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Index to all lecture notes …Previous lecture
Construct truth tables to convince yourself of these!
proof #1 (semantic, using the def of logical equiv):
A | B | (A → B) | ¬(A → B) | ¬B | (A ∧ ¬B) |
---|---|---|---|---|---|
T | T | T | F | F | F |
T | F | F | T | T | T |
F | T | T | F | F | F |
F | F | T | F | T | F |
Because the 4th and 6th columns are identical,
we can conclude that the 2 propositions are logically equivalent.
QED
proof #2 (syntactic, using symbol manipulation):
Theorem (Law of Substitutivity of Logical Equivalents):
Now, here's the syntactic proof that ¬(A → B) ≡ (A ∧ ¬B):
≡ (¬¬A ∧ ¬B), by DeMorgan & SL
≡ (A ∧ ¬B), by Double Negation & SL
QED
Note that this symbol manipulation is a little bit like the Transformer toys that turn trucks into monsters, and vice versa.
Here, we "transformed" ¬(A → B)
into (A ∧ ¬B)
by manipulating the symbols according to the rules of Shakespeare's
Law and propositional equivalences.
A semantic "proof" would be needed to decide the truth value of "John gave a book to Mary".
But we only need a syntactic "proof" to decide that "John gave a book to Mary" means the same thing as "Mary received a book from John".
Clearly, you could write a computer program to do this.
(For more info, see
"Automated Theorem Proving")
In propositional logic, these would be represented by p, q, r