Where grammar dealt with things in existence or nonexistence that we can communicate about and apply meaning towards, logic deals with the interconnection between these things. Logic is also an art. Logic is the art of thinking.
Of all the things we perceive, there are correspondences and correlations can be made. We in perceive links and connections between things. These connections on their own can bring about a greater understanding, but we can also combine these connections to interconnect at a larger level and form larger order concepts to help us understand existence.
Through logic we can improve our thinking by recognizing contradictions. Logic is about arriving at non-contradictory identification. By removing contradictions we arrive at a more accurate representation of what is happening in reality.
By noncontradictory thinking we begin to think correctly. Thinking is correct when it is congruent and harmonious with reality. This allows us to see reality and arrive at various truths.
Through logic we learn and know more about the things we first perceive and gain meaning through grammar. We can gain a deeper understanding of what something means. We can validate and verify the definitions and identity of things in existence.