Definition:Scope (Logic)

Definition

Let $\LL_0$ be the language of propositional logic.

Let $\circ$ be a connective of $\LL_0$.

Let $\mathbf W$ be a well-formed formula of $\LL_0$.

Connective of Propositional Logic

The scope of an occurrence of $\circ$ in $\mathbf W$ is defined as:

the smallest well-formed part of $\mathbf W$ containing this occurrence of $\circ$.

Quantifier of Predicate Logic

Let $\mathbf A$ be a WFF of the language of predicate logic.

Let $Q$ be an occurrence of a quantifier in $\mathbf A$.

Let $\mathbf B$ be a well-formed part of $\mathbf A$ such that $\mathbf B$ begins (omitting outer parentheses) with $Q x$.

That is, such that $\mathbf B = \paren {Q x: \mathbf C}$ for some WFF $\mathbf C$.

$\mathbf B$ is called the scope of the quantifier $Q$.

Examples

Arbitrary Example $1$

Let $\circ$ be a binary logical connective in a compound statement $p \circ q$.

The scope of $\circ$ is $p$ and $q$.

Arbitrary Example $2$

Consider the statement:

$\paren {p \land \paren {q \lor r} } \implies \paren {s \iff \neg \, t}$

We have:

$(1): \quad$ The scope of $\land$ is $p$ and $\paren {q \lor r}$.
$(2): \quad$ The scope of $\lor$ is $q$ and $r$.
$(3): \quad$ The scope of $\implies$ is $\paren {p \land \paren {q \lor r} }$ and $\paren {s \iff \neg \, t}$.
$(4): \quad$ The scope of $\iff$ is $s$ and $\neg \, t$.
$(5): \quad$ The scope of $\neg$ is $t$.

Arbitrary Example $3$

Consider the statement:

$\exists x: \paren {x < y} \lor y = 0$

We have:

$(1): \quad$ The scope of $\exists$ is $x$.
$(2): \quad$ The scope of $\exists x$ is $\exists x: \paren {x < y}$.
$(3): \quad$ The scope of $=$ is $y$ and $0$.
$(4): \quad$ The scope of $\lor$ is $\exists x: \paren {x < y}$ and $y = 0$.
Scopes for Connectives and Quantifiers in the formula: $\exists x: \paren { x < y} \lor y = 0$
Scope (Logic)/Connective
Scope (Logic)/Connective Scope (Logic)/Connective Scope 1 Scope 2
$=$ $y=0$ $y$ $0$
$\lor$ $\exists x: \paren { x < y } \lor y=0$ $\exists x: \paren { x < y }$ $y=0$
Quantifier Scope (Logic)/Quantifier Scope
$\exists x$ $\exists x: \paren{ x < y }$ $x < y$

Also see

• Results about scope in the context of Logic can be found here.
The text passage provides definitions and examples related to the concept of "scope" in logic, specifically in propositional and predicate logic. It includes several examples that illustrate the scope of various logical connectives and quantifiers.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
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system
Definition:Scope (Logic)

Definition

Let $\LL_0$ be the language of propositional logic.

Let $\circ$ be a connective of $\LL_0$.

Let $\mathbf W$ be a well-formed formula of $\LL_0$.

Connective of Propositional Logic

The scope of an occurrence of $\circ$ in $\mathbf W$ is defined as:

the smallest well-formed part of $\mathbf W$ containing this occurrence of $\circ$.

Quantifier of Predicate Logic

Let $\mathbf A$ be a WFF of the language of predicate logic.

Let $Q$ be an occurrence of a quantifier in $\mathbf A$.

Let $\mathbf B$ be a well-formed part of $\mathbf A$ such that $\mathbf B$ begins (omitting outer parentheses) with $Q x$.

That is, such that $\mathbf B = \paren {Q x: \mathbf C}$ for some WFF $\mathbf C$.

$\mathbf B$ is called the scope of the quantifier $Q$.

Examples

Arbitrary Example $1$

Let $\circ$ be a binary logical connective in a compound statement $p \circ q$.

The scope of $\circ$ is $p$ and $q$.

Arbitrary Example $2$

Consider the statement:

$\paren {p \land \paren {q \lor r} } \implies \paren {s \iff \neg \, t}$

We have:

$(1): \quad$ The scope of $\land$ is $p$ and $\paren {q \lor r}$.
$(2): \quad$ The scope of $\lor$ is $q$ and $r$.
$(3): \quad$ The scope of $\implies$ is $\paren {p \land \paren {q \lor r} }$ and $\paren {s \iff \neg \, t}$.
$(4): \quad$ The scope of $\iff$ is $s$ and $\neg \, t$.
$(5): \quad$ The scope of $\neg$ is $t$.

Arbitrary Example $3$

Consider the statement:

$\exists x: \paren {x < y} \lor y = 0$

We have:

$(1): \quad$ The scope of $\exists$ is $x$.
$(2): \quad$ The scope of $\exists x$ is $\exists x: \paren {x < y}$.
$(3): \quad$ The scope of $=$ is $y$ and $0$.
$(4): \quad$ The scope of $\lor$ is $\exists x: \paren {x < y}$ and $y = 0$.
Scopes for Connectives and Quantifiers in the formula: $\exists x: \paren { x < y} \lor y = 0$
Scope (Logic)/Connective
Scope (Logic)/Connective Scope (Logic)/Connective Scope 1 Scope 2
$=$ $y=0$ $y$ $0$
$\lor$ $\exists x: \paren { x < y } \lor y=0$ $\exists x: \paren { x < y }$ $y=0$
Quantifier Scope (Logic)/Quantifier Scope
$\exists x$ $\exists x: \paren{ x < y }$ $x < y$

Also see

• Results about scope in the context of Logic can be found here.
user
The text passage provides definitions and examples related to the concept of "scope" in logic, specifically in propositional and predicate logic. It includes several examples that illustrate the scope of various logical connectives and quantifiers.

TYPE: NO_QUESTION