Without quite answering your question let us remember that propositional logic (or sentential logic) deals with the truth or falsity of propositions. Here we can think of a proposition as anything about which one can say
it is the case that or
it is not the case that.
The definition for synonymy that you give can be read:
if p characterises a situation and p is synonymous with q then q also characterises that same situation in the same way
if p does not characterise a situation and p is synonymous with q then q also does not characterise that same situation in the same way
if q characterises a situation and p is synonymous with q then p characterises that same situation
if q does not characterise a situation and p is synonymous with q then p does not characterise that same situation in the same way
It is being extra verbose in order to show that the relation is symmetric.
Saeed would appear to be trying to capture semantic relations other than truth-functional relations -- as truth-functional relations are well studied. He is using [T]ruth and [F]alsity here to capture the notion of preservation (or not) of meaning across/between lexical/phrasal/sentential units.
So let's take the phrases
the printer is on fire and
the printer is in flames. Both of these are propositions and if it is the case that an electrical fault has caused your printer to catch fire then both these propositions will be true. Propositions seem to be always of the form
subject (S) copula (c) predicate (P) -- See here for a discussion of the copula -- Are they always cast in this form? In truth I am unsure :) Anyway, we cannot say
It is the case that on fire or
It is the case that in flames but we can say
It is the case that on fire is synonymous with in flames which is how the notion of [T]ruth / [F]alsity is being tied with the notion of synonymy.