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Simple sentences which are true or false are basic propositions. Larger and more complex sentences are constructed from basic propositions by combining them with connectives. Thus propositions and connectives are the basic elements of propositional logic. Though there are many connectives, we are going to use the following five basic connectives here:
NOT, AND, OR, IF_THEN (or IMPLY), IF_AND_ONLY_IF.
They are also denoted by the symbols: ¬, ⋀,⋁,→,↔ ,
respectively.
Often we want to discuss properties/relations common to all propositions. In such a case rather than stating them for each individual proposition we use variables representing an arbitrary proposition and state properties/relations in terms of those variables. Those variables are called a propositional variable. Propositional variables are also considered a proposition and called a proposition since they represent a proposition hence they behave the same way as propositions. A proposition in general contains a number of variables. For example (P ⋁Q) contains variables P and Q each of which represents an arbitrary proposition. Thus a proposition takes different values depending on the values of the constituent variables. This relationship of the value of a proposition and those of its constituent variables can be represented by a table. It tabulates the value of a proposition for all possible values of its variables and it is called a truth table.
For example the following table shows the relationship between the values of P, Q and P⋁Q:
OR | ||
P | Q | (P ⋁Q) |
F | F | F |
F | T | T |
T | F | T |
T | T | T |
Let us define the meaning of the five connectives by showing the relationship between the truth value (i.e. true or false) of composite propositions and those of their component propositions. They are going to be shown using truth table. In the tables P and Q represent arbitrary propositions, and true and false are represented by T and F, respectively.
NOT | |
P | ¬P |
T | F |
F | T |
AND | ||
P | Q | (P ⋀Q) |
F | F | F |
F | T | F |
T | F | F |
T | T | T |
OR | ||
P | Q | (P ⋁Q) |
F | F | F |
F | T | T |
T | F | T |
T | T | T |
IMPLIES | ||
P | Q | (P→Q) |
F | F | T |
F | T | T |
T | F | F |
T | T | T |
IF AND ONLY IF | ||
P | Q | ( P ↔Q ) |
F | F | T |
F | T | F |
T | F | F |
T | T | T |
First it is informally shown how complex propositions are constructed from simple ones. Then more general way of constructing propositions is given.
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