# Definition:Binding Priority/Propositional Logic

## Contents

## 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

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 4.2$: The Construction of an Axiom System - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 2.1$: Formation Rules - 1972: Patrick Suppes:
*Axiomatic Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.2$ Logic and Notation - 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.2$: Propositional formulas - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.4$: Main Connective - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.1$: Declarative sentences: Convention $1.3$ - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): $\S 2.1.3$