Kevin Walkner
University of Vienna
With the emergence of formal methods in philosophy, the traditional study of definitions was largely replaced by a modern, mathematical theory of definitions. In this context, conditions under which the extension of a formal theory and its signature by a constant and a formula counts as a definitional theory extension were studied. Explications of two criteria for definitional theory extension were posited. These two criteria are called “”eliminability”” and “”non-creativity”” (the latter also often “”conservativeness””).
In addition to that, a definition for the term ‘explicit definition’ was proposed. Essentially, it provides syntactic conditions for formulas. Joseph Shoenfield was able to show that whenever an explicit definition is added to a formal theory and thereby introduces a new descriptive constant to the theory’s signature, the thereby extended theory is a non-creative extension and one, in which the newly introduced constant is eliminable – roughly put: he has shown that extension by explicit definitions implies non-creativity and eliminability.
A few years later, Reinhard Kleinknecht was able to prove that every constant which can be eliminated in a theory can also be explicitly defined in that theory – roughly put: he was able to show that eliminability implies explicit definability. Combined with Shoenfield’s result, it is purported to follow that – again roughly put – eliminability also implies non-creativity.
This paper has two main parts. The first consists of a critique of Kleinknecht’s result. It will be argued that his proof could only be established because the meaning of the term ‘non-creative’ was subtly changed – in a way that makes the modified, new concept inadequate.
The second provides a proof that every closed formula that non-creatively extends a theory and its signature, such that this constant is then eliminable is equivalent to an explicit definition of this constant – i.e. modulo closure and equivalence the converse to Shoenfield’s theorem .
Chair: Marvin Thinschmidt
Time: September 13th, 12:00 – 12:30
Location: SR 1.005, online
SOPhiA 2026 