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I was learning knowledge representation and reasoning in AI, in which till now, I have studied predicate logic and first order logic. Forward chaining, Backward chaining, Refutation resolution method etc. for proofs.

What I didn't understand is, is there any real world example where this kind of AI is being used ?

By AI, I have always heard of Mainly machine learning (with its subclasses defined as supervised, unsupervised and reinforcement learning).

Can anyone give an example if there is any knowledge based AI system which uses this type of algorithms / rules for inference. Or is this field just outdated for today's scenerios ?

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Knowledge representation is used extensively in the bioinformatics community where ontologies are used to infer classification hierarchies in the annotation of biological data. See for example the Ontology Lookup Service (OLS) (Disclaimer: I am a maintainer of OLS).

As an example, look at liver disease. On the righthand side you will see some of the axioms used in its definition. These are axioms used in inferring the hierarchical structure of the ontology and the placement of liver disease in the hierarchy :

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How is this useful? When a user sees that some data is annotated with EFO_0001421 (liver disease) they can get additional context information from the ontology. I.e., that liver disease is considered to be a digestive system disease and an endocrine system disease. Moreover, by looking at the children of liver disease, they can find all the diseases that are considered to be liver diseases.

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