What exactly is a first-order logic?Is First Order Logic (FOL) the only fundamental logic?Is First Order Logic (FOL) the only fundamental logic?What are the advantages to intentionally omit (small) parts from first order logic?Has anyone developed a “restricted English” notation for logic (first-order or otherwise)?How important are equality, functions and constants for first order logic?Is there any relation beetwen justification logic and type theory?Concerning the logical projection: How to express nullary and unary operations as binary operations?Calculus for Higher Order LogicWhat exactly are the identity rules in logic?Ontological status of variablesClassical logic, symbolic logic, higher-order logic, First-order logic? Learning from scratch
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What exactly is a first-order logic?
Is First Order Logic (FOL) the only fundamental logic?Is First Order Logic (FOL) the only fundamental logic?What are the advantages to intentionally omit (small) parts from first order logic?Has anyone developed a “restricted English” notation for logic (first-order or otherwise)?How important are equality, functions and constants for first order logic?Is there any relation beetwen justification logic and type theory?Concerning the logical projection: How to express nullary and unary operations as binary operations?Calculus for Higher Order LogicWhat exactly are the identity rules in logic?Ontological status of variablesClassical logic, symbolic logic, higher-order logic, First-order logic? Learning from scratch
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Can someone explain in simple terms what exactly is a first-order logic?
From my amateur standpoint, I think that first-order logic is a some kind of a system of symbols and general logical rules and operations defined on that set of symbols in such a way that a first-order logic has some expressional "power" (that is, some statements can be represented in first-order logic and some theorems about first-order logic can be deduced).
However, when it comes to theorems, that is where I am stuck, because, basically, I do not know what exactly can be proved in first-order logic, including theorems about statements in first-order logic and about compound statements, and also theorems about first-order logic itself.
So, can someone here give, in as simple as possible terms, an explanation and description of a first-order logic? Preferably, as short as possible one.
Also, is there only one first-order logic or there are many first-order logics, each differing from all the other in axioms that are used to build such a theory?
logic symbolic-logic philosophy-of-logic
edited Jun 15 at 8:20
Mauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
asked Jun 15 at 6:04
GrešnikGrešnik
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Grešnik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
Can someone explain in simple terms what exactly is a first-order logic?
From my amateur standpoint, I think that first-order logic is a some kind of a system of symbols and general logical rules and operations defined on that set of symbols in such a way that a first-order logic has some expressional "power" (that is, some statements can be represented in first-order logic and some theorems about first-order logic can be deduced).
However, when it comes to theorems, that is where I am stuck, because, basically, I do not know what exactly can be proved in first-order logic, including theorems about statements in first-order logic and about compound statements, and also theorems about first-order logic itself.
So, can someone here give, in as simple as possible terms, an explanation and description of a first-order logic? Preferably, as short as possible one.
Also, is there only one first-order logic or there are many first-order logics, each differing from all the other in axioms that are used to build such a theory?
logic symbolic-logic philosophy-of-logic
edited Jun 15 at 8:20
Mauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
asked Jun 15 at 6:04
GrešnikGrešnik
1474 bronze badges
New contributor
Grešnik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
First order logic is a logic equivalent to a predicate calculus, a formal system with connectives and quantifiers, where one can only quantify over non-logical variables, but not over predicates. Some logical laws and rules of inference govern possible deductions. More broadly, systems built over it (by adding non-logical axioms) are called first order, e.g. Peano arithmetic and ZFC set theory.
– Conifold
Jun 15 at 6:38
See Is First Order Logic (FOL) the only fundamental logic ?
– Mauro ALLEGRANZA
Jun 15 at 7:55
add a comment |
Can someone explain in simple terms what exactly is a first-order logic?
From my amateur standpoint, I think that first-order logic is a some kind of a system of symbols and general logical rules and operations defined on that set of symbols in such a way that a first-order logic has some expressional "power" (that is, some statements can be represented in first-order logic and some theorems about first-order logic can be deduced).
However, when it comes to theorems, that is where I am stuck, because, basically, I do not know what exactly can be proved in first-order logic, including theorems about statements in first-order logic and about compound statements, and also theorems about first-order logic itself.
