rule of substitution of equivalents

logic

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July 13, 2026

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The rule of substitution of equivalents is a principle in logic that allows for the replacement of a part of a logical formula with an equivalent expression. This replacement results in a new formula that is also equivalent to the original. Two formulas are considered equivalents if they have identical truth tables, meaning they produce the same truth values under all possible conditions. This rule is fundamental in propositional calculus, the simplest branch of logic, which deals with complete propositions and their combinations using operators like 'not,' 'and,' 'or,' 'implies,' and 'is equivalent to'. The application of this rule is crucial for transforming and simplifying logical expressions while maintaining their logical value.12

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formal logic
Logical manipulations in LPC
Because the rule of substitution of equivalents can be shown to hold in LPC, it follows that (∃x) may be replaced anywhere in a wff by ∼(∀x)∼, or (∀x) by ∼(∃x)∼, and the resulting wff will be equivalent to the original. Similarly, because the law of double…
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Because the rule of substitution of equivalents can be shown to hold in LPC, it follows that (∃x) may be replaced anywhere in a wff by ∼(∀x)∼, or (∀x) by ∼(∃x)∼, and the resulting wff will be equivalent to the original. Similarly, because the law of double…
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formal logic
Validity in PC
…further important principle is the rule of substitution of equivalents. Two wffs, α and β, are said to be equivalents when α ≡ β is valid. (The wffs α and β are equivalents if and only if they have identical truth tables.) The rule states that, if any part of…
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…further important principle is the rule of substitution of equivalents. Two wffs, α and β, are said to be equivalents when α ≡ β is valid. (The wffs α and β are equivalents if and only if they have identical truth tables.) The rule states that, if any part of…
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