Final answer:
The given statement does not validate the hypothesis since being popular does not necessarily follow from using shining toothpaste. It represents a non-sequitur and does not conform to valid deductive reasoning principles like the Law of Detachment or the Law of Syllogism.
Step-by-step explanation:
The statement “If you use shining toothpaste, then you'll be popular” is a hypothesis that sets up a conditional relationship between using shining toothpaste and being popular.
The conclusion, “You are popular,” however, does not validate the hypothesis on its own because it does not confirm whether the popularity is a result of using shining toothpaste or not. Therefore, while someone may be popular, it cannot be determined from the given statement alone that shining toothpaste is the cause of their popularity.
This type of argument is a fallacy as it assumes that being popular must necessarily follow from using shining toothpaste, which is a non-sequitur. The logic does not adhere to the principles of valid deductive inferences such as the Law of Detachment or the Law of Syllogism.
The Law of Detachment would require a premise stating that a specific person is indeed using shining toothpaste to then conclude they're popular. The Law of Syllogism would need two conditional statements linking shining toothpaste with being popular, and being popular with another outcome.