*   starting our in-class discussion of Chapter 10 -
    a Little Propositional Logic

*   definition of propositional logic:
    *   a branch of logic dealing with PROPOSITIONS
        (things that can be labeled as TRUE or FALSE,
	 such as logical statements or expressions in a
         computer programming language),
        their COMBINATIONS,
	and the CONNECTIVES between them

        *   examples of basic connectives:
	    AND, OR, NOT, IF-THEN (also called IMPLIES)

*   Why BOTHER with propositional logic?

    *   because it strips down an argument to allow for
        a logical analysis of its structure;
	"First the Logic, Then the Facts"

*   it has SOME similarities to algbra:
    *   propositions can be represented by letters
    *   logical operations (such as and, or not)
        are then performed on the letters to
	simplify and analyze arguments

*   SIMPLE and COMPOUND statements
    *   a "simple" statement is an "elemental" fact/statement
        without ANY ANDs, ORs (or NOTs)

    *   a "compound" statement is one or more simple
        statements joined together with connectives
	such as AND, OR, etc.

   *   A basic question to ask is:
       how do the True/False values of the PARTS of
       a compound statement affect the statement overall?

*   a USEFUL tool in propositional logic is:
    TRUTH TABLES

    these make it easier to determine the truth or falsehood of
    a compound statement

    *   tend to have a column for each simple statement
        and a column for each desired operation on those
	and then fill in the columns with ALL possible
	combinations of True and False for the simple
	statements involved,

	AND then complete the columns for the operations
	involved

*   let's quickly demo this!

    here's the truth table for the AND operation 
    *   (P and Q) should ONLY be true if *both* 
        P is true *and* Q is true

    P AND Q (where P and Q are propositions or simple statements)

    P    Q    (P and Q)
   --------------------
    T    T        T
    T    F        F
    F    T        F
    F    F        F

    ...(P and Q) is true ONLY if both P is true AND Q is true;
       any other combination makes it false.

*   HOW do truth tables relate to valid arguments?
    *   remember: in a VALID *deductive* argument,
        it is IMPOSSIBLE for all the premises to
        be true and the Conclusion *possibly* be false

    *   SO - consider a truth table for an argument --

        *   say you were to write the argument in symbolic
	    form with the proposition letters for simple statements
	    and these connective/logical operators

	*   THEN say you made a truth table with all of these 

        *   if you were to remove all lines in the truth table
            where ANY premise is false,
	    and then look at the values in the conclusion column
            they had better all be True,
	         OR the argument is INVALID
                 (there's a CHANCE the premises could be all true
		 but the conclusion false...!)

*   SYMBOLS used in truth tables:
    *   AND operation:   AND  &  ^
    *   OR operation:    OR   |  v
    *   NOT operation:   NOT  ~  ¬  
                                    
    *   IF-Then/Implies: IF..THEN ->

*   rewriting our truth table for the AND operation using a symbol:

    P    Q       P^Q
   --------------------
    T    T        T
    T    F        F
    F    T        F
    F    F        F

*   let's try a truth table for the OR operation
    PvQ

    *   PvQ should be true if P is true *or* Q is true
        (and also if both are true)

    P    Q      PvQ
   -----------------
    T    T       T
    T    F       T
    F    T       T
    F    F       F

*   and let's try a truth table for the NOT operation
    ~P

    *   the NOT operation, negation, makes ~P have the
        OPPOSITE value from P -- if P is true, ~P is false,
	and if P is false, then ~P is true

    P    ~P
   ---------
    T     F
    F     T

*   quick-n-sleazy example using a truth table to judge
    a deductive argument's validity:

    Premise 1: Tina is tall.
    Premise 2: Sarah is tall.
    Conclusion: Therefore, Tina is tall AND Sarah is tall.

    P = Tina is tall
    Q = Sarah is tall

    conclusion: P ^ Q

    P    Q       P^Q
   --------------------
    T    T        T
    T    F        F  <-- cross out! has a false premise!
    F    T        F  <-- cross out! has a false premise!
    F    F        F  <-- cross out! has 2 false premises!

    in the only row(s) left,
    if both premises are true, the conclusion is true,
    so this argument IS valid;

*   continuing with this on WEDNESDAY;