Logical Form in language

So, for today's class in phil. language we will be discussing what a logical form of a language is. The definition that was set forth by the author of the book that we are reading is as follows:

The logical form of a language is what the grammatical structured demands of logical rules, e.g. the logical implications that result from the use of a language. (This is a loose definition and paraphrase.)

So, this seems similarly related to the problem of negative existentials. If we say that pegasusi don't exist, how can we say that? If we refer to a non-existent object it seems that the statement is frivolous. If we attempt to deny that an object exists, it seems that it must exist in some sense. I think that there is a difference between 'there is' and 'there exists'. Otherwise how can we talk about objects such as pegasusi or squared-circles? This is really a tricky topic. I just woke up, so I'll have to compose better thoughts on this after class.