Lecture Notes, 10 Sep 2010

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1. "Recursive" Definition of ("Well-Formed" or Grammatically Correct) Proposition:
   1. Base Case:
      Atomic propositions (represented by p, q, r, etc.) are (well-formed (wf), grammatically correct) propositions.

   2. "Recursive" Cases:
      If A, B are (wf, grammatically correct) propositions (either atomic or molecular),
      then:
      1. ¬A
      2. (A ∧ B)
      3. (A ∨ B)
      4. (A → B)
      5. (A ↔ B)
      6. (A ⊕ B)

      are (wf, grammatically correct) "molecular" (or "compound") propositions.

   3. "Closure" clause: Nothing else is a (wf, grammatically correct) proposition.

   ---

2. Computing truth tables for molecular propositions:
   1. Base case: The truth value of an atomic proposition is either T or else it is F.

   2. Recursive case:
      1. The recursive case is based on the Principle of Compositionality:

         The truth value of a molecular proposition depends on (i.e., is a function of) (i.e., is computed from)

         1. the truth values of its "constituent parts"
         2. and its "principal connective"
            Let * be any of the logical connectives.
            Then:
            • The (constituent) parts of (A * B) are A and B.
            • The principal connective of (A * B) is *.

      2. So, to compute the t-val of a molecular proposition of the form (A * B),
         first compute the t-val of A,
         then compute the t-val of B,
         then use the t-val for * to combine these t-vals into the t-val for (A * B).

      3. 
         How do you compute the t-vals of A and of B?
         If they are molecular, follow rule B
         (but aren't we in the middle of applying rule B?
         Yes!
         This is called a "recursive" application of the rule,
         because the same rule "recurs" (= re-occurs) inside of its own application)

         If they are atomic, then apply the Base Case, rule A:

         the t-val is either T or else F.

   3. 
      In the t-table for ((p ∨ q) ∧ ¬(p ∧ q)) from last lecture,
      the "input" columns are the tvals of the atomic propositions,
      the "intermediate columns" are the tvals of the constituent parts,
      and the "output" column is the final t-val for the molecular proposition.

   ---

3. Translating from English to Logic:
   1. This is a hard problem, because there are lots of different ways to say the same thing in English;

   2. It is equivalent to the Natural-Language Understanding Problem of computational linguistics:
      How do you represent English sentences so that computers can understand them?
      1. Trivial way: input English sentence S₁; output atomic proposition p₁
      2. For better ways, see websites, esp. Suber's guide

   ---

4. Propositional Equivalences (§1.2)
   1. Consider the t-table for (p ∨ ¬p)
      (I won't type it in here; you should do it!)

      Note that, for all possible t-vals of its atomic proposition, the t-val of the molecular proposition is T.

   2. Def:
      Let A be a proposition.
      Then A is a tautology
      =def
      for all possible t-vals of A's atomic propositions (i.e., for all rows of A's t-table), t-val(A)=T.

   3. Consider the t-table for (p ∧ ¬p)
      (I won't type it in here; you should do it!)

      Note that, for all possible t-vals of its atomic proposition, the t-val of the molecular proposition is F.

   4. Def:
      Let A be a proposition.
      Then A is a contradiction
      =def
      for all possible t-vals of A's atomic propositions (i.e., for all rows of A's t-table), t-val(A)=F.

   5. Consider the t-table for (p ∧ q)
      (I won't type it in here; you should do it!)

      
      Note that, there are some possible t-vals of its atomic proposition such that the t-val of the molecular proposition is T,
      but there are some other possible t-vals of its atomic proposition such that the t-val of the molecular proposition is F.

   6. 
      Def:
      Let A be a proposition.
      Then A is contingent
      =def
      there are some possible t-vals of A's atomic propositions (i.e., there are some rows of A's t-table) such that t-val(A)=T
      (so it's not a contradiction),
      but there are some (other) possible t-vals of A's atomic propositions (i.e., there are some other rows of A's t-table) such that t-val(A)=F
      (so it's not a tautology).

   7. Recall that the input-output columns of the t-table for ((p ∨ q) ∧ ¬(p ∧ q)) matched those of the t-table for (p ⊕ q).
      1. Def:
         Let A, B be propositions.
         Then A is (logically) equivalent to B
         =def
         for all rows of A's and B's t-tables, t-val(A) = t-val(B)

      2. Notation: A ≡ B for: A is logically equivalent to B
         • So, we can say:
           ((p ∨ q) ∧ ¬(p ∧ q)) ≡ (p ⊕ q)

         • Note that ≡ is not one of our proposition-forming binary connectives
           (it's not part of the recursive definition of "proposition").
           And "A ≡ B" is not a proposition of propositional logic.
           Instead, it's a sentence of "mathematical English" that is about propositional logic.