Propositional and First-order Logic

Propositions

What is a proposition? A proposition is a statement that is either true or false. Examples of propositions: 1. 2 + 3 = 5 2. 1 + 1 = 3 The first statement " 2 + 3 = 5 " is a proposition whose value is true. The second statement " 1 + 1 = 3 " is a proposition whose value is false since 1 + 1 is not equal to 3.  

Statements that are not proposition

1. X + Y > 4 2. X = 3 3. Are you leaving? 4. Buy four books X and Y are variables. So, any operation on variables cannot be a proposition, since we don't know if the statement is true or false. A question is also not a proposition, since it does not have a true or false value.  Similarly, orders such as "Buy four books" is not a proposition as it does not have a true or false value.

Propositional Variables

Propositional logic is the study of how propositions are combined and related.  A propositional variable is a letter such as P, Q, R that represents a proposition. For example, P is the proposition: All birds can fly. Propositional variables can have only the values: True or False. Such variables are also called Boolean variables.

Compound Propositions

We can combine, modify and relate propositions using words such "not", "and", "or", "implies" and "if-then". For example, we can combine three propositions like this: If all humans are mortal and all Greeks are human, then all Greeks are mortal. Here, there are three propositions which have been combined to form a compound proposition. Let's denote each proposition with a propositional variable. A: All humans are mortal B: All Greeks are human C: All Greeks are mortal.   The compound proposition can then be written as: If A and B, then C.

Logical Connectives

The words connecting propositions such as "and", "or", "if-then" are called logical connectives. There are several logical connectives. NOT, AND, OR, IF-THEN, IMPLIES, IF-AND-ONLY-IF First, we shall study the three logical connectives: "NOT", "AND" and "OR". We can represent compound propositions using logical connectives. Examples are: P AND Q IF P THEN Q