So, can someone here give, in as simple as possible terms, an explanation and description of a first-order logic? Preferably, as short as possible one.
Also, is there only one first-order logic or there are many first-order logics, each differing from all the other in axioms that are used to build such a theory?
logic symbolic-logic philosophy-of-logic
edited Jun 15 at 8:20
Mauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
asked Jun 15 at 6:04
GrešnikGrešnik
1474 bronze badges
New contributor
Grešnik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Can someone explain in simple terms what exactly is a first-order logic?
From my amateur standpoint, I think that first-order logic is a some kind of a system of symbols and general logical rules and operations defined on that set of symbols in such a way that a first-order logic has some expressional "power" (that is, some statements can be represented in first-order logic and some theorems about first-order logic can be deduced).
However, when it comes to theorems, that is where I am stuck, because, basically, I do not know what exactly can be proved in first-order logic, including theorems about statements in first-order logic and about compound statements, and also theorems about first-order logic itself.
So, can someone here give, in as simple as possible terms, an explanation and description of a first-order logic? Preferably, as short as possible one.
Also, is there only one first-order logic or there are many first-order logics, each differing from all the other in axioms that are used to build such a theory?
logic symbolic-logic philosophy-of-logic
logic symbolic-logic philosophy-of-logic
edited Jun 15 at 8:20
Mauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
asked Jun 15 at 6:04
GrešnikGrešnik
1474 bronze badges
New contributor
Grešnik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jun 15 at 8:20
Mauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
asked Jun 15 at 6:04
GrešnikGrešnik
1474 bronze badges
New contributor
Grešnik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jun 15 at 8:20
Mauro ALLEGRANZA
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edited Jun 15 at 8:20
Mauro ALLEGRANZA
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edited Jun 15 at 8:20
Mauro ALLEGRANZA
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28.4k2 gold badges20 silver badges68 bronze badges
asked Jun 15 at 6:04
GrešnikGrešnik
1474 bronze badges
New contributor
Grešnik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jun 15 at 6:04
GrešnikGrešnik
1474 bronze badges
asked Jun 15 at 6:04
GrešnikGrešnik
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1474 bronze badges
New contributor
Grešnik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
Grešnik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
First order logic is a logic equivalent to a predicate calculus, a formal system with connectives and quantifiers, where one can only quantify over non-logical variables, but not over predicates. Some logical laws and rules of inference govern possible deductions. More broadly, systems built over it (by adding non-logical axioms) are called first order, e.g. Peano arithmetic and ZFC set theory.
– Conifold
Jun 15 at 6:38
See Is First Order Logic (FOL) the only fundamental logic ?
– Mauro ALLEGRANZA
Jun 15 at 7:55
add a comment |
First order logic is a logic equivalent to a predicate calculus, a formal system with connectives and quantifiers, where one can only quantify over non-logical variables, but not over predicates. Some logical laws and rules of inference govern possible deductions. More broadly, systems built over it (by adding non-logical axioms) are called first order, e.g. Peano arithmetic and ZFC set theory.
– Conifold
Jun 15 at 6:38
See Is First Order Logic (FOL) the only fundamental logic ?
– Mauro ALLEGRANZA
Jun 15 at 7:55
First order logic is a logic equivalent to a predicate calculus, a formal system with connectives and quantifiers, where one can only quantify over non-logical variables, but not over predicates. Some logical laws and rules of inference govern possible deductions. More broadly, systems built over it (by adding non-logical axioms) are called first order, e.g. Peano arithmetic and ZFC set theory.
– Conifold
Jun 15 at 6:38
First order logic is a logic equivalent to a predicate calculus, a formal system with connectives and quantifiers, where one can only quantify over non-logical variables, but not over predicates. Some logical laws and rules of inference govern possible deductions. More broadly, systems built over it (by adding non-logical axioms) are called first order, e.g. Peano arithmetic and ZFC set theory.
– Conifold
Jun 15 at 6:38
See Is First Order Logic (FOL) the only fundamental logic ?
– Mauro ALLEGRANZA
Jun 15 at 7:55
See Is First Order Logic (FOL) the only fundamental logic ?
