Scope of The Concept
A statement is “vacuously true” if it resembles the statement, where is known to be false.
Statements that can be reduced (with suitable transformations) to this basic form include the following:
- , where it is the case that .
- , where the set is empty.
- , where the symbol is restricted to a type that has no representatives.
Vacuous truth is usually applied in classical logic, which in particular is two-valued, and most of the arguments in the next section will be based on this assumption. However, vacuous truth also appears in, for example, intuitionistic logic in the same situations given above. Indeed, the first two forms above will yield vacuous truth in any logic that uses material conditional, but there are other logics which do not.
Read more about this topic: Vacuous Truth
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“For it is not the bare words but the scope of the writer that gives the true light, by which any writing is to be interpreted; and they that insist upon single texts, without considering the main design, can derive no thing from them clearly.”
—Thomas Hobbes (1579–1688)
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“Every new concept first comes to the mind in a judgment.”
—Charles Sanders Peirce (1839–1914)