Question: Which alphabet symbols always have the same meaning?

It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation. For example, the logical symbol

∧

{\displaystyle \land }

 always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

Answer: logical symbols

<hl> It is common to divide the symbols of the alphabet into logical symbols , which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation. <hl> For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

It is common to divide the symbols of the alphabet into <hl> logical symbols <hl>, which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation. For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

It is common to divide the symbols of the alphabet into <hl> logical symbols <hl> , which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation.


Question: What is the property of a logical symbol?

Answer: always have the same meaning

<hl> It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning , and non-logical symbols, whose meaning varies by interpretation. <hl> For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

It is common to divide the symbols of the alphabet into logical symbols, which <hl> always have the same meaning <hl>, and non-logical symbols, whose meaning varies by interpretation. For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

It is common to divide the symbols of the alphabet into logical symbols, which <hl> always have the same meaning <hl> , and non-logical symbols, whose meaning varies by interpretation.


Question: Which symbols of the alphabet has meanings that vary?

Answer: non-logical symbols

<hl> It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols , whose meaning varies by interpretation. <hl> For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and <hl> non-logical symbols <hl>, whose meaning varies by interpretation. For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and <hl> non-logical symbols <hl> , whose meaning varies by interpretation.


Question: What is the property of a non-logical symbol?

Answer: meaning varies by interpretation

<hl> It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation . <hl> For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols, whose <hl> meaning varies by interpretation <hl>. For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols, whose <hl> meaning varies by interpretation <hl> .


Question: What word does the logical symbol ^  represents?

Answer: and

<hl> It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation. <hl> For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.

It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, <hl> and <hl> non-logical symbols, whose meaning varies by interpretation. For example, the logical symbol ∧ {\displaystyle \land } always represents "and"; it is never interpreted as "or". On the other hand, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand.


Question: If the definition has parentheses what does it mean for each formula?

The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, there is a unique parse tree for each formula). This property is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a proof of unique readability.

Answer: there is a unique parse tree

<hl> The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, there is a unique parse tree for each formula). <hl> This property is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a proof of unique readability.

The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, <hl> there is a unique parse tree <hl> for each formula). This property is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a proof of unique readability.


Question: What is it known as if formula has unique parse tree?

Answer: readability of formulas

The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, there is a unique parse tree for each formula). <hl> This property is known as unique readability of formulas . <hl> There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a proof of unique readability.

The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, there is a unique parse tree for each formula). This property is known as unique <hl> readability of formulas <hl>. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a proof of unique readability.

This property is known as unique <hl> readability of formulas <hl> .


Question: What is one convention used instead of parentheses in formulas?

Answer: colons

The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, there is a unique parse tree for each formula). This property is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. <hl> For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. <hl> Each author's particular definition must be accompanied by a proof of unique readability.

The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, there is a unique parse tree for each formula). This property is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use <hl> colons <hl> or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a proof of unique readability.


Question: What must accompany an authors particular definition?

Answer: proof of unique readability

a proof of unique readability .

The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, there is a unique parse tree for each formula). This property is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by <hl> a proof of unique readability . <hl>

The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition (in other words, there is a unique parse tree for each formula). This property is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a <hl> proof of unique readability <hl>.

a <hl> proof of unique readability <hl> .


Question: What is a less common notation convention used instead of infix?

The definitions above use infix notation for binary connectives such as

→

{\displaystyle \to }

. A less common convention is Polish notation, in which one writes

→

{\displaystyle \rightarrow }

,

∧

{\displaystyle \wedge }

, and so on in front of their arguments rather than between them. This convention allows all punctuation symbols to be discarded. Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read it. In Polish notation, the formula

Answer: Polish notation

The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . <hl> A less common convention is Polish notation , in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. <hl> This convention allows all punctuation symbols to be discarded. Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read it. In Polish notation, the formula

The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . A less common convention is <hl> Polish notation <hl>, in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. This convention allows all punctuation symbols to be discarded. Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read it. In Polish notation, the formula

A less common convention is <hl> Polish notation <hl> , in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them.


Question: What can be discarded when using Polish notation?

Answer: punctuation symbols

The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . A less common convention is Polish notation, in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. <hl> This convention allows all punctuation symbols to be discarded. <hl> Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read it. In Polish notation, the formula

The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . A less common convention is Polish notation, in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. This convention allows all <hl> punctuation symbols <hl> to be discarded. Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read it. In Polish notation, the formula


Question: Why is Polish notation rarely used?

Answer: it is hard for humans to read

The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . A less common convention is Polish notation, in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. This convention allows all punctuation symbols to be discarded. <hl> Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read it. <hl> In Polish notation, the formula

The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . A less common convention is Polish notation, in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. This convention allows all punctuation symbols to be discarded. Polish notation is compact and elegant, but rarely used in practice because <hl> it is hard for humans to read <hl> it. In Polish notation, the formula


Question: What is two attractive features of Polish notation?