– Mauro ALLEGRANZA
Jun 15 at 7:55
add a comment |
1 Answer
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FOL is the natural logic environment to formalize mathematical theories.
The basic characteristic of predicate calculus is the use of quantifiers : first-order logic is predicate calculus where quantification is restricted to individual variables (variables ranging over "objects") and quantification over predicate variables (i.e. variables ranging over "properties") is not allowed.
Propositional calculus, instead, is only a "toy": it is based on a very simplified model of language that is not useful to develop interesting theories, but can be used efficiently to study the basic properties of a formal system : consistency, completeness, etc.
With FOL we have the "logical engine", i.e. the syntax of the language with axioms and rules, and we usually study it in a similar way to the study of propositional calculus, in order to understand the basic meta-logic properties.
When we study "pure" FOL, we define the derivability relation (⊢), where :
⊢ φ means : "formula φ is derivable in the calculus", and Γ ⊢ φ means : "formula φ is derivable in the calculus form the set Γ of assumptions".
With it we prove the fundamental Soundness and Completeness Theorem :
Γ ⊢ φ iff Γ ⊨ φ, where the symbol ⊨ means semantic consequence.
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.
Unfortunately, not all interesting mathematical properties are expressible with FOL; see Second-order and Higher-order Logic.
answered Jun 15 at 8:08
Mauro ALLEGRANZAMauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
You might want to define the ⊨ symbol before using it.
– Kevin
Jun 18 at 16:59
add a comment |
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FOL is the natural logic environment to formalize mathematical theories.
The basic characteristic of predicate calculus is the use of quantifiers : first-order logic is predicate calculus where quantification is restricted to individual variables (variables ranging over "objects") and quantification over predicate variables (i.e. variables ranging over "properties") is not allowed.
Propositional calculus, instead, is only a "toy": it is based on a very simplified model of language that is not useful to develop interesting theories, but can be used efficiently to study the basic properties of a formal system : consistency, completeness, etc.
With FOL we have the "logical engine", i.e. the syntax of the language with axioms and rules, and we usually study it in a similar way to the study of propositional calculus, in order to understand the basic meta-logic properties.
When we study "pure" FOL, we define the derivability relation (⊢), where :
⊢ φ means : "formula φ is derivable in the calculus", and Γ ⊢ φ means : "formula φ is derivable in the calculus form the set Γ of assumptions".
With it we prove the fundamental Soundness and Completeness Theorem :
Γ ⊢ φ iff Γ ⊨ φ, where the symbol ⊨ means semantic consequence.
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.
Unfortunately, not all interesting mathematical properties are expressible with FOL; see Second-order and Higher-order Logic.
answered Jun 15 at 8:08
Mauro ALLEGRANZAMauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
You might want to define the ⊨ symbol before using it.
– Kevin
Jun 18 at 16:59
add a comment |
FOL is the natural logic environment to formalize mathematical theories.
The basic characteristic of predicate calculus is the use of quantifiers : first-order logic is predicate calculus where quantification is restricted to individual variables (variables ranging over "objects") and quantification over predicate variables (i.e. variables ranging over "properties") is not allowed.
Propositional calculus, instead, is only a "toy": it is based on a very simplified model of language that is not useful to develop interesting theories, but can be used efficiently to study the basic properties of a formal system : consistency, completeness, etc.
With FOL we have the "logical engine", i.e. the syntax of the language with axioms and rules, and we usually study it in a similar way to the study of propositional calculus, in order to understand the basic meta-logic properties.
When we study "pure" FOL, we define the derivability relation (⊢), where :
⊢ φ means : "formula φ is derivable in the calculus", and Γ ⊢ φ means : "formula φ is derivable in the calculus form the set Γ of assumptions".
With it we prove the fundamental Soundness and Completeness Theorem :
Γ ⊢ φ iff Γ ⊨ φ, where the symbol ⊨ means semantic consequence.
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.
Unfortunately, not all interesting mathematical properties are expressible with FOL; see Second-order and Higher-order Logic.
answered Jun 15 at 8:08
Mauro ALLEGRANZAMauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
You might want to define the ⊨ symbol before using it.
