Lecture Notes, 15 Sep 2010

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1. Propositional Equivalences:
   1. There are a whole host of famous propositional equivalences;
      see the charts on pp. 24–25.

   2. Among the most important are:
      • (A → B) ≡ (¬A ∨ B) ≡ (¬B → ¬A) ≡ ¬(A ∧ ¬B)
      • the distributive laws (look them up!)
      • De Morgan's Laws (look them up!)

      Construct truth tables to convince yourself of these!

   3. Also note that, because (A → B) ≡ (¬A ∨ B),
      it's also the case that ¬(A → B) ≡ ¬(¬A ∨ B)
      (You should convince yourself of this by constructing another truth table!)

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2. Syntactic vs. Semantic Proofs:
   1. There are 2 different ways of proving things:
      1. semantically (in terms of truth values)
      2. syntactically (by means of symbol manipulation)
         • This is the way that computers would have to do it—by manipulating symbols,
           which is what they do best

   2. Here's a theorem for which I'll give 2 proofs, one semantic, one syntactic:

   3. Theorem:
      ¬(A → B) ≡ (A ∧ ¬B)

      proof #1 (semantic, using the def of logical equiv):

      Show: For all rows of their t-tables, tval(¬(A → B)) = tval(A ∧ ¬B)

      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):

      To do a syntactic proof, we need to appeal to "Shakespeare's Law" ("a rose by any other name would smell as sweet", Romeo & Juliet):

      Theorem (Law of Substitutivity of Logical Equivalents):

      Let A, B be logically equivalent propositions.
      Then A, B can be substituted for each other in any "larger" molecular (or compound) proposition C without changing tval(C).

      Now, here's the syntactic proof that ¬(A → B) ≡ (A ∧ ¬B):

      ¬(A → B) ≡ ¬(¬A ∨ B), by an earlier t-table and Shakespeare's Law

                      ≡ (¬¬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".

      We need to look at the world to see if that sentence is true.

      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".

      We can determine that they are "equivalent in meaning" just by looking at the syntax of the sentences;
      we don't have to look at the world

      Clearly, you could write a computer program to do this.
      (For more info, see "Automated Theorem Proving")

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3. §1.3: Predicates & Quantifiers
   1. The language of propositional logic is very "coarse grained"
      1. It allows us to talk about The True and The False
         but not about objects and their properties & relations

      2. E.g., can't show the relationship between:
         • My Mac is a networked computer.
         • Some Macs are networked computers.
         • All Macs are networked computers.

         In propositional logic, these would be represented by p, q, r

   2. So we need an "object-oriented", finer-grained logical language
      • It's called first-order predicate logic (or sometimes "first-order predicate calculus",
        which has nothing to do with MTH 141)

      • We'll call it "first-order logic", or FOL, for short.