Answer: compact and elegant

The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . A less common convention is Polish notation, in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. This convention allows all punctuation symbols to be discarded. <hl> Polish notation is compact and elegant , but rarely used in practice because it is hard for humans to read it. <hl> In Polish notation, the formula

The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . A less common convention is Polish notation, in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. This convention allows all punctuation symbols to be discarded. Polish notation is <hl> compact and elegant <hl>, but rarely used in practice because it is hard for humans to read it. In Polish notation, the formula

Polish notation is <hl> compact and elegant <hl> , but rarely used in practice because it is hard for humans to read it.


Question: Which is a common notation for binary connectives?

Answer: infix

<hl> The definitions above use infix notation for binary connectives such as → {\displaystyle \to } . <hl> A less common convention is Polish notation, in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. This convention allows all punctuation symbols to be discarded. Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read it. In Polish notation, the formula

The definitions above use <hl> infix <hl> notation for binary connectives such as → {\displaystyle \to } . A less common convention is Polish notation, in which one writes → {\displaystyle \rightarrow } , ∧ {\displaystyle \wedge } , and so on in front of their arguments rather than between them. This convention allows all punctuation symbols to be discarded. Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read it. In Polish notation, the formula


Question: What logic can the term everyone loves someone use?

English sentences like "everyone loves someone" can be formalized by first-order logic formulas like ∀x∃y L(x,y). This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantifiers ∀ and ∃ and the loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with 8 different meanings can be built. The following diagrams show models for each of them, assuming that there are exactly five individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x and column y indicates L(x,y). Only for the formulas 9 and 10 is the model unique, all other formulas may be satisfied by several models.

Answer: first-order

<hl> English sentences like "everyone loves someone" can be formalized by first-order logic formulas like ∀x∃y L(x,y). <hl> This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantifiers ∀ and ∃ and the loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with 8 different meanings can be built. The following diagrams show models for each of them, assuming that there are exactly five individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x and column y indicates L(x,y). Only for the formulas 9 and 10 is the model unique, all other formulas may be satisfied by several models.

English sentences like "everyone loves someone" can be formalized by <hl> first-order <hl> logic formulas like ∀x∃y L(x,y). This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantifiers ∀ and ∃ and the loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with 8 different meanings can be built. The following diagrams show models for each of them, assuming that there are exactly five individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x and column y indicates L(x,y). Only for the formulas 9 and 10 is the model unique, all other formulas may be satisfied by several models.


Question: How would everyone loves someone be abbreviated in first order logic?

Answer: ∀x∃y L(x,y)

<hl> English sentences like "everyone loves someone" can be formalized by first-order logic formulas like ∀x∃y L(x,y) . <hl> This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantifiers ∀ and ∃ and the loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with 8 different meanings can be built. The following diagrams show models for each of them, assuming that there are exactly five individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x and column y indicates L(x,y). Only for the formulas 9 and 10 is the model unique, all other formulas may be satisfied by several models.

English sentences like "everyone loves someone" can be formalized by first-order logic formulas like <hl> ∀x∃y L(x,y) <hl>. This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantifiers ∀ and ∃ and the loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with 8 different meanings can be built. The following diagrams show models for each of them, assuming that there are exactly five individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x and column y indicates L(x,y). Only for the formulas 9 and 10 is the model unique, all other formulas may be satisfied by several models.

English sentences like "everyone loves someone" can be formalized by first-order logic formulas like <hl> ∀x∃y L(x,y) <hl> .


Question: How many meanings can be built from L(x,y)

Answer: 8

English sentences like "everyone loves someone" can be formalized by first-order logic formulas like ∀x∃y L(x,y). This is accomplished by abbreviating the relation "x loves y" by L(x,y). <hl> Using just the two quantifiers ∀ and ∃ and the loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with 8 different meanings can be built. <hl> The following diagrams show models for each of them, assuming that there are exactly five individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x and column y indicates L(x,y). Only for the formulas 9 and 10 is the model unique, all other formulas may be satisfied by several models.

English sentences like "everyone loves someone" can be formalized by first-order logic formulas like ∀x∃y L(x,y). This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantifiers ∀ and ∃ and the loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with <hl> 8 <hl> different meanings can be built. The following diagrams show models for each of them, assuming that there are exactly five individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x and column y indicates L(x,y). Only for the formulas 9 and 10 is the model unique, all other formulas may be satisfied by several models.


Question: Which two formulas are unique models?

