- #1
AmagicalFishy
- 50
- 1
Hello, folks.
I'm taking my first formal logic class and some of the things seem contradictory; I know it's because I'm not fully understanding something, but I don't know what I'm not fully understanding—I hope someone can help me. The problem begins:
The statement I'm having trouble with is...
{ p(a), p(b), p(f(a)), p(f(b)) }⊢Fitch∀x.p(x)
... which I marked true. I'm able to prove that ∀x.p(x) while using only p(a) as a premise, even. The answer is false, and I'm told "p may not hold for terms like f(f(a)), f(f(b)), and so forth." But how could it not? Why would p(f(f(a))) not hold if ∀x.p(x)?
What I think of as I finish typing this that I'm misunderstanding what ⊢Fitch really means, which is "Prove using the Fitch system and no aspects of Herbrand logic." The only way to prove ∀x.p(x) is by using Universal Introduction and Elimination—which... is not encompassed by the provable operator ⊢Fitch?
I'm taking my first formal logic class and some of the things seem contradictory; I know it's because I'm not fully understanding something, but I don't know what I'm not fully understanding—I hope someone can help me. The problem begins:
Problem said:Let L2 be the language consisting of object constants a, b, a unary function constant f, and unary relation constants p, q.
For each of the following statements, state whether it is true or false under the language L2.
The statement I'm having trouble with is...
{ p(a), p(b), p(f(a)), p(f(b)) }⊢Fitch∀x.p(x)
... which I marked true. I'm able to prove that ∀x.p(x) while using only p(a) as a premise, even. The answer is false, and I'm told "p may not hold for terms like f(f(a)), f(f(b)), and so forth." But how could it not? Why would p(f(f(a))) not hold if ∀x.p(x)?
What I think of as I finish typing this that I'm misunderstanding what ⊢Fitch really means, which is "Prove using the Fitch system and no aspects of Herbrand logic." The only way to prove ∀x.p(x) is by using Universal Introduction and Elimination—which... is not encompassed by the provable operator ⊢Fitch?