Limitation of Aristotelian logic
One serious limitation of Aristotelian system is its narrow conception of proposition. He restricted it to subject predicate form. Though class-relation is implicit in this theory of syllogism, Aristotle ignored it. There is little wonder that Aristotle did not think of any other relations.
The tools used to test arguments or to construct arguments by Aristotelian system are insufficient. Modern logic further augmented the tools not only in number but also in variety.
Importance of symbolic logic
Symbolic logic is also called modern logic. In symbolic logic symbols replaces words to a great extent. Symbolic logic and mathematics do not stand sundered; so much so, modern logic is also called mathematical logic.
So what is the importance of symbolic logic over classical logic? Language is, generally, ambiguous. It is so for two reasons. In the first place, a significant number of words are equivocal and secondly, many times the construction of sentences and their juxtaposition are misleading so much so they convey meaning very different from what the speaker or author intends.
Replacement of words by symbols and application of logical syntax different from grammatical syntax completely eliminates ambiguity. The meaning of logical syntax becomes clear in due course when sentences are represented by symbols. It is possible to test the validity of arguments only when the statements are unambiguous.
Further, use of symbols saves time and effort required to test the validity of arguments.