Scope (Logic)/Examples

Examples of Scope in the context of Logic

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 $

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