Project 1 #69
Project 1 #69
Conversation
| T F F T | ||
| T F T T | ||
| T T F T | ||
| T T T T |
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Please fix the spacing.
State whether each is a tautology, a contradiction, or contingent.
| (a) (A→B)→C | ||
| (A ∨ C) ∧ (¬B ∨ C) | ||
| (b) (A→(B∨C))∨(C→¬A) | ||
| (A→(B∨C))∨(C→¬A) |
| (b) (A→(B∨C))∨(C→¬A) | ||
| (A→(B∨C))∨(C→¬A) | ||
| (c) (¬A∧¬B∧C)∨(¬A∧¬C)∨(B∧C)∨A | ||
| (¬A∧¬B∧C)∨(¬A∧¬C)∨(B∧C)∨A |
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You should have conjunctions of disjunction pairs...
| (b) A(x,y) that says that x is an aunt of y | ||
| (c) C(x,y) that says that x and y are cousins | ||
| (d) O(x) that says that x is an only child | ||
| (e) T(x) that says that x has exactly two brothers |
| 6. Explain the difference - using natural language - between the first-order prefixes: | ||
| ``` | ||
| (a) ∃x∀y and ∀x∃y | ||
| There exists for every such that for all y and For all x there exists a y |
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This makes no sense. There exists for every what?
| (a) ∃x∀y and ∀x∃y | ||
| There exists for every such that for all y and For all x there exists a y | ||
| (b) ∃x∀y∃z and ∀x∃y∀z | ||
| There exists an x such that for all y, there exists an z and For all x there exists a y such that for all z |
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You should say why these are different, not just write out in natural language what the symbols mean.
| ``` | ||
| (a) ∀x∀y(¬(Px ∧ Qx) → (¬Px ∨ ¬Qx)) | ||
| 1. ¬∀x∀y(¬(Px ∧ Qx) → (¬Px ∨ ¬Qx)) | ||
| 2. ¬∀y(¬(Pa ∧ Qa) → (¬Pa ∨ ¬Qa))(1) |
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Why does line 2 follow from line 1? It involves dropping a negation without altering anything within its scope.
| assume -Px. | ||
| -Px + -Qx. | ||
| F. | ||
| therefore —Px. |
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?????? If you assume ~Px, derive ~Px + ~Qx, which contradicts a previous line, then what follows is Px, rather than ~Px.
| -Px + -Qx. | ||
| F. | ||
| therefore —Px. | ||
| assume Q. |
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This makes no sense.
So you know, I've stopped reading the proofs in detail at this point. They all need to be revised.
| therefore -(Qx + Px). | ||
| therefore -Px & -Qx => -(Qx + Px). | ||
| 10. Compare and contrast the proofs provided for (a) in your answers to questions 8 and 9. Explain the different assumptions, strategies, etc. exhibited in tree proofs vs natural deduction proofs. | ||
| The natural deduction proof uses double negation to complete the proof while a tree proof attempts to cancel out the assumptions it uses to get the proof. |
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This is incorrect and superficial. Say more.
Here is my project1