Lecture 2, Tuesday Sept. 5

Lecture Notes, propositional equivalence.

0. Propositions

Propositions are statements with a truth value --- they are either true or false. For example,

"7 + 4 = 10", is a proposition (it is false), and
"George W. Bush is president", is a proposition (it is true),

while statements such as:

"Do your homework", while good advice, is not a proposition, and
"This statement is false", is not a proposition because we do not allow self-referential sentences,

We often use the letters P,Q,R as propositional variables, and they represent some proposition (although we may not know if they are true or false). We can make logical expression from propositional variables through the use of connectives. The most common of these connectives are:

NOT, with symbol ~,
AND, with symbol ^,
OR, with symbol v,
IMPLIES, with symbol -->.

slightly less common are the following connectives

IF AND ONLY IF, sometimes shortened as IFF, with symbol <-->
EXCLUSIVE OR, with symbol (plus with a circle around it).
NAND
NOR

Information on translating between logic and english can be found at the following fantastic web page: link, alththough note that they use a different notation than we do. What does "but" translate as? What are necessary and sufficient conditions?

1. Truth tables.

A logical expression, such as ((p AND q) or r), has a truth value that depends on the truth values of the propositions (the p's, q's, and r's) that make up the expression. A truth table is an organized way of writing down the truth value of an expression, by exhaustively considering every possible set of truth values for the propositions that make up the expression.

Complete truth tables

in a complete truth table, every new column is made as the "and", "or", "implies", "if and only if", or "not" of earlier columns.

How many rows are there in a complete truth table?

2^{number of variables}

How many columns are there in a complete truth table?

number of connectives in the formula, (perhaps less if there is
duplication of clauses in the formula).

Two expressions are called logically equivalent if their truth value is the same in EVERY condition, that is, if their truth value is the same in every row.

2. Logical Equivalences

Some of these "logically equivalent" expressions have special names


Identity           P ^ T       <==>  P         % Rules w/ one propositional variables
Identity           P v F       <==>  P
Domination         P ^ F       <==>  F
Domination         P v T       <==>  T
Absorption         P ^ F       <==>  F
Absorption         P v T       <==>  T
Negation           P ^ ~P      <==>  F
Negation           P v ~P      <==>  T
Idempotent         P ^ P       <==>  P 
Idempotent         P v P       <==>  P 
Commutative Law    P ^ Q       <==>  Q ^ P     % Rules w/ two propositional variables
Commutative Law    P v Q       <==>  Q v P
DeMorgans Law    ~(P v Q)      <==>  (~P ^ ~Q)
DeMorgans Law    ~(P ^ Q)      <==>  (~P v ~Q)
Implies Rule       P --> Q     <==>  (~P v Q)
Contrapositive:    P --> Q     <==>  ~Q --> ~P
Associative Law    P v (Q v R) <==>  (P v Q) v R  % Rules w/ 3 propositional variables
Associative Law    P ^ (Q ^ R) <==>  (P ^ Q) ^ R
Distributive Law   P ^ (Q v R) <==>  (P ^ Q) v (P ^ R) 
Distributive Law   P v (Q ^ R) <==>  (P v Q) ^ (P v R)

3. Common Logic Errors

There are also some common errors, expressions that are not logically equivalent. One of these is that an expression is not equivalent to its "Converse". The converse of P --> Q is Q --> P. Are these two expressions equivalent? To show they are not, we need to find an "assignment of truth values to P,Q" which gives the two expressions different truth values.

4. Proofs

Using the "real" logical equivalences, we can PROVE other equivalences. We can use the above rules to replace all or parts of an expression without changing its truth value. One proof that two formulas are equivalent is to show the chain of replacements that goes from one formula to the other. A fully justified proof includes an annotation of which rule is used in each step.

5. Tautology

When a propositional expression is equivalent to TRUE, we call it a TAUTOLOGY. One can also call the expressions valid.

in a tautology, every possible possible way of assigning truth values to the propositions leads to an expression which is true.

5.5 Satisfiable

When there is some way of assigning truth values so that a propositional expression evaluates to TRUE, we call it satisfiable. A satisfiable expression is valid if every possible way of assigning variables makes the expression evaluate to TRUE.

6. Contradictions

When a propositional expression is equivalent to FALSE, we call it a CONTRADICTION, or unsatisfiable

in a contradiction, every possible possible way of assigning truth values to the propositions leads to an expression which is false.

7. Equivalence

Expressions A, B are equivalent iff A <--> B is a tautology.

8. Completeness of And, OR, NOT

Can define an arbitrary propositional function, by specifying its truth values for a whole column of a truth table. We now constructively prove that we can express any function using just and, or, and not.

9. Optimization of logical circuitry.

There are two things that one might want to minimize:

Criteria one: length of the formula. If we define new connectives, we could make the length of every possible formula with two propositional variables be just three characters long. How many connectives would we need to add?
Criteria two: number of different connectives. If we use fewer different kinds of connectives, we can perhaps simplify the manufacturing process and make it easier to build chips. Can we make up our logical formulas without using some of our connectives?

What do we have to do to prove that we don't need a specific connective?

Why do we use the set of connectives that we do?