Descriptive versus primitive definitions

As we have explained before, our intention is to switch over from λC to an extended formal system with definitions that can be fruitfully used for the formalisation of mathematical texts (including logic).

In the previous chapter we have defined the system λD0, an extension of λC with definitions as ‘first class citizens’. We have based λD0 on so-called descriptive definitions. The word ‘descriptive’ means that each defined constant is connected to an explicit definiens, giving a formal description of what the constant represents. The new name (the constant), so to say, ‘stands for’ the ‘describing’ expression to which it has been coupled in its definition.

When it comes to mathematics (and logic) in general, there is still one thing that we miss: the possibility to express so-called primitive notions, necessary for the incorporation of axioms and axiomatic notions. These appear as soon as we go beyond the so-called constructive logic (cf. Sections 7.4 and 11.8), or when we incorporate mathematics in a style based on axioms, as often happens.

The constants introduced in primitive definitions – as opposed to those in descriptive definitions – are not accompanied by a descriptive expression. These so-called primitive constants are only provided with a type to which the constant belongs, but there is no further restriction or characterisation. Consequently, primitive constants cannot be unfolded, simply because there is nothing to unfold them to.