Last Update: 15 October 2010
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FOL representation:
Translation into FOL:
Formal proof of validity {with strategy shown in comment-braces like this}:
1. | ∃xMiss(x) | : P3 |
2. | Miss(c) | : 1, EI |
3. | ∀x[Exam(x) → ¬Miss(x)] | : P2 |
4. | Exam(c) → ¬Miss(c) | : 3, UI |
5. | ¬Exam(c) | : 2,4; MT |
6. | ∀x[Pass(x) → Exam(x)] | : P1 |
7. | Pass(c) → Exam(c) | : 6, UI |
8. | ¬Pass(c) | : 5,7; MT |
9. | ∃x¬Pass(x) | : 8, EG |
Consider another version of the argument:
Propositional-Logic Representation:
Propositional-Logic Translation:
Formal proof of validity (2 versions):
1. | m | : P3 | or: | p → e | : P1 | ||
2. | e → ¬m | : P2 | e → ¬m | : P2 | |||
3. | ¬e | : 1,2; MT | p → ¬m | : 1,2; HS | |||
4. | p → e | : P1 | m | : P3 | |||
5. | ¬p | : 4,5; MT | ¬p | : 3,4; MT |
Truth-Table Analysis:
P3 | P1 | P2 | C | ||||||||||||||||||
e | p | m | ¬m | p→e | e→¬m | ¬p | |||||||||||||||
T | T | T | F | T | F | F | |||||||||||||||
T | T | F | T | T | T | F | |||||||||||||||
T | F | T | F | T | F | T | |||||||||||||||
T | F | F | T | T | T | T | |||||||||||||||
F | T | T | F | F | T | F | |||||||||||||||
F | T | F | T | F | T | F | |||||||||||||||
F | F | T | F | T | T | T | ⇐ only row with all T premises; has T conclusion /∴ valid argument | ||||||||||||||
F | F | F | T | T | T | T |
Given that we can express the argument in both propositional logic and
FOL,
which should we use?
In general, use the simplest logic that enables you to prove validity,
which, in this case, would be propositional logic.
However, there are some arguments that are invalid if expressed in
propositional logic
but valid if expressed in FOL;
these have to be
expressed in FOL to show that they are, indeed, valid.
Here's one example (which goes back to Aristotle):
In propositional logic, this would be:
which is clearly invalid.
In FOL, we might have:
P1. ∀x[Horse(x) → Animal(x)]
C . ∴ ∀x∀y[(Horse(x) ∧ Head(x,y)) → (Animal(x) ∧ Head(x,y))]
proof:
1. | Horse(a) ∧ Head(b,a) | : temporary assumption to show (Animal(a) ∧ Head(b,a)) by Dir. Pf. |
2. | ∀x[Horse(x) → Animal(x)] | : P1 |
3. | Horse(a) → Animal(a) | : 2, UI |
4. | Horse(a) | : 1, Simp |
5. | Animal(a) | : 3,4; MP |
6. | Head(b,a) | : 1, Simp |
7. | Animal(a) ∧ Head(b,a) | : 5,6; Conj |
8. | (Horse(a) ∧ Head(b,a)) → (Animal(a) ∧ Head(b,a)) | : 1,7; Direct Proof |
9. | ∀x∀y[(Horse(x) ∧ Head(x,y)) → (Animal(x) ∧ Head(x,y))] | : 8, UG |