Final answer:
When determining if a conclusion follows from given statements, the Law of Detachment and the Law of Syllogism are key. Valid deductive inferences guarantee the conclusion's truth if the premises are true, as seen in arguments like disjunctive syllogism.
Step-by-step explanation:
The Law of Detachment and the Law of Syllogism are used to determine the validity of deductive inferences and conclusions drawn from given statements. The Law of Detachment states that if a conditional statement ('If p, then q') is accepted as true, and the antecedent (p) occurs, then the consequent (q) must also occur. The Law of Syllogism, on the other hand, enables the drawing of a conclusion from two conditional statements when the consequent of one is the antecedent of the other.
A conclusion follows by the Law of Detachment when we have a true conditional statement and the affirmation of its hypothesis; we can then conclude that the conclusion of that conditional statement must be true. The Law of Syllogism allows us to deduce a new conclusion from two premises in the form of 'If p, then q' and 'If q, then r' — leading to a new valid statement 'If p, then r'.
Valid deductive inferences are those where the truth of the premises guarantees the truth of the conclusion. A disjunctive syllogism is one such valid argument form; for example: 'Either it will rain today or it will not rain today. It is not raining. Therefore, it will not rain today.' This uses the law of the excluded middle and noncontradiction to justify the truth of the conclusion given the initial premises.