Final answer:
The question asks to assess the validity of a logical argument using deductive reasoning, where the correct answer is that the argument is valid via disjunctive syllogism.Therefore, the correct answer is 'c. DD—valid'
Step-by-step explanation:
The question posed relates to propositional logic, which is a key area in mathematical reasoning. It asks to evaluate the validity of a logical argument presented. The argument uses symbols representing logical operators and statements:
- (\sim H \supset K) means 'if not H, then K'.
- (H \supset \sim T) means 'if H, then not T'.
- H \vee \sim H is the law of excluded middle, stating that either H or not H must be true.
It's then concluded that (\sim T \vee K) must be true, meaning 'either not T or K is true'. This conclusion is a result of valid deductive inferences, specifically a form of inference called disjunctive syllogism.
A disjunctive syllogism allows one to conclude that if we have a disjunction (a statement formed using 'or') and one of the disjuncts is false, then the other must be true. Given the premises in the argument, and using the law of excluded middle, we can deduce that the conclusion follows necessarily.
Therefore, the correct answer is 'c. DD—valid', as the argument is a valid instance of a disjunctive syllogism.