Inference in First Order Logic (FOL)-Eliminating Quantifiers-Universal & Existential Instantiation

Unit – 2 – Logical Reasoning Inferences in First Order Logic Eliminating Quantifiers - Universal Instantiation & Existential Instantiation Inference in FOL is used to generate new sentences from existing sentences. Definition: An expression X logically follows form a set S, if every interpretation that satisfies S also satisfies X The function of logical inference is to produce new sentence that logically follow a given set of FOL sentence. Universal Instantiation (UI) /Universal Elimination UI says that we can infer (produce) any sentence obtained by substituting a ground term for the variable. we use the notion of Substitutions for these instantiations. Let SUBST(θ,α) denote the result of applying the substitution θ to the sentence α v αSUBST({v/g}, α) for any variable v and ground term g. Substitutions E.g., KB contains “all greedy kings are evil” x King(x)  Greedy(x)  Evil(x) yields (eliminate ) SUBST(x/John) King(John)  Greedy(John)  Evil(John) SUBST(x/Richard) King(Richard)  Greedy(Richard)  Evil(Richard) SUBST(x/Father(John)) King(Father(John))  Greedy(Father(John))  Evil(Father(John)) Existential Instantiation (EI) / Existential Elimination For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base: v α SUBST({v/k}, α) E.g., x Crown(x)  OnHead(x, John) yields: (eliminate ) Crown(C1)  OnHead(C1,John) Skolem Constant provided C1 is a new constant symbol which is not in KB but satisfy all properties of ‘x’, called a Skolem constant (skolemization – replacing variables with ground terms) Reduction to Propositional Inference Suppose the KB contains just the following: x King(x)  Greedy(x)  Evil(x) King(John) Greedy(John) Brother(Richard, John) Instantiating the universal sentence in all possible ways, we have: King(John)  Greedy(John)  Evil(John) King(Richard)  Greedy(Richard)  Evil(Richard) King(John) Greedy(John) Brother(Richard, John) The new KB is Propositionalized: proposition symbols are King(John), Greedy(John), Evil(John), King(Richard), etc. Problems with Propositionalization Propositionalization seems to generate lots of irrelevant sentences. with function symbols, there are infinitely many ground terms, e.g., Father(Father(Father(John))) Subscribe this channel, comment and share with your friends. For Syllabus, Text Books, Materials and Previous University Question Papers and important questions Follow me on Blog : https://dsumathi.blogspot.com/ Facebook Page : https://www.facebook.com/profile.php?... Instagram :   / dsumathiphd