Propositions are statements that are only either true or false. They cannot be both, nor can they be any other value. They are built from one or more atoms (logical connectives are used to connect 2 or more such atoms).
All atoms are propositions, because they are either true or false, but the opposite is not true. A proposition may be able to be broken down into atoms, but an atom cannot be broken down any further.
Consider the example:
This is a proposition: I either am a human and have 2 legs, or I don’t. But it can be broken down into the atoms:
which are also either true or false (hence are propositions) but cannot be broken down any further. So, we have constructed a proposition using those two atoms (and the logical connective conjunction
Properties
A proposition can also be:
Tautology
A tautology is a statement that always holds true. For example :
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | F | T | T | T |
The last column is a tautology.
Logical Equivalence
Two statements are said to be logically equivalent if they both have the same truth tables. if:
| T | T |
| T | T |
| F | F |
| F | F |
Contradiction
A logical statement is a contradiction if it is always false. For example, is a contradiction:
| T | F | F |
| T | F | F |
| F | T | F |
| F | T | F |