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 2 added 282 characters in body edited Jun 26 at 9:32 Mauro ALLEGRANZA 28.9k22 gold badges2020 silver badges6868 bronze badges The "rules of logic" are the object of study of formal logic and mathematical logic. They define languages and proof systems, like e.g. predicate calculus, that are "applicable" to any topics whatever. The so-called "laws of logic" are formulas that are true irrespective of any possible interpretation, i.e. they hold in every interpretation. In this sense, the laws of "pure" logic are applicable to every domain of discourse consisting of objects whatever. In addition to the study of "pure" predicate logic, we are interested to add to the "logical engine" suitable non-logical constants, like ∈ ("in"), the binary relation of set theory, or + and × ("plus" and "times"), the basic arithmetical operations, with suitable axioms that govern their behaviour. Thus, according to the specific mathematical symbols and axioms introduced, we have different mathematical theories; when the collection of axioms is the first-order version of Peano's axioms, we have PA, i.e. first-order theory of arithmetic. The same for ZF, i.e. Zermelo-Fraenkel Set Theory. In this sense, first-order theory of arithmetic and set theory are specific "applications" of logic to specific domains of inquiry (or universes). We may say that formal mathematical theories are the study of topics where the universe is "constrained" to some specific properties : those defined by the non-logical symbol used (∈, +) and by the specific mathematical axioms postulated for the theory. The "rules of logic" are the object of study of formal logic and mathematical logic. They define languages and proof systems, like e.g. predicate calculus, that are "applicable" to any topics whatever. The so-called "laws of logic" are formulas that are true irrespective of any possible interpretation, i.e. they hold in every interpretation. In this sense, the laws of "pure" logic are applicable to every domain of discourse consisting of objects whatever. In addition to the study of "pure" predicate logic, we are interested to add to the "logical engine" suitable non-logical constants, like ∈ ("in"), the binary relation of set theory, or + and × ("plus" and "times"), the basic arithmetical operations, with suitable axioms that govern their behaviour. Thus, according to the specific mathematical symbols and axioms introduced, we have different mathematical theories; when the collection of axioms is the first-order version of Peano's axioms, we have PA, i.e. first-order theory of arithmetic. The same for ZF, i.e. Zermelo-Fraenkel Set Theory. In this sense, first-order theory of arithmetic and set theory are specific "applications" of logic to domains of inquiry. The "rules of logic" are the object of study of formal logic and mathematical logic. They define languages and proof systems, like e.g. predicate calculus, that are "applicable" to any topics whatever. The so-called "laws of logic" are formulas that are true irrespective of any possible interpretation, i.e. they hold in every interpretation. In this sense, the laws of "pure" logic are applicable to every domain of discourse consisting of objects whatever. In addition to the study of "pure" predicate logic, we are interested to add to the "logical engine" suitable non-logical constants, like ∈ ("in"), the binary relation of set theory, or + and × ("plus" and "times"), the basic arithmetical operations, with suitable axioms that govern their behaviour. Thus, according to the specific mathematical symbols and axioms introduced, we have different mathematical theories; when the collection of axioms is the first-order version of Peano's axioms, we have PA, i.e. first-order theory of arithmetic. The same for ZF, i.e. Zermelo-Fraenkel Set Theory. In this sense, first-order theory of arithmetic and set theory are "applications" of logic to specific domains of inquiry (or universes). We may say that formal mathematical theories are the study of topics where the universe is "constrained" to some specific properties : those defined by the non-logical symbol used (∈, +) and by the specific mathematical axioms postulated for the theory. 1 answered Jun 26 at 9:21 Mauro ALLEGRANZA 28.9k22 gold badges2020 silver badges6868 bronze badges The "rules of logic" are the object of study of formal logic and mathematical logic. They define languages and proof systems, like e.g. predicate calculus, that are "applicable" to any topics whatever. The so-called "laws of logic" are formulas that are true irrespective of any possible interpretation, i.e. they hold in every interpretation. In this sense, the laws of "pure" logic are applicable to every domain of discourse consisting of objects whatever. In addition to the study of "pure" predicate logic, we are interested to add to the "logical engine" suitable non-logical constants, like ∈ ("in"), the binary relation of set theory, or + and × ("plus" and "times"), the basic arithmetical operations, with suitable axioms that govern their behaviour. Thus, according to the specific mathematical symbols and axioms introduced, we have different mathematical theories; when the collection of axioms is the first-order version of Peano's axioms, we have PA, i.e. first-order theory of arithmetic. The same for ZF, i.e. Zermelo-Fraenkel Set Theory. In this sense, first-order theory of arithmetic and set theory are specific "applications" of logic to domains of inquiry.