I need help on Quantifier logic problems?
Using the symbolization key given, symbolize each sentence in QL.
Universal Discourse: animals
Dx: x is a puppy
Hx: x is lost
Tx: is a Terrier
Fxy: x can run faster than y
r: Tim
1.No puppy can run faster than Tim.
2. No animal is faster than itself
3. Some puppies are Terriers and some are not.
4. There is a puppy who is not a Terrier but is faster than any Terrier
5. For every puppy, there is some animal that is faster.
Here is my attempt:I'm uncertain about question 1 and 4 and I need more clarification on those.
1. ~∀x(Dx --> Frx)
2. ~∃xFxx
3. ∃x[(Dx & Tx) & (Dx & ~Tx)]
4. ∃x[(Dx & ~Tx) & ∀yFxy]
5. x(Dx--> ∃yFxy)
1. No puppy can run faster than Tim.
~∀x(Dx → Fxr)
2. No animal is faster than itself.
~∀x(Ax → Fxx)
3. Some puppies are Terriers and some are not.
∃x[(Dx & Tx) ∧ (Dx & ~Tx)]
4. There is a puppy who is not a Terrier but is faster than any Terrier.
∃x[(Dx & ~Tx) ∧ ∀y(Ty → Fxy)]
5. For every puppy, there is some animal that is faster.
∀x(Dx → ∃y(Ay ∧ Fxy))
To symbolize the sentences in quantifier logic, we can go through each sentence and determine the appropriate quantifiers and predicates to use.
1. "No puppy can run faster than Tim."
To symbolize this sentence, we can use a universal quantifier (∀) to express "no puppy." We also need to use predicates to represent "is a puppy" (Dx) and "can run faster than" (Fxy). Finally, we need to specify the individual Tim using a constant symbol (r).
The sentence can be symbolized as: ∀x(Dx → ¬Fxr)
Explanation:
- ∀x: Universal quantifier indicating "for all x."
- (Dx → ¬Fxr): The predicate Dx represents "x is a puppy," ¬Fxr represents "x cannot run faster than Tim." The arrow → denotes "implies."
2. "No animal is faster than itself."
This sentence also involves a universal quantifier (∀) and predicates for "is an animal" (Ax) and "is faster than" (Fxy).
The sentence can be symbolized as: ∀x(Ax → ¬Fxx)
Explanation:
- ∀x: Universal quantifier indicating "for all x."
- (Ax → ¬Fxx): The predicate Ax represents "x is an animal," ¬Fxx represents "x is not faster than itself."
3. "Some puppies are Terriers and some are not."
This sentence involves existential quantifiers (∃) and predicates for "is a puppy" (Dx) and "is a Terrier" (Tx).
The sentence can be symbolized as: ∃x(Dx ∧ Tx) ∧ ∃y(Dy ∧ ¬Ty)
Explanation:
- ∃x(Dx ∧ Tx): Existential quantifier indicating "there exists x" such that Dx is true (x is a puppy) and Tx is true (x is a Terrier).
- ∃y(Dy ∧ ¬Ty): Existential quantifier indicating "there exists y" such that Dy is true (y is a puppy) and ¬Ty is true (y is not a Terrier).
4. "There is a puppy who is not a Terrier but is faster than any Terrier."
This sentence involves an existential quantifier (∃), a universal quantifier (∀), and predicates for "is a puppy" (Dx), "is a Terrier" (Tx), and "can run faster than" (Fxy).
The sentence can be symbolized as: ∃x(Dx ∧ ¬Tx ∧ ∀y(Ty → Fxy))
Explanation:
- ∃x(Dx ∧ ¬Tx): Existential quantifier indicating "there exists x" such that Dx is true (x is a puppy) and ¬Tx is true (x is not a Terrier).
- ∀y(Ty → Fxy): Universal quantifier indicating "for all y," if Ty is true (y is a Terrier), then Fxy is true (x can run faster than y).
5. "For every puppy, there is some animal that is faster."
This sentence involves a universal quantifier (∀), an existential quantifier (∃), and predicates for "is a puppy" (Dx) and "can run faster than" (Fxy).
The sentence can be symbolized as: ∀x(Dx → ∃y(Ay ∧ Fxy))
Explanation:
- ∀x(Dx → ∃y(Ay ∧ Fxy)): Universal quantifier indicating "for all x," if Dx is true (x is a puppy), then there exists y such that Ay is true (y is an animal) and Fxy is true (x can run faster than y).
I hope this helps you understand how to symbolize the given sentences in quantifier logic. Let me know if you have any further questions or need more clarification!