Answer: 9 and 10

English sentences like "everyone loves someone" can be formalized by first-order logic formulas like ∀x∃y L(x,y). This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantifiers ∀ and ∃ and the loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with 8 different meanings can be built. The following diagrams show models for each of them, assuming that there are exactly five individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x and column y indicates L(x,y). <hl> Only for the formulas 9 and 10 is the model unique, all other formulas may be satisfied by several models. <hl>

English sentences like "everyone loves someone" can be formalized by first-order logic formulas like ∀x∃y L(x,y). This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantifiers ∀ and ∃ and the loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with 8 different meanings can be built. The following diagrams show models for each of them, assuming that there are exactly five individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x and column y indicates L(x,y). Only for the formulas <hl> 9 and 10 <hl> is the model unique, all other formulas may be satisfied by several models.


Question: What is the dedctive system used for ?

A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely.  These finite deductions themselves are often called derivations in proof theory. They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.

Answer: used to demonstrate, on a purely syntactic basis

<hl> A deductive system is used to demonstrate, on a purely syntactic basis , that one formula is a logical consequence of another formula. <hl> There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.

A deductive system is <hl> used to demonstrate, on a purely syntactic basis <hl>, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.

A deductive system is <hl> used to demonstrate, on a purely syntactic basis <hl> , that one formula is a logical consequence of another formula.


Question: How the deductive system works ?

Answer: one formula is a logical consequence of another formula.

<hl> A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. <hl> These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.

A deductive system is used to demonstrate, on a purely syntactic basis, that <hl> one formula is a logical consequence of another formula. <hl> There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.


Question: What are the examples of the dedctive system ?

Answer: Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method

A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. <hl> There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method , and resolution. <hl> These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.

A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including <hl> Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method <hl>, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.

There are many such systems for first-order logic, including <hl> Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method <hl> , and resolution.


Question: What are finite deductions are called ?

Answer: are often called derivations in proof theory

A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. <hl> These finite deductions themselves are often called derivations in proof theory . <hl> They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.

A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves <hl> are often called derivations in proof theory <hl>. They are also often called proofs, but are completely formalized unlike natural-language mathematical proofs.

These finite deductions themselves <hl> are often called derivations in proof theory <hl> .


Question: What they are also called oftenly ?

Answer: They are also often called proofs

A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. <hl> They are also often called proofs , but are completely formalized unlike natural-language mathematical proofs. <hl>

A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. <hl> They are also often called proofs <hl>, but are completely formalized unlike natural-language mathematical proofs.

<hl> They are also often called proofs <hl> , but are completely formalized unlike natural-language mathematical proofs.


Question: What is the common rule of inference ?

For example, one common rule of inference is the rule of substitution. If t is a term and φ is a formula possibly containing the variable x, then φ[t/x] (often denoted φ[x/t]) is the result of replacing all free instances of x by t in φ. The substitution rule states that for any φ and any term t, one can conclude φ[t/x] from φ provided that no free variable of t becomes bound during the substitution process. (If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ to differ from the free variables of t.)

Answer: one common rule of inference is the rule of substitution

<hl> For example, one common rule of inference is the rule of substitution . <hl> If t is a term and φ is a formula possibly containing the variable x, then φ[t/x] (often denoted φ[x/t]) is the result of replacing all free instances of x by t in φ. The substitution rule states that for any φ and any term t, one can conclude φ[t/x] from φ provided that no free variable of t becomes bound during the substitution process. (If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ to differ from the free variables of t.)

For example, <hl> one common rule of inference is the rule of substitution <hl>. If t is a term and φ is a formula possibly containing the variable x, then φ[t/x] (often denoted φ[x/t]) is the result of replacing all free instances of x by t in φ. The substitution rule states that for any φ and any term t, one can conclude φ[t/x] from φ provided that no free variable of t becomes bound during the substitution process. (If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ to differ from the free variables of t.)

For example, <hl> one common rule of inference is the rule of substitution <hl> .


Question: What is the result of replacing all free instances of X by t in Q ?

Answer: then φ[t/x] (often denoted φ[x/t])

For example, one common rule of inference is the rule of substitution. <hl> If t is a term and φ is a formula possibly containing the variable x, then φ[t/x] (often denoted φ[x/t]) is the result of replacing all free instances of x by t in φ. <hl> The substitution rule states that for any φ and any term t, one can conclude φ[t/x] from φ provided that no free variable of t becomes bound during the substitution process. (If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ to differ from the free variables of t.)

For example, one common rule of inference is the rule of substitution. If t is a term and φ is a formula possibly containing the variable x, <hl> then φ[t/x] (often denoted φ[x/t]) <hl> is the result of replacing all free instances of x by t in φ. The substitution rule states that for any φ and any term t, one can conclude φ[t/x] from φ provided that no free variable of t becomes bound during the substitution process. (If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ to differ from the free variables of t.)


