[itex](\forall x)(\forall y)[Dx\rightarrow(Gx\wedge Fx)][Py\rightarrow(Gy\wedge Fy)][/itex]
Quantificational logic is a formal system used in mathematical logic to represent and analyze statements involving quantifiers, such as "for all" and "there exists". It is used to reason about the properties and relationships of objects and sets.
Quantificational logic uses symbols to represent quantifiers and logical connectives. The universal quantifier "for all" is represented by the symbol ∀ and the existential quantifier "there exists" is represented by the symbol ∃. Logical connectives such as "and", "or", and "not" are represented by the symbols ∧, ∨, and ¬, respectively.
The first step in translating an English sentence into quantificational logic is to identify the quantifiers and logical connectives present. Then, replace each quantifier with the appropriate symbol, and each logical connective with its corresponding symbol. Finally, the sentence can be simplified and rearranged using the rules of quantificational logic.
In quantificational logic, negation can be represented using the symbol ¬. When translating a negated sentence, the negation should be placed immediately before the quantifier or statement that is being negated. For example, the negation of "All dogs are friendly" would be represented as ¬(∀x)(Dog(x)→Friendly(x)).
Quantificational logic has many applications in mathematics and computer science. It is used to prove theorems, formulate logical statements, and analyze the properties of mathematical structures. In computer science, it is used in the design and analysis of algorithms, as well as in database and artificial intelligence systems.