Practice Set 5.1

I. Prove the following theorems:
1. ∴ ~A ⊃ ~(A · B)
2. ∴ A ≡ ~ ~ A [The Law of Double Negation]
3. ∴ (A ⊃ B) ≡ (~B ⊃ ~A) [The Law of Transposition]
4. ∴ (A ∨ B) ∨ ~A

II. Find the following proofs for the following arguments.
1. ~D ⊃ C / C ⊃ (~F ⊃ D) / ~F ∴ D
2. ~(A · ~B) ∴ A ⊃ B
3. ~L ⊃ (M · N) / M ⊃ (~L ⊃ ~O) / O ∴ L
4. (R ⊃ S) · (F ⊃ W) / ~(S ∨ W) ∴ ~(F ∨ R)
5. A ∴ ~A ⊃ G
6. A ∨ B ∴ ~(~A · ~B) [Hint: you will use CD and RAA, and the order in which you make your subproofs will determine whether the proof is manageable or really long]
7. ~(~A · ~B) ∴ A ∨ B [Hint: the basic strategy is RAA, but you will need to use RAA twice more, inside the main RAA strategy]

III. If p and q are both theorems in our propositional logic, is the sentence p · q also a theorem? Explain.

IV. Show that the following argument is SL valid (hint: be very aware of what your atomic sentences are!):
Russell’s Paradox:
If the usual assumptions of intuitive set theory are all true, then the set of all sets that are not members of themselves is a member of the set of all sets that are not members of themselves if and only if the set of all sets that are not members of themselves is not a member of the set of all sets that are not members of themselves. Therefore, the usual assumptions of intuitive set theory are not all true.

Practice Set 4.3

Proofs for Practice Set 4.3

(1) 1. F ∨ F A Deduce: F


(2) 1. ~A ⊃ B A
2. C ⊃ ~D A
3. ~A ∨ C A Deduce: B ∨ ~D


(3) 1. (H ∨ ~B) ⊃ R A
2. (H ∨ ~M) ⊃ P A
3. H A Deduce: P · R


(4) 1. (D ⊃ B) · (C ⊃ D) A
2. (B ⊃ D) · (E ⊃ C) A
3. B ∨ E A Deduce: D ∨ B


(5) 1. D ∨ (E ∨ G) A Deduce: E ∨ (D ∨ G)


(6) 1. T ∨ (P · ~S) A
2. ~S ⊃ T A
3. ~P ⊃ ~T A Deduce: P ≡ T


(7) 1. D ⊃ (B ≡ A) A
2. D ∨ C A
3. ~A ⊃ ~C A Deduce: B ⊃ A

Practice Set 4.2

Proofs section of 4.2 (and all that is due for Exam 2)

Construct proofs for the following arguments.
(1) 1. A ⊃ B A Deduce: ~B ⊃ ~A

(2) 1. D ⊃ (E ⊃ F) A Deduce: E ⊃ (D ⊃ F)

(3) 1. A ⊃ B A Deduce: (~A ⊃ ~C) ⊃ (C ⊃ B)

(4) 1. (D ∨ A) ⊃ B A
2. A A Deduce: B

(5) 1. C ≡ D A
2. D A Deduce: C

Practice 4.1

Shortened Practice set 4.1 (this is the proof section for set 4.1 and all that is due for exam 2]

III. Construct proofs for the following arguments.
(1) 1. A · B A
2. C · D A Deduce: A · D

(2) 1. D ⊃ E A
2. E ⊃ F A
3. ~F A Deduce: ~D

(3) 1. A ⊃ B A
2. C · A A Deduce: B

(4) 1. D ⊃ (A · B) A
2. E ⊃ ~A A
3. D A Deduce: ~E

(5) 1. (B · A) ⊃ E A
2. ~ ~A · B A Deduce: E

(6) 1. D ⊃ (B ⊃ A) A
2. D · ~A A Deduce: ~B

(7) 1. E · (D ∨ B) A
2. (D ∨ B) ⊃ ~ ~A A Deduce: A

(8) 1. D ⊃ (B · A) A
2. ~E · D A
3. ~C ⊃ ~B A Deduce: C

Logic Puzzle #1: The Surprise Execution

The following is a logic puzzle that is widely circulated and has many versions. A popular variation makes the puzzle less macabre by replacing the prisoner with a student and the surprise execution with a surprise quiz (or is this more macabre?). The version that appears below is from Gustafson and Ulrich's Elementary Symbolic Logic.

One Saturday, a prisoner was sentenced to be hanged. "The hanging will take place at 8:00AM," the judge told him, "on Monday, Tuesday, Wednesday, Thursday, or Friday of next week. But you will not have any idea which day it will be until you are so informed at dawn on that day."

Alone later in his cell, the prisoner began to think about the judge's remarks. Obviously they could not wait until the last day to hang him, without violating the judge's decree. "If the first four mornings went past and I were still alive," he thought to himself, "I'd know for sure by that fourth afternoon when the hanging would be--there would only be one day left when it could be! But the judge said that I would not be able to tell in advance like that. So the last day is absolutely ruled out as the day for the hanging; it can only take place on Monday, Tuesday, Wednesday or Thursday." So Thursday, he knew, was the last day they could pick for the hanging. But of course they could not wait until the last day.... "Thursday's not a real possibility either, then," thought the prisoner. "Were I still alive at dusk on Wednesday, I would know that Thursday had to be the day. Friday has already been ruled out."

"So Wednesday seems to be the last day the judge could pick. But for the same reasons, Wednesday can be eliminated as a real possibility... and then Tuesday, and finally Monday. They cannot possibly hang me in accordance with the judge's decree!"

Is the prisoner really saved? Has he, through his logical prowess, discovered that his execution cannot be carried out, barring a violation of the judge's own decree? Or, has he missed something? Will the execution still pay him a visit and on what day will his execution be?

Homework Problems?

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