Question: what can be concluded from the substitution rule ?

Answer: one can conclude φ[t/x] from φ provided that no free variable of t becomes bound

For example, one common rule of inference is the rule of substitution. If t is a term and φ is a formula possibly containing the variable x, then φ[t/x] (often denoted φ[x/t]) is the result of replacing all free instances of x by t in φ. <hl> The substitution rule states that for any φ and any term t, one can conclude φ[t/x] from φ provided that no free variable of t becomes bound during the substitution process. <hl> (If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ to differ from the free variables of t.)

For example, one common rule of inference is the rule of substitution. If t is a term and φ is a formula possibly containing the variable x, then φ[t/x] (often denoted φ[x/t]) is the result of replacing all free instances of x by t in φ. The substitution rule states that for any φ and any term t, <hl> one can conclude φ[t/x] from φ provided that no free variable of t becomes bound <hl> during the substitution process. (If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ to differ from the free variables of t.)


Question: what is necessary to do if the variables of t becomes bound ?

Answer: If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ

For example, one common rule of inference is the rule of substitution. If t is a term and φ is a formula possibly containing the variable x, then φ[t/x] (often denoted φ[x/t]) is the result of replacing all free instances of x by t in φ. The substitution rule states that for any φ and any term t, one can conclude φ[t/x] from φ provided that no free variable of t becomes bound during the substitution process. <hl> ( If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ to differ from the free variables of t.) <hl>

For example, one common rule of inference is the rule of substitution. If t is a term and φ is a formula possibly containing the variable x, then φ[t/x] (often denoted φ[x/t]) is the result of replacing all free instances of x by t in φ. The substitution rule states that for any φ and any term t, one can conclude φ[t/x] from φ provided that no free variable of t becomes bound during the substitution process. (<hl> If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ <hl> to differ from the free variables of t.)

( <hl> If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ <hl> to differ from the free variables of t.)


Question: How the restrictions on bound variables to be seen ?

To see why the restriction on bound variables is necessary, consider the logically valid formula φ given by

∃
x
(
x
=
y
)

{\displaystyle \exists x(x=y)}

, in the signature of (0,1,+,×,=) of arithmetic. If t is the term "x + 1", the formula φ[t/y] is

∃
x
(
x
=
x
+
1
)

{\displaystyle \exists x(x=x+1)}

, which will be false in many interpretations. The problem is that the free variable x of t  became bound during the substitution. The intended replacement can be obtained by renaming the bound variable x of φ to something else, say z, so that the formula after substitution is

∃
z
(
z
=
x
+
1
)

{\displaystyle \exists z(z=x+1)}

, which is again logically valid.

Answer: by renaming the bound variable x of φ to something else, say z

To see why the restriction on bound variables is necessary, consider the logically valid formula φ given by ∃ x ( x = y ) {\displaystyle \exists x(x=y)} , in the signature of (0,1,+,×,=) of arithmetic. If t is the term "x + 1", the formula φ[t/y] is ∃ x ( x = x + 1 ) {\displaystyle \exists x(x=x+1)} , which will be false in many interpretations. The problem is that the free variable x of t became bound during the substitution. <hl> The intended replacement can be obtained by renaming the bound variable x of φ to something else, say z , so that the formula after substitution is ∃ z ( z = x + 1 ) {\displaystyle \exists z(z=x+1)} , which is again logically valid. <hl>

To see why the restriction on bound variables is necessary, consider the logically valid formula φ given by ∃ x ( x = y ) {\displaystyle \exists x(x=y)} , in the signature of (0,1,+,×,=) of arithmetic. If t is the term "x + 1", the formula φ[t/y] is ∃ x ( x = x + 1 ) {\displaystyle \exists x(x=x+1)} , which will be false in many interpretations. The problem is that the free variable x of t became bound during the substitution. The intended replacement can be obtained <hl> by renaming the bound variable x of φ to something else, say z <hl>, so that the formula after substitution is ∃ z ( z = x + 1 ) {\displaystyle \exists z(z=x+1)} , which is again logically valid.

The intended replacement can be obtained <hl> by renaming the bound variable x of φ to something else, say z <hl> , so that the formula after substitution is ∃ z ( z = x + 1 ) {\displaystyle \exists z(z=x+1)} , which is again logically valid.


Question: What is the main problem with the equation E(X)X=X+1 ?

Answer: which will be false in many interpretations. The problem is that the free variable x of t  became bound during the substitution.

