Final answer:
The arguments presented for both 'Connie is healthy' and 'Tom is friendly' are valid when using resolution in FOPL. The conclusions logically follow from their respective premises using disjunctive syllogism and modus ponens.
Step-by-step explanation:
The question asks us to determine if certain arguments in the First-Order Predicate Logic (FOPL) are valid using resolution. We will analyze each argument separately.
a. Connie is healthy.
Premises:
- All Italian mothers can cook. (M ⇒ C)
- All cooks are healthy. (C ⇒ H)
- Either Connie or Jing Jing is an Italian mother. (MConnie ∨ MJingJing)
- Jing Jing is not an Italian mother. (¬MJingJing)
Conclusion: Therefore, Connie is healthy. (HConnie)
To determine if the argument is valid, we will use the disjunctive syllogism and modus ponens:
- If Jing Jing is not an Italian mother, then Connie is. (¬MJingJing ⇒ MConnie)
- If Connie is an Italian mother, then she can cook. (MConnie ⇒ CConnie)
- If Connie can cook, then she is healthy. (CConnie ⇒ HConnie)
- Therefore, Connie is healthy. (HConnie)
This argument is valid, as the conclusion logically follows from the premises.
b. Tom is friendly.
Premises:
- All New Yorkers are cosmopolitan. (N ⇒ C)
- All cosmopolitan people are friendly. (C ⇒ F)
- Either Tom or Nick is a New Yorker. (NTom ∨ NNick)
- Nick is not a New Yorker. (¬NNick)
Conclusion: Tom is friendly. (FTom)
To determine if the argument is valid, we follow a similar process of disjunctive syllogism and modus ponens:
- If Nick is not a New Yorker, then Tom is. (¬NNick ⇒ NTom)
- If Tom is a New Yorker, then he is cosmopolitan. (NTom ⇒ CTom)
- If Tom is cosmopolitan, then he is friendly. (CTom ⇒ FTom)
- Therefore, Tom is friendly. (FTom)
This argument is also valid, as its conclusion follows logically from the premises.