Definition:Binding Priority/Propositional Logic

From ProofWiki
Jump to navigation Jump to search

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 priority of one over the other is rarely a question.


Also known as

The binding priority of a system of connectives can also be seen referred to as their precedence.

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


Also defined as

Some sources do impose a priority of $\land$ over $\lor$, but this is not a universal convention.

Similarly artificial is any priority imposed of $\implies$ over $\impliedby$.


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