To see why the restriction on bound variables is necessary, consider the logically valid formula φ given by ∃ x ( x = y ) {\displaystyle \exists x(x=y)} , in the signature of (0,1,+,×,=) of arithmetic. <hl> If t is the term "x + 1", the formula φ[t/y] is ∃ x ( x = x + 1 ) {\displaystyle \exists x(x=x+1)} , which will be false in many interpretations. The problem is that the free variable x of t became bound during the substitution. The intended replacement can be obtained by renaming the bound variable x of φ to something else, say z, so that the formula after substitution is ∃ z ( z <hl> = x + 1 ) {\displaystyle \exists z(z=x+1)} , which is again logically valid.

To see why the restriction on bound variables is necessary, consider the logically valid formula φ given by ∃ x ( x = y ) {\displaystyle \exists x(x=y)} , in the signature of (0,1,+,×,=) of arithmetic. If t is the term "x + 1", the formula φ[t/y] is ∃ x ( x = x + 1 ) {\displaystyle \exists x(x=x+1)} , <hl> which will be false in many interpretations. The problem is that the free variable x of t became bound during the substitution. <hl> The intended replacement can be obtained by renaming the bound variable x of φ to something else, say z, so that the formula after substitution is ∃ z ( z = x + 1 ) {\displaystyle \exists z(z=x+1)} , which is again logically valid.


Question: what to replace in the equation to avoid the previous problem ?

Answer: The intended replacement can be obtained by renaming the bound variable x of φ to something else, say z

To see why the restriction on bound variables is necessary, consider the logically valid formula φ given by ∃ x ( x = y ) {\displaystyle \exists x(x=y)} , in the signature of (0,1,+,×,=) of arithmetic. If t is the term "x + 1", the formula φ[t/y] is ∃ x ( x = x + 1 ) {\displaystyle \exists x(x=x+1)} , which will be false in many interpretations. The problem is that the free variable x of t became bound during the substitution. <hl> The intended replacement can be obtained by renaming the bound variable x of φ to something else, say z , so that the formula after substitution is ∃ z ( z = x + 1 ) {\displaystyle \exists z(z=x+1)} , which is again logically valid. <hl>

To see why the restriction on bound variables is necessary, consider the logically valid formula φ given by ∃ x ( x = y ) {\displaystyle \exists x(x=y)} , in the signature of (0,1,+,×,=) of arithmetic. If t is the term "x + 1", the formula φ[t/y] is ∃ x ( x = x + 1 ) {\displaystyle \exists x(x=x+1)} , which will be false in many interpretations. The problem is that the free variable x of t became bound during the substitution. <hl> The intended replacement can be obtained by renaming the bound variable x of φ to something else, say z <hl>, so that the formula after substitution is ∃ z ( z = x + 1 ) {\displaystyle \exists z(z=x+1)} , which is again logically valid.

<hl> The intended replacement can be obtained by renaming the bound variable x of φ to something else, say z <hl> , so that the formula after substitution is ∃ z ( z = x + 1 ) {\displaystyle \exists z(z=x+1)} , which is again logically valid.


Question: What rule demonetrates the common aspects of the rule of inference ?

The substitution rule demonstrates several common aspects of rules of inference. It is entirely syntactical; one can tell whether it was correctly applied without appeal to any interpretation. It has (syntactically defined) limitations on when it can be applied, which must be respected to preserve the correctness of derivations. Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.

Answer: The substitution rule

<hl> The substitution rule demonstrates several common aspects of rules of inference. <hl> It is entirely syntactical; one can tell whether it was correctly applied without appeal to any interpretation. It has (syntactically defined) limitations on when it can be applied, which must be respected to preserve the correctness of derivations. Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.

<hl> The substitution rule <hl> demonstrates several common aspects of rules of inference. It is entirely syntactical; one can tell whether it was correctly applied without appeal to any interpretation. It has (syntactically defined) limitations on when it can be applied, which must be respected to preserve the correctness of derivations. Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.


Question: Why a one can tell wheather if the rule is correctly applied without the appeal of any interpretion ?

Answer: It is entirely syntactical

The substitution rule demonstrates several common aspects of rules of inference. <hl> It is entirely syntactical ; one can tell whether it was correctly applied without appeal to any interpretation. <hl> It has (syntactically defined) limitations on when it can be applied, which must be respected to preserve the correctness of derivations. Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.

The substitution rule demonstrates several common aspects of rules of inference. <hl> It is entirely syntactical <hl>; one can tell whether it was correctly applied without appeal to any interpretation. It has (syntactically defined) limitations on when it can be applied, which must be respected to preserve the correctness of derivations. Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.

<hl> It is entirely syntactical <hl> ; one can tell whether it was correctly applied without appeal to any interpretation.


Question: What must be respected to preserve the correctness of divertions, when applied?

