $\neg q\to\neg r$ be true premises. From $p$ and $p\to\neg q$, Modus Ponens gives $\neg q$. From $\neg q$ and $\neg q\to\neg r$, Modus Ponens gives $\neg r$. Hence, the conclusion follows from the premises. ::: :::solved-problem Establish the validity of the argument $p\to r,\ r\to s,\ t\lor\neg s,\ \neg t\lor u,\ \neg u\ \therefore\ \neg p$. ::: :::solution Let the premises be true. From $p\to r$ and $r\to s$, the Law of the Syllogism gives $p\to s$. From $\neg t\lor u$ and $\neg u$, Disjunctive Syllogism gives $\neg t$. From $t\lor\neg s$ and $\neg t$, Disjunctive Syllogism gives $\neg s$. From $p\to s$ and $\neg s$, Modus Tollens gives $\neg p$. Hence, the argument is valid. ::: :::exercise Use Modus Ponens or Modus Tollens to complete each valid argument: (1) If Janice has trouble starting her car, then Angela will check Janice’s spark plugs. Janice had trouble starting her car. (2) If Brady solved the first problem correctly, then the answer is $137$. Brady’s answer is not $137$. (3) If this is a repeat-until loop, then the body of the loop is executed at least once. This is a repeat-until loop. (4) If Tim plays basketball in the afternoon, then he will not watch television in the evening. Tim watched television in the evening. (5) If Mary Lou does not tear up George’s photographs, then she will have to display them on his bulletin board. Mary Lou did not display George’s photographs on his bulletin board. ::: :::answer (1) Angela will check Janice’s spark plugs. (2) Brady did not solve the first problem correctly. (3) The body of the loop is executed at least once. (4) Tim did not play basketball in the afternoon. (5) Mary Lou tore up George’s photographs. ::: :::faq[Frequently Asked Questions] Q: Why is Modus Ponens called direct? A: It moves from $p$ and $p\to q$ directly to $q$. Q: How is Modus Tollens different from Modus Ponens? A: Modus Tollens uses the denial of the conclusion of a conditional to deny its hypothesis. Q: Can the order of premises be changed? A: Yes. The rule depends on the logical form, not on the order in which the premises are written. ::: :::call-to-action[Apply Direct Inference Rules] subtitle: Practise matching arguments to Modus Ponens, Modus Tollens, and the Law of the Syllogism before studying common fallacies. button: Next Lesson | /discrete-mathematics/mathematical-logic/logical-implication-and-rules-of-inference/invalid-conditional-reasoning button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/logical-implication-and-rules-of-inference/valid-arguments-and-logical-implication :::