Argument Forms

An argument is a sequence of propositions, where the last proposition is the conclusion. An argument form is a sequence of logical forms, called premises, where the last form is the conclusion. An argument can be valid or invalid. Please note, that the validity of an argument does not depend on the propositions involved, but rather on the structure of the argument itself.

We can use truth tables and rules of inference to deduce the validity of an argument.

A common rule of inference is Generalization. It states that if we have a statement $p$ that is true, then we can conclude that $p \lor q$ is also true, for any proposition $q$. The form of Generalization is as follows: $p$ therefore $p \lor q$. We can also inference $q$ therefore $p \lor q$. The inclusive or allows us to infer the disjunction from either of its components as the statement will be true as long as at least one of the components is true.

Another common rule of inference is Specialization. It states that if we have a statement $p \land q$ that is true, then we can conclude that $p$ and $q$ is also true. The form of Specialization is as follows: $p \land q$ therefore $p$, and $p \land q$ therefore $q$. This rule allows us to infer the truth values of the individual components of a conjunction from the conjunction itself. As we know that for a conjunction to be true, both components must be true.

If have a disjunction $p \lor q$ and we know that one of the components is false, we can conclude that the other component must be true. This rule is known as Elimination. The form of Elimination is as follows: $p \lor q$, not $p$ therefore $q$, and $p \lor q$, not $q$ therefore $p$. This rule allows us to infer the truth of one component of a disjunction when we know that the other component is false, as for the disjunction to be true, at least one of the components must be true.

If $p \to q$ and $q \to r$, we can conclude that $p \to r$. This rule is known as Transposition. The form of Transposition is as follows: If $p$ then $q$, if $q$ then $r$ therefore if $p$ then $r$. This rule allows us to chain conditionals together to derive new conditionals.

Suppose we know that $p \lor q$ is true and we also know that if $p$ is true then $r$ is true. Suppose we also know that if $q$ is true then $r$ is true. We can conclude that $r$ is true. This rule is known as Division by Cases. The form of Division by Cases is as follows: $p \lor q$, if $p$ then $r$, if $q$ then $r$ therefore $r$. This rule allows us to infer the truth of a statement based on the truth of a disjunction and the implications of its components. As we know that either $p$ or $q$ is true and both lead to $r$ being true, we can conclude that $r$ must be true.

One of the most common rules of inference is Modus Ponens. It states that if we have a conditional statement $p \to q$ and we know that $p$ is true, then we can conclude that $q$ is also true. The form of Modus Ponens is as follows: If $p$ then $q$, $p$ therefore $q$.

$$\begin{array}{|c|c|c|c|} \hline p & q & p \to q & \text{Conclusion } (q) \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \end{array}$$

In this truth table, we can see that in the first row, where both $p$ and $q$ are true, the conclusion is also true. In the second row, where $p$ is true and $q$ is false, the conclusion is false. In the third and fourth rows, where $p$ is false, the conclusion is not applicable. Therefore, Modus Ponens is a valid rule of inference.

Another important rule of inference is Modus Tollens. It states that if we have a conditional statement $p \to q$ and we know that $q$ is false, then we can conclude that $p$ is also false. The form of Modus Tollens is as follows: If $p$ then $q$, not $q$ therefore not $p$. This follows from the contrapositive of the conditional statement.

$$\begin{array}{|c|c|c|c|} \hline p & q & p \to q & \text{Conclusion } (\neg p) \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \end{array}$$