There is the idea of setting up a code or convention or dictionary between atomic propositions and capital letters.
There are compound propositions, each of which has a main connective which connects its components.
There are five propositional logical connectives:
'∼' which translates back to 'it is not the case that...'
'∧' which translates back to '... and ...'
'∨' which translates back to '... or ...'
'⊃' which translates back to 'if... then ...'
'≡' which translates back to '... if and only if ...'
Not all propositions are atomic propositions. Consider the proposition asserted by 'It is not the case that in 2011 the United States had a female President'. This is a true proposition, yet it is not an atomic one. It is made up of the atomic proposition 'in 2012 the United States had a female President' (which is false) and negation (expressed by 'It is not the case that...'), and the resulting compound proposition, which is the negation of a false proposition, is true.
There are several types of compound proposition.
Thus far we have considered only 'monadic' predicates-- our atomic formulas consist of a predicate followed by only one term-- for example, Fx. But in English we regularly encounter dyadic predicates or relations. For example, 'Arthur is taller than Bert' cannot be symbolized with the tools we have used so far; what is needed is a relation to represent '...is taller than ...' Txy, say, and then the proposition would be symbolized Tab.
In Predicate Logic there are two new logical connectives, the Universal Quantifier (∀x) and the Existential Quantifier (∃x). These are used for symbolizing certain English constructions (they also have their own rules of inference and their own semantics, which we will learn about later).