Propositional Logic --- Level A
Syntax of propositions
First it is informally shown how complex propositions are constructed from simple ones.
Then more general way of constructing propositions is given.
In everyday life we often combine propositions to form more complex propositions without paying much attention to them. For example combining "Grass is green", and "The sun is red" we say something like "Grass is green and the sun is red", "If the sun is red, grass is green", "The sun is red and the grass is not green" etc. Here "Grass is green", and "The sun is red" are propositions, and form them using connectives "and", "if... then ..." and "not" a little more complex propositions are formed. These new propositions can in turn be combined with other propositions to construct more complex propositions. They then can be combined to form even more complex propositions. This process of obtaining more and more complex propositions can be described more generally as follows:
Let X and Y represent arbitrary propositions. Then
[ X], [X Y], [X Y], [X Y], and [X Y]
Note that X and Y here represent an arbitrary proposition.
This is actually a part of more rigorous definition of proposition which we see
Example : [ P -> [Q V R] ] is a proposition and it is obtained by first constructing [Q V R] by applying [X V Y] to propositions Q and R considering them as X and Y, respectively, then by applying [ X -> Y ] to the two propositions P and [Q V R] considering them as X and Y, respectively.
Note: Rigorously speaking X and Y above are place holders for propositions, and so they are not exactly a proposition. They are called a propositional variable, and propositions formed from them using connectives are called a propositional form. However, we are not going to distinguish them here, and both specific propositions such as "2 is greater than 1" and propositional forms such as (P Q) are going to be called a proposition.