rule of substitution

logic

Also known as: rule of uniform substitution

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July 13, 2026

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In logic, the rule of substitution is a principle that dictates what expressions can replace one another. This is crucial for manipulating and deriving new statements from existing ones within logical systems.1

In the context of the predicate calculus, a fundamental area of formal logic, this rule is applied to propositional functions. A propositional function is a statement containing a variable, like "x is wise," which doesn't have a truth value until the variable is assigned a specific value. When a specific value or constant is substituted for a variable in a statement form, the result is called a substitution instance. For example, replacing x with "Socrates" in "x is wise" yields the proposition "Socrates is wise," which is either true or false.234

The rule of substitution ensures that these replacements are consistent and meaningful, allowing for the construction of valid arguments and the exploration of logical relationships.41

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formal logic
Axiomatization of LPC
Rules of uniform substitution for predicate calculi, though formulable, are mostly very complicated, and, to avoid the necessity for these rules, axioms for these systems are therefore usually given by axiom schemata in the sense explained earlier (see above Axiomatization of PC). Given the formation…
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Rules of uniform substitution for predicate calculi, though formulable, are mostly very complicated, and, to avoid the necessity for these rules, axioms for these systems are therefore usually given by axiom schemata in the sense explained earlier (see above Axiomatization of PC). Given the formation…
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formal logic
Axiomatization of PC
…on the other hand, the rule of substitution would no longer be needed, and modus ponens could be the only transformation rule. This method makes no difference to the theorems that can be derived, but, in some branches of logic (though not in PC), it is simpler to work with…
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…on the other hand, the rule of substitution would no longer be needed, and modus ponens could be the only transformation rule. This method makes no difference to the theorems that can be derived, but, in some branches of logic (though not in PC), it is simpler to work with…
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formal logic
Validity in PC
…it (the rule of [uniform] substitution).
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…it (the rule of [uniform] substitution).
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