Inference Rule with Quantifier and implication

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SUMMARY

The discussion centers on the logical implications of the statements regarding healthy and tasty food, specifically using quantifiers and propositional logic. The participant correctly identifies that the argument structure does not allow for the conclusion that hamburgers are tasty, as the premises indicate that all healthy food is not tasty. The conclusion that Duncan does not eat spinach is valid based on the premises provided. The analysis confirms that the relationship between healthiness and taste does not imply that unhealthy food is tasty.

PREREQUISITES
  • Understanding of propositional logic and quantifiers
  • Familiarity with logical implications and argument structures
  • Basic knowledge of logical notation (e.g., d(x), h(x), t(x))
  • Ability to analyze logical statements and derive conclusions
NEXT STEPS
  • Study the principles of propositional logic and quantifiers in detail
  • Learn about logical implications and their applications in argumentation
  • Explore examples of logical fallacies related to health and taste
  • Practice translating everyday statements into logical propositions
USEFUL FOR

Students of logic, philosophy enthusiasts, and anyone interested in understanding the nuances of logical reasoning and argumentation.

master cherundo
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Homework Statement


All healthy food does not taste good. "
Spinach is a healthy food. Duncan only want to eat tasty food. Duncan does not eat spinach. Hamburger is not a healthy food.
Write all possible conclusion.
I try to translate it into proposition with quantifier, such as [tex]t(x)=[/tex]x is tasty, [tex]h(x)=[/tex]x is healthy, [tex]d(x)=[/tex] Doddy eats x.

The Attempt at a Solution



I think [tex]d(hamburger)[/tex] is not a conclusion because the argument is [tex]d(x) \rightarrow t(x)[/tex]. We cannot conclude [tex]d(hamburger)[/tex] or [tex]t(hamburger)[/tex], because proposition said [tex]h(x) \rightarrow ~t(x)[/tex] and [tex]d(x) \rightarrow t(x)[/tex]. Is it right?
 
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Yes, that is correct. (Assuming that "Doddy" is a nickname for "Duncan"!) Saying "All healthy food does not taste good" does NOT imply that unhealthy food does taste good. Of course, "not d(spinach)" would be a conclusion from the first two statements but that is given as the third statement.
 

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