Answer: It has (syntactically defined) limitations on when it can be applied

The substitution rule demonstrates several common aspects of rules of inference. It is entirely syntactical; one can tell whether it was correctly applied without appeal to any interpretation. <hl> It has (syntactically defined) limitations on when it can be applied , which must be respected to preserve the correctness of derivations. <hl> Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.

The substitution rule demonstrates several common aspects of rules of inference. It is entirely syntactical; one can tell whether it was correctly applied without appeal to any interpretation. <hl> It has (syntactically defined) limitations on when it can be applied <hl>, which must be respected to preserve the correctness of derivations. Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.

<hl> It has (syntactically defined) limitations on when it can be applied <hl> , which must be respected to preserve the correctness of derivations.


Question: Why are these limitations are necessary ?

Answer: because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.

The substitution rule demonstrates several common aspects of rules of inference. It is entirely syntactical; one can tell whether it was correctly applied without appeal to any interpretation. It has (syntactically defined) limitations on when it can be applied, which must be respected to preserve the correctness of derivations. <hl> Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule. <hl>

The substitution rule demonstrates several common aspects of rules of inference. It is entirely syntactical; one can tell whether it was correctly applied without appeal to any interpretation. It has (syntactically defined) limitations on when it can be applied, which must be respected to preserve the correctness of derivations. Moreover, as is often the case, these limitations are necessary <hl> because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule. <hl>


Question: Describe the dedction in the Hillbert-style system ?

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.

Answer: a list of formulas, each of which is a logical axiom

<hl> A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom , a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. <hl> The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.

A deduction in a Hilbert-style deductive system is <hl> a list of formulas, each of which is a logical axiom <hl>, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.

A deduction in a Hilbert-style deductive system is <hl> a list of formulas, each of which is a logical axiom <hl> , a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference.


Question: What assumptions to be told about the Hillbert system ?

Answer: a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference

<hl> A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference . <hl> The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, <hl> a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference <hl>. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, <hl> a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference <hl> .


Question: What the logical axiom consists of ?

Answer: The logical axioms consist of several axiom schemas of logically valid formulas

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. <hl> The logical axioms consist of several axiom schemas of logically valid formulas ; these encompass a significant amount of propositional logic. <hl> The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. <hl> The logical axioms consist of several axiom schemas of logically valid formulas <hl>; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.

<hl> The logical axioms consist of several axiom schemas of logically valid formulas <hl> ; these encompass a significant amount of propositional logic.


Question: What the rule of inference enables ?

Answer: The rules of inference enable the manipulation of quantifiers.

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. <hl> The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. <hl> It is common to have only modus ponens and universal generalization as rules of inference.

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. <hl> The rules of inference enable the manipulation of quantifiers. <hl> Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.


Question: What makes the relation between te rule of inference and the HIllbert system typical ? ?

Answer: Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms.

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. <hl> Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference. <hl>

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. <hl> Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. <hl> It is common to have only modus ponens and universal generalization as rules of inference.


Question: what is a collection of formal systems used in mathematics, philosophy,linguistics and computer sciences

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects.  It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

Answer: First-order logic

<hl> First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. <hl> It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

<hl> First-order logic <hl> is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.


Question: first order is also known as what?

Answer: first-order predicate calculus

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. <hl> It is also known as first-order predicate calculus , the lower predicate calculus, quantification theory, and predicate logic. <hl> First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as <hl> first-order predicate calculus <hl>, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

It is also known as <hl> first-order predicate calculus <hl> , the lower predicate calculus, quantification theory, and predicate logic.


Question: what does first order logic uses quantified variables over

Answer: (non-logical) objects

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. <hl> First-order logic uses quantified variables over (non-logical) objects . <hl> It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over <hl> (non-logical) objects <hl>. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic uses quantified variables over <hl> (non-logical) objects <hl> .


Question: what distinguishes first order logic from propositional logic

Answer: the use of sentences that contain variables

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. <hl> It allows the use of sentences that contain variables , so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. <hl> This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows <hl> the use of sentences that contain variables <hl>, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

It allows <hl> the use of sentences that contain variables <hl> , so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable.


Question: give one theory used in first order logic

Answer: quantification theory

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. <hl> It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory , and predicate logic. <hl> First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, <hl> quantification theory <hl>, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

It is also known as first-order predicate calculus, the lower predicate calculus, <hl> quantification theory <hl> , and predicate logic.


Question: What type of variable does first-order logic use?

Answer: quantified variables

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. <hl> First-order logic uses quantified variables over (non-logical) objects. <hl> It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses <hl> quantified variables <hl> over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.


Question: What is the name for a collection of formal systems, used in math and other scientific fields?

Answer: First-order logic

<hl> First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. <hl> It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

<hl> First-order logic <hl> is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.


Question: What are the two other names for first-order logic that both contain the word "calculus?"

