The binding priority is the convention defining the order of binding strength of the individual connectives in a logical formula.
Binding priorities can be overridden by using parenthesis in appropriate places. Parenthesis always takes priority over conventional binding priorities.
Binding Priority of Connectives of Propositional Logic
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
- Precedence: a higher precedence is the same thing as a tighter binding priority.
- Rank: a higher rank is the same thing as a tighter binding priority.
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $1$ Formation Rules
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.1$: Simple and Compound Statements
- 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.2$: Propositional formulas
- 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
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.1.3$