Arguments and Validity

Now that we know how to state things precisely, we are ready to think about putting statements together to form arguments. A rigorous argument that is valid constitutes a proof. We need to put the statements together using valid rules.

For example, given the premises:

a conclusion might be "it will rain". Intuitively, this seems valid.

An argument is valid if the truth of the premises implies the conclusion. Given premises $p_1, p_2, \ldots, p_n$, and conclusion $c$, the argument is valid if and only if $(p_1 \wedge p_2 \wedge \cdots \wedge p_n) \rightarrow c$. Note that false premises can lead to a false conclusion!

Rules of Inference

How do we show validity? We use the rules of inference: To show that the premises imply the conclusion, we apply the rules of inference to the premises until we get the conclusion. Here are some examples of how to show an argument is valid: It is also valid to replace premises with others that are logically equivalent. For example, an implication can be replaced with its contrapositive.

Not all arguments are valid! To show an argument is invalid, find truth values for each proposition that make all of the premises true, but the conclusion false.

This works because proving an argument is valid is just showing that an implication is true. Therefore, to show an argument is invalid, we need to show that the implication is false. An implication is false only when the hypothesis is true and the conclusion is false. Since the hypothesis is the conjunction of the premises, this means that each premise is true and the conclusion is false. $$ \underbrace{(p_1 \wedge p_2 \wedge \cdots \wedge p_n)}_{\text{all $T$}} \rightarrow \underbrace{c}_{F} $$

In the above example, it happens that there is only one truth setting that results in all premises being true and the conclusion being false. In general, there could be many different such truth settings.

Arguments with Quantified Statements

Until now, we have restricted our attention to propositional logic. Recall that $P(x)$ is a propositional function, and so when $x$ is an element of the universe of discourse, we simply have a proposition that can be dealt with using the rules of inference for propositions. For predicate logic, we need a few more rules of inference that will allow us to deal with quantified statements. Here are some examples of arguments with quantified statements: The next example will help to illustrate when Universal Generalization may not be applied.