Answer: first-order predicate calculus, the lower predicate calculus

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. <hl> It is also known as first-order predicate calculus, the lower predicate calculus , quantification theory, and predicate logic. <hl> First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as <hl> first-order predicate calculus, the lower predicate calculus <hl>, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

It is also known as <hl> first-order predicate calculus, the lower predicate calculus <hl> , quantification theory, and predicate logic.


Question: What are the two other names for first-order logic that do not contain the word "calculus?"

Answer: quantification theory, and predicate logic

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. <hl> It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic . <hl> First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, <hl> quantification theory, and predicate logic <hl>. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

It is also known as first-order predicate calculus, the lower predicate calculus, <hl> quantification theory, and predicate logic <hl> .


Question: How does first-order logic differ from propositional logic?

Answer: does not use quantifiers

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. <hl> This distinguishes it from propositional logic, which does not use quantifiers . <hl>

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which <hl> does not use quantifiers <hl>.

This distinguishes it from propositional logic, which <hl> does not use quantifiers <hl> .


Question: First-order logic is also known as first-order predicate calculus, the lower predicate calculus, predicate logic, and what?

Answer: quantification theory

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. <hl> It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory , and predicate logic. <hl> First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, <hl> quantification theory <hl>, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

It is also known as first-order predicate calculus, the lower predicate calculus, <hl> quantification theory <hl> , and predicate logic.


Question: The First-order logic distinguishes itself from this logic because this logic does not use quantifiers.

Answer: propositional logic

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. <hl> This distinguishes it from propositional logic , which does not use quantifiers. <hl>

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from <hl> propositional logic <hl>, which does not use quantifiers.

This distinguishes it from <hl> propositional logic <hl> , which does not use quantifiers.


Question: Under the First-order logic, sentences can be used that contain what?

Answer: variables

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. <hl> First-order logic uses quantified variables over (non-logical) objects. <hl> It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified <hl> variables <hl> over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.


Question: Except for mathematics, philosophy, and computer science, where else is the First-order logic used in?

Answer: linguistics

<hl> First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics , <hl> and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, <hl> linguistics <hl>, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, <hl> linguistics <hl> ,


Question: What does First-order uses over (non-biological) objects?

Answer: quantified variables

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. <hl> First-order logic uses quantified variables over (non-logical) objects. <hl> It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses <hl> quantified variables <hl> over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.


Question: What is "First-order logic"?

Answer: a collection of formal systems

<hl> First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. <hl> It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is <hl> a collection of formal systems <hl> used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.


Question: What is first order logic used in?

Answer: mathematics, philosophy, linguistics, and computer science

<hl> First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science . <hl> It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in <hl> mathematics, philosophy, linguistics, and computer science <hl>. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in <hl> mathematics, philosophy, linguistics, and computer science <hl> .


Question: What is first order logic also knows as?

Answer: first-order predicate calculus

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. <hl> It is also known as first-order predicate calculus , the lower predicate calculus, quantification theory, and predicate logic. <hl> First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as <hl> first-order predicate calculus <hl>, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

It is also known as <hl> first-order predicate calculus <hl> , the lower predicate calculus, quantification theory, and predicate logic.


Question: What does First-order logic allow?

Answer: It allows the use of sentences that contain variables

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. <hl> It allows the use of sentences that contain variables , so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. <hl> This distinguishes it from propositional logic, which does not use quantifiers.

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. <hl> It allows the use of sentences that contain variables <hl>, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which does not use quantifiers.

<hl> It allows the use of sentences that contain variables <hl> , so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable.


Question: How does propositional logic differ from First-order logic?

Answer: does not use quantifiers

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. <hl> This distinguishes it from propositional logic, which does not use quantifiers . <hl>

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. It allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form X is a man where X is a variable. This distinguishes it from propositional logic, which <hl> does not use quantifiers <hl>.

This distinguishes it from propositional logic, which <hl> does not use quantifiers <hl> .


Question: What are matrices that satisfy the antecedent known as?

Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion (RHS) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation). In contrast, only some matrices, which satisfy formula 2, happen to satisfy also formula 5, whereas others, also satisfying formula 2, do not; therefore formula 5 is not a logical consequence of formula 2.

Answer: LHS

<hl> Some formulas imply others, i.e. all matrices satisfying the antecedent ( LHS ) also satisfy the conclusion (RHS) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation). <hl> In contrast, only some matrices, which satisfy formula 2, happen to satisfy also formula 5, whereas others, also satisfying formula 2, do not; therefore formula 5 is not a logical consequence of formula 2.

