Q.15 Explain Inference in fuzzy logic.

Translate the following First Order Logic in English sentence. (Attached)

Show the difference in the interaction of rules, data and answers in machine learning and classical programming.

Provide an example of a logical equivalence involving quantifiers in first-order logic. Your answer: Please answer the questions in a few lines or paragraphs.

3. Prove ⊢ (¬A → A) → A in Hilbert deductive system. Note: In addition to the axioms and rule of inference of H, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook). You may not use theorem 3.31, as this is precisely that theorem.Theorem 3.20 ⊢ ¬A → (A → B). Theorem 3.21 ⊢ A → (¬A → B). Theorem 3.22 ⊢ ¬¬A → A. Theorem3.23 ⊢A→¬¬A.Theorem3.25 ⊢(A→B)→(¬B→¬A).Theorem3.28 ⊢(¬A→false)→A.Rule 3.29 (Reductio ad absurdum) Theorem3.30 ⊢(A→¬A)→¬A.   Prove {¬A} ⊢ (¬B → A) → B in H. Note: In addition to the axioms and rule of inference of Hilbert deductive system, you may use any of the derived rules and/or theorems 3.20-3.30 (as numbered in the textbook).

In modal logic, what is the duality between necessity and possibility? Provide examples of modal dualities.

Specify the interconnections among things and explain the idea of heredity.