Having tested tautologies, contradictions, and valid arguments by final truth-table columns, the next step is to express argument validity as logical implication. Valid arguments and logical implication give a precise way to say that a conclusion is forced by given premises. In ordinary reasoning, students often accept a conclusion because it sounds plausible; in mathematical logic, plausibility is not enough. In this lesson, valid arguments and logical implication will be studied through implication forms, truth conditions, and short formal deductions. :::definition[Premises and Conclusion] Let $p_1,p_2,p_3,\ldots,p_n$ and $q$ be statements. In an argument of the form $$\begin{equation*} p_1,\ p_2,\ p_3,\ldots,p_n\ \therefore\ q, \end{equation*}$$ the statements $p_1,p_2,p_3,\ldots,p_n$ are called the premises, and the statement $q$ is called the conclusion. ::: :::definition[Implication Form of an Argument] Let $p_1,p_2,\ldots,p_n$ be premises and let $q$ be a conclusion. The implication form corresponding to the argument $$\begin{equation*} p_1,\ p_2,\ldots,p_n\ \therefore\ q \end{equation*}$$ is the conditional statement $$\begin{equation*} (p_1\land p_2\land\cdots\land p_n)\to q. \end{equation*}$$ ::: :::definition[Valid Argument] Let $p_1,p_2,\ldots,p_n$ be premises and let $q$ be a conclusion. The argument $$\begin{equation*} p_1,\ p_2,\ldots,p_n\ \therefore\ q \end{equation*}$$ is called a valid argument if whenever all the premises are true, the conclusion is also true. Equivalently, the argument is valid if $$\begin{equation*} (p_1\land p_2\land\cdots\land p_n)\to q \end{equation*}$$ is a tautology. ::: A valid argument does not require that the premises are actually true in the real world. It requires that no truth-value assignment makes all premises true while making the conclusion false. Thus, to disprove validity, one searches for a row where every premise has value $1$ and the conclusion has value $0$. To prove validity, one may show directly that such a row cannot occur. This is why valid arguments and logical implication are closely connected. A valid argument fails only if one row makes every premise true while the conclusion is false. Use the controls to choose truth values for the statements in the Roger studies argument. The display evaluates each premise and the conclusion for that single row. Notice that a row with a false premise does not disprove validity, but a row with all true premises and a false conclusion would be a counterexample. :::scientific-preview[Validity Row Explorer]