Using the same defifinitions as in the previous question, translate each of the following predicate logic statements into English. Try to make your English translations as natural sounding as possible. These statements appear long, but try to break them into smaller cohesive pieces. ∀w ∈ K,(T(w) ∧ W(w)) →(∃p ∈ K, P(p) ∧ S(p) ∧ I(p) ∧ A(p, w)). 2. ∃k ∈ K, ∃r ∈ K, k =r ∧ T(k) ∧ T(r) ∧ F(k, r) ∧ F(r, k) ∧ (∀z ∈ K, Z(z) → A(z, k)). 3. ∃t ∈ K, T(t) ∧ W(t) ∧ ∃x ∈ K, ∃y ∈ K, x = y ∧ Z(x) ∧ W(x) ∧ Z(y) ∧ W(y) ∧ A(t, x) ∧ A(t, y) ∧ (∀w ∈ K,(x = w ∧ y = w ∧ Z(w) ∧ W(w)) → ∼ A(t, w))

Construct proof for the following argument within the system of sentential logic: 1. (A & B) ⊃ (C V D)    Premise2. ~(C V (B ⊃ X))         Premise3. ~[D ≡ ~(X & Y)]        Premise4. ~A ⊃ ~Z                     Premise     /: .  ~Z

This question is from course Advanced Logic in graduate level. Please give the answer, if yoy know, in a clear and understandable way, as soon as you can.

This exercise uses the function MapColor and predicates In(x,y), Borders(x,y), and Country(x), whose arguments are geographical regions, along with constant symbols for various regions. In each of the following we give an English sentence and a number of candidate logical expressions. For each of the logical expressions, state whether it (1) correctly expresses the English sentence; (2) is syntactically invalid and therefore meaningless; or (3) is syntactically valid but does not express the meaning of the English sentence.1. Paris and Marseilles are both in France.1.  In(Paris∧Marseilles,France).2.  In(Paris,France)∧In(Marseilles,France).3.  In(Paris,France)∨In(Marseilles,France).2. There is a country that borders both Iraq and Pakistan.1.  ∃c Country(c)∧Border(c,Iraq)∧Border(c,Pakistan).2.  ∃c Country(c)⇒[Border(c,Iraq)∧Border(c,Pakistan)].3.  [∃c Country(c)]⇒[Border(c,Iraq)∧Border(c,Pakistan)].4.  ∃c Border(Country(c),Iraq∧Pakistan).3. All countries that border Ecuador are in South…