Definition:Binding Priority/Propositional Logic

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Definition

The binding priority convention which is almost universally used for the connectives of propositional logic is:

$(1): \quad \neg$ binds more tightly than $\lor$ and $\land$
$(2): \quad \lor$ and $\land$ bind more tightly than $\implies$ and $\impliedby$
$(3): \quad \implies$ and $\impliedby$ bind more tightly than $\iff$

Note that there is no overall convention defining which of $\land$ and $\lor$ bears a higher binding priority, and therefore we consider them to have equal priority.

Because of this fact, unless specifically defined, expressions such as $p \land q \lor r$ can not be interpreted unambiguously, and parenthesis must be used to determine the exact priorities which are to be used to interpret particular statements which may otherwise be ambiguous.

Most sources do not recognise the use of $\impliedby$ as a separate connective from $\implies$, so the binding priority of one over the other is rarely a question.


Also defined as

When defining binding priority for propositional logic, some sources do impose a priority of $\land$ over $\lor$.

Similarly, some sources also impose a priority of $\implies$ over $\iff$.

However, these are not a universal convention, and such a binding priority is artificial.


Some sources express the binding priority rules by saying that $\iff$ and $\implies$ dominate $\land$ and $\lor$.


Also known as

Binding priority is also known as:


Examples

Conjunction over Conditional

The convention for binding priority states that:

$\paren {x < y \land y < z} \implies x < z$

can be written as:

$x < y \land y < z \implies x < z$

as $\land$ has a higher binding priority than $\implies$.


Disjunction over Biconditional

The convention for binding priority states that:

$x + y \ne 0 \iff \paren {x \ne 0 \lor y \ne 0}$

can be written as:

$x + y \ne 0 \iff x \ne 0 \lor y \ne 0$

as $\lor$ has a higher binding priority than $\iff$.


Sources

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