Propositional logic and truth table (CSCI 2824 Spring 2015)
In this lecture, we will cover the following concepts:
Propositional Logic, Connectives and Truth Tables.
The truth table to analyze situations.
Equivalence of Formulae, Tautologies and Fallacies.
This lecture corresponds to section 1.3 in Ensley and Crawley's book.
Logic
In a very crude sense, logic is the assembly language of mathematics (or philosophy). Logic can be defined as the study of reasoning itself or the study of techniques for drawing valid conclusions from premises. Understanding Logic is very important to computing at many levels. For instance, propositional Logic and propositional connectives such as AND, OR and NOT form the basic building block of digital circuits from which we build modern computers. Logic is also key to areas of CS such as automata theory, programming language semantics and artificial intelligence.
Propositional Logic
What are propositions? Propositions are simply declarative statements that are either true or false, but not both.
Examples of propositions:
“Socrates is mortal”.
“Jenny went to lunch with Craig”.
“1+1=2”.
“I am telling the truth”.
“The quick brown fox jumped over the lazy brown dog”.
“If 2+2=5, then I am the richest man on earth”.
While studying propositional logic, we do not really care about what the proposition itself “means” just that it is either true or it is false.
Therefore, we simply use propositional variables (also called Boolean variables) to represent propositions. Why this? Just makes it nice and algebra like. Proposition variables can take on boolean values, i.e., T (true) or F (false).
We use letters like 
and so on to represent proposition variables.
Propositional Logic Formulae
Formulae in propositional logic are defined as follows:
Any propositional variable is a formula.
Conjunction: If
and 
are formulas, then 
(read as AND ) is a formula. Disjunction: If
and are formulas, then 
(read as OR ) is a formula. Negation: If
is a formula then 
(read as NOT ) is a formula.
Caution on “OR” operator
When we say ‘‘Either 
OR 
is true’’, we mean that
at least one of the two propositions , is true.
It could be that both of and are true.
(See the definition in the textbook.)
This is different from the exclusive OR (written as XOR or
). For any propositions and ,
means “either or , but not both”.
Let us try some examples. Read the following propositional formulae aloud:
-
.
Example
Consider the following two propositions:
- : It is rainy today.
- : It is sunny today.
Then:
| | It is rainy and sunny today. |
 | It is rainy or sunny today. |
 | It is rainy and not sunny today. |
Truth table
We evaluate propositional formulae using truth tables. For any given proposition formula depending on several propositional variables, we can draw a truth table considering all possible combinations of boolean values that the variables can take, and in the table we evaluate the resulting boolean value of the proposition formula for each combination of boolean values.
Truth Table for AND
| | |  |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Each row represents some kind of a situation. For example, the top most row represents the
situation when propositions and are both ’'true’’. Then we conclude that the
formula in this situation is also ’'true’’.
Logicians call these situations models. In this case, a Logician would
say that the truth assignment 
is a model
of the formula 
. In other words, if you imagine
a situation wherein the proposition is true and the proposition is true then
in that situation formula is true.
To avoid confusion let us use the term “situations” and “models”. We will formalize models later for first-order (predicate) logic.
Truth Table for OR
| | |  |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Can you write down all the models of (read OR )?
Does this correspond to your conception of 
(the logical connective OR)?
Truth Table for NOT
| |  |
| T | F |
| F | T |
Truth Table for Compound Formulae
XOR is an important derived connective that is defined in terms of
. It has the interpretation of “either-or”: i.e, either p or q, but not both.
XOR is the formula 
.
Its truth table can be written as below:
| | |  |  | XOR |
| T | F | T | F | T |
| T | T | F | F | F |
| F | T | F | T | T |
| F | F | F | F | F |
We will go through few more examples of truth tables in the book.
Other examples of derived connectives are:
Implication
is defined as 
(we will study this in detail next week). Equivalence
is defined as 
. Nand (a.k.a not of and)
is defined as 
. NOR (a.k.a not of or)
is defined as 
XOR (exclusive or)
is defined as
.
Let us write the truth table for the 
connective (equivalence connective):
| | | |  |  |
| T | F | F | F | F |
| T | T | T | F | T |
| F | T | F | F | F |
| F | F | F | T | T |
Tautology, Fallacies and Equivalence
Tautology
A formula is a tautology if and only if it is true no matter what value one gives to the propositions involved in the formula.
Example is 
.
| | | |
| T | F | T |
| F | T | T |
No matter what you value one gives , the formula is always true.
Other examples of tautology are
(De Morgan's Law). 
(also written as 
). 
(do not even try expanding this :-)
Fallacies
Fallacies are the opposite of tautologies. These are formulae that are false no matter what the truth values of the propositions in them.
Example: 
.
| | | |
| T | F | F |
| F | T | F |
If we take a tautology and negate it then it becomes a fallacy.
Therefore 
is a fallacy.
Logical Equivalence
Two formulae are logically equivalent if and only if they have the same truth value in each row
of the (joint) truth table.
When two propositional formulae and are logically equivalent,
we use the notation 
or 
.
Example:
The formulae 
and 
are logically equivalent.
To see why let us write their truth tables (we tack them together for convenience).
| | | |  |
| T | T | F | F |
| F | T | T | T |
| T | F | T | T |
| F | F | T | T |
Notice that for all the truth table rows, 
coincide.
Examples of equivalent formulae include
- and
. - and
. - and (De-Morgan's law).
Technically, you need not assume that the formulae have the same set of propositions. For example, and
are not logically equivalent. However, 
and are logically equivalent.
Similarly, 
and 
are logically equivalent since they are both fallacies.
Example: De Morgan's laws
The De Morgan's laws are good examples of logically equivalent
formulae: for any two proposition formulae and ,
Again, the equivalence can be provde using truth tables.
Theorem:
Statement of Theorem:
Whenever two formulae 
and 
are equivalent then 
is a tautology.
Proof: Consider the truth table for the formula generated from a truth table
that has a column for and a column for (as shown below).
 |  |  |  | | |  |
| T | T | | F | F | F | T |
 |  |  | | | | |
| F | F | | T | T | T | T |
Note that for column to have a false entry at some row, and must have different
values at that row. But since is logically equivalent to , they always have the same value at each row.
As a result, the column must be all true, for all the rows in the truth table. Therefore
it is a tautology.