Some formulas imply others, i.e. all matrices satisfying the antecedent (<hl> LHS <hl>) also satisfy the conclusion (RHS) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation). In contrast, only some matrices, which satisfy formula 2, happen to satisfy also formula 5, whereas others, also satisfying formula 2, do not; therefore formula 5 is not a logical consequence of formula 2.

Some formulas imply others, i.e. all matrices satisfying the antecedent ( <hl> LHS <hl> ) also satisfy the conclusion (RHS) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation).


Question: What are matrices that satisfy the conclusion known as?

Answer: RHS

<hl> Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion ( RHS ) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation). <hl> In contrast, only some matrices, which satisfy formula 2, happen to satisfy also formula 5, whereas others, also satisfying formula 2, do not; therefore formula 5 is not a logical consequence of formula 2.

Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion (<hl> RHS <hl>) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation). In contrast, only some matrices, which satisfy formula 2, happen to satisfy also formula 5, whereas others, also satisfying formula 2, do not; therefore formula 5 is not a logical consequence of formula 2.

Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion ( <hl> RHS <hl> ) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation).


Question: What do some matrices that satisfy formula 2 also satisfy?

Answer: satisfy also formula 5

Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion (RHS) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation). <hl> In contrast, only some matrices, which satisfy formula 2, happen to satisfy also formula 5 , whereas others, also satisfying formula 2, do not; therefore formula 5 is not a logical consequence of formula 2. <hl>

Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion (RHS) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation). In contrast, only some matrices, which satisfy formula 2, happen to <hl> satisfy also formula 5 <hl>, whereas others, also satisfying formula 2, do not; therefore formula 5 is not a logical consequence of formula 2.

In contrast, only some matrices, which satisfy formula 2, happen to <hl> satisfy also formula 5 <hl> , whereas others, also satisfying formula 2, do not; therefore formula 5 is not a logical consequence of formula 2.


Question: Is formula 5 a logical consequence of formula 2?

Answer: formula 5 is not a logical consequence of formula 2

Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion (RHS) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation). <hl> In contrast, only some matrices, which satisfy formula 2, happen to satisfy also formula 5, whereas others, also satisfying formula 2, do not; therefore formula 5 is not a logical consequence of formula 2 . <hl>

Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion (RHS) of the implication — e.g. formula 3 implies formula 1, i.e.: each matrix fulfilling formula 3 also fulfills formula 1, but not vice versa (see the Hasse diagram for this ordering relation). In contrast, only some matrices, which satisfy formula 2, happen to satisfy also formula 5, whereas others, also satisfying formula 2, do not; therefore <hl> formula 5 is not a logical consequence of formula 2 <hl>.

In contrast, only some matrices, which satisfy formula 2, happen to satisfy also formula 5, whereas others, also satisfying formula 2, do not; therefore <hl> formula 5 is not a logical consequence of formula 2 <hl> .


Question: What does an interpretation of a first-order language assign?

An interpretation of a first-order language assigns a denotation to all non-logical constants in that language. It also determines a domain of discourse that specifies the range of the quantifiers. The result is that each term is assigned an object that it represents, and each sentence is assigned a truth value. In this way, an interpretation provides semantic meaning to the terms and formulas of the language. The study of the interpretations of formal languages is called formal semantics. What follows is a description of the standard or Tarskian semantics for first-order logic. (It is also possible to define game semantics for first-order logic, but aside from requiring the axiom of choice, game semantics agree with Tarskian semantics for first-order logic, so game semantics will not be elaborated herein.)

Answer: denotation to all non-logical constants in that language

<hl> An interpretation of a first-order language assigns a denotation to all non-logical constants in that language . <hl> It also determines a domain of discourse that specifies the range of the quantifiers. The result is that each term is assigned an object that it represents, and each sentence is assigned a truth value. In this way, an interpretation provides semantic meaning to the terms and formulas of the language. The study of the interpretations of formal languages is called formal semantics. What follows is a description of the standard or Tarskian semantics for first-order logic. (It is also possible to define game semantics for first-order logic, but aside from requiring the axiom of choice, game semantics agree with Tarskian semantics for first-order logic, so game semantics will not be elaborated herein.)

An interpretation of a first-order language assigns a <hl> denotation to all non-logical constants in that language <hl>. It also determines a domain of discourse that specifies the range of the quantifiers. The result is that each term is assigned an object that it represents, and each sentence is assigned a truth value. In this way, an interpretation provides semantic meaning to the terms and formulas of the language. The study of the interpretations of formal languages is called formal semantics. What follows is a description of the standard or Tarskian semantics for first-order logic. (It is also possible to define game semantics for first-order logic, but aside from requiring the axiom of choice, game semantics agree with Tarskian semantics for first-order logic, so game semantics will not be elaborated herein.)

An interpretation of a first-order language assigns a <hl> denotation to all non-logical constants in that language <hl> .