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 $y = 0$.
Scope (Logic)/Connective
|
---|
Work In Progress In particular: Sorry, I'm working on this You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code. |
Also see
Sources
This page may be the result of a refactoring operation. As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. In particular: Will need to check carefully whether these are definition 1 or 2 If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{SourceReview}} from the code. |
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{I}: 5$: Using Brackets
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): scope