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Section 1.3 Valid Arguments
In the previous sections of this chapter we looked at statements from the point of view of truth and falsity. We verified the truth or falsity of statements via truth tables, which allowed us to consider all possible ways in which various component statements might be true or false. This approach, while the most basic way to treat the truth or falsity of statements, does not appear to resemble the way mathematicians prove theorems, which is by starting with the hypotheses, and then writing one new statement at a time, each of which is implied by the previous statements, until the conclusion is reached. In this section we look at the analogous construction in logic, that is, the rules of logical argumentation, and we will see the relation of this approach to what was discussed in the previous sections of this chapter.
Subsection Logical Arguments
Definition 1.46 .
A logical argument is a collection of statements, the last of which is the conclusion of the argument, and the rest of which are the premises of the argument.
Example 1.48 .
Consider the following collection of statements, which has a number of premises together with a conclusion.
If the poodle-o-matic is cheap or is energy efficient, then it will not make money for the manufacturer. If the poodle-o-matic is painted red, then it will make money for the manufacturer. The poodle-o-matic is cheap. Therefore the poodle-o-matic is not painted red.
An argument is a collection of statements that are broken up into premises and a conclusion. However, a random collection of statements, in which there is no inherent connection between those designated as premises and the one designated as conclusion, will not be of much use.
Definition 1.49 .
An argument is valid if the conclusion necessarily follows from the premises.
To a mathematician, what logicians call an argument would simply correspond to the statement of a theorem; the justification that an argument is valid would correspond to what mathematicians call the proof of the theorem.