– Kevin
Jun 18 at 16:59
add a comment |
FOL is the natural logic environment to formalize mathematical theories.
The basic characteristic of predicate calculus is the use of quantifiers : first-order logic is predicate calculus where quantification is restricted to individual variables (variables ranging over "objects") and quantification over predicate variables (i.e. variables ranging over "properties") is not allowed.
Propositional calculus, instead, is only a "toy": it is based on a very simplified model of language that is not useful to develop interesting theories, but can be used efficiently to study the basic properties of a formal system : consistency, completeness, etc.
With FOL we have the "logical engine", i.e. the syntax of the language with axioms and rules, and we usually study it in a similar way to the study of propositional calculus, in order to understand the basic meta-logic properties.
When we study "pure" FOL, we define the derivability relation (⊢), where :
⊢ φ means : "formula φ is derivable in the calculus", and Γ ⊢ φ means : "formula φ is derivable in the calculus form the set Γ of assumptions".
With it we prove the fundamental Soundness and Completeness Theorem :
Γ ⊢ φ iff Γ ⊨ φ, where the symbol ⊨ means semantic consequence.
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.
Unfortunately, not all interesting mathematical properties are expressible with FOL; see Second-order and Higher-order Logic.
answered Jun 15 at 8:08
Mauro ALLEGRANZAMauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
FOL is the natural logic environment to formalize mathematical theories.
The basic characteristic of predicate calculus is the use of quantifiers : first-order logic is predicate calculus where quantification is restricted to individual variables (variables ranging over "objects") and quantification over predicate variables (i.e. variables ranging over "properties") is not allowed.
Propositional calculus, instead, is only a "toy": it is based on a very simplified model of language that is not useful to develop interesting theories, but can be used efficiently to study the basic properties of a formal system : consistency, completeness, etc.
With FOL we have the "logical engine", i.e. the syntax of the language with axioms and rules, and we usually study it in a similar way to the study of propositional calculus, in order to understand the basic meta-logic properties.
When we study "pure" FOL, we define the derivability relation (⊢), where :
⊢ φ means : "formula φ is derivable in the calculus", and Γ ⊢ φ means : "formula φ is derivable in the calculus form the set Γ of assumptions".
With it we prove the fundamental Soundness and Completeness Theorem :
Γ ⊢ φ iff Γ ⊨ φ, where the symbol ⊨ means semantic consequence.
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.
Unfortunately, not all interesting mathematical properties are expressible with FOL; see Second-order and Higher-order Logic.
answered Jun 15 at 8:08
Mauro ALLEGRANZAMauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
edited 2 days ago
answered Jun 15 at 8:08
Mauro ALLEGRANZAMauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
answered Jun 15 at 8:08
Mauro ALLEGRANZAMauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
answered Jun 15 at 8:08
Mauro ALLEGRANZAMauro ALLEGRANZA
28.4k2 gold badges20 silver badges68 bronze badges
28.4k2 gold badges20 silver badges68 bronze badges
You might want to define the ⊨ symbol before using it.
– Kevin
Jun 18 at 16:59
add a comment |
You might want to define the ⊨ symbol before using it.
– Kevin
Jun 18 at 16:59
You might want to define the ⊨ symbol before using it.
– Kevin
Jun 18 at 16:59
You might want to define the ⊨ symbol before using it.
– Kevin
Jun 18 at 16:59
add a comment |
Grešnik is a new contributor. Be nice, and check out our Code of Conduct.
Grešnik is a new contributor. Be nice, and check out our Code of Conduct.
Grešnik is a new contributor. Be nice, and check out our Code of Conduct.
Grešnik is a new contributor. Be nice, and check out our Code of Conduct.
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First order logic is a logic equivalent to a predicate calculus, a formal system with connectives and quantifiers, where one can only quantify over non-logical variables, but not over predicates. Some logical laws and rules of inference govern possible deductions. More broadly, systems built over it (by adding non-logical axioms) are called first order, e.g. Peano arithmetic and ZFC set theory.
– Conifold
Jun 15 at 6:38
See Is First Order Logic (FOL) the only fundamental logic ?
– Mauro ALLEGRANZA
Jun 15 at 7:55