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Fitch Proofs: Logic Exercises, Study notes of Computer Science

University of Pittsburgh (Pitt) - Medical Center-Health SystemComputer Science

A collection of fitch proofs for various logic exercises, including propositional logic and first-order logic. Each proof is presented step by step, with the goal, assumptions, and inference rules clearly stated.

Typology: Study notes

Pre 2010

Uploaded on 09/02/2009

koofers-user-5eh
koofers-user-5eh 🇺🇸

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Fr i Feb 27 22:27:47 EST 2004 Fi tch: probl emMi nus2.prf 1
1. P ∨ Q ; reminder of proof of resolution step
2. ¬P
3. P
4. ⊥⊥ Intro 2,3
5. Q⊥ Elim 4
6. Q
7. QReit 6
8. Q∨ Elim 1,3-5,6-7
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

Partial preview of the text

Download Fitch Proofs: Logic Exercises and more Study notes Computer Science in PDF only on Docsity!

Fri Feb 27 22:27:47 EST 2004 Fitch: problemMinus2.prf 1

P

Q ; reminder of proof of resolution step

¬P

P

Intro

2,

Q

Elim

4

Q

Q

Reit

6

Q

Elim

1,3-5,6-

Fri Feb 27 22:12:55 EST 2004 Fitch: Untitled 3 1

¬(P

Q)

¬( ¬P

¬Q)

P

P

Q

Intro

3

Intro

1,

¬P

¬ Intro

3-

Q

P

Q

Intro

7

Intro

1,

¬Q

¬ Intro

7-

¬P

¬Q

Intro

6,

Intro

2,

¬P

¬Q

¬ Intro

2-

Mon Mar 01 18:29:53 EST 2004 Fitch: Proof Quantifier Strategy 1.prf 1

x P(x)

x ¬P(x)

  1. c

¬P(c)

x ¬P(x)

Intro

4

Intro

2,

¬¬P(c)

¬ Intro

4-

P(c)

¬ Elim

7

x P(x)

Intro

3-

Intro

1,

x ¬P(x)

¬ Intro

2-

x ¬P(x)

¬ Elim

11

Goals

x ¬P(x)

Tue Feb 24 20:52:37 EST 2004 Fitch: Untitled 1 1

∃

x (P(x) ∧

Q(x))

  1. a P(a) ∧

Q(a)

  1. P(a) ∧

Elim 2

  1. Q(a) ∧

Elim 2

∃

x P(x) ∃

Intro 3

∃

x Q(x) ∃

Intro 4

∃

x P(x) ∧ ∃

x Q(x) ∧

Intro 5,

∃

x P(x) ∧ ∃

x Q(x) ∃

Elim 1,2-

Tue Feb 24 22:10:51 EST 2004 Fitch: Untitled 3 1

x F(x) ∨ ∀

x G(x)) ; premise not needed. goal is logically nec

∀

x F(x) ∨ ∀

x G(x)

∀

x F(x)

  1. a
  2. F(a) ∀

Elim 3

  1. F(a) ∨

G(a) ∨

Intro 5

∀

x (F(x) ∨

G(x)) ∀

Intro 4-

∀

x G(x)

  1. b
  2. G(b) ∀

Elim 8

  1. F(b) ∨

G(b) ∨

Intro 10

∀

x (F(x) ∨

G(x)) ∀

Intro 9-

∀

x (F(x) ∨

G(x)) ∨

Elim 2,3-7,8-

x F(x) ∨ ∀

x G(x)) → ∀

x (F(x) ∨

G(x)) →

Intro 2-

Tue Feb 24 22:44:04 EST 2004 Fitch: Untitled 4 1

∀

x (F(x) →

¬(G(x) ∨

H(x)))

∃

x (F(x) ∧

H(x))

  1. a F(a) ∧

H(a)

  1. F(a) →

¬(G(a) ∨

H(a)) ∀

Elim 1

  1. F(a) ∧

Elim 3

  1. ¬(G(a) ∨

H(a)) →

Elim 4,

  1. ¬G(a) ∧

¬H(a) Taut Con 6

  1. H(a) ∧

Elim 3

  1. ¬H(a) ∧

Elim 7

⊥ ⊥

Intro 9,

⊥ ∃

Elim 2,3-

x (F(x) ∧

H(x)) ¬ Intro 2-

Fri Feb 27 21:11:10 EST 2004 Fitch: Problem6.prf 1

x F(x)

x(G(x)

H(x))

x (F(x)

¬G(x))

x H(x)

x F(x)

x (G(x)

H(x))

FO Con

1

x F(x)

x F(x)

Reit

5

x (G(x)

H(x))

  1. a

G(a)

H(a)

Elim

7

x ¬H(x)

FO Con

3

¬H(a)

Elim

10

G(a)

Taut Con

9,

F(a)

¬G(a)

Elim

2

¬F(a)

¬G(a)

FO Con

13

¬F(a)

Taut Con

12,

x ¬F(x)

Intro

8-

x F(x)

FO Con

16

x F(x)

Elim

4,5-6,7-

x ¬F(x)

FO Con

18

x H(x)

x ¬F(x)

Intro

3-

Tue Mar 02 10:55:21 EST 2004 Fitch: problem7.prf 1

∃

x (Teacher(x) ∧

Scholar(x))

∀

x (Scholar(x) →

¬(HasTime(x,football) ∨

HasTime(x,basketball)))

  1. a Teacher(a) ∧

Scholar(a)

  1. Scholar(a) ∧

Elim 3

  1. Scholar(a) →

¬(HasTime(a, football) ∨

HasTime(a, basketball))

Elim 2

  1. ¬(HasTime(a, football) ∨

HasTime(a, basketball)) →

Elim 4,

  1. ¬HasTime(a,football) ∧

¬HasTime(a,basketball) Taut Con 6

  1. ¬HasTime(a,basketball) ∧

Elim 7

  1. Teacher(a) ∧

Elim 3

  1. Teacher(a) ∧

¬HasTime(a, basketball) ∧

Intro 8,

∃

x (Teacher(x) ∧

¬HasTime(x, basketball)) ∃

Intro 10

∃

x (Teacher(x) ∧

¬HasTime(x,basketball)) ∃

Elim 1,3-

  • problem8world.wld Page 1 of 1 03/02/

problem9sentences.sen Page 1 of 1 03/02/

T 1. ∀x (∀y LeftOf(x, y) → Large(x))

T 2. ∀x (∃y ¬LeftOf(x, y) ∨ Large(x)) ; 2 is FO-equivalent to 1 F 3. ∀x ∀y (LeftOf(x, y) → Large(x))

Thu Mar 04 15:03:19 EST 2004 Fitch: bicondallintro.prf 1

∀

x (P(x) →

Q(x))

∀

x (Q(x) →

P(x))

  1. a
    1. P(a)
    2. P(a) →

Q(a) ∀

Elim 1

  1. Q(a) →

Elim 4,

  1. Q(a)
  2. Q(a) →

P(a) ∀

Elim 2

  1. P(a) →

Elim 7,

  1. P(a) ↔

Q(a) ↔

Intro 4-6,7-

∀

x (P(x) ↔

Q(x)) ∀

Intro 3-