well-formed formula

logic

Also known as: wff

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July 15, 2026

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A well-formed formula (wff) in formal logic is a sequence of symbols that adheres to specific formation rules, ensuring it is syntactically correct within a given logical system. The propositional calculus (PC) is a fundamental branch of logic that deals with complete propositions and their combinations using operators like "not" (\u223c), "and" (\u00b7), "or" (\u2228), "if...then" (\u2283), and "if and only if" (\u2261).12

Formation rules dictate how wffs are constructed:
* A variable (e.g., p, q) is a wff.1
* If \u03b1 is a wff, then \u223c\u03b1 is a wff.1
* If \u03b1 and \u03b2 are wffs, then combinations like (\u03b1 \u00b7 \u03b2) are wffs.1

These rules ensure that formulas are unambiguous and meaningful within the logical framework. The predicate calculus extends this by including individual variables and predicates, allowing for more complex statements about objects and their properties.13

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set theory
Schemas for generating well-formed formulas
The ZFC “axiom of extension” conveys the idea that, as in naive set theory, a set is determined solely by its members. It should be noted that this is not merely a logically necessary property of equality but an assumption about the membership…
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The ZFC “axiom of extension” conveys the idea that, as in naive set theory, a set is determined solely by its members. It should be noted that this is not merely a logically necessary property of equality but an assumption about the membership…
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metalogic
Syntax and semantics
…expressions that shall count as well-formed formulas (sentences or meaningful expressions), applicable mechanically, in the sense that a machine could check whether a candidate satisfies the requirements. This specification usually contains three parts: (1) a list of primitive symbols (basic units) given mechanically, (2) certain combinations of these symbols, singled…
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…expressions that shall count as well-formed formulas (sentences or meaningful expressions), applicable mechanically, in the sense that a machine could check whether a candidate satisfies the requirements. This specification usually contains three parts: (1) a list of primitive symbols (basic units) given mechanically, (2) certain combinations of these symbols, singled…
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formal logic
The predicate calculus
…of individual variables is a wff of the predicate calculus. Such a wff is known as an atomic formula, and the predicate variable in it is said to be of degree n, if n is the number of individual variables following it. The degree of a predicate variable is sometimes…
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…of individual variables is a wff of the predicate calculus. Such a wff is known as an atomic formula, and the predicate variable in it is said to be of degree n, if n is the number of individual variables following it. The degree of a predicate variable is sometimes…
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formal logic
Formation rules for PC
…as they are usually called, well-formed formulas (wffs). Rules that specify this are called formation rules. From an intuitive point of view, it is desirable that the wffs of PC be just those sequences of PC symbols that, in terms of the interpretation given above, make sense and are unambiguous;…
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…as they are usually called, well-formed formulas (wffs). Rules that specify this are called formation rules. From an intuitive point of view, it is desirable that the wffs of PC be just those sequences of PC symbols that, in terms of the interpretation given above, make sense and are unambiguous;…
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